From 3666310253e437893e7bb70c85e619a8ffa5616f Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Thu, 14 Sep 2023 13:30:40 +0200 Subject: [PATCH 01/33] start linear elasticity tutorial --- docs/src/literate-tutorials/heat_equation.jl | 4 +- .../literate-tutorials/linear_elasticity.jl | 149 +++++++++++++++++- 2 files changed, 148 insertions(+), 5 deletions(-) diff --git a/docs/src/literate-tutorials/heat_equation.jl b/docs/src/literate-tutorials/heat_equation.jl index 4a82dac7b7..bd2d6f0117 100644 --- a/docs/src/literate-tutorials/heat_equation.jl +++ b/docs/src/literate-tutorials/heat_equation.jl @@ -173,9 +173,7 @@ end # # !!! note "Notation" # Comparing again with [Introduction to FEM](@ref), `f` and `u` correspond to -# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized -# versions. However, through the code we use `f` and `u` instead to reflect the strong -# connection between the weak form and the Ferrite implementation. +# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized versions. However, through the code we use `f` and `u` instead to reflect the strong connection between the weak form and the Ferrite implementation. function assemble_global(cellvalues::CellValues, K::SparseMatrixCSC, dh::DofHandler) ## Allocate the element stiffness matrix and element force vector diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 3381be54a6..753c3d25b6 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -1,3 +1,148 @@ -# # Linear elasticity +# # [Linear elasticity](@id tutorial-linear-elasticity) +#- +#md # !!! tip +#md # This example is also available as a Jupyter notebook: +#md # [`heat_equation.ipynb`](@__NBVIEWER_ROOT_URL__/examples/heat_equation.ipynb). +#- # -# TBW +# ## Introduction +# +# The heat equation is the "Hello, world!" equation of finite elements. +# Here we solve the equation on a unit square, with a uniform internal source. +# The strong form of the (linear) heat equation is given by +# +# ```math +# -\boldsymbol{\sigma} \cdot \boldsymbol{\nabla} = \boldsymbol{b} \quad \textbf{x} \in \Omega, +# ``` +# +# where $\boldsymbol{\sigma}$ is the stress tensor, $\boldsymbol{b}$ is the body force and +# $\Omega$ the domain. +# +# In this example, we use linear elasticity, such that +# ```math +# \boldsymbol{\sigma} = \boldsymbol{E} : \boldsymbol \varepsilon +# ``` +# where $\boldsymbol{E}$ is the elastic stiffness tensor and $\boldsymbol{\varepsilon}$ is +# the small strain tensor that is computed from the displacement field $\boldsymbol{u}$ as +# ```math +# \boldsymbol{\varepsilon} = \frac{1}{2} \left( +# \boldsymbol{\nabla} \otimes \boldsymbol{u} +# + +# \boldsymbol{u} \otimes \boldsymbol{\nabla} +# \right) +# ``` +# +# The resulting weak form is given given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that +# ```math +# \int_\Omega +# \boldsymbol{\sigma} : \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) +# \, \mathrm{d}V +# = +# \int_{\partial\Omega} +# \boldsymbol{t}^\ast \cdot \delta \boldsymbol{u} +# \, \mathrm{d}A +# \int_\Omega +# \boldsymbol{b} \cdot \delta \boldsymbol{u} +# \, \mathrm{d}V +# \quad \forall \, \delta \boldsymbol{u} \in \mathbb{T}, +# ``` +# where $\delta \boldsymbol{u}$ is a vector valued test function, and where $\mathbb{U}$ and +# $\mathbb{T}$ are suitable trial and test function sets, respectively. +#- + +function assemble_cell!(ke, re, cellvalues, cell, material, ue) + fill!(ke, 0.0) + fill!(re, 0.0) + + n_basefuncs = getnbasefunctions(cellvalues) + reinit!(cellvalues, cell) + + for q_point in 1:getnquadpoints(cellvalues) + ## For each integration point, compute stress and material stiffness + ε = function_symmetric_gradient(cellvalues, q_point, ue) # Total strain + σ, ∂σ∂ε = material_routine(ε, material) + + dΩ = getdetJdV(cellvalues, q_point) + for i in 1:n_basefuncs + Nᵢ∇ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) + re[i] += σ ⊡ Nᵢ∇ * dΩ # add internal force to residual + for j in 1:i # loop only over lower half + ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) + ke[i, j] += Nᵢ∇ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ * dΩ + end + end + end + # symmetrize_lower!(ke) # needed? what does assembly in symmetric global matrix does? +end + + +# #### Global assembly +# We define the function `assemble_global` to loop over the elements and do the global +# assembly. The function takes our `cellvalues`, the sparse matrix `K`, and our DofHandler +# as input arguments and returns the assembled global stiffness matrix, and the assembled +# global force vector. We start by allocating `Ke`, `fe`, and the global force vector `f`. +# We also create an assembler by using `start_assemble`. The assembler lets us assemble into +# `K` and `f` efficiently. We then start the loop over all the elements. In each loop +# iteration we reinitialize `cellvalues` (to update derivatives of shape functions etc.), +# compute the element contribution with `assemble_element!`, and then assemble into the +# global `K` and `f` with `assemble!`. +# +# !!! note "Notation" +# Comparing again with [Introduction to FEM](@ref), `f` and `u` correspond to +# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized +# versions. However, through the code we use `f` and `u` instead to reflect the strong +# connection between the weak form and the Ferrite implementation. + +function assemble_global(cellvalues::CellValues, K::SparseMatrixCSC, dh::DofHandler) + ## Allocate the element stiffness matrix and element force vector + n_basefuncs = getnbasefunctions(cellvalues) + ke = zeros(n_basefuncs, n_basefuncs) + fe = zeros(n_basefuncs) + ## Allocate global force vector f + f = zeros(ndofs(dh)) + ## Create an assembler + assembler = start_assemble(K, f) + ## Loop over all cels + for cell in CellIterator(dh) + ## Reinitialize cellvalues for this cell + reinit!(cellvalues, cell) + ## Compute element contribution + @views ue = a[celldofs(cell)] + assemble_cell!(ke, re, cellvalues, get_cell_coordinates(cell), material, ue) + ## Assemble ke and fe into K and f + assemble!(assembler, celldofs(cell), ke, fe) + end + return K, f +end +#md nothing # hide + +# ### Solution of the system +# The last step is to solve the system. First we call `assemble_global` +# to obtain the global stiffness matrix `K` and force vector `f`. +K, f = assemble_global(cellvalues, K, dh); + +# To account for the boundary conditions we use the `apply!` function. +# This modifies elements in `K` and `f` respectively, such that +# we can get the correct solution vector `u` by using `\`. +apply!(K, f, ch) +u = K \ f; + +# ### Exporting to VTK +# To visualize the result we export the grid and our field `u` +# to a VTK-file, which can be viewed in e.g. [ParaView](https://www.paraview.org/). +vtk_grid("heat_equation", dh) do vtk + vtk_point_data(vtk, dh, u) +end + +## test the result #src +using Test #src +@test norm(u) ≈ 3.307743912641305 #src + +#md # ## [Plain program](@id heat_equation-plain-program) +#md # +#md # Here follows a version of the program without any comments. +#md # The file is also available here: [`heat_equation.jl`](heat_equation.jl). +#md # +#md # ```julia +#md # @__CODE__ +#md # ``` From 5c855f11a3a42bd0f7cffb5e0bc07b3a4ce7f9c1 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Sat, 23 Sep 2023 14:25:27 +0200 Subject: [PATCH 02/33] start linear elasticity tutorial --- .../literate-tutorials/linear_elasticity.jl | 46 +++++++++++++++++-- docs/src/literate-tutorials/logo.geo | 43 +++++++++++++++++ 2 files changed, 84 insertions(+), 5 deletions(-) create mode 100644 docs/src/literate-tutorials/logo.geo diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 753c3d25b6..98c96d1e9b 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -41,6 +41,7 @@ # \int_{\partial\Omega} # \boldsymbol{t}^\ast \cdot \delta \boldsymbol{u} # \, \mathrm{d}A +# + # \int_\Omega # \boldsymbol{b} \cdot \delta \boldsymbol{u} # \, \mathrm{d}V @@ -50,6 +51,42 @@ # $\mathbb{T}$ are suitable trial and test function sets, respectively. #- +# First we load Ferrite, and some other packages we need +using Ferrite, FerriteGmsh, SparseArrays +# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by `FerriteGmsh` here. +grid = togrid("logo.geo") # TODO: where to store correctly and how to refer to this? +# By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. +# Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. +addfaceset!(grid, "right", x->x[1] ≈ 1.0) +addfaceset!(grid, "top", x->x[2] ≈ 1.0) +addfaceset!(grid, "left", x->x[1] < 1e-6) +addfaceset!(grid, "bottom", x->x[2] < 1e-6) + +# ### Trial and test functions +# We use linear Lagrange functions as test and trial functions. The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). +# Since the displacement field $\boldsymbol{u}$ is a vector valued field, we vectorize the interpolation by the spatial dimension `dim`. +dim = 2 +order = 1 # linear interpolation +ip = Lagrange{RefTriangle, order}()^dim +qr = QuadratureRule{RefQuadrilateral}(2) # 2 quadrature points per spatial dimension +cellvalues = CellValues(qr, ip) + +# ### Degrees of freedom +# For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that `u` has two degrees of freedom per node because we vectorized the interpolation above. +dh = DofHandler(grid) +add!(dh, :u, ip) +close!(dh); + +# ### Boundary conditions +# Now, we add boundary conditions. We simply support the bottom and the left side and prescribe a displacement upwards on the top edge. +# The last argument to `Dirichlet` determines which components of the field should be constrained. +ch = ConstraintHandler(dh) +add!(ch, Dirichlet(:u, getfaceset(grid, "bottom"), (x, t) -> 0.0, 2)) +add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) +add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) +close!(ch) + +# ### Element routine function assemble_cell!(ke, re, cellvalues, cell, material, ue) fill!(ke, 0.0) fill!(re, 0.0) @@ -64,15 +101,14 @@ function assemble_cell!(ke, re, cellvalues, cell, material, ue) dΩ = getdetJdV(cellvalues, q_point) for i in 1:n_basefuncs - Nᵢ∇ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) - re[i] += σ ⊡ Nᵢ∇ * dΩ # add internal force to residual - for j in 1:i # loop only over lower half + ∇Nᵢ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) + re[i] += σ ⊡ ∇Nᵢ * dΩ # add internal force to residual + for j in 1:n_basefuncs ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) - ke[i, j] += Nᵢ∇ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ * dΩ + ke[i, j] += ∇Nᵢ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ * dΩ end end end - # symmetrize_lower!(ke) # needed? what does assembly in symmetric global matrix does? end diff --git a/docs/src/literate-tutorials/logo.geo b/docs/src/literate-tutorials/logo.geo new file mode 100644 index 0000000000..b08ec37afc --- /dev/null +++ b/docs/src/literate-tutorials/logo.geo @@ -0,0 +1,43 @@ +Point (1) = {1.000000000000,1.000000000000,0.000000000000}; +Point (2) = {1.000000000000,0.379757979263,-0.000000000000}; +Point (3) = {0.801534880751,0.454963545532,-0.000000000000}; +Point (4) = {0.656107331955,1.000000000000,-0.000000000000}; +Point (5) = {0.600672553037,0.775245709538,-0.000000000000}; +Point (6) = {0.000000000000,1.000000000000,0.000000000000}; +Point (7) = {0.392825178821,0.672136259831,-0.000000000000}; +Point (8) = {1.000000000000,0.000000000000,0.000000000000}; +Point (9) = {0.547800422194,-0.000000000000,-0.000000000000}; +Point (10) = {0.488710023938,0.224380304618,-0.000000000000}; +Point (11) = {0.000000000000,0.000000000000,0.000000000000}; +Point (12) = {-0.000000000000,0.324566579562,-0.000000000000}; +Point (13) = {0.172066723668,0.367888021869,-0.000000000000}; +Line (1) = {2,1}; +Line (2) = {1,4}; +Line (3) = {2,3}; +Line (4) = {3,5}; +Line (5) = {5,4}; +Line (6) = {4,6}; +Line (7) = {7,6}; +Line (8) = {5,7}; +Line (9) = {8,2}; +Line (10) = {9,8}; +Line (11) = {9,10}; +Line (12) = {10,3}; +Line (13) = {12,11}; +Line (14) = {11,9}; +Line (15) = {12,13}; +Line (16) = {13,10}; +Line (17) = {6,12}; +Line (18) = {7,13}; +Line Loop (1) = {-1,3,4,5,-2}; +Plane Surface (1) = {1}; Physical Surface (1) = {1}; +Line Loop (2) = {7,-6,-5,8}; +Plane Surface (2) = {2}; Physical Surface (2) = {2}; +Line Loop (3) = {-10,11,12,-3,-9}; +Plane Surface (3) = {3}; Physical Surface (3) = {3}; +Line Loop (4) = {-11,-14,-13,15,16}; +Plane Surface (4) = {4}; Physical Surface (4) = {4}; +Line Loop (5) = {-7,18,-15,-17}; +Plane Surface (5) = {5}; Physical Surface (5) = {5}; +Line Loop (6) = {-16,-18,-8,-4,-12}; +Plane Surface (6) = {6}; Physical Surface (6) = {6}; From 77734043e73d8302a593239cab180bdf551609cf Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 27 Sep 2023 15:08:13 +0200 Subject: [PATCH 03/33] runnable linear elasticity example --- .../literate-tutorials/linear_elasticity.jl | 155 ++++++++++++------ .../literate-tutorials/linear_elasticity.png | Bin 0 -> 121183 bytes docs/src/literate-tutorials/logo.geo | 2 + 3 files changed, 109 insertions(+), 48 deletions(-) create mode 100644 docs/src/literate-tutorials/linear_elasticity.png diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 98c96d1e9b..2c961e030f 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -1,5 +1,9 @@ # # [Linear elasticity](@id tutorial-linear-elasticity) -#- +# +# ![](linear_elasticity.png) +# +# *Figure 1*: Linear elastically deformed Ferrite logo. +# #md # !!! tip #md # This example is also available as a Jupyter notebook: #md # [`heat_equation.ipynb`](@__NBVIEWER_ROOT_URL__/examples/heat_equation.ipynb). @@ -48,27 +52,50 @@ # \quad \forall \, \delta \boldsymbol{u} \in \mathbb{T}, # ``` # where $\delta \boldsymbol{u}$ is a vector valued test function, and where $\mathbb{U}$ and -# $\mathbb{T}$ are suitable trial and test function sets, respectively. -#- +# $\mathbb{T}$ are suitable trial and test function sets, respectively. The boundary traction +# is denoted $\boldsymbol{t}^\ast$ and body forces are denoted $\boldsymbol{b}$. +# +# However, for this example we will neglect all external loads and thus the weak form reads: +# ```math +# \int_\Omega +# \boldsymbol{\sigma} : \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) +# \, \mathrm{d}V +# = +# 0 \,. +# ``` -# First we load Ferrite, and some other packages we need +# First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays # Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by `FerriteGmsh` here. -grid = togrid("logo.geo") # TODO: where to store correctly and how to refer to this? +grid = togrid("src/literate-tutorials/logo.geo") # TODO: where to store correctly and how to refer to this? # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. -addfaceset!(grid, "right", x->x[1] ≈ 1.0) addfaceset!(grid, "top", x->x[2] ≈ 1.0) addfaceset!(grid, "left", x->x[1] < 1e-6) addfaceset!(grid, "bottom", x->x[2] < 1e-6) # ### Trial and test functions # We use linear Lagrange functions as test and trial functions. The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). -# Since the displacement field $\boldsymbol{u}$ is a vector valued field, we vectorize the interpolation by the spatial dimension `dim`. +# +# Since the displacement field $\boldsymbol{u}$ is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ to approximate the test and trial functions: +# ```math +# \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i +# \qquad +# \delta \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \delta \hat{u}_i +# ``` +# Here $N$ is the number of nodal variables and $\hat{u}_i$ / $\delta\hat{u}_i$ represent the i-th nodal value. +# Using the Einstein summation convention, we can write this in short form as +# $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \boldsymbol{N}_i \, \delta\hat{u}_i$. +# +# Here we use linear triangular elements (also called constant strain triangles) with a single +# quadrature point. +# The vector valued shape functions are constructed by raising the interpolation +# to the power `dim` (the dimension) since the displacement field has one component in each +# spatial dimension. dim = 2 order = 1 # linear interpolation -ip = Lagrange{RefTriangle, order}()^dim -qr = QuadratureRule{RefQuadrilateral}(2) # 2 quadrature points per spatial dimension +ip = Lagrange{RefTriangle, order}()^dim # vector valued interpolation +qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point cellvalues = CellValues(qr, ip) # ### Degrees of freedom @@ -86,26 +113,70 @@ add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) close!(ch) +# ### Material routine +# For the sake of structuring the program, we introduce a material routine here. +# We define a data structure that holds the material constants, here it stores the +# shear modulus $G$ and the bulk modulus $K$. The material routine then computes the stress +# $\boldsymbol{\sigma}$ and its tangent $\frac{\partial\boldsymbol{\sigma}}{\partial\boldsymbol{\varepsilon}}$ +# based on the input strain $\boldsymbol{\varepsilon}$. Here, we use automatic differentiation +# for computing the stress tangent. You can read more about automatic differentiation on tensor +# operations in the [`Tensors.jl` docs](https://ferrite-fem.github.io/Tensors.jl/stable/man/automatic_differentiation/). +struct Elasticity + G::Float64 + K::Float64 +end + +function material_routine(material::Elasticity, ε::SymmetricTensor{2}) + (; G, K) = material + stress(ε) = 2G * dev(ε) + K * tr(ε) * one(ε) + ∂σ∂ε, σ = gradient(stress, ε, :all) + return σ, ∂σ∂ε +end + +E = 200e3 # Young's modulus [MPa] +ν = 0.3 # Poisson's ratio [-] +material = Elasticity(E/2(1+ν), E/3(1-2ν)) + # ### Element routine -function assemble_cell!(ke, re, cellvalues, cell, material, ue) +# The residual vector and stiffness matrix follow from the weak form such that +# ```math +# \left(\underline{R}\right)_i +# = +# \int_\Omega +# \boldsymbol{\sigma} : \boldsymbol{\nabla} \boldsymbol{N}_i +# \, \mathrm{d}V +# \,, \qquad +# \left(\underline{\underline{K}}\right)_{ij} +# = +# \int_\Omega +# \left( +# \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} +# : +# \boldsymbol{\nabla}^\mathrm{sym} \boldsymbol{N}_j +# \right) +# : +# \boldsymbol{\nabla} \boldsymbol{N}_i +# \, \mathrm{d}V +# ``` +# The element routine computes the local residual vector `re` and stiffness matrix `ke` +# for a single element. `ke` and `re` are pre-allocated and reused for all elements. +function assemble_cell!(ke, fe, cellvalues, material, ue) fill!(ke, 0.0) - fill!(re, 0.0) + fill!(fe, 0.0) n_basefuncs = getnbasefunctions(cellvalues) - reinit!(cellvalues, cell) - for q_point in 1:getnquadpoints(cellvalues) - ## For each integration point, compute stress and material stiffness - ε = function_symmetric_gradient(cellvalues, q_point, ue) # Total strain - σ, ∂σ∂ε = material_routine(ε, material) + ## For each integration point, compute strain, stress and material stiffness + ε = function_symmetric_gradient(cellvalues, q_point, ue) + σ, ∂σ∂ε = material_routine(material, ε) dΩ = getdetJdV(cellvalues, q_point) for i in 1:n_basefuncs ∇Nᵢ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) - re[i] += σ ⊡ ∇Nᵢ * dΩ # add internal force to residual + fe[i] += σ ⊡ ∇Nᵢ * dΩ # add internal force to residual for j in 1:n_basefuncs ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) - ke[i, j] += ∇Nᵢ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ * dΩ + ke[i, j] += (∂σ∂ε ⊡ ∇ˢʸᵐNⱼ) ⊡ ∇Nᵢ * dΩ end end end @@ -114,37 +185,24 @@ end # #### Global assembly # We define the function `assemble_global` to loop over the elements and do the global -# assembly. The function takes our `cellvalues`, the sparse matrix `K`, and our DofHandler -# as input arguments and returns the assembled global stiffness matrix, and the assembled -# global force vector. We start by allocating `Ke`, `fe`, and the global force vector `f`. -# We also create an assembler by using `start_assemble`. The assembler lets us assemble into -# `K` and `f` efficiently. We then start the loop over all the elements. In each loop -# iteration we reinitialize `cellvalues` (to update derivatives of shape functions etc.), -# compute the element contribution with `assemble_element!`, and then assemble into the -# global `K` and `f` with `assemble!`. -# -# !!! note "Notation" -# Comparing again with [Introduction to FEM](@ref), `f` and `u` correspond to -# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized -# versions. However, through the code we use `f` and `u` instead to reflect the strong -# connection between the weak form and the Ferrite implementation. - -function assemble_global(cellvalues::CellValues, K::SparseMatrixCSC, dh::DofHandler) +# assembly. The function takes the sparse matrix `K`, a guess of the displacement vector +# (this would be more relevant for non-linear problems), our DofHandler, our `cellvalues` and +# the material definition that we want to use for the stress computation +# as input arguments and fills the assembled global stiffness matrix, and the assembled +# global force vector. +function assemble_global!(K, f, a, dh, cellvalues, material) ## Allocate the element stiffness matrix and element force vector n_basefuncs = getnbasefunctions(cellvalues) ke = zeros(n_basefuncs, n_basefuncs) fe = zeros(n_basefuncs) - ## Allocate global force vector f - f = zeros(ndofs(dh)) ## Create an assembler assembler = start_assemble(K, f) - ## Loop over all cels + ## Loop over all cells for cell in CellIterator(dh) - ## Reinitialize cellvalues for this cell - reinit!(cellvalues, cell) - ## Compute element contribution + reinit!(cellvalues, cell) # update spatial derivatives based on element coordinates @views ue = a[celldofs(cell)] - assemble_cell!(ke, re, cellvalues, get_cell_coordinates(cell), material, ue) + ## Compute element contribution + assemble_cell!(ke, fe, cellvalues, material, ue) ## Assemble ke and fe into K and f assemble!(assembler, celldofs(cell), ke, fe) end @@ -155,9 +213,12 @@ end # ### Solution of the system # The last step is to solve the system. First we call `assemble_global` # to obtain the global stiffness matrix `K` and force vector `f`. -K, f = assemble_global(cellvalues, K, dh); +K = create_sparsity_pattern(dh) +f = zeros(ndofs(dh)) +a = zeros(ndofs(dh)) +assemble_global!(K, f, a, dh, cellvalues, material); -# To account for the boundary conditions we use the `apply!` function. +# To account for the Dirichlet boundary conditions we use the `apply!` function. # This modifies elements in `K` and `f` respectively, such that # we can get the correct solution vector `u` by using `\`. apply!(K, f, ch) @@ -166,13 +227,11 @@ u = K \ f; # ### Exporting to VTK # To visualize the result we export the grid and our field `u` # to a VTK-file, which can be viewed in e.g. [ParaView](https://www.paraview.org/). -vtk_grid("heat_equation", dh) do vtk +vtk_grid("linear_elasticity", dh) do vtk vtk_point_data(vtk, dh, u) + vtk_cellset(vtk, grid) end - -## test the result #src -using Test #src -@test norm(u) ≈ 3.307743912641305 #src +#md nothing # hide #md # ## [Plain program](@id heat_equation-plain-program) #md # diff --git a/docs/src/literate-tutorials/linear_elasticity.png b/docs/src/literate-tutorials/linear_elasticity.png new file mode 100644 index 0000000000000000000000000000000000000000..5a03ba219eb243ba8859dc1387c60a4a85b33a15 GIT binary patch literal 121183 zcmY&;1ymeO)9x-BED+ow5F|iwcXxMpcXwIbgS)%CEf5@nyIUZ*L$JlUdEfv0{(EoF z>FS!Ar=IHSb7rQery~{RB~TFY5di=Iij<_NG5`Qn{aeF6!2Pv+G4)FL+W=jZC4>RB zQ-sHVPeRNzrOf5z05pH~4**yoIsp0~$ln71#0S9ss{;ViK!X3(m4Vd%!9W23;Z^{c z|6sKLw*M5#zxA*8e_H5#;QwXJhx#uXsG1M`zk1j|Zb??h{C^t+M@cOg000T&p9%zI zX5;>4)3Q?4bk&rT;Qvo&{wLFat$$PHN91Mr-(llN#MA!`4*&=Pq(p^O zJ%MLk8u5C|8JBIKDXyZf@cTK=@mu6ksxBgN`t|?v zsc|TIe_Bdgit>@R=la&x>x@MAkJ#IQqIH=v;tJl!*0c}&M*wXC%>?y`Ka78?-+&tb zhNgr>2W>5i(w;r}9~V04lWW}nfCD464e=Dotw;84Q*{j6VgCv|#kAr7vWqT2jf3!& zMCdJ={^9zE5Y)j7_a7)}A+{#mLIl1uSJr={!!7JH{TueL_|($NoG-o*Kfd~JPyY($ zeEPV5`}kKxl*1wxjf6kYj5DbhT8{4X{!2<+jIG(uOgNpvk`Gl(t6p8d9K3)w-uzzz z>XCk~lU;ZB;8RXV|B+IzCBO;qhkdX(;UJbnNI?!ke2!|KcvofmQhWJU@{K z{>5rX7RbUhZS(=XlbKW2lg0I)i{q;%!~2UECJ|bVhlDoQX2qVN+j3Mfg+6IinZBd3 zL)@E*?I(CJ?z5H0nbH4X`6DEbN66gv0(|q`E5o$3biAGF~{o7Q;gjvo9oeGK$3=QhQ2F?4qEN z+McwV)QW#u15h^67)ECKe#Cxc0>Fyw&Qf8yxIO|u3sjA7T6j=yls2Nbm4au$LIDP% zLt$e-(@-SV`1p41zsrN)OR87&($k&e)HQI=kwulSI8^TwHg&0=;4I8H_Y<5TBpS7j z7(0cu6cGIRkH>##W-GQXPTDSM9~&%n%Pn&V@tha;<2ouT{M*|7%c>Y0Tv;3=onXYJ zoP2-HKsheBV}Ud0UTYnGtlQbF>{rTIZ=Q9niZzQ$h1VI=@$DHR5! z+RUl=1d$vsI^jqyd{*JNM1=dp=Tre(>r*{&?)eN_pZT%>)<6FO6`bIbnVE~tp?JPl z$=<@J6#}U5x1~~KG4++!<#t*EP0T7=q@fKSLtY7x=&AlIXa&8yv*9y z3r5*_IA&8b`7LP`3%Z~xmph*}#$(-TjeM6

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z-_j5>6_lYjt-M1G;efPF)SEHfU*D4Hi{m=Aca)PMFH2cc1sdJB2!0RyAFk~fBnF^@ zW-*+|k#maqNGu~G6$AMVQhLH#v6u8ZWe#mGe#*FW-ynbd*;4vr^VnULcS|Za_@1?* w0vH3D@P+07Z Date: Wed, 27 Sep 2023 15:11:10 +0200 Subject: [PATCH 04/33] suppress output after code blocks --- docs/src/literate-tutorials/linear_elasticity.jl | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 2c961e030f..12e2aabf2b 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -72,7 +72,7 @@ grid = togrid("src/literate-tutorials/logo.geo") # TODO: where to store correctl # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. addfaceset!(grid, "top", x->x[2] ≈ 1.0) addfaceset!(grid, "left", x->x[1] < 1e-6) -addfaceset!(grid, "bottom", x->x[2] < 1e-6) +addfaceset!(grid, "bottom", x->x[2] < 1e-6); # ### Trial and test functions # We use linear Lagrange functions as test and trial functions. The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). @@ -96,7 +96,7 @@ dim = 2 order = 1 # linear interpolation ip = Lagrange{RefTriangle, order}()^dim # vector valued interpolation qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point -cellvalues = CellValues(qr, ip) +cellvalues = CellValues(qr, ip); # ### Degrees of freedom # For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that `u` has two degrees of freedom per node because we vectorized the interpolation above. @@ -111,7 +111,7 @@ ch = ConstraintHandler(dh) add!(ch, Dirichlet(:u, getfaceset(grid, "bottom"), (x, t) -> 0.0, 2)) add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) -close!(ch) +close!(ch); # ### Material routine # For the sake of structuring the program, we introduce a material routine here. @@ -135,7 +135,7 @@ end E = 200e3 # Young's modulus [MPa] ν = 0.3 # Poisson's ratio [-] -material = Elasticity(E/2(1+ν), E/3(1-2ν)) +material = Elasticity(E/2(1+ν), E/3(1-2ν)); # ### Element routine # The residual vector and stiffness matrix follow from the weak form such that @@ -181,7 +181,7 @@ function assemble_cell!(ke, fe, cellvalues, material, ue) end end end - +#md nothing # hide # #### Global assembly # We define the function `assemble_global` to loop over the elements and do the global From 9e148c9517d8293efbe21d3920415ef160d07e3c Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 27 Sep 2023 16:35:04 +0200 Subject: [PATCH 05/33] load mesh from assets --- docs/download_resources.jl | 1 + .../literate-tutorials/linear_elasticity.jl | 4 +- docs/src/literate-tutorials/logo.geo | 45 ------------------- 3 files changed, 3 insertions(+), 47 deletions(-) delete mode 100644 docs/src/literate-tutorials/logo.geo diff --git a/docs/download_resources.jl b/docs/download_resources.jl index bfd507f8a7..f6de512c4e 100644 --- a/docs/download_resources.jl +++ b/docs/download_resources.jl @@ -5,6 +5,7 @@ const directory = joinpath(@__DIR__, "src", "tutorials") mkpath(directory) for (file, url) in [ + "logo.geo" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/logo.geo", "periodic-rve.msh" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/periodic-rve.msh", "periodic-rve-coarse.msh" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/periodic-rve-coarse.msh", "transient_heat.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/transient_heat.gif", diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 12e2aabf2b..094f3be021 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -66,8 +66,8 @@ # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays -# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by `FerriteGmsh` here. -grid = togrid("src/literate-tutorials/logo.geo") # TODO: where to store correctly and how to refer to this? +# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by loading [`logo.geo`](logo.geo) with [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl) here. +grid = togrid("logo.geo") # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. addfaceset!(grid, "top", x->x[2] ≈ 1.0) diff --git a/docs/src/literate-tutorials/logo.geo b/docs/src/literate-tutorials/logo.geo deleted file mode 100644 index 22434c9631..0000000000 --- a/docs/src/literate-tutorials/logo.geo +++ /dev/null @@ -1,45 +0,0 @@ -Point (1) = {1.000000000000,1.000000000000,0.000000000000}; -Point (2) = {1.000000000000,0.379757979263,-0.000000000000}; -Point (3) = {0.801534880751,0.454963545532,-0.000000000000}; -Point (4) = {0.656107331955,1.000000000000,-0.000000000000}; -Point (5) = {0.600672553037,0.775245709538,-0.000000000000}; -Point (6) = {0.000000000000,1.000000000000,0.000000000000}; -Point (7) = {0.392825178821,0.672136259831,-0.000000000000}; -Point (8) = {1.000000000000,0.000000000000,0.000000000000}; -Point (9) = {0.547800422194,-0.000000000000,-0.000000000000}; -Point (10) = {0.488710023938,0.224380304618,-0.000000000000}; -Point (11) = {0.000000000000,0.000000000000,0.000000000000}; -Point (12) = {-0.000000000000,0.324566579562,-0.000000000000}; -Point (13) = {0.172066723668,0.367888021869,-0.000000000000}; -Line (1) = {2,1}; -Line (2) = {1,4}; -Line (3) = {2,3}; -Line (4) = {3,5}; -Line (5) = {5,4}; -Line (6) = {4,6}; -Line (7) = {7,6}; -Line (8) = {5,7}; -Line (9) = {8,2}; -Line (10) = {9,8}; -Line (11) = {9,10}; -Line (12) = {10,3}; -Line (13) = {12,11}; -Line (14) = {11,9}; -Line (15) = {12,13}; -Line (16) = {13,10}; -Line (17) = {6,12}; -Line (18) = {7,13}; -Line Loop (1) = {-1,3,4,5,-2}; -Plane Surface (1) = {1}; Physical Surface (1) = {1}; -Line Loop (2) = {7,-6,-5,8}; -Plane Surface (2) = {2}; Physical Surface (2) = {2}; -Line Loop (3) = {-10,11,12,-3,-9}; -Plane Surface (3) = {3}; Physical Surface (3) = {3}; -Line Loop (4) = {-11,-14,-13,15,16}; -Plane Surface (4) = {4}; Physical Surface (4) = {4}; -Line Loop (5) = {-7,18,-15,-17}; -Plane Surface (5) = {5}; Physical Surface (5) = {5}; -Line Loop (6) = {-16,-18,-8,-4,-12}; -Plane Surface (6) = {6}; Physical Surface (6) = {6}; - -ReverseMesh Surface{1,2,3,4,5,6}; From de13a3860756b03fb423f3bc0bf4c2cebbf78625 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 27 Sep 2023 16:38:59 +0200 Subject: [PATCH 06/33] revert unindented changes on heat equation tutorial --- docs/src/literate-tutorials/heat_equation.jl | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/heat_equation.jl b/docs/src/literate-tutorials/heat_equation.jl index bd2d6f0117..4a82dac7b7 100644 --- a/docs/src/literate-tutorials/heat_equation.jl +++ b/docs/src/literate-tutorials/heat_equation.jl @@ -173,7 +173,9 @@ end # # !!! note "Notation" # Comparing again with [Introduction to FEM](@ref), `f` and `u` correspond to -# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized versions. However, through the code we use `f` and `u` instead to reflect the strong connection between the weak form and the Ferrite implementation. +# $\underline{\hat{f}}$ and $\underline{\hat{u}}$, since they represent the discretized +# versions. However, through the code we use `f` and `u` instead to reflect the strong +# connection between the weak form and the Ferrite implementation. function assemble_global(cellvalues::CellValues, K::SparseMatrixCSC, dh::DofHandler) ## Allocate the element stiffness matrix and element force vector From 02443577ef5e41be0cefcf604caa14fb7d9f9e74 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 27 Sep 2023 17:13:46 +0200 Subject: [PATCH 07/33] an actual introduction for linear elasticity --- docs/src/literate-tutorials/linear_elasticity.jl | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 094f3be021..9619a827d6 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -11,14 +11,14 @@ # # ## Introduction # -# The heat equation is the "Hello, world!" equation of finite elements. -# Here we solve the equation on a unit square, with a uniform internal source. -# The strong form of the (linear) heat equation is given by +# The classical first finite element problem to solve in solid mechanics is a linear balance +# of momentum problem. We will use this to introduce a vector valued field, as well as the +# [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox. # +# The strong form of the balance of momentum is given by # ```math # -\boldsymbol{\sigma} \cdot \boldsymbol{\nabla} = \boldsymbol{b} \quad \textbf{x} \in \Omega, # ``` -# # where $\boldsymbol{\sigma}$ is the stress tensor, $\boldsymbol{b}$ is the body force and # $\Omega$ the domain. # From ce6fb00347d7bbc2e4e7d4f5bfe0bd75b57c3bbd Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 27 Sep 2023 17:15:10 +0200 Subject: [PATCH 08/33] forgotten semicolon --- docs/src/literate-tutorials/linear_elasticity.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 9619a827d6..03fc12f1b3 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -67,7 +67,7 @@ # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays # Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by loading [`logo.geo`](logo.geo) with [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl) here. -grid = togrid("logo.geo") +grid = togrid("logo.geo"); # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. addfaceset!(grid, "top", x->x[2] ≈ 1.0) From d1a7aad19fedf78a26330071398bf92e2429a008 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Thu, 28 Sep 2023 11:30:54 +0200 Subject: [PATCH 09/33] Update docs/src/literate-tutorials/linear_elasticity.jl MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Maximilian Köhler --- docs/src/literate-tutorials/linear_elasticity.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 03fc12f1b3..7049391901 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -6,7 +6,7 @@ # #md # !!! tip #md # This example is also available as a Jupyter notebook: -#md # [`heat_equation.ipynb`](@__NBVIEWER_ROOT_URL__/examples/heat_equation.ipynb). +#md # [`linear_elasticity.ipynb`](@__NBVIEWER_ROOT_URL__/examples/linear_elasticity.ipynb). #- # # ## Introduction From 82ad1641a3bcf59ad7bb4b7d14a1a8d6dd2bb253 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 1 Nov 2023 12:00:25 +0100 Subject: [PATCH 10/33] remove internal force vector computation --- .../literate-tutorials/linear_elasticity.jl | 110 ++++++++---------- 1 file changed, 48 insertions(+), 62 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 7049391901..3ba61d7989 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -63,11 +63,13 @@ # = # 0 \,. # ``` - +# +# Finally, we choose to operate on a 2-dimensional problem under plain strain conditions. # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays # Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by loading [`logo.geo`](logo.geo) with [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl) here. grid = togrid("logo.geo"); +#md nothing # hide # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. addfaceset!(grid, "top", x->x[2] ≈ 1.0) @@ -99,53 +101,43 @@ qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point cellvalues = CellValues(qr, ip); # ### Degrees of freedom -# For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that `u` has two degrees of freedom per node because we vectorized the interpolation above. +# For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that +# `u` has two degrees of freedom per node because we vectorized the interpolation above. dh = DofHandler(grid) add!(dh, :u, ip) close!(dh); # ### Boundary conditions -# Now, we add boundary conditions. We simply support the bottom and the left side and prescribe a displacement upwards on the top edge. -# The last argument to `Dirichlet` determines which components of the field should be constrained. +# Now, we add boundary conditions. We simply support the bottom and the left side and +# prescribe a displacement upwards on the top edge. +# The last argument to `Dirichlet` determines which components of the field should be +# constrained. ch = ConstraintHandler(dh) add!(ch, Dirichlet(:u, getfaceset(grid, "bottom"), (x, t) -> 0.0, 2)) add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) close!(ch); -# ### Material routine -# For the sake of structuring the program, we introduce a material routine here. -# We define a data structure that holds the material constants, here it stores the -# shear modulus $G$ and the bulk modulus $K$. The material routine then computes the stress -# $\boldsymbol{\sigma}$ and its tangent $\frac{\partial\boldsymbol{\sigma}}{\partial\boldsymbol{\varepsilon}}$ -# based on the input strain $\boldsymbol{\varepsilon}$. Here, we use automatic differentiation -# for computing the stress tangent. You can read more about automatic differentiation on tensor -# operations in the [`Tensors.jl` docs](https://ferrite-fem.github.io/Tensors.jl/stable/man/automatic_differentiation/). -struct Elasticity - G::Float64 - K::Float64 -end - -function material_routine(material::Elasticity, ε::SymmetricTensor{2}) - (; G, K) = material - stress(ε) = 2G * dev(ε) + K * tr(ε) * one(ε) - ∂σ∂ε, σ = gradient(stress, ε, :all) - return σ, ∂σ∂ε -end - +# ### Material behavior +# Next, we need to define the material behavior. Since we use linear elasticity here, +# we have a linear problem and only need to assemble the stiffness matrix, but not the +# residual vector. Consequently we only need the tangent of the stress $\boldsymbol{\sigma}$ +# with respect to the small strain tensor $\boldsymbol{\varepsilon}$ for our element routine. +# +# We also operate on a 2-dimensional problem under plain strain conditions here, keep in +# mind that the plane stress stiffness tensor is defined differently. E = 200e3 # Young's modulus [MPa] ν = 0.3 # Poisson's ratio [-] -material = Elasticity(E/2(1+ν), E/3(1-2ν)); + +λ = E*ν / ((1 + ν) * (1 - 2ν)) # 1st Lamé parameter +μ = E / (2(1 + ν)) # 2nd Lamé parameter +I = one(SymmetricTensor{2, dim}) # 2nd order unit tensor +II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor +∂σ∂ε = 2μ * II + λ * (I ⊗ I) # elastic stiffness tensor # ### Element routine -# The residual vector and stiffness matrix follow from the weak form such that +# The stiffness matrix follows from the weak form such that # ```math -# \left(\underline{R}\right)_i -# = -# \int_\Omega -# \boldsymbol{\sigma} : \boldsymbol{\nabla} \boldsymbol{N}_i -# \, \mathrm{d}V -# \,, \qquad # \left(\underline{\underline{K}}\right)_{ij} # = # \int_\Omega @@ -158,22 +150,19 @@ material = Elasticity(E/2(1+ν), E/3(1-2ν)); # \boldsymbol{\nabla} \boldsymbol{N}_i # \, \mathrm{d}V # ``` -# The element routine computes the local residual vector `re` and stiffness matrix `ke` -# for a single element. `ke` and `re` are pre-allocated and reused for all elements. -function assemble_cell!(ke, fe, cellvalues, material, ue) +# The element routine computes the local stiffness matrix `ke` +# for a single element. `ke` is pre-allocated and reused for all elements. +# +# Note that the elastic stiffness tensor is constant. Thus is needs to be computed and once +# and can then be used for all integration points. +function assemble_cell!(ke, cellvalues, ∂σ∂ε) fill!(ke, 0.0) - fill!(fe, 0.0) n_basefuncs = getnbasefunctions(cellvalues) for q_point in 1:getnquadpoints(cellvalues) - ## For each integration point, compute strain, stress and material stiffness - ε = function_symmetric_gradient(cellvalues, q_point, ue) - σ, ∂σ∂ε = material_routine(material, ε) - dΩ = getdetJdV(cellvalues, q_point) for i in 1:n_basefuncs ∇Nᵢ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) - fe[i] += σ ⊡ ∇Nᵢ * dΩ # add internal force to residual for j in 1:n_basefuncs ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) ke[i, j] += (∂σ∂ε ⊡ ∇ˢʸᵐNⱼ) ⊡ ∇Nᵢ * dΩ @@ -185,38 +174,35 @@ end # #### Global assembly # We define the function `assemble_global` to loop over the elements and do the global -# assembly. The function takes the sparse matrix `K`, a guess of the displacement vector -# (this would be more relevant for non-linear problems), our DofHandler, our `cellvalues` and -# the material definition that we want to use for the stress computation -# as input arguments and fills the assembled global stiffness matrix, and the assembled -# global force vector. -function assemble_global!(K, f, a, dh, cellvalues, material) - ## Allocate the element stiffness matrix and element force vector +# assembly. The function takes the preallocated sparse matrix `K`, our DofHandler `dh`, our +# `cellvalues` and the elastic stiffness tensor `∂σ∂ε` as input arguments and computes the +# global stiffness matrix `K`. +function assemble_global!(K, dh, cellvalues, ∂σ∂ε) + ## Allocate the element stiffness matrix n_basefuncs = getnbasefunctions(cellvalues) ke = zeros(n_basefuncs, n_basefuncs) - fe = zeros(n_basefuncs) ## Create an assembler - assembler = start_assemble(K, f) + assembler = start_assemble(K) ## Loop over all cells for cell in CellIterator(dh) reinit!(cellvalues, cell) # update spatial derivatives based on element coordinates - @views ue = a[celldofs(cell)] ## Compute element contribution - assemble_cell!(ke, fe, cellvalues, material, ue) + assemble_cell!(ke, cellvalues, ∂σ∂ε) ## Assemble ke and fe into K and f - assemble!(assembler, celldofs(cell), ke, fe) + assemble!(assembler, celldofs(cell), ke) end - return K, f + return K end #md nothing # hide # ### Solution of the system -# The last step is to solve the system. First we call `assemble_global` -# to obtain the global stiffness matrix `K` and force vector `f`. +# The last step is to solve the system. First we allocate the global stiffness matrix `K` +# and assemble it. K = create_sparsity_pattern(dh) -f = zeros(ndofs(dh)) -a = zeros(ndofs(dh)) -assemble_global!(K, f, a, dh, cellvalues, material); +assemble_global!(K, dh, cellvalues, ∂σ∂ε); +# Then we allocate the external force vector. Since we don't apply any external forces, +# it is a zero vector in this case. +f = zeros(ndofs(dh)); # To account for the Dirichlet boundary conditions we use the `apply!` function. # This modifies elements in `K` and `f` respectively, such that @@ -229,14 +215,14 @@ u = K \ f; # to a VTK-file, which can be viewed in e.g. [ParaView](https://www.paraview.org/). vtk_grid("linear_elasticity", dh) do vtk vtk_point_data(vtk, dh, u) - vtk_cellset(vtk, grid) + vtk_cellset(vtk, grid) # export cellsets of grains for logo-coloring end #md nothing # hide -#md # ## [Plain program](@id heat_equation-plain-program) +#md # ## [Plain program](@id linear_elasticity-plain-program) #md # #md # Here follows a version of the program without any comments. -#md # The file is also available here: [`heat_equation.jl`](heat_equation.jl). +#md # The file is also available here: [`linear_elasticity.jl`](linear_elasticity). #md # #md # ```julia #md # @__CODE__ From 5bde8aa217faa15a014ad43d130c8a8d6d39b47d Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Wed, 1 Nov 2023 13:00:25 +0100 Subject: [PATCH 11/33] suppress gmsh output --- docs/src/literate-tutorials/linear_elasticity.jl | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 3ba61d7989..49592565f8 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -68,6 +68,8 @@ # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays # Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by loading [`logo.geo`](logo.geo) with [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl) here. +FerriteGmsh.Gmsh.initialize() # hide +FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide grid = togrid("logo.geo"); #md nothing # hide # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. @@ -128,12 +130,13 @@ close!(ch); # mind that the plane stress stiffness tensor is defined differently. E = 200e3 # Young's modulus [MPa] ν = 0.3 # Poisson's ratio [-] +#md nothing # hide λ = E*ν / ((1 + ν) * (1 - 2ν)) # 1st Lamé parameter μ = E / (2(1 + ν)) # 2nd Lamé parameter I = one(SymmetricTensor{2, dim}) # 2nd order unit tensor II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor -∂σ∂ε = 2μ * II + λ * (I ⊗ I) # elastic stiffness tensor +∂σ∂ε = 2μ * II + λ * (I ⊗ I); # elastic stiffness tensor # ### Element routine # The stiffness matrix follows from the weak form such that From 97e3d6d384a45a07913bb249b63b15f7337853eb Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Thu, 2 Nov 2023 13:40:46 +0100 Subject: [PATCH 12/33] add comment on addfaceset --- docs/src/literate-tutorials/linear_elasticity.jl | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 49592565f8..02a4d14881 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -73,8 +73,9 @@ FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide grid = togrid("logo.geo"); #md nothing # hide # By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. +# [`addfaceset!`](@ref) allows to add facesets to the grid based on coordinates. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. -addfaceset!(grid, "top", x->x[2] ≈ 1.0) +addfaceset!(grid, "top", x->x[2] ≈ 1.0) # faces for which x[2] ≈ 1.0 for all nodes addfaceset!(grid, "left", x->x[1] < 1e-6) addfaceset!(grid, "bottom", x->x[2] < 1e-6); From ea3cffed7435cbbbce2e535baf06353025ec4048 Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Thu, 2 Nov 2023 14:51:55 +0100 Subject: [PATCH 13/33] add neumann bc --- .../literate-tutorials/linear_elasticity.jl | 38 +++++++++++++++---- 1 file changed, 30 insertions(+), 8 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 02a4d14881..a4369b011f 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -101,7 +101,9 @@ dim = 2 order = 1 # linear interpolation ip = Lagrange{RefTriangle, order}()^dim # vector valued interpolation qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point -cellvalues = CellValues(qr, ip); +cellvalues = CellValues(qr, ip) +qr_face = FaceQuadratureRule{RefTriangle}(1) +facevalues = FaceValues(qr_face, ip); # ### Degrees of freedom # For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that @@ -118,7 +120,7 @@ close!(dh); ch = ConstraintHandler(dh) add!(ch, Dirichlet(:u, getfaceset(grid, "bottom"), (x, t) -> 0.0, 2)) add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) -add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) +#add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) close!(ch); # ### Material behavior @@ -161,7 +163,6 @@ II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor # and can then be used for all integration points. function assemble_cell!(ke, cellvalues, ∂σ∂ε) fill!(ke, 0.0) - n_basefuncs = getnbasefunctions(cellvalues) for q_point in 1:getnquadpoints(cellvalues) dΩ = getdetJdV(cellvalues, q_point) @@ -173,6 +174,7 @@ function assemble_cell!(ke, cellvalues, ∂σ∂ε) end end end + return ke end #md nothing # hide @@ -199,20 +201,40 @@ function assemble_global!(K, dh, cellvalues, ∂σ∂ε) end #md nothing # hide +function assemble_external_forces!(f_ext, dh, faceset, facevalues, prescribed_traction) + fe_ext = zeros(getnbasefunctions(facevalues)) + for face in FaceIterator(dh, faceset) + reinit!(facevalues, face) + fill!(fe_ext, 0.0) + for qp in 1:getnquadpoints(facevalues) + X = spatial_coordinate(facevalues, qp, getcoordinates(face)) + tₚ = prescribed_traction(X) + dΓ = getdetJdV(facevalues, qp) + for i in 1:getnbasefunctions(facevalues) + Nᵢ = shape_value(facevalues, qp, i) + fe_ext[i] += tₚ ⋅ Nᵢ * dΓ + end + end + assemble!(f_ext, celldofs(face), fe_ext) + end + return f_ext +end +#md nothing # hide + # ### Solution of the system # The last step is to solve the system. First we allocate the global stiffness matrix `K` # and assemble it. K = create_sparsity_pattern(dh) assemble_global!(K, dh, cellvalues, ∂σ∂ε); -# Then we allocate the external force vector. Since we don't apply any external forces, -# it is a zero vector in this case. -f = zeros(ndofs(dh)); +# Then we allocate the external force vector. +f_ext = zeros(ndofs(dh)); +assemble_external_forces!(f_ext, dh, getfaceset(grid, "top"), facevalues, x->Vec(0.0, 20e3*x[1])) # To account for the Dirichlet boundary conditions we use the `apply!` function. # This modifies elements in `K` and `f` respectively, such that # we can get the correct solution vector `u` by using `\`. -apply!(K, f, ch) -u = K \ f; +apply!(K, f_ext, ch) +u = K \ f_ext; # ### Exporting to VTK # To visualize the result we export the grid and our field `u` From 38ca05671a6a0a09b43c20d2573b11bd82825d9c Mon Sep 17 00:00:00 2001 From: Kim Louisa Auth Date: Tue, 9 Apr 2024 16:04:41 +0200 Subject: [PATCH 14/33] changes I don't remember --- .../literate-tutorials/linear_elasticity.jl | 36 +++++++++++-------- 1 file changed, 21 insertions(+), 15 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index a4369b011f..0366eb5718 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -55,13 +55,15 @@ # $\mathbb{T}$ are suitable trial and test function sets, respectively. The boundary traction # is denoted $\boldsymbol{t}^\ast$ and body forces are denoted $\boldsymbol{b}$. # -# However, for this example we will neglect all external loads and thus the weak form reads: +# However, for this example we will neglect body foces and thus the weak form reads: # ```math # \int_\Omega # \boldsymbol{\sigma} : \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) # \, \mathrm{d}V # = -# 0 \,. +# \int_{\partial\Omega} +# \boldsymbol{t}^\ast \cdot \delta \boldsymbol{u} +# \, \mathrm{d}A \,. # ``` # # Finally, we choose to operate on a 2-dimensional problem under plain strain conditions. @@ -124,22 +126,26 @@ add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) close!(ch); # ### Material behavior -# Next, we need to define the material behavior. Since we use linear elasticity here, -# we have a linear problem and only need to assemble the stiffness matrix, but not the -# residual vector. Consequently we only need the tangent of the stress $\boldsymbol{\sigma}$ -# with respect to the small strain tensor $\boldsymbol{\varepsilon}$ for our element routine. +# Next, we need to define the material behavior. In this example, we use linear elasticity, +# such that +# ```math +# \boldsymbol{\sigma} = \boldsymbol{\mathrm{E}} : \boldsymbol \varepsilon +# ``` +# where $\boldsymbol{\mathrm{E}}$ is the elastic stiffness tensor and +# $\boldsymbol{\varepsilon}$ is the small strain tensor. +# Consequently, we need to set up the elastic stiffness tensor $\boldsymbol{\mathrm{E}}$. # -# We also operate on a 2-dimensional problem under plain strain conditions here, keep in -# mind that the plane stress stiffness tensor is defined differently. -E = 200e3 # Young's modulus [MPa] +# Here, we operate in plane strain conditions, keep in mind that the plane stress stiffness +# tensor is defined differently. +Emod = 200e3 # Young's modulus [MPa] ν = 0.3 # Poisson's ratio [-] #md nothing # hide -λ = E*ν / ((1 + ν) * (1 - 2ν)) # 1st Lamé parameter -μ = E / (2(1 + ν)) # 2nd Lamé parameter +λ = Emod*ν / ((1 + ν) * (1 - 2ν)) # 1st Lamé parameter +μ = Emod / (2(1 + ν)) # 2nd Lamé parameter I = one(SymmetricTensor{2, dim}) # 2nd order unit tensor II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor -∂σ∂ε = 2μ * II + λ * (I ⊗ I); # elastic stiffness tensor +𝐄 = 2μ * II + λ * (I ⊗ I); # elastic stiffness tensor # ### Element routine # The stiffness matrix follows from the weak form such that @@ -159,8 +165,8 @@ II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor # The element routine computes the local stiffness matrix `ke` # for a single element. `ke` is pre-allocated and reused for all elements. # -# Note that the elastic stiffness tensor is constant. Thus is needs to be computed and once -# and can then be used for all integration points. +# Note that the elastic stiffness tensor $\boldsymbol{\mathrm{E}}$ is constant. +# Thus is needs to be computed and once and can then be used for all integration points. function assemble_cell!(ke, cellvalues, ∂σ∂ε) fill!(ke, 0.0) n_basefuncs = getnbasefunctions(cellvalues) @@ -178,7 +184,7 @@ function assemble_cell!(ke, cellvalues, ∂σ∂ε) end #md nothing # hide -# #### Global assembly +# ### Global assembly # We define the function `assemble_global` to loop over the elements and do the global # assembly. The function takes the preallocated sparse matrix `K`, our DofHandler `dh`, our # `cellvalues` and the elastic stiffness tensor `∂σ∂ε` as input arguments and computes the From e1bde02b4edb42ba200cbaf97058470e164ac380 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 4 Aug 2024 12:52:41 +0200 Subject: [PATCH 15/33] Elasticity tutorial update and fixups --- .../literate-tutorials/linear_elasticity.jl | 219 +++++++++++------- 1 file changed, 133 insertions(+), 86 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 0366eb5718..8b39e34ed8 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -6,28 +6,34 @@ # #md # !!! tip #md # This example is also available as a Jupyter notebook: -#md # [`linear_elasticity.ipynb`](@__NBVIEWER_ROOT_URL__/examples/linear_elasticity.ipynb). +#md # [`linear_elasticity.ipynb`](@__NBVIEWER_ROOT_URL__/tutorials/linear_elasticity.ipynb). #- # # ## Introduction # # The classical first finite element problem to solve in solid mechanics is a linear balance -# of momentum problem. We will use this to introduce a vector valued field, as well as the -# [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox. +# of momentum problem. We will use this to introduce a vector valued field, the displacements +# $\boldsymbol{u}$. In addition, some features of the +# [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox is demonstrated. # -# The strong form of the balance of momentum is given by +# The strong form of the balance of momentum for quasi-static loading is given by # ```math -# -\boldsymbol{\sigma} \cdot \boldsymbol{\nabla} = \boldsymbol{b} \quad \textbf{x} \in \Omega, +# \begin{align*} +# \boldsymbol{\sigma} \cdot \boldsymbol{\nabla} + \boldsymbol{b} &= 0 \quad \boldsymbol{x} \in \Omega \\ +# \boldsymbol{u} &= \boldsymbol{u}_\mathrm{D} \quad \boldsymbol{x} \in \Gamma_\mathrm{D} \\ +# \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad \boldsymbol{x} \in \Gamma_\mathrm{N} \\ +# \end{align*} # ``` -# where $\boldsymbol{\sigma}$ is the stress tensor, $\boldsymbol{b}$ is the body force and -# $\Omega$ the domain. +# where $\boldsymbol{\sigma}$ is the (Cauchy) stress tensor and $\boldsymbol{b}$ the body force. +# The domain, $\Omega$, has the boundary $\Gamma$, consisting of a Dirichlet part, $\Gamma_\mathrm{D}$, and +# a Neumann part, $\Gamma_\mathrm{N}$. # # In this example, we use linear elasticity, such that # ```math -# \boldsymbol{\sigma} = \boldsymbol{E} : \boldsymbol \varepsilon +# \boldsymbol{\sigma} = \boldsymbol{\mathsf{E}} : \boldsymbol{\varepsilon} # ``` -# where $\boldsymbol{E}$ is the elastic stiffness tensor and $\boldsymbol{\varepsilon}$ is -# the small strain tensor that is computed from the displacement field $\boldsymbol{u}$ as +# where $\boldsymbol{\mathsf{E}}$ is the elastic stiffness tensor and $\boldsymbol{\varepsilon}$ is +# the small strain tensor, # ```math # \boldsymbol{\varepsilon} = \frac{1}{2} \left( # \boldsymbol{\nabla} \otimes \boldsymbol{u} @@ -38,53 +44,62 @@ # # The resulting weak form is given given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that # ```math -# \int_\Omega -# \boldsymbol{\sigma} : \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) +# \int_\Omega +# \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} # \, \mathrm{d}V # = # \int_{\partial\Omega} -# \boldsymbol{t}^\ast \cdot \delta \boldsymbol{u} +# \delta \boldsymbol{u} \cdot \boldsymbol{t} # \, \mathrm{d}A # + # \int_\Omega -# \boldsymbol{b} \cdot \delta \boldsymbol{u} +# \delta \boldsymbol{u} \cdot \boldsymbol{b} # \, \mathrm{d}V # \quad \forall \, \delta \boldsymbol{u} \in \mathbb{T}, # ``` # where $\delta \boldsymbol{u}$ is a vector valued test function, and where $\mathbb{U}$ and # $\mathbb{T}$ are suitable trial and test function sets, respectively. The boundary traction -# is denoted $\boldsymbol{t}^\ast$ and body forces are denoted $\boldsymbol{b}$. +# is denoted $\boldsymbol{t}$ and body forces are denoted $\boldsymbol{b}$. # # However, for this example we will neglect body foces and thus the weak form reads: # ```math -# \int_\Omega -# \boldsymbol{\sigma} : \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) +# \int_\Omega +# \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} # \, \mathrm{d}V # = # \int_{\partial\Omega} -# \boldsymbol{t}^\ast \cdot \delta \boldsymbol{u} +# \delta \boldsymbol{u} \cdot \boldsymbol{t} # \, \mathrm{d}A \,. # ``` # # Finally, we choose to operate on a 2-dimensional problem under plain strain conditions. # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays -# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! This is done by loading [`logo.geo`](logo.geo) with [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl) here. +# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! +# This is done by downloading [`logo.geo`](logo.geo) and loading it using [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl), +using Downloads: download +logo_mesh = "logo.geo" +asset_url = "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/" +isfile(logo_mesh) || download(string(asset_url, logo_mesh), logo_mesh) + FerriteGmsh.Gmsh.initialize() # hide FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide -grid = togrid("logo.geo"); +grid = togrid(logo_mesh); #md nothing # hide -# By default the grid lacks the facesets for the boundaries, so we add them by Ferrite here. -# [`addfaceset!`](@ref) allows to add facesets to the grid based on coordinates. +# By default the grid lacks the facetsets for the boundaries, so we add them by Ferrite here. +# [`addfacetset!`](@ref) allows to add facetsets to the grid based on coordinates. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. -addfaceset!(grid, "top", x->x[2] ≈ 1.0) # faces for which x[2] ≈ 1.0 for all nodes -addfaceset!(grid, "left", x->x[1] < 1e-6) -addfaceset!(grid, "bottom", x->x[2] < 1e-6); +addfacetset!(grid, "top", x -> x[2] ≈ 1.0) # faces for which x[2] ≈ 1.0 for all nodes +addfacetset!(grid, "left", x -> abs(x[1]) < 1e-6) +addfacetset!(grid, "bottom", x -> abs(x[2]) < 1e-6); # ### Trial and test functions -# We use linear Lagrange functions as test and trial functions. The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). +# We use linear Lagrange functions as test and trial functions. +# The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. +# All currently available interpolations can be found under [`Interpolation`](@ref). # -# Since the displacement field $\boldsymbol{u}$ is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ to approximate the test and trial functions: +# Since the displacement field $\boldsymbol{u}$ is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ +# to approximate the test and trial functions: # ```math # \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i # \qquad @@ -93,7 +108,7 @@ addfaceset!(grid, "bottom", x->x[2] < 1e-6); # Here $N$ is the number of nodal variables and $\hat{u}_i$ / $\delta\hat{u}_i$ represent the i-th nodal value. # Using the Einstein summation convention, we can write this in short form as # $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \boldsymbol{N}_i \, \delta\hat{u}_i$. -# +# # Here we use linear triangular elements (also called constant strain triangles) with a single # quadrature point. # The vector valued shape functions are constructed by raising the interpolation @@ -104,8 +119,8 @@ order = 1 # linear interpolation ip = Lagrange{RefTriangle, order}()^dim # vector valued interpolation qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point cellvalues = CellValues(qr, ip) -qr_face = FaceQuadratureRule{RefTriangle}(1) -facevalues = FaceValues(qr_face, ip); +qr_face = FacetQuadratureRule{RefTriangle}(1) +facetvalues = FacetValues(qr_face, ip); # ### Degrees of freedom # For distributing degrees of freedom, we define a `DofHandler`. The `DofHandler` knows that @@ -115,37 +130,70 @@ add!(dh, :u, ip) close!(dh); # ### Boundary conditions -# Now, we add boundary conditions. We simply support the bottom and the left side and -# prescribe a displacement upwards on the top edge. -# The last argument to `Dirichlet` determines which components of the field should be -# constrained. +# We set Dirichlet boundary conditions by fixing the motion normal to the bottom and left +# boundaries. The last argument to `Dirichlet` determines which components of the field should be +# constrained. If no argument is given, all components are constrained by default. ch = ConstraintHandler(dh) -add!(ch, Dirichlet(:u, getfaceset(grid, "bottom"), (x, t) -> 0.0, 2)) -add!(ch, Dirichlet(:u, getfaceset(grid, "left"), (x, t) -> 0.0, 1)) -#add!(ch, Dirichlet(:u, getfaceset(grid, "top"), (x, t) -> 0.1, 2)) +add!(ch, Dirichlet(:u, getfacetset(grid, "bottom"), (x, t) -> 0.0, 2)) +add!(ch, Dirichlet(:u, getfacetset(grid, "left"), (x, t) -> 0.0, 1)) close!(ch); +# In addition, we will use Neumann boundary conditions on the top surface, where +# we add a traction vector of the form +# ```math +# \boldsymbol{t}_\mathrm{N}(\boldsymbol{x}) = (20e3) x_1 \boldsymbol{e}_2 +# ``` +# On the right boundary, we don't do anything. This becomes a Neumann boundary +# with zero traction. +traction(x) = Vec(0.0, 20e3 * x[1]) + +function assemble_external_forces!(f_ext, dh, facetset, facetvalues, prescribed_traction) + fe_ext = zeros(getnbasefunctions(facetvalues)) + for face in FacetIterator(dh, facetset) + reinit!(facetvalues, face) + fill!(fe_ext, 0.0) + for qp in 1:getnquadpoints(facetvalues) + X = spatial_coordinate(facetvalues, qp, getcoordinates(face)) + tₚ = prescribed_traction(X) + dΓ = getdetJdV(facetvalues, qp) + for i in 1:getnbasefunctions(facetvalues) + Nᵢ = shape_value(facetvalues, qp, i) + fe_ext[i] += tₚ ⋅ Nᵢ * dΓ + end + end + assemble!(f_ext, celldofs(face), fe_ext) + end + return f_ext +end +#md nothing # hide + # ### Material behavior -# Next, we need to define the material behavior. In this example, we use linear elasticity, -# such that +# Next, we need to define the material behavior. +# In this example, we use plane strain linear isotropic elasticity, with Hooke's law as # ```math -# \boldsymbol{\sigma} = \boldsymbol{\mathrm{E}} : \boldsymbol \varepsilon +# \boldsymbol{\sigma} = 2G \boldsymbol{\varepsilon}^\mathrm{dev} + 3K \boldsymbol{\varepsilon}^\mathrm{vol} +# ``` +# where $G$ is the shear modulus and $K$ the bulk modulus. The volumetric, $\boldsymbol{\varepsilon}^\mathrm{vol}$, +# and deviatoric, $\boldsymbol{\varepsilon}^\mathrm{dev}$ strains, are defined as +# ```math +# \begin{align*} +# \boldsymbol{\varepsilon}^\mathrm{vol} &= \frac{\mathrm{tr}(\boldsymbol{\varepsilon})}{3}\boldsymbol{I} +# \boldsymbol{\varepsilon}^\mathrm{dev} &= \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^\mathrm{vol} +# \end{align*} # ``` -# where $\boldsymbol{\mathrm{E}}$ is the elastic stiffness tensor and -# $\boldsymbol{\varepsilon}$ is the small strain tensor. -# Consequently, we need to set up the elastic stiffness tensor $\boldsymbol{\mathrm{E}}$. # -# Here, we operate in plane strain conditions, keep in mind that the plane stress stiffness -# tensor is defined differently. +# Starting from Young's modulus, $E$, and Poisson's ratio, $\nu$, the shear and bulk modulus are +# ```math +# G = \frac{E}{2(1 + \nu)}, \quad K = \frac{E}{3(1 - 2\nu)} +# ``` +# Finally, the stiffness tensor can be calculated as the derivative of stress wrt. strain. +# Due to the linearity, we can calculate this at any point, and choose zero strain. Emod = 200e3 # Young's modulus [MPa] -ν = 0.3 # Poisson's ratio [-] -#md nothing # hide +ν = 0.3 # Poisson's ratio [-] -λ = Emod*ν / ((1 + ν) * (1 - 2ν)) # 1st Lamé parameter -μ = Emod / (2(1 + ν)) # 2nd Lamé parameter -I = one(SymmetricTensor{2, dim}) # 2nd order unit tensor -II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor -𝐄 = 2μ * II + λ * (I ⊗ I); # elastic stiffness tensor +Gmod = Emod / (2(1 + ν)) # Shear modulus +Kmod = Emod / (3(1 - 2ν)) # Bulk modulus +E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); # ### Element routine # The stiffness matrix follows from the weak form such that @@ -153,16 +201,16 @@ II = one(SymmetricTensor{4, dim}) # 4th order symmetric unit tensor # \left(\underline{\underline{K}}\right)_{ij} # = # \int_\Omega +# \boldsymbol{\nabla} \boldsymbol{N}_i # \left( # \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} # : # \boldsymbol{\nabla}^\mathrm{sym} \boldsymbol{N}_j # \right) -# : -# \boldsymbol{\nabla} \boldsymbol{N}_i +# # \, \mathrm{d}V # ``` -# The element routine computes the local stiffness matrix `ke` +# The element routine computes the local stiffness matrix `ke` # for a single element. `ke` is pre-allocated and reused for all elements. # # Note that the elastic stiffness tensor $\boldsymbol{\mathrm{E}}$ is constant. @@ -197,7 +245,8 @@ function assemble_global!(K, dh, cellvalues, ∂σ∂ε) assembler = start_assemble(K) ## Loop over all cells for cell in CellIterator(dh) - reinit!(cellvalues, cell) # update spatial derivatives based on element coordinates + ## Update the shape function gradients based on the cell coordinates + reinit!(cellvalues, cell) ## Compute element contribution assemble_cell!(ke, cellvalues, ∂σ∂ε) ## Assemble ke and fe into K and f @@ -207,50 +256,48 @@ function assemble_global!(K, dh, cellvalues, ∂σ∂ε) end #md nothing # hide -function assemble_external_forces!(f_ext, dh, faceset, facevalues, prescribed_traction) - fe_ext = zeros(getnbasefunctions(facevalues)) - for face in FaceIterator(dh, faceset) - reinit!(facevalues, face) - fill!(fe_ext, 0.0) - for qp in 1:getnquadpoints(facevalues) - X = spatial_coordinate(facevalues, qp, getcoordinates(face)) - tₚ = prescribed_traction(X) - dΓ = getdetJdV(facevalues, qp) - for i in 1:getnbasefunctions(facevalues) - Nᵢ = shape_value(facevalues, qp, i) - fe_ext[i] += tₚ ⋅ Nᵢ * dΓ - end - end - assemble!(f_ext, celldofs(face), fe_ext) - end - return f_ext -end -#md nothing # hide - # ### Solution of the system # The last step is to solve the system. First we allocate the global stiffness matrix `K` # and assemble it. -K = create_sparsity_pattern(dh) -assemble_global!(K, dh, cellvalues, ∂σ∂ε); -# Then we allocate the external force vector. -f_ext = zeros(ndofs(dh)); -assemble_external_forces!(f_ext, dh, getfaceset(grid, "top"), facevalues, x->Vec(0.0, 20e3*x[1])) +K = allocate_matrix(dh) +assemble_global!(K, dh, cellvalues, E4); +# Then we allocate and assemble the external force vector. +f_ext = zeros(ndofs(dh)) +assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, x->Vec(0.0, 20e3*x[1])); # To account for the Dirichlet boundary conditions we use the `apply!` function. -# This modifies elements in `K` and `f` respectively, such that -# we can get the correct solution vector `u` by using `\`. +# This modifies elements in `K` and `f` respectively, such that we can get the +# correct solution vector `u` by using solving the linear equation system $K_{ij} u_j = f^\mathrm{ext}_i$, apply!(K, f_ext, ch) u = K \ f_ext; # ### Exporting to VTK # To visualize the result we export the grid and our field `u` # to a VTK-file, which can be viewed in e.g. [ParaView](https://www.paraview.org/). -vtk_grid("linear_elasticity", dh) do vtk - vtk_point_data(vtk, dh, u) - vtk_cellset(vtk, grid) # export cellsets of grains for logo-coloring +# For fun, to color the logo, we create cell data with numbers according to the color. +color_data = zeros(Int, getncells(grid)) +colors = Dict( + "1" => 1, "5" => 1, # purple + "2" => 2, "3" => 2, # red + "4" => 3, # blue + "6" => 4 # green + ) +for (key, color) in colors + for i in getcellset(grid, key) + color_data[i] = color + end +end + +VTKFile("linear_elasticity", dh) do vtk + write_solution(vtk, dh, u) + write_cell_data(vtk, color_data, "colors") end #md nothing # hide +using Test #hide +@test norm(u) ≈ 0.3179358535755679 #hide +nothing #hide + #md # ## [Plain program](@id linear_elasticity-plain-program) #md # #md # Here follows a version of the program without any comments. From 3b8c26e472563735064bcbb5a7acd66ad99c868d Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 4 Aug 2024 13:16:33 +0200 Subject: [PATCH 16/33] Further improvements, update figure --- docs/download_resources.jl | 1 - .../literate-tutorials/linear_elasticity.jl | 17 +++++++++-------- .../literate-tutorials/linear_elasticity.png | Bin 121183 -> 211452 bytes 3 files changed, 9 insertions(+), 9 deletions(-) diff --git a/docs/download_resources.jl b/docs/download_resources.jl index b82ff846d0..2dfbeffb2f 100644 --- a/docs/download_resources.jl +++ b/docs/download_resources.jl @@ -5,7 +5,6 @@ const directory = joinpath(@__DIR__, "src", "tutorials") mkpath(directory) for (file, url) in [ - "logo.geo" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/logo.geo", "periodic-rve.msh" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/periodic-rve.msh", "periodic-rve-coarse.msh" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/periodic-rve-coarse.msh", "transient_heat.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/transient_heat.gif", diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 8b39e34ed8..6a5b657cfe 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -98,14 +98,14 @@ addfacetset!(grid, "bottom", x -> abs(x[2]) < 1e-6); # The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. # All currently available interpolations can be found under [`Interpolation`](@ref). # -# Since the displacement field $\boldsymbol{u}$ is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ +# Since the displacement field, $\boldsymbol{u}$, is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ # to approximate the test and trial functions: # ```math # \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i # \qquad # \delta \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \delta \hat{u}_i # ``` -# Here $N$ is the number of nodal variables and $\hat{u}_i$ / $\delta\hat{u}_i$ represent the i-th nodal value. +# Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal value. # Using the Einstein summation convention, we can write this in short form as # $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \boldsymbol{N}_i \, \delta\hat{u}_i$. # @@ -198,14 +198,14 @@ E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTenso # ### Element routine # The stiffness matrix follows from the weak form such that # ```math -# \left(\underline{\underline{K}}\right)_{ij} +# K_{ij} # = # \int_\Omega -# \boldsymbol{\nabla} \boldsymbol{N}_i # \left( +# (\boldsymbol{N}_i \otimes \boldsymbol{\nabla}) : # \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} # : -# \boldsymbol{\nabla}^\mathrm{sym} \boldsymbol{N}_j +# (\boldsymbol{N}_j \otimes \boldsymbol{\nabla})^\mathrm{sym} # \right) # # \, \mathrm{d}V @@ -213,7 +213,7 @@ E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTenso # The element routine computes the local stiffness matrix `ke` # for a single element. `ke` is pre-allocated and reused for all elements. # -# Note that the elastic stiffness tensor $\boldsymbol{\mathrm{E}}$ is constant. +# Note that the elastic stiffness tensor $\boldsymbol{\mathsf{E}}$ is constant. # Thus is needs to be computed and once and can then be used for all integration points. function assemble_cell!(ke, cellvalues, ∂σ∂ε) fill!(ke, 0.0) @@ -272,9 +272,10 @@ apply!(K, f_ext, ch) u = K \ f_ext; # ### Exporting to VTK -# To visualize the result we export the grid and 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z-_j5>6_lYjt-M1G;efPF)SEHfU*D4Hi{m=Aca)PMFH2cc1sdJB2!0RyAFk~fBnF^@ zW-*+|k#maqNGu~G6$AMVQhLH#v6u8ZWe#mGe#*FW-ynbd*;4vr^VnULcS|Za_@1?* w0vH3D@P+07Z Date: Sun, 4 Aug 2024 15:46:22 +0200 Subject: [PATCH 17/33] Fix test on linux --- docs/src/literate-tutorials/linear_elasticity.jl | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 6a5b657cfe..30aa033bf3 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -289,15 +289,18 @@ for (key, color) in colors end end +# And then we save to the vtk file. + VTKFile("linear_elasticity", dh) do vtk write_solution(vtk, dh, u) write_cell_data(vtk, color_data, "colors") end -#md nothing # hide -using Test #hide -@test norm(u) ≈ 0.3179358535755679 #hide -nothing #hide +using Test #hide +if Sys.islinux() # gmsh not os stable #hide + @test norm(u) ≈ 0.31742879147646924 #hide +end #hide +nothing #hide #md # ## [Plain program](@id linear_elasticity-plain-program) #md # From a906ec3078a577443321c5938271ffcebf373163 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 4 Aug 2024 16:30:19 +0200 Subject: [PATCH 18/33] Minor formatting --- docs/src/literate-tutorials/linear_elasticity.jl | 1 - 1 file changed, 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 30aa033bf3..11efd64937 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -290,7 +290,6 @@ for (key, color) in colors end # And then we save to the vtk file. - VTKFile("linear_elasticity", dh) do vtk write_solution(vtk, dh, u) write_cell_data(vtk, color_data, "colors") From c6783e4277a9fab896e20b769c2c70d15787756b Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 4 Aug 2024 16:39:40 +0200 Subject: [PATCH 19/33] Disable test that fails seemingly unrelated --- docs/src/literate-tutorials/ns_vs_diffeq.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/ns_vs_diffeq.jl b/docs/src/literate-tutorials/ns_vs_diffeq.jl index 28db2ea391..c8baef8e2f 100644 --- a/docs/src/literate-tutorials/ns_vs_diffeq.jl +++ b/docs/src/literate-tutorials/ns_vs_diffeq.jl @@ -586,7 +586,7 @@ close(pvd); using Test #hide -if IS_CI #hide +if false # Disable to avoid sporadic doc build failures "if IS_CI" #hide function compute_divergence(dh, u, cellvalues_v) #hide divv = 0.0 #hide for cell in CellIterator(dh) #hide From 26ceb4acb7a9acce0f2780fb2017bacb4bd48fa3 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 4 Aug 2024 17:10:23 +0200 Subject: [PATCH 20/33] Fix some traction-related formatting --- docs/src/literate-tutorials/linear_elasticity.jl | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 11efd64937..f57c336853 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -26,7 +26,9 @@ # ``` # where $\boldsymbol{\sigma}$ is the (Cauchy) stress tensor and $\boldsymbol{b}$ the body force. # The domain, $\Omega$, has the boundary $\Gamma$, consisting of a Dirichlet part, $\Gamma_\mathrm{D}$, and -# a Neumann part, $\Gamma_\mathrm{N}$. +# a Neumann part, $\Gamma_\mathrm{N}$, with outward pointing normal vector $\boldsymbol{n}$. +# $\boldsymbol{u}_\mathrm{D}$ denotes prescribed displacements on $\Gamma_\mathrm{D}$, +# while $\boldsymbol{t}_\mathrm{N}$ the known tractions on $\Gamma_\mathrm{N}$. # # In this example, we use linear elasticity, such that # ```math @@ -48,7 +50,7 @@ # \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} # \, \mathrm{d}V # = -# \int_{\partial\Omega} +# \int_{\Gamma} # \delta \boldsymbol{u} \cdot \boldsymbol{t} # \, \mathrm{d}A # + @@ -67,7 +69,7 @@ # \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} # \, \mathrm{d}V # = -# \int_{\partial\Omega} +# \int_{\Gamma} # \delta \boldsymbol{u} \cdot \boldsymbol{t} # \, \mathrm{d}A \,. # ``` From bc41b54315e3ad24dc9c1369a68def034b194bfd Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 5 Aug 2024 17:09:11 +0200 Subject: [PATCH 21/33] Apply solution by @termi-official Co-authored-by: Dennis Ogiermann --- docs/src/literate-tutorials/linear_elasticity.jl | 1 + docs/src/literate-tutorials/ns_vs_diffeq.jl | 2 +- 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index f57c336853..5a04223ec9 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -87,6 +87,7 @@ isfile(logo_mesh) || download(string(asset_url, logo_mesh), logo_mesh) FerriteGmsh.Gmsh.initialize() # hide FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide grid = togrid(logo_mesh); +FerriteGmsh.Gmsh.finalize(); #md nothing # hide # By default the grid lacks the facetsets for the boundaries, so we add them by Ferrite here. # [`addfacetset!`](@ref) allows to add facetsets to the grid based on coordinates. diff --git a/docs/src/literate-tutorials/ns_vs_diffeq.jl b/docs/src/literate-tutorials/ns_vs_diffeq.jl index c8baef8e2f..28db2ea391 100644 --- a/docs/src/literate-tutorials/ns_vs_diffeq.jl +++ b/docs/src/literate-tutorials/ns_vs_diffeq.jl @@ -586,7 +586,7 @@ close(pvd); using Test #hide -if false # Disable to avoid sporadic doc build failures "if IS_CI" #hide +if IS_CI #hide function compute_divergence(dh, u, cellvalues_v) #hide divv = 0.0 #hide for cell in CellIterator(dh) #hide From 1038b51f1e45b9ffff6fd6638cefd8f8d05e3df7 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 5 Aug 2024 18:01:18 +0200 Subject: [PATCH 22/33] Use VTKGridFile --- docs/src/literate-tutorials/linear_elasticity.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 5a04223ec9..b1e7a67b2d 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -293,7 +293,7 @@ for (key, color) in colors end # And then we save to the vtk file. -VTKFile("linear_elasticity", dh) do vtk +VTKGridFile("linear_elasticity", dh) do vtk write_solution(vtk, dh, u) write_cell_data(vtk, color_data, "colors") end From d123875489faaf25166289b01c2e0dfc89448d44 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Thu, 8 Aug 2024 13:39:32 +0200 Subject: [PATCH 23/33] Hopefully complete example --- .../literate-tutorials/linear_elasticity.jl | 197 +++++++++++------- 1 file changed, 118 insertions(+), 79 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index b1e7a67b2d..3d6a7f74f2 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -13,15 +13,16 @@ # # The classical first finite element problem to solve in solid mechanics is a linear balance # of momentum problem. We will use this to introduce a vector valued field, the displacements -# $\boldsymbol{u}$. In addition, some features of the +# $\boldsymbol{u}(\boldsymbol{x})$. In addition, some features of the # [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox is demonstrated. # +# ### Strong form # The strong form of the balance of momentum for quasi-static loading is given by # ```math # \begin{align*} # \boldsymbol{\sigma} \cdot \boldsymbol{\nabla} + \boldsymbol{b} &= 0 \quad \boldsymbol{x} \in \Omega \\ # \boldsymbol{u} &= \boldsymbol{u}_\mathrm{D} \quad \boldsymbol{x} \in \Gamma_\mathrm{D} \\ -# \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad \boldsymbol{x} \in \Gamma_\mathrm{N} \\ +# \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad \boldsymbol{x} \in \Gamma_\mathrm{N} # \end{align*} # ``` # where $\boldsymbol{\sigma}$ is the (Cauchy) stress tensor and $\boldsymbol{b}$ the body force. @@ -32,49 +33,66 @@ # # In this example, we use linear elasticity, such that # ```math -# \boldsymbol{\sigma} = \boldsymbol{\mathsf{E}} : \boldsymbol{\varepsilon} +# \boldsymbol{\sigma} = \boldsymbol{\mathsf{E}} : \boldsymbol{\varepsilon}, \quad +# \boldsymbol{\varepsilon} = \left(\boldsymbol{u} \otimes \boldsymbol{\nabla}\right)^\mathrm{sym} # ``` # where $\boldsymbol{\mathsf{E}}$ is the elastic stiffness tensor and $\boldsymbol{\varepsilon}$ is -# the small strain tensor, -# ```math -# \boldsymbol{\varepsilon} = \frac{1}{2} \left( -# \boldsymbol{\nabla} \otimes \boldsymbol{u} -# + -# \boldsymbol{u} \otimes \boldsymbol{\nabla} -# \right) -# ``` +# the small strain tensor. # +# ### Weak form # The resulting weak form is given given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that # ```math # \int_\Omega # \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} -# \, \mathrm{d}V +# \, \mathrm{d}\Omega # = # \int_{\Gamma} # \delta \boldsymbol{u} \cdot \boldsymbol{t} -# \, \mathrm{d}A +# \, \mathrm{d}\Gamma # + # \int_\Omega # \delta \boldsymbol{u} \cdot \boldsymbol{b} -# \, \mathrm{d}V +# \, \mathrm{d}\Omega # \quad \forall \, \delta \boldsymbol{u} \in \mathbb{T}, # ``` -# where $\delta \boldsymbol{u}$ is a vector valued test function, and where $\mathbb{U}$ and -# $\mathbb{T}$ are suitable trial and test function sets, respectively. The boundary traction -# is denoted $\boldsymbol{t}$ and body forces are denoted $\boldsymbol{b}$. -# -# However, for this example we will neglect body foces and thus the weak form reads: +# where $\delta \boldsymbol{u}$ is a vector valued test function and +# $\boldsymbol{t} = \boldsymbol{\sigma}\cdot\boldsymbol{n}$ is the boundary traction. +# $\mathbb{U}$ and $\mathbb{T}$ denote suitable trial and test function spaces. +# In this example, we will neglect body foces and the weak form reduces to # ```math # \int_\Omega # \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} -# \, \mathrm{d}V +# \, \mathrm{d}\Omega # = # \int_{\Gamma} # \delta \boldsymbol{u} \cdot \boldsymbol{t} -# \, \mathrm{d}A \,. +# \, \mathrm{d}\Gamma \,. # ``` # -# Finally, we choose to operate on a 2-dimensional problem under plain strain conditions. +# ### Finite element form +# Finally, the finite element form is obtained by introducing the finite element shape functions. +# Since the displacement field, $\boldsymbol{u}$, is vector valued, we use vector valued shape functions +# $\delta\boldsymbol{N}_i$ and $\boldsymbol{N}_i$ to approximate the test and trial functions: +# ```math +# \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i +# \qquad +# \delta \boldsymbol{u} \approx \sum_{i=1}^N \delta\boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \delta \hat{u}_i +# ``` +# Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal values. +# Using the Einstein summation convention, we can write this in short form as +# $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \delta\boldsymbol{N}_i \, \delta\hat{u}_i$. +# +# Inserting the these into the weak form, and noting that that the equation should hold for all $\delta \hat{u}_i$, we get +# ```math +# \underbrace{\int_\Omega \left(\delta \boldsymbol{N}_i \otimes \boldsymbol{\nabla}\right) : \boldsymbol{\sigma}\ \mathrm{d}\Omega}_{f_i^\mathrm{int}} = \underbrace{\int_\Gamma \delta \boldsymbol{N}_i \cdot \boldsymbol{t}\ \mathrm{d}\Gamma}_{f_i^\mathrm{ext}} +# ``` +# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathsf{E}}:\boldsymbol{\varepsilon}$, +# in the internal force vector, $f_i^\mathrm{int}$, yields the linear equation +# ```math +# \underbrace{\left(\int_\Omega \left(\delta \boldsymbol{N}_i \otimes \boldsymbol{\nabla}\right) : \boldsymbol{\mathsf{E}} : \left(\boldsymbol{N}_j \otimes \boldsymbol{\nabla}\right)^\mathrm{sym}\ \mathrm{d}\Omega\right)}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} +# ``` +# +# ## Implementation # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays # Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! @@ -88,7 +106,7 @@ FerriteGmsh.Gmsh.initialize() # hide FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide grid = togrid(logo_mesh); FerriteGmsh.Gmsh.finalize(); -#md nothing # hide +#md nothing #hide # By default the grid lacks the facetsets for the boundaries, so we add them by Ferrite here. # [`addfacetset!`](@ref) allows to add facetsets to the grid based on coordinates. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. @@ -97,32 +115,28 @@ addfacetset!(grid, "left", x -> abs(x[1]) < 1e-6) addfacetset!(grid, "bottom", x -> abs(x[2]) < 1e-6); # ### Trial and test functions -# We use linear Lagrange functions as test and trial functions. -# The grid is composed of triangular elements, thus we need the Lagrange functions defined on `RefTriangle`. -# All currently available interpolations can be found under [`Interpolation`](@ref). +# In this tutorial, we use linear Lagrange functions, and apply the same functions as test and trial, +# $\delta\boldsymbol{N}_i = \boldsymbol{N}_i$, following Galerkin's method. +# As our grid is composed of triangular elements, we need the Lagrange functions defined +# on a `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). # -# Since the displacement field, $\boldsymbol{u}$, is vector valued, we use vector valued shape functions $\boldsymbol{N}_i$ -# to approximate the test and trial functions: -# ```math -# \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i -# \qquad -# \delta \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \delta \hat{u}_i -# ``` -# Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal value. -# Using the Einstein summation convention, we can write this in short form as -# $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \boldsymbol{N}_i \, \delta\hat{u}_i$. -# -# Here we use linear triangular elements (also called constant strain triangles) with a single -# quadrature point. +# Here we use linear triangular elements (also called constant strain triangles). # The vector valued shape functions are constructed by raising the interpolation # to the power `dim` (the dimension) since the displacement field has one component in each # spatial dimension. dim = 2 order = 1 # linear interpolation -ip = Lagrange{RefTriangle, order}()^dim # vector valued interpolation +ip = Lagrange{RefTriangle, order}()^dim; # vector valued interpolation + +# In order to evaluate the integrals, we need to specify the quadrature rules to use. Due to the +# linear interpolation, a single quadrature point suffices, both inside the cell and on the facet. +# In 2d, a facet is the edge of the element. qr = QuadratureRule{RefTriangle}(1) # 1 quadrature point +qr_face = FacetQuadratureRule{RefTriangle}(1); + +# Finally, we collect the interpolations and quadrature rules into the `CellValues` and `FacetValues` +# buffers, which we will later use to evaluate the integrals over the cells and facets. cellvalues = CellValues(qr, ip) -qr_face = FacetQuadratureRule{RefTriangle}(1) facetvalues = FacetValues(qr_face, ip); # ### Degrees of freedom @@ -146,29 +160,37 @@ close!(ch); # ```math # \boldsymbol{t}_\mathrm{N}(\boldsymbol{x}) = (20e3) x_1 \boldsymbol{e}_2 # ``` -# On the right boundary, we don't do anything. This becomes a Neumann boundary -# with zero traction. traction(x) = Vec(0.0, 20e3 * x[1]) +# On the right boundary, we don't do anything, resulting in a zero traction Neumann boundary. +# In order to assemble the external forces, $f_i^\mathrm{ext}$, we need to iterate over all +# facets in the relevant facetset. We do this by using the [`FacetIterator`](@ref). + function assemble_external_forces!(f_ext, dh, facetset, facetvalues, prescribed_traction) + ## Create a temporary array for the facet's local contributions to the external force vector fe_ext = zeros(getnbasefunctions(facetvalues)) for face in FacetIterator(dh, facetset) + ## Update the facetvalues to the correct facet number reinit!(facetvalues, face) + ## Reset the temporary array for the next facet fill!(fe_ext, 0.0) for qp in 1:getnquadpoints(facetvalues) - X = spatial_coordinate(facetvalues, qp, getcoordinates(face)) - tₚ = prescribed_traction(X) + ## Calculate the global coordinate of the quadrature point. + x = spatial_coordinate(facetvalues, qp, getcoordinates(face)) + tₚ = prescribed_traction(x) + ## Get the integration weight for the current quadrature point. dΓ = getdetJdV(facetvalues, qp) for i in 1:getnbasefunctions(facetvalues) Nᵢ = shape_value(facetvalues, qp, i) fe_ext[i] += tₚ ⋅ Nᵢ * dΓ end end + ## Add the local contributions to the correct indices in the global external force vector assemble!(f_ext, celldofs(face), fe_ext) end return f_ext end -#md nothing # hide +#md nothing #hide # ### Material behavior # Next, we need to define the material behavior. @@ -180,7 +202,7 @@ end # and deviatoric, $\boldsymbol{\varepsilon}^\mathrm{dev}$ strains, are defined as # ```math # \begin{align*} -# \boldsymbol{\varepsilon}^\mathrm{vol} &= \frac{\mathrm{tr}(\boldsymbol{\varepsilon})}{3}\boldsymbol{I} +# \boldsymbol{\varepsilon}^\mathrm{vol} &= \frac{\mathrm{tr}(\boldsymbol{\varepsilon})}{3}\boldsymbol{I}, \quad # \boldsymbol{\varepsilon}^\mathrm{dev} &= \boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^\mathrm{vol} # \end{align*} # ``` @@ -198,43 +220,54 @@ Gmod = Emod / (2(1 + ν)) # Shear modulus Kmod = Emod / (3(1 - 2ν)) # Bulk modulus E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); +#md # !!! details "Plane stress instead of plane strain?" +#md # In order to change this example to consider plane stress instead of plane strain, +#md # the stiffness tensor should be changed to reflect this. For example, the plane stress +#md # elasticity stiffness in voigt notation, using engineering strains, is given as +#md # ```math +#md # \underline{\underline{\boldsymbol{E}}} = \frac{E}{1 - \nu^2}\begin{bmatrix} +#md # 1 & \nu & 0 \\ +#md # \nu & 1 & 0 \\ +#md # 0 & 0 & (1 - \nu)/2 +#md # \end{bmatrix} +#md # ``` +#md # Which can be converted into a 4th order tensor as +#md # ```julia +#md # E_voigt = Emod * [1.0 ν 0.0; ν 1.0 0.0; 0.0 0.0 (1-ν)/2] / (1 - ν^2) +#md # E4 = fromvoigt(SymmetricTensor{4,2}, E_voigt) +#md # ``` +#md # # ### Element routine -# The stiffness matrix follows from the weak form such that -# ```math -# K_{ij} -# = -# \int_\Omega -# \left( -# (\boldsymbol{N}_i \otimes \boldsymbol{\nabla}) : -# \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} -# : -# (\boldsymbol{N}_j \otimes \boldsymbol{\nabla})^\mathrm{sym} -# \right) -# -# \, \mathrm{d}V -# ``` -# The element routine computes the local stiffness matrix `ke` -# for a single element. `ke` is pre-allocated and reused for all elements. +# To calculate the global stiffness matrix, $K_{ij}$, the element routine computes the +# local stiffness matrix `ke` for a single element and assembles it into the global matrix. +# `ke` is pre-allocated and reused for all elements. # # Note that the elastic stiffness tensor $\boldsymbol{\mathsf{E}}$ is constant. # Thus is needs to be computed and once and can then be used for all integration points. function assemble_cell!(ke, cellvalues, ∂σ∂ε) - fill!(ke, 0.0) - n_basefuncs = getnbasefunctions(cellvalues) for q_point in 1:getnquadpoints(cellvalues) + ## Get the integration weight for the quadrature point dΩ = getdetJdV(cellvalues, q_point) - for i in 1:n_basefuncs - ∇Nᵢ = shape_gradient(cellvalues, q_point, i)# shape_symmetric_gradient(cellvalues, q_point, i) - for j in 1:n_basefuncs + for i in 1:getnbasefunctions(cellvalues) + ## Gradient of the test function + ∇Nᵢ = shape_gradient(cellvalues, q_point, i) + for j in 1:getnbasefunctions(cellvalues) + ## Symmetric gradient of the trial function ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) - ke[i, j] += (∂σ∂ε ⊡ ∇ˢʸᵐNⱼ) ⊡ ∇Nᵢ * dΩ + ke[i, j] += (∇Nᵢ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ) * dΩ end end end return ke end -#md nothing # hide - +#md nothing #hide +# +#md # !!! details "Performance tips" +#md # 1. In this example, $\mathsf{\boldsymbol{E}}$, is symmetric, and we could have used +#md # the symmetric part of test function gradient above. +#md # 2. The product `∇Nᵢ ⊡ ∂σ∂ε` can be done outside the "j"-loop, but is kept inside here +#md # to highlight the similarity to the finite element form. +#md # # ### Global assembly # We define the function `assemble_global` to loop over the elements and do the global # assembly. The function takes the preallocated sparse matrix `K`, our DofHandler `dh`, our @@ -250,14 +283,16 @@ function assemble_global!(K, dh, cellvalues, ∂σ∂ε) for cell in CellIterator(dh) ## Update the shape function gradients based on the cell coordinates reinit!(cellvalues, cell) + ## Reset the element stiffness matrix + fill!(ke, 0.0) ## Compute element contribution assemble_cell!(ke, cellvalues, ∂σ∂ε) - ## Assemble ke and fe into K and f + ## Assemble ke into K assemble!(assembler, celldofs(cell), ke) end return K end -#md nothing # hide +#md nothing #hide # ### Solution of the system # The last step is to solve the system. First we allocate the global stiffness matrix `K` @@ -266,7 +301,7 @@ K = allocate_matrix(dh) assemble_global!(K, dh, cellvalues, E4); # Then we allocate and assemble the external force vector. f_ext = zeros(ndofs(dh)) -assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, x->Vec(0.0, 20e3*x[1])); +assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, traction); # To account for the Dirichlet boundary conditions we use the `apply!` function. # This modifies elements in `K` and `f` respectively, such that we can get the @@ -291,6 +326,7 @@ for (key, color) in colors color_data[i] = color end end +#md nothing #hide # And then we save to the vtk file. VTKGridFile("linear_elasticity", dh) do vtk @@ -298,11 +334,14 @@ VTKGridFile("linear_elasticity", dh) do vtk write_cell_data(vtk, color_data, "colors") end -using Test #hide -if Sys.islinux() # gmsh not os stable #hide - @test norm(u) ≈ 0.31742879147646924 #hide -end #hide -nothing #hide +# The mesh produced by gmsh is not stable between different OS #hide +# For linux, we therefore test carefully, and for other OS we provide #hide +# a coarse test to indicate introduced errors before running CI. #hide +using Test #hide +linux_result = 0.31742879147646924 #hide +@test abs(norm(u) - linux_result) < 0.01 #hide +Sys.islinux() && @test norm(u) ≈ linux_result #hide +nothing #hide #md # ## [Plain program](@id linear_elasticity-plain-program) #md # From 95517a7fbeeab79a23dcc2fd78cfbbb293ed82a2 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Thu, 8 Aug 2024 14:20:36 +0200 Subject: [PATCH 24/33] Fix links and test hiding --- docs/src/literate-tutorials/linear_elasticity.jl | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 3d6a7f74f2..37621e042e 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -118,7 +118,8 @@ addfacetset!(grid, "bottom", x -> abs(x[2]) < 1e-6); # In this tutorial, we use linear Lagrange functions, and apply the same functions as test and trial, # $\delta\boldsymbol{N}_i = \boldsymbol{N}_i$, following Galerkin's method. # As our grid is composed of triangular elements, we need the Lagrange functions defined -# on a `RefTriangle`. All currently available interpolations can be found under [`Interpolation`](@ref). +# on a `RefTriangle`. All currently available interpolations can be found under +# [`Interpolation`](@ref reference-interpolation). # # Here we use linear triangular elements (also called constant strain triangles). # The vector valued shape functions are constructed by raising the interpolation @@ -334,9 +335,9 @@ VTKGridFile("linear_elasticity", dh) do vtk write_cell_data(vtk, color_data, "colors") end -# The mesh produced by gmsh is not stable between different OS #hide -# For linux, we therefore test carefully, and for other OS we provide #hide -# a coarse test to indicate introduced errors before running CI. #hide +# The mesh produced by gmsh is not stable between different OS #src +# For linux, we therefore test carefully, and for other OS we provide #src +# a coarse test to indicate introduced errors before running CI. #src using Test #hide linux_result = 0.31742879147646924 #hide @test abs(norm(u) - linux_result) < 0.01 #hide @@ -346,7 +347,7 @@ nothing #hide #md # ## [Plain program](@id linear_elasticity-plain-program) #md # #md # Here follows a version of the program without any comments. -#md # The file is also available here: [`linear_elasticity.jl`](linear_elasticity). +#md # The file is also available here: [`linear_elasticity.jl`](linear_elasticity.jl). #md # #md # ```julia #md # @__CODE__ From 30fbaf8f732ec6df480f3d6854b97838d3bc02a7 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sat, 10 Aug 2024 12:40:32 +0200 Subject: [PATCH 25/33] Improve figure and fix spelling/grammar/formatting --- docs/download_resources.jl | 1 + .../literate-tutorials/linear_elasticity.jl | 48 +++++++++---------- 2 files changed, 25 insertions(+), 24 deletions(-) diff --git a/docs/download_resources.jl b/docs/download_resources.jl index 2dfbeffb2f..d3ebca71ef 100644 --- a/docs/download_resources.jl +++ b/docs/download_resources.jl @@ -13,6 +13,7 @@ for (file, url) in [ "porous_media_0p25.inp" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/porous_media_0p25.inp", "reactive_surface.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/reactive_surface.gif", "nsdiffeq.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/nsdiffeq.gif", + "linear_elasticity.svg" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/bbc742bf3f04352df5ba941e3097c1ea19b43807/assets/linear_elasticity.svg", ] afile = joinpath(directory, file) if !isfile(afile) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 37621e042e..e01ca0e9f1 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -1,11 +1,11 @@ # # [Linear elasticity](@id tutorial-linear-elasticity) # -# ![](linear_elasticity.png) +# ![](linear_elasticity.svg) # # *Figure 1*: Linear elastically deformed Ferrite logo. # #md # !!! tip -#md # This example is also available as a Jupyter notebook: +#md # This tutorial is also available as a Jupyter notebook: #md # [`linear_elasticity.ipynb`](@__NBVIEWER_ROOT_URL__/tutorials/linear_elasticity.ipynb). #- # @@ -14,16 +14,16 @@ # The classical first finite element problem to solve in solid mechanics is a linear balance # of momentum problem. We will use this to introduce a vector valued field, the displacements # $\boldsymbol{u}(\boldsymbol{x})$. In addition, some features of the -# [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox is demonstrated. +# [`Tensors.jl`](https://github.com/Ferrite-FEM/Tensors.jl) toolbox are demonstrated. # # ### Strong form # The strong form of the balance of momentum for quasi-static loading is given by # ```math -# \begin{align*} -# \boldsymbol{\sigma} \cdot \boldsymbol{\nabla} + \boldsymbol{b} &= 0 \quad \boldsymbol{x} \in \Omega \\ -# \boldsymbol{u} &= \boldsymbol{u}_\mathrm{D} \quad \boldsymbol{x} \in \Gamma_\mathrm{D} \\ -# \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad \boldsymbol{x} \in \Gamma_\mathrm{N} -# \end{align*} +# \begin{alignat*}{2} +# \boldsymbol{\sigma} \cdot \boldsymbol{\nabla} + \boldsymbol{b} &= 0 \quad &&\boldsymbol{x} \in \Omega \\ +# \boldsymbol{u} &= \boldsymbol{u}_\mathrm{D} \quad &&\boldsymbol{x} \in \Gamma_\mathrm{D} \\ +# \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad &&\boldsymbol{x} \in \Gamma_\mathrm{N} +# \end{alignat*} # ``` # where $\boldsymbol{\sigma}$ is the (Cauchy) stress tensor and $\boldsymbol{b}$ the body force. # The domain, $\Omega$, has the boundary $\Gamma$, consisting of a Dirichlet part, $\Gamma_\mathrm{D}$, and @@ -31,7 +31,7 @@ # $\boldsymbol{u}_\mathrm{D}$ denotes prescribed displacements on $\Gamma_\mathrm{D}$, # while $\boldsymbol{t}_\mathrm{N}$ the known tractions on $\Gamma_\mathrm{N}$. # -# In this example, we use linear elasticity, such that +# In this tutorial, we use linear elasticity, such that # ```math # \boldsymbol{\sigma} = \boldsymbol{\mathsf{E}} : \boldsymbol{\varepsilon}, \quad # \boldsymbol{\varepsilon} = \left(\boldsymbol{u} \otimes \boldsymbol{\nabla}\right)^\mathrm{sym} @@ -58,7 +58,7 @@ # where $\delta \boldsymbol{u}$ is a vector valued test function and # $\boldsymbol{t} = \boldsymbol{\sigma}\cdot\boldsymbol{n}$ is the boundary traction. # $\mathbb{U}$ and $\mathbb{T}$ denote suitable trial and test function spaces. -# In this example, we will neglect body foces and the weak form reduces to +# In this tutorial, we will neglect body foces and the weak form reduces to # ```math # \int_\Omega # \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} @@ -95,20 +95,20 @@ # ## Implementation # First we load Ferrite, and some other packages we need. using Ferrite, FerriteGmsh, SparseArrays -# Like for the Heat Equation example, we will use a unit square - but here we'll load the grid of the Ferrite logo! +# As in the heat equation tutorial, we will use a unit square - but here we'll load the grid of the Ferrite logo! # This is done by downloading [`logo.geo`](logo.geo) and loading it using [`FerriteGmsh.jl`](https://github.com/Ferrite-FEM/FerriteGmsh.jl), using Downloads: download logo_mesh = "logo.geo" asset_url = "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/" isfile(logo_mesh) || download(string(asset_url, logo_mesh), logo_mesh) -FerriteGmsh.Gmsh.initialize() # hide +FerriteGmsh.Gmsh.initialize() #hide FerriteGmsh.Gmsh.gmsh.option.set_number("General.Verbosity", 2) #hide grid = togrid(logo_mesh); -FerriteGmsh.Gmsh.finalize(); +FerriteGmsh.Gmsh.finalize(); #hide #md nothing #hide -# By default the grid lacks the facetsets for the boundaries, so we add them by Ferrite here. -# [`addfacetset!`](@ref) allows to add facetsets to the grid based on coordinates. +# The generated grid lacks the facetsets for the boundaries, so we add them by using Ferrite's +# [`addfacetset!`](@ref). It allows us to add facetsets to the grid based on coordinates. # Note that approximate comparison to 0.0 doesn't work well, so we use a tolerance instead. addfacetset!(grid, "top", x -> x[2] ≈ 1.0) # faces for which x[2] ≈ 1.0 for all nodes addfacetset!(grid, "left", x -> abs(x[1]) < 1e-6) @@ -161,7 +161,7 @@ close!(ch); # ```math # \boldsymbol{t}_\mathrm{N}(\boldsymbol{x}) = (20e3) x_1 \boldsymbol{e}_2 # ``` -traction(x) = Vec(0.0, 20e3 * x[1]) +traction(x) = Vec(0.0, 20e3 * x[1]); # On the right boundary, we don't do anything, resulting in a zero traction Neumann boundary. # In order to assemble the external forces, $f_i^\mathrm{ext}$, we need to iterate over all @@ -195,7 +195,7 @@ end # ### Material behavior # Next, we need to define the material behavior. -# In this example, we use plane strain linear isotropic elasticity, with Hooke's law as +# In this tutorial, we use plane strain linear isotropic elasticity, with Hooke's law as # ```math # \boldsymbol{\sigma} = 2G \boldsymbol{\varepsilon}^\mathrm{dev} + 3K \boldsymbol{\varepsilon}^\mathrm{vol} # ``` @@ -222,9 +222,9 @@ Kmod = Emod / (3(1 - 2ν)) # Bulk modulus E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); #md # !!! details "Plane stress instead of plane strain?" -#md # In order to change this example to consider plane stress instead of plane strain, -#md # the stiffness tensor should be changed to reflect this. For example, the plane stress -#md # elasticity stiffness in voigt notation, using engineering strains, is given as +#md # In order to change this tutorial to consider plane stress instead of plane strain, +#md # the stiffness tensor should be changed to reflect this. The plane stress elasticity +#md # stiffness matrix in Voigt notation for engineering shear strains, is given as #md # ```math #md # \underline{\underline{\boldsymbol{E}}} = \frac{E}{1 - \nu^2}\begin{bmatrix} #md # 1 & \nu & 0 \\ @@ -232,7 +232,7 @@ E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTenso #md # 0 & 0 & (1 - \nu)/2 #md # \end{bmatrix} #md # ``` -#md # Which can be converted into a 4th order tensor as +#md # This matrix can be converted into the 4th order stiffness tensor as #md # ```julia #md # E_voigt = Emod * [1.0 ν 0.0; ν 1.0 0.0; 0.0 0.0 (1-ν)/2] / (1 - ν^2) #md # E4 = fromvoigt(SymmetricTensor{4,2}, E_voigt) @@ -264,7 +264,7 @@ end #md nothing #hide # #md # !!! details "Performance tips" -#md # 1. In this example, $\mathsf{\boldsymbol{E}}$, is symmetric, and we could have used +#md # 1. In this tutorial, `∂σ∂ε`, is minor symmetric, and we could have used #md # the symmetric part of test function gradient above. #md # 2. The product `∇Nᵢ ⊡ ∂σ∂ε` can be done outside the "j"-loop, but is kept inside here #md # to highlight the similarity to the finite element form. @@ -305,8 +305,8 @@ f_ext = zeros(ndofs(dh)) assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, traction); # To account for the Dirichlet boundary conditions we use the `apply!` function. -# This modifies elements in `K` and `f` respectively, such that we can get the -# correct solution vector `u` by using solving the linear equation system $K_{ij} u_j = f^\mathrm{ext}_i$, +# This modifies elements in `K` and `f`, such that we can get the +# correct solution vector `u` by using solving the linear equation system $K_{ij} \hat{u}_j = f^\mathrm{ext}_i$, apply!(K, f_ext, ch) u = K \ f_ext; From 6103a8148e34b587643324ed2324ac9c9670bf19 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 18 Aug 2024 19:31:59 +0200 Subject: [PATCH 26/33] Apply suggestions from reviews --- .../literate-tutorials/linear_elasticity.jl | 168 ++++++++++++------ 1 file changed, 112 insertions(+), 56 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index e01ca0e9f1..40f0b85fef 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -2,7 +2,7 @@ # # ![](linear_elasticity.svg) # -# *Figure 1*: Linear elastically deformed Ferrite logo. +# *Figure 1*: Linear elastically deformed 1mm $\times$ 1mm Ferrite logo. # #md # !!! tip #md # This tutorial is also available as a Jupyter notebook: @@ -20,7 +20,7 @@ # The strong form of the balance of momentum for quasi-static loading is given by # ```math # \begin{alignat*}{2} -# \boldsymbol{\sigma} \cdot \boldsymbol{\nabla} + \boldsymbol{b} &= 0 \quad &&\boldsymbol{x} \in \Omega \\ +# \mathrm{div}(\boldsymbol{\sigma}) + \boldsymbol{b} &= 0 \quad &&\boldsymbol{x} \in \Omega \\ # \boldsymbol{u} &= \boldsymbol{u}_\mathrm{D} \quad &&\boldsymbol{x} \in \Gamma_\mathrm{D} \\ # \boldsymbol{n} \cdot \boldsymbol{\sigma} &= \boldsymbol{t}_\mathrm{N} \quad &&\boldsymbol{x} \in \Gamma_\mathrm{N} # \end{alignat*} @@ -33,17 +33,20 @@ # # In this tutorial, we use linear elasticity, such that # ```math -# \boldsymbol{\sigma} = \boldsymbol{\mathsf{E}} : \boldsymbol{\varepsilon}, \quad -# \boldsymbol{\varepsilon} = \left(\boldsymbol{u} \otimes \boldsymbol{\nabla}\right)^\mathrm{sym} +# \boldsymbol{\sigma} = \boldsymbol{\mathsf{C}} : \boldsymbol{\varepsilon}, \quad +# \boldsymbol{\varepsilon} = \left[\mathrm{grad}(\boldsymbol{u})\right]^\mathrm{sym} # ``` -# where $\boldsymbol{\mathsf{E}}$ is the elastic stiffness tensor and $\boldsymbol{\varepsilon}$ is -# the small strain tensor. +# where $\boldsymbol{\mathsf{C}}$ is the 4th order elastic stiffness tensor and +# $\boldsymbol{\varepsilon}$ the small strain tensor. +# The colon, $:$, represents the double contraction, +# $\sigma_{ij} = \mathsf{C}_{ijkl} \varepsilon_{kl}$, and the superscript $\mathrm{sym}$ +# denotes the symmetric part. # # ### Weak form # The resulting weak form is given given as follows: Find ``\boldsymbol{u} \in \mathbb{U}`` such that # ```math # \int_\Omega -# \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} +# \mathrm{grad}(\delta \boldsymbol{u}) : \boldsymbol{\sigma} # \, \mathrm{d}\Omega # = # \int_{\Gamma} @@ -55,13 +58,13 @@ # \, \mathrm{d}\Omega # \quad \forall \, \delta \boldsymbol{u} \in \mathbb{T}, # ``` -# where $\delta \boldsymbol{u}$ is a vector valued test function and -# $\boldsymbol{t} = \boldsymbol{\sigma}\cdot\boldsymbol{n}$ is the boundary traction. -# $\mathbb{U}$ and $\mathbb{T}$ denote suitable trial and test function spaces. -# In this tutorial, we will neglect body foces and the weak form reduces to +# where $\mathbb{U}$ and $\mathbb{T}$ denote suitable trial and test function spaces. +# $\delta \boldsymbol{u}$ is a vector valued test function and +# $\boldsymbol{t} = \boldsymbol{\sigma}\cdot\boldsymbol{n}$ is the traction vector on +# the boundary. In this tutorial, we will neglect body foces and the weak form reduces to # ```math # \int_\Omega -# \left(\delta \boldsymbol{u} \otimes \boldsymbol{\nabla} \right) : \boldsymbol{\sigma} +# \mathrm{grad}(\delta \boldsymbol{u}) : \boldsymbol{\sigma} # \, \mathrm{d}\Omega # = # \int_{\Gamma} @@ -74,9 +77,9 @@ # Since the displacement field, $\boldsymbol{u}$, is vector valued, we use vector valued shape functions # $\delta\boldsymbol{N}_i$ and $\boldsymbol{N}_i$ to approximate the test and trial functions: # ```math -# \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \hat{u}_i +# \boldsymbol{u} \approx \sum_{i=1}^N \boldsymbol{N}_i (\boldsymbol{x}) \, \hat{u}_i # \qquad -# \delta \boldsymbol{u} \approx \sum_{i=1}^N \delta\boldsymbol{N}_i \left(\boldsymbol{x}\right) \, \delta \hat{u}_i +# \delta \boldsymbol{u} \approx \sum_{i=1}^N \delta\boldsymbol{N}_i (\boldsymbol{x}) \, \delta \hat{u}_i # ``` # Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal values. # Using the Einstein summation convention, we can write this in short form as @@ -84,12 +87,12 @@ # # Inserting the these into the weak form, and noting that that the equation should hold for all $\delta \hat{u}_i$, we get # ```math -# \underbrace{\int_\Omega \left(\delta \boldsymbol{N}_i \otimes \boldsymbol{\nabla}\right) : \boldsymbol{\sigma}\ \mathrm{d}\Omega}_{f_i^\mathrm{int}} = \underbrace{\int_\Gamma \delta \boldsymbol{N}_i \cdot \boldsymbol{t}\ \mathrm{d}\Gamma}_{f_i^\mathrm{ext}} +# \underbrace{\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\sigma}\ \mathrm{d}\Omega}_{f_i^\mathrm{int}} = \underbrace{\int_\Gamma \delta \boldsymbol{N}_i \cdot \boldsymbol{t}\ \mathrm{d}\Gamma}_{f_i^\mathrm{ext}} # ``` -# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathsf{E}}:\boldsymbol{\varepsilon}$, +# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathsf{C}}:\boldsymbol{\varepsilon}$, # in the internal force vector, $f_i^\mathrm{int}$, yields the linear equation # ```math -# \underbrace{\left(\int_\Omega \left(\delta \boldsymbol{N}_i \otimes \boldsymbol{\nabla}\right) : \boldsymbol{\mathsf{E}} : \left(\boldsymbol{N}_j \otimes \boldsymbol{\nabla}\right)^\mathrm{sym}\ \mathrm{d}\Omega\right)}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} +# \underbrace{\left[\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\mathsf{C}} : \left[\mathrm{grad}(\boldsymbol{N}_j)\right]^\mathrm{sym}\ \mathrm{d}\Omega\right]}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} # ``` # # ## Implementation @@ -115,8 +118,8 @@ addfacetset!(grid, "left", x -> abs(x[1]) < 1e-6) addfacetset!(grid, "bottom", x -> abs(x[2]) < 1e-6); # ### Trial and test functions -# In this tutorial, we use linear Lagrange functions, and apply the same functions as test and trial, -# $\delta\boldsymbol{N}_i = \boldsymbol{N}_i$, following Galerkin's method. +# In this tutorial, we use the same linear Lagrange shape functions to approximate both the +# test and trial spaces, i.e. $\delta\boldsymbol{N}_i = \boldsymbol{N}_i$. # As our grid is composed of triangular elements, we need the Lagrange functions defined # on a `RefTriangle`. All currently available interpolations can be found under # [`Interpolation`](@ref reference-interpolation). @@ -159,7 +162,7 @@ close!(ch); # In addition, we will use Neumann boundary conditions on the top surface, where # we add a traction vector of the form # ```math -# \boldsymbol{t}_\mathrm{N}(\boldsymbol{x}) = (20e3) x_1 \boldsymbol{e}_2 +# \boldsymbol{t}_\mathrm{N}(\boldsymbol{x}) = (20e3) x_1 \boldsymbol{e}_2\ \mathrm{N}/\mathrm{mm}^2 # ``` traction(x) = Vec(0.0, 20e3 * x[1]); @@ -175,9 +178,11 @@ function assemble_external_forces!(f_ext, dh, facetset, facetvalues, prescribed_ reinit!(facetvalues, face) ## Reset the temporary array for the next facet fill!(fe_ext, 0.0) + ## Access the cell's coordinates + cell_coordinates = getcoordinates(face) for qp in 1:getnquadpoints(facetvalues) ## Calculate the global coordinate of the quadrature point. - x = spatial_coordinate(facetvalues, qp, getcoordinates(face)) + x = spatial_coordinate(facetvalues, qp, cell_coordinates) tₚ = prescribed_traction(x) ## Get the integration weight for the current quadrature point. dΓ = getdetJdV(facetvalues, qp) @@ -194,12 +199,18 @@ end #md nothing #hide # ### Material behavior -# Next, we need to define the material behavior. +# Next, we need to define the material behavior, specifically the elasticity tensor $\mathsf{C}$. # In this tutorial, we use plane strain linear isotropic elasticity, with Hooke's law as # ```math # \boldsymbol{\sigma} = 2G \boldsymbol{\varepsilon}^\mathrm{dev} + 3K \boldsymbol{\varepsilon}^\mathrm{vol} # ``` -# where $G$ is the shear modulus and $K$ the bulk modulus. The volumetric, $\boldsymbol{\varepsilon}^\mathrm{vol}$, +# where $G$ is the shear modulus and $K$ the bulk modulus. The linearity of this expression implies that +# can be represented by the general expression, $\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$, +# from above, with +# ```math +# \mathsf{C} := \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} +# ``` +# The volumetric, $\boldsymbol{\varepsilon}^\mathrm{vol}$, # and deviatoric, $\boldsymbol{\varepsilon}^\mathrm{dev}$ strains, are defined as # ```math # \begin{align*} @@ -212,14 +223,15 @@ end # ```math # G = \frac{E}{2(1 + \nu)}, \quad K = \frac{E}{3(1 - 2\nu)} # ``` -# Finally, the stiffness tensor can be calculated as the derivative of stress wrt. strain. -# Due to the linearity, we can calculate this at any point, and choose zero strain. Emod = 200e3 # Young's modulus [MPa] ν = 0.3 # Poisson's ratio [-] Gmod = Emod / (2(1 + ν)) # Shear modulus Kmod = Emod / (3(1 - 2ν)) # Bulk modulus -E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); + +# Finally, we demonstrate `Tensors.jl`'s automatic differentiation capabilities when +# calculating the stiffness tensor +C = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); #md # !!! details "Plane stress instead of plane strain?" #md # In order to change this tutorial to consider plane stress instead of plane strain, @@ -234,8 +246,8 @@ E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTenso #md # ``` #md # This matrix can be converted into the 4th order stiffness tensor as #md # ```julia -#md # E_voigt = Emod * [1.0 ν 0.0; ν 1.0 0.0; 0.0 0.0 (1-ν)/2] / (1 - ν^2) -#md # E4 = fromvoigt(SymmetricTensor{4,2}, E_voigt) +#md # C_voigt = Emod * [1.0 ν 0.0; ν 1.0 0.0; 0.0 0.0 (1-ν)/2] / (1 - ν^2) +#md # C = fromvoigt(SymmetricTensor{4,2}, E_voigt) #md # ``` #md # # ### Element routine @@ -243,9 +255,9 @@ E4 = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTenso # local stiffness matrix `ke` for a single element and assembles it into the global matrix. # `ke` is pre-allocated and reused for all elements. # -# Note that the elastic stiffness tensor $\boldsymbol{\mathsf{E}}$ is constant. +# Note that the elastic stiffness tensor $\boldsymbol{\mathsf{C}}$ is constant. # Thus is needs to be computed and once and can then be used for all integration points. -function assemble_cell!(ke, cellvalues, ∂σ∂ε) +function assemble_cell!(ke, cellvalues, C) for q_point in 1:getnquadpoints(cellvalues) ## Get the integration weight for the quadrature point dΩ = getdetJdV(cellvalues, q_point) @@ -255,20 +267,16 @@ function assemble_cell!(ke, cellvalues, ∂σ∂ε) for j in 1:getnbasefunctions(cellvalues) ## Symmetric gradient of the trial function ∇ˢʸᵐNⱼ = shape_symmetric_gradient(cellvalues, q_point, j) - ke[i, j] += (∇Nᵢ ⊡ ∂σ∂ε ⊡ ∇ˢʸᵐNⱼ) * dΩ + ke[i, j] += (∇Nᵢ ⊡ C ⊡ ∇ˢʸᵐNⱼ) * dΩ end end end return ke end #md nothing #hide +# *Performance tip:* The product `∇Nᵢ ⊡ ∂σ∂ε` can be done outside the "j"-loop, +# but is kept inside here to highlight the similarity to the finite element form. # -#md # !!! details "Performance tips" -#md # 1. In this tutorial, `∂σ∂ε`, is minor symmetric, and we could have used -#md # the symmetric part of test function gradient above. -#md # 2. The product `∇Nᵢ ⊡ ∂σ∂ε` can be done outside the "j"-loop, but is kept inside here -#md # to highlight the similarity to the finite element form. -#md # # ### Global assembly # We define the function `assemble_global` to loop over the elements and do the global # assembly. The function takes the preallocated sparse matrix `K`, our DofHandler `dh`, our @@ -299,7 +307,7 @@ end # The last step is to solve the system. First we allocate the global stiffness matrix `K` # and assemble it. K = allocate_matrix(dh) -assemble_global!(K, dh, cellvalues, E4); +assemble_global!(K, dh, cellvalues, C); # Then we allocate and assemble the external force vector. f_ext = zeros(ndofs(dh)) assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, traction); @@ -310,31 +318,79 @@ assemble_external_forces!(f_ext, dh, getfacetset(grid, "top"), facetvalues, trac apply!(K, f_ext, ch) u = K \ f_ext; -# ### Exporting to VTK -# To visualize the result we export the grid and our field, `u`, -# to a VTK-file, which can be viewed in e.g. [ParaView](https://www.paraview.org/). -# For fun we'll color the logo, and thus create cell data with numbers according -# to the logo colors. -color_data = zeros(Int, getncells(grid)) -colors = Dict( - "1" => 1, "5" => 1, # purple - "2" => 2, "3" => 2, # red - "4" => 3, # blue - "6" => 4 # green - ) -for (key, color) in colors - for i in getcellset(grid, key) - color_data[i] = color +# ### Postprocessing +# In this case, we want to analyze the displacements, as well as the stress field. +# We calculate the stress in each quadrature point, and then export it in two different +# ways: +# 1) Constant in each cell (matching the approximation of constant strains in each element). +# Note that a current limitation is that cell data for second order tensors must be exported +# component-wise (see issue #768) +# 2) Interpolated using the linear lagrange ansatz functions via the [`L2Projector`](@ref). + +function calculate_stresses(grid, dh, cv, u, C) + qp_stresses = [ + [zero(SymmetricTensor{2,2}) for _ in 1:getnquadpoints(cv)] + for _ in 1:getncells(grid)] + avg_cell_stresses = tuple((zeros(getncells(grid)) for _ in 1:3)...) + for cell in CellIterator(dh) + reinit!(cv, cell) + cell_stresses = qp_stresses[cellid(cell)] + for q_point in 1:getnquadpoints(cv) + ε = function_symmetric_gradient(cv, q_point, u, celldofs(cell)) + cell_stresses[q_point] = C ⊡ ε + end + σ_avg = sum(cell_stresses) / getnquadpoints(cv) + avg_cell_stresses[1][cellid(cell)] = σ_avg[1, 1] + avg_cell_stresses[2][cellid(cell)] = σ_avg[2, 2] + avg_cell_stresses[3][cellid(cell)] = σ_avg[1, 2] end + return qp_stresses, avg_cell_stresses end -#md nothing #hide -# And then we save to the vtk file. +qp_stresses, avg_cell_stresses = calculate_stresses(grid, dh, cellvalues, u, C) + +# We now use the the L2Projector to project the stress-field onto the piecewise linear +# finite element space that we used to solve the problem. +proj = L2Projector(Lagrange{RefTriangle, 1}(), grid) +stress_field = project(proj, qp_stresses, qr) + +# Export also values for each color in the logo. #src +color_data = zeros(Int, getncells(grid)) #hide +colors = [ #hide + "1" => 1, "5" => 1, # purple #hide + "2" => 2, "3" => 2, # red #hide + "4" => 3, # blue #hide + "6" => 4 # green #hide + ] #hide +for (key, color) in colors #hide + for i in getcellset(grid, key) #hide + color_data[i] = color #hide + end #hide +end #hide +#md nothing #hide + +# To visualize the result we export to a VTK-file. Specifically, an unstructured +# grid file, `.vtu`, is created, which can be viewed in e.g. +# [ParaView](https://www.paraview.org/). VTKGridFile("linear_elasticity", dh) do vtk write_solution(vtk, dh, u) - write_cell_data(vtk, color_data, "colors") + for (i, key) in enumerate(("11", "22", "12")) + write_cell_data(vtk, avg_cell_stresses[i], "sigma_" * key) + end + write_projection(vtk, proj, stress_field, "stress field") + Ferrite.write_cellset(vtk, grid) + write_cell_data(vtk, color_data, "colors") #hide end +# We used the displacement field to visualize the deformed logo in *Figure 1*, +# and in *Figure 2*, we demonstrate the difference between the interpolated stress +# field and the constant stress in each cell. + +# ![](linear_elasticity_stress.png) +# +# *Figure 2*: Vertical normal stresses (MPa) exported using the `L2Projector` (left) +# and constant stress in each cell (right). + # The mesh produced by gmsh is not stable between different OS #src # For linux, we therefore test carefully, and for other OS we provide #src # a coarse test to indicate introduced errors before running CI. #src From c8936ae5f86b5449e6bb93fad16513f257262066 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 18 Aug 2024 19:55:14 +0200 Subject: [PATCH 27/33] Download stress results to elasticity tutorial --- docs/download_resources.jl | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/download_resources.jl b/docs/download_resources.jl index d3ebca71ef..1a78b46f7f 100644 --- a/docs/download_resources.jl +++ b/docs/download_resources.jl @@ -13,7 +13,8 @@ for (file, url) in [ "porous_media_0p25.inp" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/porous_media_0p25.inp", "reactive_surface.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/reactive_surface.gif", "nsdiffeq.gif" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/nsdiffeq.gif", - "linear_elasticity.svg" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/bbc742bf3f04352df5ba941e3097c1ea19b43807/assets/linear_elasticity.svg", + "linear_elasticity.svg" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/linear_elasticity.svg", + "linear_elasticity_stress.png" => "https://raw.githubusercontent.com/Ferrite-FEM/Ferrite.jl/gh-pages/assets/linear_elasticity_stress.png", ] afile = joinpath(directory, file) if !isfile(afile) From e9913bda8c1af2aa80306d4eadc1f4960d9c7bfa Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Sun, 18 Aug 2024 20:13:25 +0200 Subject: [PATCH 28/33] Supress outputs --- docs/src/literate-tutorials/linear_elasticity.jl | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 40f0b85fef..9042ddc788 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -347,12 +347,12 @@ function calculate_stresses(grid, dh, cv, u, C) return qp_stresses, avg_cell_stresses end -qp_stresses, avg_cell_stresses = calculate_stresses(grid, dh, cellvalues, u, C) +qp_stresses, avg_cell_stresses = calculate_stresses(grid, dh, cellvalues, u, C); # We now use the the L2Projector to project the stress-field onto the piecewise linear # finite element space that we used to solve the problem. proj = L2Projector(Lagrange{RefTriangle, 1}(), grid) -stress_field = project(proj, qp_stresses, qr) +stress_field = project(proj, qp_stresses, qr); # Export also values for each color in the logo. #src color_data = zeros(Int, getncells(grid)) #hide @@ -368,7 +368,7 @@ for (key, color) in colors #hide end #hide end #hide #md nothing #hide - +# # To visualize the result we export to a VTK-file. Specifically, an unstructured # grid file, `.vtu`, is created, which can be viewed in e.g. # [ParaView](https://www.paraview.org/). From fd58706b410bd03f5d1db81aadea8f4a538c2142 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 19 Aug 2024 15:41:27 +0200 Subject: [PATCH 29/33] Remove performance tips completely --- docs/src/literate-tutorials/linear_elasticity.jl | 2 -- 1 file changed, 2 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 9042ddc788..19dc7a01aa 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -274,8 +274,6 @@ function assemble_cell!(ke, cellvalues, C) return ke end #md nothing #hide -# *Performance tip:* The product `∇Nᵢ ⊡ ∂σ∂ε` can be done outside the "j"-loop, -# but is kept inside here to highlight the similarity to the finite element form. # # ### Global assembly # We define the function `assemble_global` to loop over the elements and do the global From dc3067b7da9acaa69fb383c9a6149f3f4ed706d7 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 19 Aug 2024 17:05:00 +0200 Subject: [PATCH 30/33] Apply suggestions from code review Co-authored-by: Dennis Ogiermann --- docs/src/literate-tutorials/linear_elasticity.jl | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 19dc7a01aa..ff53a1299d 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -61,7 +61,7 @@ # where $\mathbb{U}$ and $\mathbb{T}$ denote suitable trial and test function spaces. # $\delta \boldsymbol{u}$ is a vector valued test function and # $\boldsymbol{t} = \boldsymbol{\sigma}\cdot\boldsymbol{n}$ is the traction vector on -# the boundary. In this tutorial, we will neglect body foces and the weak form reduces to +# the boundary. In this tutorial, we will neglect body forces (i.e. $\boldsymbol{b} = \boldsymbol{0}$) and the weak form reduces to # ```math # \int_\Omega # \mathrm{grad}(\delta \boldsymbol{u}) : \boldsymbol{\sigma} @@ -81,7 +81,7 @@ # \qquad # \delta \boldsymbol{u} \approx \sum_{i=1}^N \delta\boldsymbol{N}_i (\boldsymbol{x}) \, \delta \hat{u}_i # ``` -# Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal values. +# Here $N$ is the number of nodal variables, with $\hat{u}_i$ and $\delta\hat{u}_i$ representing the $i$-th nodal value. # Using the Einstein summation convention, we can write this in short form as # $\boldsymbol{u} \approx \boldsymbol{N}_i \, \hat{u}_i$ and $\delta\boldsymbol{u} \approx \delta\boldsymbol{N}_i \, \delta\hat{u}_i$. # From d22365994fb27369a609055f3f427ea1e9eed681 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 19 Aug 2024 17:17:15 +0200 Subject: [PATCH 31/33] Adress review comments --- .../literate-tutorials/linear_elasticity.jl | 33 +++++++++---------- 1 file changed, 16 insertions(+), 17 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index ff53a1299d..27c5d842e8 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -33,13 +33,13 @@ # # In this tutorial, we use linear elasticity, such that # ```math -# \boldsymbol{\sigma} = \boldsymbol{\mathsf{C}} : \boldsymbol{\varepsilon}, \quad +# \boldsymbol{\sigma} = \boldsymbol{\mathbb{C}} : \boldsymbol{\varepsilon}, \quad # \boldsymbol{\varepsilon} = \left[\mathrm{grad}(\boldsymbol{u})\right]^\mathrm{sym} # ``` -# where $\boldsymbol{\mathsf{C}}$ is the 4th order elastic stiffness tensor and +# where $\boldsymbol{\mathbb{C}}$ is the 4th order elastic stiffness tensor and # $\boldsymbol{\varepsilon}$ the small strain tensor. # The colon, $:$, represents the double contraction, -# $\sigma_{ij} = \mathsf{C}_{ijkl} \varepsilon_{kl}$, and the superscript $\mathrm{sym}$ +# $\sigma_{ij} = \mathbb{C}_{ijkl} \varepsilon_{kl}$, and the superscript $\mathrm{sym}$ # denotes the symmetric part. # # ### Weak form @@ -89,10 +89,10 @@ # ```math # \underbrace{\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\sigma}\ \mathrm{d}\Omega}_{f_i^\mathrm{int}} = \underbrace{\int_\Gamma \delta \boldsymbol{N}_i \cdot \boldsymbol{t}\ \mathrm{d}\Gamma}_{f_i^\mathrm{ext}} # ``` -# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathsf{C}}:\boldsymbol{\varepsilon}$, +# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathbb{C}}:\boldsymbol{\varepsilon}$, # in the internal force vector, $f_i^\mathrm{int}$, yields the linear equation # ```math -# \underbrace{\left[\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\mathsf{C}} : \left[\mathrm{grad}(\boldsymbol{N}_j)\right]^\mathrm{sym}\ \mathrm{d}\Omega\right]}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} +# \underbrace{\left[\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\mathbb{C}} : \left[\mathrm{grad}(\boldsymbol{N}_j)\right]^\mathrm{sym}\ \mathrm{d}\Omega\right]}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} # ``` # # ## Implementation @@ -199,16 +199,15 @@ end #md nothing #hide # ### Material behavior -# Next, we need to define the material behavior, specifically the elasticity tensor $\mathsf{C}$. +# Next, we need to define the material behavior, specifically the elastic stiffness tensor, $\mathbb{C}$. # In this tutorial, we use plane strain linear isotropic elasticity, with Hooke's law as # ```math # \boldsymbol{\sigma} = 2G \boldsymbol{\varepsilon}^\mathrm{dev} + 3K \boldsymbol{\varepsilon}^\mathrm{vol} # ``` -# where $G$ is the shear modulus and $K$ the bulk modulus. The linearity of this expression implies that -# can be represented by the general expression, $\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$, -# from above, with +# where $G$ is the shear modulus and $K$ the bulk modulus. This expression can be written as +# $\boldsymbol{\sigma} = \mathbb{C}:\boldsymbol{\varepsilon}$, with # ```math -# \mathsf{C} := \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} +# \mathbb{C} := \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} # ``` # The volumetric, $\boldsymbol{\varepsilon}^\mathrm{vol}$, # and deviatoric, $\boldsymbol{\varepsilon}^\mathrm{dev}$ strains, are defined as @@ -230,12 +229,12 @@ Gmod = Emod / (2(1 + ν)) # Shear modulus Kmod = Emod / (3(1 - 2ν)) # Bulk modulus # Finally, we demonstrate `Tensors.jl`'s automatic differentiation capabilities when -# calculating the stiffness tensor +# calculating the elastic stiffness tensor C = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor{2,2})); #md # !!! details "Plane stress instead of plane strain?" #md # In order to change this tutorial to consider plane stress instead of plane strain, -#md # the stiffness tensor should be changed to reflect this. The plane stress elasticity +#md # the elastic stiffness tensor should be changed to reflect this. The plane stress elasticity #md # stiffness matrix in Voigt notation for engineering shear strains, is given as #md # ```math #md # \underline{\underline{\boldsymbol{E}}} = \frac{E}{1 - \nu^2}\begin{bmatrix} @@ -244,7 +243,7 @@ C = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor #md # 0 & 0 & (1 - \nu)/2 #md # \end{bmatrix} #md # ``` -#md # This matrix can be converted into the 4th order stiffness tensor as +#md # This matrix can be converted into the 4th order elastic stiffness tensor as #md # ```julia #md # C_voigt = Emod * [1.0 ν 0.0; ν 1.0 0.0; 0.0 0.0 (1-ν)/2] / (1 - ν^2) #md # C = fromvoigt(SymmetricTensor{4,2}, E_voigt) @@ -255,7 +254,7 @@ C = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor # local stiffness matrix `ke` for a single element and assembles it into the global matrix. # `ke` is pre-allocated and reused for all elements. # -# Note that the elastic stiffness tensor $\boldsymbol{\mathsf{C}}$ is constant. +# Note that the elastic stiffness tensor $\boldsymbol{\mathbb{C}}$ is constant. # Thus is needs to be computed and once and can then be used for all integration points. function assemble_cell!(ke, cellvalues, C) for q_point in 1:getnquadpoints(cellvalues) @@ -278,9 +277,9 @@ end # ### Global assembly # We define the function `assemble_global` to loop over the elements and do the global # assembly. The function takes the preallocated sparse matrix `K`, our DofHandler `dh`, our -# `cellvalues` and the elastic stiffness tensor `∂σ∂ε` as input arguments and computes the +# `cellvalues` and the elastic stiffness tensor `C` as input arguments and computes the # global stiffness matrix `K`. -function assemble_global!(K, dh, cellvalues, ∂σ∂ε) +function assemble_global!(K, dh, cellvalues, C) ## Allocate the element stiffness matrix n_basefuncs = getnbasefunctions(cellvalues) ke = zeros(n_basefuncs, n_basefuncs) @@ -293,7 +292,7 @@ function assemble_global!(K, dh, cellvalues, ∂σ∂ε) ## Reset the element stiffness matrix fill!(ke, 0.0) ## Compute element contribution - assemble_cell!(ke, cellvalues, ∂σ∂ε) + assemble_cell!(ke, cellvalues, C) ## Assemble ke into K assemble!(assembler, celldofs(cell), ke) end From 551dca37d07e46718aec9b0198015fe038bf5df4 Mon Sep 17 00:00:00 2001 From: Knut Andreas Meyer Date: Mon, 19 Aug 2024 20:03:41 +0200 Subject: [PATCH 32/33] 4th order tensor to mathsf --- .../literate-tutorials/linear_elasticity.jl | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/docs/src/literate-tutorials/linear_elasticity.jl b/docs/src/literate-tutorials/linear_elasticity.jl index 27c5d842e8..09e778f181 100644 --- a/docs/src/literate-tutorials/linear_elasticity.jl +++ b/docs/src/literate-tutorials/linear_elasticity.jl @@ -33,13 +33,13 @@ # # In this tutorial, we use linear elasticity, such that # ```math -# \boldsymbol{\sigma} = \boldsymbol{\mathbb{C}} : \boldsymbol{\varepsilon}, \quad +# \boldsymbol{\sigma} = \mathsf{C} : \boldsymbol{\varepsilon}, \quad # \boldsymbol{\varepsilon} = \left[\mathrm{grad}(\boldsymbol{u})\right]^\mathrm{sym} # ``` -# where $\boldsymbol{\mathbb{C}}$ is the 4th order elastic stiffness tensor and +# where $\mathsf{C}$ is the 4th order elastic stiffness tensor and # $\boldsymbol{\varepsilon}$ the small strain tensor. # The colon, $:$, represents the double contraction, -# $\sigma_{ij} = \mathbb{C}_{ijkl} \varepsilon_{kl}$, and the superscript $\mathrm{sym}$ +# $\sigma_{ij} = \mathsf{C}_{ijkl} \varepsilon_{kl}$, and the superscript $\mathrm{sym}$ # denotes the symmetric part. # # ### Weak form @@ -89,10 +89,10 @@ # ```math # \underbrace{\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\sigma}\ \mathrm{d}\Omega}_{f_i^\mathrm{int}} = \underbrace{\int_\Gamma \delta \boldsymbol{N}_i \cdot \boldsymbol{t}\ \mathrm{d}\Gamma}_{f_i^\mathrm{ext}} # ``` -# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \boldsymbol{\mathbb{C}}:\boldsymbol{\varepsilon}$, +# Inserting the linear constitutive relationship, $\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$, # in the internal force vector, $f_i^\mathrm{int}$, yields the linear equation # ```math -# \underbrace{\left[\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \boldsymbol{\mathbb{C}} : \left[\mathrm{grad}(\boldsymbol{N}_j)\right]^\mathrm{sym}\ \mathrm{d}\Omega\right]}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} +# \underbrace{\left[\int_\Omega \mathrm{grad}(\delta \boldsymbol{N}_i) : \mathsf{C} : \left[\mathrm{grad}(\boldsymbol{N}_j)\right]^\mathrm{sym}\ \mathrm{d}\Omega\right]}_{K_{ij}}\ \hat{u}_j = f_i^\mathrm{ext} # ``` # # ## Implementation @@ -199,15 +199,15 @@ end #md nothing #hide # ### Material behavior -# Next, we need to define the material behavior, specifically the elastic stiffness tensor, $\mathbb{C}$. +# Next, we need to define the material behavior, specifically the elastic stiffness tensor, $\mathsf{C}$. # In this tutorial, we use plane strain linear isotropic elasticity, with Hooke's law as # ```math # \boldsymbol{\sigma} = 2G \boldsymbol{\varepsilon}^\mathrm{dev} + 3K \boldsymbol{\varepsilon}^\mathrm{vol} # ``` # where $G$ is the shear modulus and $K$ the bulk modulus. This expression can be written as -# $\boldsymbol{\sigma} = \mathbb{C}:\boldsymbol{\varepsilon}$, with +# $\boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$, with # ```math -# \mathbb{C} := \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} +# \mathsf{C} := \frac{\partial \boldsymbol{\sigma}}{\partial \boldsymbol{\varepsilon}} # ``` # The volumetric, $\boldsymbol{\varepsilon}^\mathrm{vol}$, # and deviatoric, $\boldsymbol{\varepsilon}^\mathrm{dev}$ strains, are defined as @@ -254,7 +254,7 @@ C = gradient(ϵ -> 2 * Gmod * dev(ϵ) + 3 * Kmod * vol(ϵ), zero(SymmetricTensor # local stiffness matrix `ke` for a single element and assembles it into the global matrix. # `ke` is pre-allocated and reused for all elements. # -# Note that the elastic stiffness tensor $\boldsymbol{\mathbb{C}}$ is constant. +# Note that the elastic stiffness tensor $\mathsf{C}$ is constant. # Thus is needs to be computed and once and can then be used for all integration points. function 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