ch-LSFEM.tex

%\chapter{Least-Squares Finite Element Methods}





\section{$1.2$ Least-squares mixed finite elements}
A further finite element approach, which increasingly gains attention in the last decades, is the least-squares method. Initially, the least-squares method is the standard approach for regression analysis where it is used to compute, for an arbitrary number of datapoints, a curve which fulfills the given points in a least-squares sense.

The least-squares finite element method (LSFEM) is characterized by several advantages. The LSFEM replaces, for instance, a constrained minimization problem (with saddlepoint structure) by a least-squares formulation without constraints. Thus, it is not restricted by the latter mentioned LBB-condition and it is possible to combine (more or less arbitrary) polynomial orders for the interpolation of the unknowns without losing stability properties.

Furthermore, the resulting system matrices are always positive definite which could be advantageous for the applied solver.
Specifically, positive definite

In addition to that the least-squares functional can be set up using freely selectable field variables and governing equations. 

A further advantage of the method is that the FOSLS functional serves as an inherent a posteriori error indicator (compare e.g. Cai and Starke [2004] and Bochev and Gunzburger [2009]).  The additional computational costs of error estimation are greatly reduced (evaluation), thus facilitating (for example) means of adaptive refinement (adaptive) mesh refinement and other adaptive strategies.

Unfortunately, the method, as far it has been investigated until today, contains also several disadvantages. 

LSFEM are not without disadvantages.Here the weak performance of low order elements has to be mentioned, compare e.g. Pontaza [2003], Pontaza and Reddy [2003] and Schwarz et al. [2010]. Furthermore, the used (physical) residuals respectively their weighting have a crucial impact on the accuracy of the solution and have to be balanced suitably. 

First applications of the least-squares method in the field of finite elements respectively their mathematical analysis can be found e.g. in Lynn and Arya [1973; 1974], Zienkiewicz et al. [1974], Jespersen [1977] and Fix et al. [1979]. An overview over the least-squares method and its first applications in the field of finite elements is given in Eason [1976]. Furthermore, in this publication inter alia the advantages of the method are already discussed. In the following decade the number of publications concernig this topics reduces (e.g. Aziz et al. [1985]). This is eventually due to weak approximation quality for lower order elements which where mainly used at that time. 

The beginning of the 1990's brought a boom for the method, which was mainly restricted on the field of fluid mechanics, compare e.g. Jiang and Chang [1990], Chang and Jiang [1990], Jiang [1992], Tang and Tsang [1993], Cai et al. [1994], Bochev and Gunzburger [1994; 1995], Cai et al. [1995], Chang et al. [1995], Bell and Surana [1994], Bochev [1994; 1999], Berndt et al. [1997], Bochev et al. [1998], Bochev et al. [1999], Ding and Tsang [2001; 2003], Pontaza and Reddy [2003], Kayser-Herold and Matthies [2003] and Kayser-Herold and Matthies [2007]. Here, the used first-order systems interpolate different combinations as, for instance, the flow velocity, the pressure, the stresses or the vorticity field. The approximation of these unknown quantities was mainly restricted on the Sobolev space $H^{1}(\mathcal{B})$. 
\footnote{For an overview over the developments of the least-squares method in the field of fluid mechanics the reader is referred to Jiang [1998], Bochev and Gunzburger [2009] and Kayser-Herold and Matthies [2005] and the references therein.}

In the field of solid mechanics, least-squares methods have been developed for Linear Elasticity and the generalized Stokes equations simultaneously for the cases of plane displacement and pure traction in the works of McCormick et al, Cai et al. [1995], and others.
In the context of Linear Elasticity, the authors used as a basis a div-curl-grad system of first-order with the unknown fields velocity, vorticity and pressure (VVP). 
Further investigations of this working group, also in the field of linear elastic problems, are inter alia Cai et al. $[1997 ; 1998 ; 2000$ a;b] and Kim et al. [2000]. In this publications the displacement and the displacement gradient are used as field quantities and are interpolated in $H^{1}(\mathcal{B})$. 
In the works of Cai and Starke [2004] and Cai et al. [2005] the authors used the displacement and the stresses as unknown fields. The approximation of the quantities has been executed in $H^{1}(\mathcal{B})$ via standard polynomial interpolation for the displacement field and, for the stress field, vector-valued interpolation functions in $H(\operatorname{div}, \mathcal{B})$ of Raviart-Thomas type \footnote{see Raviart and Thomas [1977]} . Schwarz [2009] and Schwarz et al. [2010] continued to use these interpolation spaces and exmained the use of the functional as an error indicator, the fulfillment of the stress symmetry, and behavior of the formulations for (nearly) incompressible materials. Furthermore, a main benefit of the formulation in Schwarz et al. [2010] is the good performance of the developed low order element. This is due to a modification of the first variation of the functional and the resulting improvement of the momentum balance. An extension on transversely isotropic elasticity can be found in Schwarz and Schröder [2007] and in Schwarz et al. [2014] a least-squares formulation with an additional (redundant) residual has been used. 

Bertrand et al. [2014] examined the application of the FOSLS method on curved boundaries. 

Elasto-plasticity and material nonlinearities were examined by Kwon et al. [2005] and Starke [2007] published results. A main issue in this context has been the non-smoothness of the constitutive relation in the case of plastic deformations and the resulting problems using standard nonlinear solver. To overcome this, Starke [2009] used a non-smooth Newton method which results in suitable convergence rates. In the work of Schwarz et al. [2009b] this issue is circumvented by a modified approach and in Schwarz et al. [2009a] the authors used a viscoplastic formulation. 

In Steeger et al. [2015] the authors investigate a displacement-stress formulation where they consider the performance using an interpolation of the stresses in $H(\operatorname{div}, \mathcal{B})$ compared to an interpolation of the stresses in $H^{1}(\mathcal{B})$. 

In the publications of Jiang and $\mathrm{Wu}$ [2002] and Jiang [2002] two different formulations are taken under consideration. In Jiang and Wu [2002] (part 1), beside the displacement and the stresses, the rotation with respect to the plane normal is used as a basis for the formulation. The second part (Jiang [2002]) considers a FOSLS formulation comprised of four quantities that exhibit optimal convergence rates in the provided examples (bending problems (thin plates)).

First investigations in the field of geometrically nonlinear problems in the field of solid mechanics are e.g. Westphal [2004] and Manteuffel et al. [2006] where a constitutive relation of St.Venant-Kirchhoff type has been considered. 

:east-squares formulations for hyperelastic problems has been developed, see e.g. Schwarz et al. [2012], Starke et al. [2012], Müller et al. [2014], Müller [2015] and Müller and Starke [2016]. 

In Schröder et al. [2016] several least-squares formulations for isotropic and anisotropic elasticity at small and large strains are given. 

In Kadapa et al. [2015] a formulation using the displacements and the pressure as field quantities, interpolated by NURBS (nonuniform rational B-splines), has been provided. Here, fluid mechanical problems as well as problems in the field of solid mechanics (hyperelasticity) are taken under consideration.

For hyperelastic materials, the three-field displacement-pressure-Jacobian formulation for hyperelastic materials whose strain energy density functions are decomposed into deviatoric and volumetric parts.

Kadapa et al [2020] later developed a two-field mixed displacement-pressure formulation based on an energy functional so as to account for relations between pressure and volumetric energy function.  The linearized consistent mixed displacement-pressure hyperelastic FOSLS formulation is applicable to both incompressible and compressible materials. \footnote{A linearized consistent mixed displacement-pressure formulation for hyperelasticity}

%We propose a novel mixed displacement-pressure formulation based on an energy functional that takes into account the relation between the pressure and the volumetric energy function. We demonstrate that the proposed  formulation is not only applicable for nearly and truly incompressible cases but also is consistent in the compressible regime. 
%Furthermore, we prove with analytical derivation and numerical results that the proposed two-field formulation is a simplified and efficient alternative for  Bayesian inversion methods for ductile phase field fracture were examined by Noii et al \cite{Bayesian inversion for unified ductile phase-field fracture}

B. Staber, J. Guilleminot et al [2018] studied random field models of anisotropic stored energy functions corresponding to arterial wall mechanics.  Uncertainty quantification in mechanics of arterial walls.
There, the stochastic framework relies on information theory, which is invoked in order to account for the mathematical
requirements raised by the functional analysis of nonlinear boundary value problems. All the samples generated
through the proposed approach are hence admissible, and the probabilistic formulation introduces a limited modeling
bias thanks to the entropy maximization. An efficient and robust computational methodology for sampling on
complex geometries defined by smooth manifolds was additionally detailed. Various numerical applications were
finally considered. In particular, the capability of the stochastic model to produce anisotropic correlation kernels and
realizations with specific signatures was assessed. While technical derivations were achieved on a particular functional
form, the proposed framework can readily be applied to any strain energy function of interest, as well as to other
complex soft biological tissues,


\section{Elliptic PDE: FOSLS Formulation}


\section{Stokes PDE: FOSLS Formulation}


\section{Navier-Stokes PDE: FOSLS Formulations}
In the light of efficient LSFEM solutions the advantage of For a given formulations, the associated FOSLS functional serves as an inherent a posteriori error estimator of LSFEM, and is used for deliberations on different marking strategies in an h-type adaptive mesh refinement.


\subsection{}
In the light of efficient LSFEM solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h-type adaptive mesh refinement.

\subsection{The Discrete Least-Squares Functional}

In this part, the least-squares formulation for the first-order weighted formulation
(2.5) is given, then the discrete version based on the deep neural network approximation
mation is introduced.
The first-order weighted least-squares formulation is to find $(u, \tau, \phi) \in\left(H^{1}(\Omega)\right)^{3}$ such that
\begin{equation*}
\Psi(u, \tau, \phi ; \mathbf{f})=\min _{(\eta, \nu, \chi) \in\left(H^{1}(\Omega)\right)^{3}} \Psi(\eta, \nu, \chi ; \mathbf{f}),
\end{equation*}
where $\mathbf{f}=(f, g, \zeta)$ and
\begin{equation*}
\begin{aligned}
\Psi(\eta, \nu, \chi ; \mathbf{f})=&\left\|\zeta\left(f+\nabla \cdot\left(\nu q_{1}+\chi q_{2}\right)\right)\right\|_{0, \Omega}^{2}+\left\|\zeta\left(\nu-\left(\Lambda \nabla \eta, q_{1}\right)\right)\right\|_{0, \Omega}^{2} \\
&+\left\|\zeta\left(\chi-\left(\Lambda \nabla \eta, q_{2}\right)\right)\right\|_{0, \Omega}^{2}+\|\eta-g\|_{1 / 2, \partial \Omega}^{2}
\end{aligned}
\end{equation*}
If three unknown functions $u, \tau, \phi$ are approximated by one deep neural network and its three outputs are denoted by $\hat{u}(x, \theta), \hat{\tau}(x, \theta), \hat{\phi}(x, \theta)$ (see Fig. 2), then the discrete formulation based on all sampling points reads
\begin{equation*}
\hat{\Psi}(\hat{u}, \hat{\tau}, \hat{\phi} ; \mathbf{f})(\theta)=\min _{\tilde{\theta} \in \mathbb{R}^{N}} \hat{\Psi}(\hat{\eta}, \hat{\nu}, \hat{\chi} ; \mathbf{f})(\tilde{\theta}),
\end{equation*}
where the discrete functional reads
\begin{equation*}
\begin{aligned}
\hat{\Psi}(\hat{\eta}, \hat{\nu}, \hat{\chi} ; \mathbf{f})(\tilde{\theta})=& \frac{1}{N_{f}} \sum_{i=1}^{N_{f}}\left(\zeta\left(\mathbf{x}_{i}\right)\left(f\left(\mathbf{x}_{i}\right)+\nabla \cdot\left(\hat{\nu}\left(\mathbf{x}_{i}, \tilde{\theta}\right) q_{1}\left(\mathbf{x}_{i}\right)+\hat{\chi}\left(\mathbf{x}_{i}, \tilde{\theta}\right) q_{2}\left(\mathbf{x}_{i}\right)\right)\right)\right)^{2} \\
&+\frac{1}{N_{f}} \sum_{i=1}^{N_{f}}\left(\zeta\left(\mathbf{x}_{i}\right)\left(\hat{\nu}\left(\mathbf{x}_{i}, \tilde{\theta}\right)-\left(\Lambda \nabla \hat{\eta}\left(\mathbf{x}_{i}, \tilde{\theta}\right), q_{1}\left(\mathbf{x}_{i}\right)\right)\right)\right)^{2} \\
&+\frac{1}{N_{f}} \sum_{i=1}^{N_{f}}\left(\zeta\left(\mathbf{x}_{i}\right)\left(\hat{\chi}\left(\mathbf{x}_{i}, \tilde{\theta}\right)-\left(\Lambda \nabla \hat{\eta}\left(\mathbf{x}_{i}, \tilde{\theta}\right), q_{2}\left(\mathbf{x}_{i}\right)\right)\right)\right)^{2} \\
&+\frac{\omega_{D}}{N_{D}} \sum_{i=1}^{N_{D}}\left(\hat{\eta}\left(\mathbf{x}_{i}, \tilde{\theta}\right)-g\left(\mathbf{x}_{i}\right)\right)^{2}
\end{aligned}
\end{equation*}
where $N_{f}$ denotes the number of collocation points in $\Omega$.  $N_{D}$ denotes the number of boundary points on $\partial \Omega$ at which Dirirchlet boundary conditions are imposed weakly (as "soft constraints").  

The parameter $\omega_{D}$ represents the boundary-weight to penalize the neural network approximations satisfying the Dirichlet boundary conditions.

%[2,64,64,64,64,64,3]
%2*64+(64*64)*(h-1)+64*3
(2+64*(h-1)+3)*64 + 3+64*h

%64*h + 3
% (2, 64),
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