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model.m
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model.m
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classdef model
properties
NVAR;
CFL;
C;
nu;
rho;
Pv;
end
methods
%% PHYSICAL METHOD
function mod=model(NVAR,CFL)
mod.NVAR=NVAR;
mod.CFL=CFL;
mod.C=vessel.empty(1,0);
mod.rho=1050.0;
mod.nu=1.*0.0045/mod.rho;
mod.Pv=3.2*133.322;
end
function mod=add(mod,v)
mod.C(length(mod.C)+1)=v;
end
function F=physicalFlux(mod,Q,i,gamma)
[a,b]=size(Q);
F(1,:)=Q(2,:);
F(2,:)=(Q(2,:).^2./Q(1,:))+gamma*Q(1,:).^(1.5);
%F(3,:)=Q(3,:).*Q(2,:)./Q(1,:);
end
function c=waveSpeed(mod,i,Q)
c=sqrt(mod.C(i).gamma*3/2*sqrt(Q(1,:)));
end
function lambda=eigenvalues(mod,i,Q)
c=mod.waveSpeed(i,Q);
lambda(1,:)=Q(2,:)./Q(1,:)-c;
lambda(2,:)=Q(2,:)./Q(1,:)+c;
%lambda(3,:)=Q(2,:)./Q(1,:);
end
function dt=timeStep(mod,i)
lambda=abs(mod.C(i).Q(2,:)./mod.C(i).Q(1,:)) + mod.waveSpeed(i,mod.C(i).Q);
maxlambda=max(lambda);
dt=mod.CFL*mod.C(i).dx/max(maxlambda);
end
function a=aFp(mod,K,gamma,a0,p)
a=(p/K+1.).^2*a0;
end
function p=pFa(mod,K,gamma,a0,Q)
p=K*((Q(1,:)/a0).^0.5-1.);
end
%% FLUXES
function fLF=numFluxLF(mod,i,QL,QR,dt,dx,gamma)
FL=physicalFlux(mod,QL,i,gamma);
FR=physicalFlux(mod,QR,i,gamma);
fLF=0.5*(FL+FR)-0.5*dx/dt*(QR-QL);
end
function fLW=numFluxLW(mod,i,QL,QR,dt,dx,gamma)
FL=physicalFlux(mod,QL,i,gamma);
FR=physicalFlux(mod,QR,i,gamma);
QLW=1/2*(QL+QR)-1/2*dt/dx*(FR-FL);
fLW=physicalFlux(mod,QLW,i,gamma);
end
function fLFORCE=numFluxLFORCE(mod,i,QL,QR,dt,dx,gamma)
fLFORCE=1/2*(numFluxLW(mod,i,QL,QR,dt,dx,gamma)+numFluxLF(mod,i,QL,QR,dt,dx,gamma));
end
function fHLLC=numFluxHLLC(mod,i,QL,QR,gamma)
aL=QL(1,:);
aR=QR(1,:);
uL=QL(2,:)./QL(1,:);
uR=QR(2,:)./QR(1,:);
% vL=QL(3,:)./QL(1,:);
% vR=QR(3,:)./QR(1,:);
cL=sqrt(mod.C(i).gamma*3/2*sqrt(aL));
cR=sqrt(mod.C(i).gamma*3/2*sqrt(aR));
uS=1/2*(uL+uR)-1/2*(cR-cL);
cS=1/2*(cL+cR)-1/8*(uR-uL);
sL=min([uL-cL,uS-cS]);
sR=max([uR+cR,uS+cS]);
sS=(sL*aR*(uR-sR)-sR*aL*(uL-sL))/(aR*(uR-sR)-aL*(uL-sL));
aS=aR*(uR-sR)/(sS-sR);
if (0<=sL)
fHLLC=physicalFlux(mod,QL,i,gamma);
elseif (sL<=0 && 0<=sS)
fHLLC=physicalFlux(mod,QL,i,gamma)+sL*(aS*[1,sS,vL]'-QL);
elseif (sS<=0 && 0<=sR)
fHLLC=physicalFlux(mod,QR,i,gamma)+sR*(aS*[1,sS,vR]'-QR);
elseif (0>=sR)
fHLLC=physicalFlux(mod,QR,i,gamma);
end
end
function fQvl=numFluxQvl(mod,i,QL,QR,gamma)
aL=QL(1,:);
aR=QR(1,:);
qL=QL(2,:);
qR=QR(2,:);
cL=sqrt(mod.C(i).gamma*3/2*sqrt(aL));
cR=sqrt(mod.C(i).gamma*3/2*sqrt(aR));
cM=sqrt(mod.C(i).gamma*3/2*sqrt(0.5*(aL+aR)));
lambda1M=(qL+qR)/(aL+aR)-cM;
lambda2M=(qL+qR)/(aL+aR)+cM;
P=[1 1; lambda1M lambda2M];
ALambda=[abs(lambda1M) 0;0 abs(lambda2M)];
MAbsM=P*ALambda*inv(P);
FL=physicalFlux(mod,QL,i,gamma);
FR=physicalFlux(mod,QR,i,gamma);
fQvl=0.5*(FL+FR)-0.5*MAbsM*(QR(1:2,:)-QL(1:2,:));
end
function fQvlLLH=numFluxQvlLLH(mod,i,QL,QR,gamma)
aL=QL(1,:);
aR=QR(1,:);
qL=QL(2,:);
qR=QR(2,:);
uL=QL(2,:)./QL(1,:);
uR=QR(2,:)./QR(1,:);
cL=sqrt(mod.C(i).gamma*3/2*sqrt(aL));
cR=sqrt(mod.C(i).gamma*3/2*sqrt(aR));
cM=sqrt(mod.C(i).gamma*3/2*sqrt(0.5*(aL+aR)));
lambda1M=(qL+qR)/(aL+aR)-cM;
lambda2M=(qL+qR)/(aL+aR)+cM;
lambda1L=qL/aL-cL;
lambda2L=qL/aL+cR;
lambda1R=qR/aR-cR;
lambda2R=qR/aR+cR;
Regalambda1R=max(abs(lambda1L),abs(lambda1R));
Regalambda2R=max(abs(lambda2L),abs(lambda2R));
P=[1 1; lambda1M lambda2M];
ALambda=[Regalambda1R 0;0 Regalambda2R];
MAbsM=P*ALambda*inv(P);
FL=physicalFlux(mod,QL,i,gamma);
FR=physicalFlux(mod,QR,i,gamma);
fQvlLLH=0.5*(FL+FR)-0.5*MAbsM*(QR(1:2,:)-QL(1:2,:));
end
function fGodEx=numFluxGodunovExact(mod,h,QL,QR,gamma)
aL=QL(1,:);
aR=QR(1,:);
uL=QL(2,:)./QL(1,:);
uR=QR(2,:)./QR(1,:);
% vL=QL(3,:)./QL(1,:);
% vR=QR(3,:)./QR(1,:);
[aS,uS]=solveERP(mod,h,aL,aR,uL,uR);
[QGod(1,1),QGod(2,1)]=sampleERP(mod,h,aL,aR,uL,uR,mod.C(h).K,mod.C(h).gamma,aS,uS,0);
%[QGod(1,1),QGod(2,1),QGod(3,1)]=sampleERP(mod,h,aL,aR,uL,uR,vL,vR,mod.C(h).K,mod.C(h).gamma,aS,uS,0);
QGod(2,1)=QGod(2,1)*QGod(1,1);
%QGod(3,1)=QGod(3,1)*QGod(1,1);
fGodEx=physicalFlux(mod,QGod,h,gamma);
end
%% EVOLVE FUNCTION
function Q=evolve(mod,h,dt,dx,gamma)
S=zeros(2,mod.C(h).NCELLS);
%S=zeros(3,mod.C(h).NCELLS);
fNum=zeros(2,mod.C(h).NCELLS+1);
%fNum=zeros(3,mod.C(h).NCELLS+1);
for i=1:(mod.C(h).NCELLS+1)
if i==1
QL=mod.C(h).Qbcs(:,1);
QR=mod.C(h).Q(:,i);
elseif i==(mod.C(h).NCELLS+1)
QL=mod.C(h).Q(:,i-1);
QR=mod.C(h).Qbcs(:,2);
else
QL=mod.C(h).Q(:,i-1);
QR=mod.C(h).Q(:,i);
end
%fNum(:,i)=numFluxHLLC(mod,h,QL,QR,gamma);
fNum(:,i)=numFluxGodunovExact(mod,h,QL,QR,gamma); % P
%fNum(:,i)=numFluxLFORCE(mod,h,QL,QR,dt,dx,gamma);
%fNum(:,i)=numFluxLW(mod,h,QL,QR,dt,dx,gamma);
%fNum(:,i)=numFluxLF(mod,h,QL,QR,dt,dx,gamma); % P
%fNum(:,i)=numFluxQvl(mod,h,QL,QR,gamma); % P
%fNum(:,i)=numFluxQvlLLH(mod,h,QL,QR,gamma); % P
end
Q=mod.C(h).Q-dt/dx*(fNum(:,2:(mod.C(h).NCELLS+1))-fNum(:,1:mod.C(h).NCELLS))+dt*S(:,1:mod.C(h).NCELLS);
end
%% BOUNDARY
function [Qbcs,BCL,BCR] = boundaryConditions(mod,h,Q,NCELLS,BCL,BCR,Qbcs,aL,aR,time)
if BCL==1 % Transparent boundary conditions
Qbcs(:,1) = Q(:,1);
elseif BCL==2 % Reflexive boundary conditions
Qbcs(1,1) = Q(1,1);
Qbcs(2,1) = -Q(2,1);
% Qbcs(3,1) = Q(3,1);
elseif BCL==3 % Imposed boundary conditions
Qbcs(1,1) = aL;
Qbcs(2,1) = UBC(mod,h,aL,-1);
% Qbcs(3,1) = Q(3,1);
elseif BCL==4
TT = 0.01; % s
T = 0.827;
qMax = 100.0*1e-6; % m^3/s
PI=pi;
t=time;
if time <= T*1.2
qFixed = qMax*max(sin(pi*time/TT),0.);
%qFixed=(3.1199+7.7982*sin(2*PI*t/T+0.5769)+4.1228*sin(4*PI*t/T-0.8738)-1.0611*sin(6*PI*t/T+0.7240)+0.7605*sin(8*PI*t/T-0.6387)-0.9148*sin(10*PI*t/T+1.1598)+0.4924*sin(12*PI*t/T-1.0905)-0.5580*sin(14*PI*t/T+1.042)+0.3280*sin(16*PI*t/T-0.5570)-0.3941*sin(18*PI*t/T+1.2685)+0.2833*sin(20*PI*t/T+0.6702)+0.2272*sin(22*PI*t/T-1.4983)+0.2249*sin(24*PI*t/T+0.9924)+0.2589*sin(26*PI*t/T-1.5616)-0.1460*sin(28*PI*t/T-1.3106)+0.2141*sin(30*PI*t/T-1.1306)-0.1253*sin(32*PI*t/T+0.1552)+0.1321*sin(34*PI*t/T-1.5595)-0.1399*sin(36*PI*t/T+0.4223)-0.0324*sin(38*PI*t/T+0.7811)-0.1211*sin(40*PI*t/T+1.0729))/1000/60;
else
%qFixed = 0.;
%mod.C(h).BCL = 2;
end
qFixed=(3.1199+7.7982*sin(2*PI*t/T+0.5769)+4.1228*sin(4*PI*t/T-0.8738)-1.0611*sin(6*PI*t/T+0.7240)+0.7605*sin(8*PI*t/T-0.6387)-0.9148*sin(10*PI*t/T+1.1598)+0.4924*sin(12*PI*t/T-1.0905)-0.5580*sin(14*PI*t/T+1.042)+0.3280*sin(16*PI*t/T-0.5570)-0.3941*sin(18*PI*t/T+1.2685)+0.2833*sin(20*PI*t/T+0.6702)+0.2272*sin(22*PI*t/T-1.4983)+0.2249*sin(24*PI*t/T+0.9924)+0.2589*sin(26*PI*t/T-1.5616)-0.1460*sin(28*PI*t/T-1.3106)+0.2141*sin(30*PI*t/T-1.1306)-0.1253*sin(32*PI*t/T+0.1552)+0.1321*sin(34*PI*t/T-1.5595)-0.1399*sin(36*PI*t/T+0.4223)-0.0324*sin(38*PI*t/T+0.7811)-0.1211*sin(40*PI*t/T+1.0729))/1000/60;
Qbcs(2,1) = qFixed;
Qbcs(1,1) = AQBC(mod,h,qFixed,-1);
% Qbcs(3,1) = Q(3,1);
elseif BCL==5
TT=0.010;
pMax=30.0*133.32;
if time<=TT*1.2
pFixed=80.*133.32+pMax*max(sin(pi*time/TT),0.);
aFixed=aFp(mod,mod.C(h).K,mod.C(h).gamma,mod.C(h).a0,pFixed);
Qbcs(1,1)=aFixed;
%rar=RiemannInv(mod,K,gamma,Q(:,1),-1);
Qbcs(2,1)=UBC(mod,h,aFixed,-1);%inflowFixedP(mod,K,gamma,aFixed,Q(:,1),rar);
% Qbcs(3,1)=Q(3,1);
else
Qbcs(1,1) = Q(1,1);
Qbcs(2,1) = 0;
% Qbcs(3,1) = Q(3,1);
mod.C(h).BCL=2;
end
end
if BCR==1 % Transparent boundary conditions
Qbcs(:,2) = Q(:,NCELLS);
elseif BCR==2 % Reflexive boundary conditions
Qbcs(1,2) = Q(1,NCELLS);
Qbcs(2,2) = -Q(2,NCELLS);
% Qbcs(3,2) = Q(3,NCELLS);
elseif BCR==3 % Reflexive boundary conditions
Qbcs(1,2) = aR;
Qbcs(2,2) = UBC(mod,h,aR,1);
% Qbcs(3,2) = Q(3,NCELLS);
elseif BCR==4
TT = 0.01; % s
qMax = 100.0*1e-6; % m^3/s
if time <= TT*1.2
qFixed = qMax*max(sin(pi*time/TT),0.);
else
qFixed = 0.;
mod.C(h).BCR = 2;
end
Qbcs(2,2) = qFixed;
Qbcs(1,2) = ABC(mod,h,qFixed,1);
% Qbcs(3,2) = Q(3,2);
elseif BCR==5
TT=0.01;
pMax=10.0*133.32;
if time<=TT*1.2
pFixed=80.*133.32+pMax*max(sin(pi*time/TT),0.);
aFixed=aFp(mod,mod.C(h).K,mod.C(h).gamma,mod.C(h).a0,pFixed);
Qbcs(1,2)=aFixed;
%rar=RiemannInv(mod,K,gamma,Q(:,1),1);
Qbcs(2,2)=UBC(mod,h,aFixed,1);
% Qbcs(3,2)=Q(3,1);
else
Qbcs(1,2) = Q(1,1);
Qbcs(2,2) = 0;
% Qbcs(3,2) = Q(3,1);
mod.C(h).BCR=2;
end
elseif BCR==6
n=mod.C(h).NCELLS;
cost=mod.RiemannInv(h,1);
options = optimoptions('fsolve','Display','off');
x0=[mod.C(h).Q(1,n)];
p=@(a)pFa(mod,mod.C(h).K,mod.C(h).gamma,mod.C(h).a0,a);
c=@(a)waveSpeed(mod,h,a);
f=@(a)cost*a*mod.C(h).Res-4*c(a)*a*mod.C(h).Res-p(a)+mod.Pv;
x=fsolve(@(a) f(a),x0,options);
Qbcs(1,2) = x;
Qbcs(2,2) = (p(x)-mod.Pv)/(mod.C(h).Res); %AQBC(mod,h,x,1);
%Qbcs(2,2) = UBC(mod,h,x,1);
% Qbcs(3,2) = 0;
end
end
%% EXACT SOLVER
function [arar,urar]=rarL(mod,i,uL,cL,xt)
urar=1/5*(uL+4*cL+4*xt);
crar=1/5*(uL+4*cL-xt);
arar=crar^4*4/9/mod.C(i).gamma^2;
%arar=(2/3/mod.C(i).gamma*(1.5.*(4*cR*uR-xt))^2)^2;
end
function [arar,urar]=rarR(mod,i,uR,cR,xt)
urar=1/5*(uR-4*cR+4*xt);
%disp(urar);
crar=1/5*(-uR+4*cR+xt);
arar=crar^4*4/9/(mod.C(i).gamma^2);
%arar=(2/3/mod.C(i).gamma*(1.5.*(4*cR*uR-xt))^2)^2;
end
function f=fK(mod,i,gamma,aK,aS,cK)
%cS=waveSpeed(mod,i,aS);
cS=sqrt(gamma*3/2*sqrt(aS));
if (aS<=aK)
f=4*(cS-cK);
else
f=sqrt(gamma*(aS-aK)*(aS^(1.5)-aK^(1.5))/(aK*aS));
end
end
function [a,u]=solveERP(mod,i,aL,aR,uL,uR)
cL=waveSpeed(mod,i,aL);
cR=waveSpeed(mod,i,aR);
tot=1e-6;
eps=1e-12;
A=1/2*(aL+aR);
err=1;
gamma=mod.C(i).gamma;
while (err>tot)
fp=(fK(mod,i,gamma,aL,A+eps,cL)+fK(mod,i,gamma,aR,A+eps,cR));
fm=(fK(mod,i,gamma,aL,A-eps,cL)+fK(mod,i,gamma,aR,A-eps,cR));
fder=(fp-fm)/(2*eps);
Aold=A;%AK
A=A-(fK(mod,i,gamma,aL,A,cL)+fK(mod,i,gamma,aR,A,cR)+uR-uL)/fder;%Ak+1
err=abs(A-Aold)/((A+Aold)/2);
end
a=A;
u=1/2*(uL+uR)+1/2*(fK(mod,i,gamma,aR,A,cR)-fK(mod,i,gamma,aL,A,cL));
end
function m = mK(mod,gamma,aK,aS)
m=sqrt(gamma*aS*aK*(aS^(1.5)-aK^(1.5))/(aS-aK));
end
function [a,u] = sampleERP(mod,i,aL,aR,uL,uR,K,gamma,aS,uS,xt)
%function [a,u,v] = sampleERP(mod,i,aL,aR,uL,uR,vL,vR,K,gamma,aS,uS,xt)
%cS=waveSpeed(mod,i,aS);
%cL=waveSpeed(mod,i,aL);
%cR=waveSpeed(mod,i,aR);
cS=sqrt(gamma*3/2*sqrt(aS));
cL=sqrt(gamma*3/2*sqrt(aL));
cR=sqrt(gamma*3/2*sqrt(aR));
mL=sqrt(gamma*aS*aL*(aS^(1.5)-aL^(1.5))/(aS-aL));
mR=sqrt(gamma*aS*aR*(aS^(1.5)-aR^(1.5))/(aS-aR));
sL=uL-mL/aL;
sR=uR+mR/aR;
if (xt<uS)
if (aS<=aL)
if (xt<(uL-cL))
a=aL;
u=uL;
elseif (xt>=(uL-cL) && xt<(uS-cS))
[a,u]=rarL(mod,i,uL,cL,xt);
elseif (xt>=(uS-cS))
a=aS;
u=uS;
end
else
if (xt<sL)
a=aL;
u=uL;
else
a=aS;
u=uS;
end
end
else
if (aS<=aR)
if (xt<(uS+cS))
a=aS;
u=uS;
elseif (xt>=(uS+cS) && xt<(uR+cR))
[a,u]=rarR(mod,i,uR,cR,xt);
elseif (xt>=(uR+cR))
a=aR;
u=uR;
end
else
if (xt<sR)
a=aS;
u=uS;
else
a=aR;
u=uR;
end
end
end
end
function Q = exactSampleERP(mod,aL,aR,uL,uR,h,aS,uS,time,gate)
%function Q = exactSampleERP(mod,aL,aR,uL,uR,vL,vR,h,aS,uS,time,gate)
NCELLS=mod.C(h).NCELLS;
xC=mod.C(h).xC;
K=mod.C(h).K;
gamma=mod.C(h).gamma;
Q=zeros(2,NCELLS);
for i = 1:NCELLS
[Q(1,i),Q(2,i)] = mod.sampleERP(h,aL,aR,uL,uR,K,gamma,aS,uS,(xC(i)-gate)/(time+1e-10));
%[Q(1,i),Q(2,i),Q(3,i)] = mod.sampleERP(h,aL,aR,uL,uR,vL,vR,K,gamma,aS,uS,(xC(i)-gate)/(time+1e-10));
end
end
end
end