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PDE1dImpl.m
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%
% This library is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 3
% of the License, or (at your option) any later version.
%
% This library is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this library; if not, visit
% http://www.gnu.org/licenses/gpl.html or write to the Free Software
% Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
%
% Copyright (C) 2016-2023 William H. Greene
classdef PDE1dImpl < handle
%% Main implementaton class for PDE solver.
properties
xmesh, tspan;
pdeFunc,icFunc,bcFunc;
numDepVars;
mCoord;
end
methods
function obj=PDE1dImpl(m, pdeFunc,icFunc,bcFunc,x,t,opts, ...
odeFunc, odeICFunc, odeMesh)
if(~isscalar(m) || ~isnumeric(m) || (m ~= 0 && m ~=1 && m ~=2))
error(['m must equal 0, 1, or 2 for Cartesian, cylindrical' ...
' or spherical coordinate systems, respectively.']);
end
obj.mCoord = m;
obj.pdeFunc = pdeFunc;
obj.icFunc = icFunc;
obj.bcFunc = bcFunc;
x=x(:)';
obj.numNodes = length(x);
t=t(:)';
obj.tspan = t;
if ~isa(opts, 'PDEOptions')
error('Argument seven must be a PDEOPtions instance.');
end
obj.pdeOpts = opts;
if isempty(opts.numIntegrationPoints)
opts.numIntegrationPoints = opts.polyOrder+1;
end
obj.intRule = GaussianIntegrationRule(GaussianIntegrationRule.Curve, ...
opts.numIntegrationPoints);
ic1 = icFunc(x(1));
obj.numDepVars = length(ic1);
obj.numElemNodes = opts.polyOrder+1;
obj.numFEMEqns = obj.numNodes*obj.numDepVars;
obj.totalNumEqns = obj.numFEMEqns;
obj.hasODE = opts.hasODE;
obj.useDiagMassMat = opts.useDiagMassMat;
obj.vectorized = opts.vectorized;
obj.numIntegrationPoints = opts.numIntegrationPoints;
obj.icDiagnostics = opts.icDiagnostics;
obj.eqnDiagnostics = opts.eqnDiagnostics;
obj.analyticalJacobian = opts.analyticalJacobian;
% FIXME!! numNodes has to be calculated carefully for this to work
% if polyorder>1!!
obj.numElems=(obj.numNodes-1)/(obj.numElemNodes-1);
gPts = obj.intRule.points;
if obj.numElemNodes==2
obj.sF = @obj.shapeLine2;
obj.dSf = @obj.dShapeLine2;
obj.N = obj.shapeLine2(gPts);
obj.dN = obj.dShapeLine2(gPts);
elseif obj.numElemNodes==3
obj.sF = @obj.shapeLine3;
obj.dSf = @obj.dShapeLine3;
obj.N = obj.shapeLine3(gPts);
obj.dN = obj.dShapeLine3(gPts);
elseif obj.numElemNodes==4
obj.sF = @obj.shapeLine4;
obj.dSf = @obj.dShapeLine4;
obj.N = obj.shapeLine4(gPts);
obj.dN = obj.dShapeLine4(gPts);
else
error('Unsupported number of element nodes.');
end
obj.y0FEM = zeros(obj.numDepVars, obj.numNodes);
for i=1:obj.numNodes
obj.y0FEM(:,i) = icFunc(x(i));
end
obj.y0 = obj.y0FEM(:);
obj.setMesh(x);
obj.NN = zeros(obj.numElemNodes, obj.numElemNodes, obj.numIntegrationPoints);
for i=1:obj.numIntegrationPoints
obj.NN(:,:,i) = obj.N(:,i)*obj.N(:,i)';
end
obj.isOctave = exist('OCTAVE_VERSION', 'builtin');
if obj.hasODE
obj.setODE(odeFunc, odeICFunc, odeMesh);
end
end
function [u,outArgs]=solveTransient(self, odeOpts)
reltol=odeOpts.RelTol;
abstol=odeOpts.AbsTol;
% flag dirichlet contraints
xl = self.xmesh(1);
xr = self.xmesh(end);
ul = self.y0FEM(:,1);
ur = self.y0FEM(:,end);
t0 = self.tspan(1);
if self.hasODE
y0Ode = self.odeImpl.getVFromSysVec(self.y0);
yp0Ode = zeros(self.odeImpl.numODEEquations,1);
else
y0Ode = [];
yp0Ode = [];
end
[pl,qLeft,pr,qRight] = callVarargFunc(self.bcFunc,...
{xl, ul,xr,ur,t0,y0Ode,yp0Ode});
self.dirConsFlagsLeft = qLeft~=0;
self.dirConsFlagsRight = qRight~=0;
%fprintf('numDepVars=%d\n', obj.numDepVars);
%prtShortVec(obj.y0, 'y0');
massFunc = @(time, u, up) self.calcMassMat(time, u, up);
rhsFunc = @(time, u, up) self.calcRHS(time, u, up);
%opts=odeset('reltol', reltol, 'abstol', abstol);
%opts=odeset(opts, 'Stats','on');
opts=odeset(odeOpts);
icdiag = self.icDiagnostics;
useDecic=true;
useOde15i=true;
if(useOde15i)
useDecic = true;
end
if(useDecic)
%icf = @(t,y,yp) massFunc(t,y,yp)*yp-rhsFunc(t,y,yp);
depVarClassType=self.y0;
icf = @(t,y,yp) self.calcResidual(t,y,yp,depVarClassType);
t0 = self.tspan(1);
n=length(self.y0);
yp0 = self.pdeOpts.initialSlope;
if isempty(yp0)
yp0 = zeros(n,1);
end
%fixed_y0 = ones(n,1);
%fixed_y0 = [ones(n-1,1);0];
%fixed_y0 = [0; ones(n-1,1)];
fixed_y0 = zeros(n,1); % all y0 free
if 0
% this strategy is too restrictive
% fix all non-BC fem dofs
fixed_y0(self.numDepVars+1:self.numFEMEqns-self.numDepVars) = 1;
% free any dofs that are obviously algebraic
[dfdy,dfdyp]=self.calcSparseJacobian(t0, self.y0, yp0);
if 0
for i=1:n
if all(dfdyp(i,:)==0) && all(dfdyp(:,i)==0)
fixed_y0(i)=0;
end
end
else
[Q,R,E] = qr(full(dfdyp));
adr=abs(diag(R));
tol=eps*max(adr);
rr=nnz(adr>tol);
ii=rr+1:n;
free_y0=logical(sum(E(:,ii)'));
fixed_y0(free_y0)=0;
end
end
%icf(0,y0,yp0)'
fixed_yp0 = zeros(n,1);
if ~isempty(self.pdeOpts.cicAbsTol)
icopts.AbsTol= self.pdeOpts.cicAbsTol;
else
icopts.AbsTol=abstol/10;
end
if ~isempty(self.pdeOpts.cicRelTol)
icopts.RelTol= self.pdeOpts.cicRelTol;
else
icopts.RelTol=reltol/10;
end
icopts.Jacobian = @self.calcJacobian;
y0=self.y0;
y0Save = y0;
yp0Save=yp0;
if self.pdeOpts.cicMethod==1
[y0,yp0]=decic(icf, t0,y0,fixed_y0,yp0,fixed_yp0,icopts);
elseif self.pdeOpts.cicMethod==0
icopts.icdiagnostics=icdiag;
icopts.maxiter=30;
[y0,yp0]=decicShampine(icf, t0,y0,fixed_y0,yp0,fixed_yp0,icopts);
end
if(icdiag)
prtShortVec(self.xmesh, 'x');
chgTol=max(icopts.RelTol*abs(self.y0), icopts.AbsTol);
if 0
[maxChg, indMax] = max(abs(y0-y0Save));
fprintf('max change, y0=%g at equation %d\n', maxChg, indMax);
end
chgY = y0-y0Save;
if self.hasODE
chgI = abs(chgY) > chgTol;
prtShortVec(chgY(chgI), 'change in y0 values');
I=1:n;
prtShortVec(I(chgI), 'changed y0 equations');
prtShortVec(y0, 'y0');
prtShortVec(yp0, 'yp0', 1, '%16.8g');
prtShortVec(icf(t0,y0,yp0), 'res');
else
R2 = @(v) reshape(v, self.numDepVars, []);
form='%.6g ';
prtMat(R2(y0Save), 'y0 before decic', 1, form);
prtMat(R2(yp0Save), 'yp0 before decic', 1, form);
prtMat(R2(y0), 'y0 after decic', 1, form);
prtMat(R2(yp0), 'yp0 after decic', 1, form);
end
end
end
abstolVec=abstol*ones(self.totalNumEqns,1);
% use large convergence tolerance on algebraic eqns
M= self.calcMassMat(t0, y0, yp0);
algEqns = abs(diag(M))<eps;
abstolVec(algEqns) = 1e6*abstol;
%prtShortVec(abstolVec, 'abstolVec');
opts=odeset(opts, 'abstol', abstolVec);
if(self.analyticalJacobian)
%disp('Using MatlabAutoDiff for Jacobian Evaluations');
jacFunc = @(time, u, up) self.calcJacobianAutoDiff(time, u, up);
opts=odeset(opts, 'jacobian', jacFunc);
elseif self.pdeOpts.useInternalNumJac
jacFunc = @(time, u, up) self.calcSparseJacobian(time, u, up);
opts=odeset(opts, 'jacobian', jacFunc);
else
% octave ode15i doesn't have JPattern option
jPat = self.calcJacPattern;
opts=odeset(opts, 'jpattern', {jPat, jPat});
end
outArgs={};
if self.pdeOpts.numberMPCycles
[u,va]=solveMP(self, icf, self.tspan, y0, yp0, opts);
outArgs=va;
else
if(useOde15i)
[outTimes,u]=ode15i(icf, self.tspan, y0, yp0, opts);
else
opts=odeset(opts,'initialslope', yp0);
opts=odeset(opts, 'Mass', massFunc);
[outTimes,u]=ode15s(rhsFunc, self.tspan, y0, opts);
end
end
if self.hasODE
uOde = u(:,self.numFEMEqns+1:self.numFEMEqns+self.odeImpl.numODEVariables);
outArgs{end+1} = uOde;
u= u(:,1:self.numFEMEqns);
end
end
function [u,varargout]=solveStatic(self, odeOpts)
reltol=odeOpts.RelTol;
abstol=odeOpts.AbsTol;
tol=1e-10;
% flag dirichlet contraints
xl = self.xmesh(1);
xr = self.xmesh(end);
ul = self.y0FEM(:,1);
ur = self.y0FEM(:,end);
t0 = self.tspan(1);
if 0
y0Ode = self.odeVFromSysVec(self.y0);
yp0Ode = 0*y0Ode;
else
y0Ode=[];
yp0Ode=[];
end
[pl,ql,pr,qr] = callVarargFunc(self.bcFunc, ...
{xl, ul,xr,ur,t0,y0Ode,yp0Ode});
self.dirConsFlagsLeft = ql~=0;
self.dirConsFlagsRight = qr~=0;
%fprintf('numDepVars=%d\n', obj.numDepVars);
%prtShortVec(obj.y0, 'y0');
u=self.y0;
up=zeros(self.totalNumEqns,1,'like', u);
maxIter = 10;
it=1;
converged = false;
debug = 1;
resFunc = @(u) self.calcResidual(0,u,up,u);
while(it<maxIter)
if debug> 1
self.printSystemVector(u, 'u');
end
r=resFunc(u);
rn = max(abs(r));
if debug
fprintf('it=%2d, rn=%g\n', it, rn);
end
if(rn < tol)
converged = true;
break;
end
if self.analyticalJacobian
J=self.calcJacobianAutoDiff(0, u, up);
else
J=self.calcJacobian(0, u, up);
end
%prtMat(J,'J');
du = J\r;
if debug> 1
self.printSystemVector(du, 'du');
end
u = u - du;
it = it + 1;
end
if ~converged
warning('pde1d:maxiter', 'Newton iteration did not converge.');
end
if(self.hasODE)
uODE = u(:,end-self.numODE+1:end);
varargout{1} = uODE;
end
u=self.femU2FromSysVec(u);
end
function testFuncs(self)
ne = self.totalNumEqns;
%u = zeros(ne, 1);
u=self.y0;
up = zeros(ne, 1);
%up=((1:ne)-1)';
self.printSystemVector(u, 'u', 1, '%16.8e');
if self.hasODE
odeIndices =self.numFEMEqns+1:ne;
v = u(odeIndices);
vDot = up(odeIndices);
uFEM = u(1:self.numFEMEqns);
upFEM = up(1:self.numFEMEqns);
else
v = [];
vDot = [];
uFEM = u;
upFEM = up;
end
dudtFunc = self.pdeOpts.initialSlope;
if ~isempty(dudtFunc)
dudt2FEM = zeros(self.numDepVars, self.numNodes);
for i=1:self.numNodes
dudt2FEM(:,i) = dudtFunc(self.xmesh(i));
end
upFEM = dudt2FEM(:);
up(1:self.numFEMEqns) = upFEM;
end
if ~isempty(self.pdeOpts.eqnDiagnosticsInitFunc)
[uFEM, upFEM] = self.pdeOpts.eqnDiagnosticsInitFunc();
u(1:self.numFEMEqns) = uFEM;
up(1:self.numFEMEqns) = upFEM;
end
self.printSystemVector(up, 'up', 1, '%16.8e');
analJac = self.analyticalJacobian;
if(analJac && self.eqnDiagnostics<1)
autoDiffF = @(u) self.calcResidual(0, u, up, u);
for i=1:10
kAD = AutoDiffJacobianAutoDiff(autoDiffF, u);
end
size(kAD)
return
end
depVarClassType=u;
[F, S,Cxd, fIntPts] = calcFEMEqns(self, 0, uFEM, upFEM, v, vDot, depVarClassType);
self.printSystemVector(S, 'S', 1, '%16.8e');
self.printSystemVector(F, 'F', 1, '%16.8e');
self.printSystemVector(Cxd, 'Cxd', 1, '%16.8e');
self.printSystemVector(S-F, 'S-F', 1, '%16.8e');
self.printSystemVector(S-F+Cxd, 'S-F+Cxd', 1, '%16.8e');
if(self.eqnDiagnostics>1)
K = self.calcJacobian(0, u, up);
prtMat(K, 'K', 1, '%13g');
fprintf('K: rank=%d, cond num=%g\n', sprank(K), condest(K));
M = self.calcMassMat(0, u, up);
prtMat(M, 'M', 1, '%13g');
Mup = M*up;
self.printSystemVector(Mup, 'Mup');
jPattern = self.calcJacPattern;
prtMat(full(jPattern), 'jPattern', 1, ' %g');
if(analJac)
autoDiffF = @(u) self.calcResidual(0, u, up, u);
kAD = AutoDiffJacobianAutoDiff(autoDiffF, u);
prtMat(kAD, 'kAD', 1, '%13g');
end
end
R = self.calcResidual(0, u, up, depVarClassType);
self.printSystemVector(R, 'R', 1, '%20.10g');
if self.hasODE
us=SysVec(u, self.numNodes, self.numDepVars);
ups=SysVec(up, self.numNodes, self.numDepVars);
self.odeImpl.testFuncs(us, ups, self.xPts, fIntPts, v, vDot);
end
end
function R = calcResidual(self, time, u, up, depVarClassType)
if nargin<5
depVarClassType=u;
end
[rhs,Cxd] = calcRHS(self, time, u, up, depVarClassType);
R = Cxd+rhs;
end
function [R,Cxd] = calcRHS(self, time, u, up, depVarClassType)
if nargin<4
depVarClassType=u;
end
if self.hasODE
v = self.odeImpl.getVFromSysVec(u);
vDot = self.odeImpl.getVFromSysVec(up);
uFEM = self.femUFromSysVec(u);
upFEM = self.femUFromSysVec(up);
else
v = [];
vDot = [];
uFEM = u;
upFEM = up;
end
[F, S,Cxd, fIntPts] = calcFEMEqns(self, time, uFEM, upFEM, v, vDot, depVarClassType);
R = F-S;
if self.hasODE
us=SysVec(u, self.numNodes, self.numDepVars);
ups=SysVec(up, self.numNodes, self.numDepVars);
[R,Cxd]=self.odeImpl.updateResiduals(time, us, ups, ...
self.xPts, fIntPts, R, Cxd);
end
% add constraints
[R,Cxd]=self.applyConstraints(R,Cxd, uFEM,v,vDot,time);
R=-R;
end
function [dfdy, dfdyp]=calcJacobian(self, time, u, up)
if(self.analyticalJacobian)
[dfdy, dfdyp]=self.calcJacobianAutoDiff(time, u, up);
else
[dfdy, dfdyp]=self.calcSparseJacobian(time, u, up);
end
end
function [dfdy, dfdyp]=calcJacobianAutoDiff(self, time, u, up)
%u=u(:);
%up=up(:);
autoDiffF = @(u) self.calcResidual(time, u, up, u);
dfdy = AutoDiffJacobianAutoDiff(autoDiffF, u);
autoDiffF = @(up) self.calcResidual(time, u, up, up);
dfdyp = AutoDiffJacobianAutoDiff(autoDiffF, up);
end
end % methods
methods(Access=private)
function setODE(self, odeFunc, odeICFunc, odeMesh)
v0=odeICFunc();
v0=v0(:);
nV = size(v0,1);
v0Dot = zeros(nV,1);
t0 = self.tspan(1);
yp0 = zeros(self.numFEMEqns, 1);
[F, S,Cxd,fIntPts] = self.calcFEMEqns(t0, self.y0, yp0, v0, v0Dot, self.y0);
u2 = self.femU2FromSysVec(self.y0);
up2 = self.femU2FromSysVec(yp0);
meshMapper = PDEMeshMapper(self.xmesh, odeMesh, self.numElemNodes, self.sF, self.dSf);
self.odeImpl = ODEImpl(meshMapper, odeFunc, odeICFunc, ...
odeMesh, t0, u2, up2, self.xPts, fIntPts, self.pdeOpts.testFunctionDOFMap);
numODEEqn = self.odeImpl.numODEEquations;
self.totalNumEqns = self.numFEMEqns + numODEEqn;
% there can be more ODE equations than ODE variables if
% some of the ODE equations are just constraints on the FEM dofs
self.y0 = [self.y0FEM(:); v0; zeros(numODEEqn-nV,1)];
end
function jac=calcJacobianFullFD(self, time, u, up)
% testing only
u=u(:);
up=up(:);
ne = length(u);
jac = zeros(ne,ne);
depVarClassType=u;
r0 = calcResidual(self, time, u, up, depVarClassType);
sqrtEps = sqrt(eps);
for i=1:ne
usave = u(i);
h = sqrtEps * max(usave, 1);
u(i) = u(i) + h;
rp = calcResidual(self, time, u, up, depVarClassType);
jac(:,i) = (rp-r0)/h;
u(i) = usave;
end
end
function jac=calcJacobianFullCD(self, time, u)
% testing only
% currently unused
jac = zeros(self.numFEMEqns, self.numFEMEqns);
disp('calc dFdu by central difference');
depVarClassType=u;
for i=1:self.numFEMEqns
usave = u(i);
delta = 1e-4;
u(i) = u(i) + delta;
rp = calcRHS(self, time, u, depVarClassType);
u(i) = usave;
u(i) = u(i) - delta;
rm = calcRHS(self, time, u, depVarClassType);
jac(:,i) = (rp-rm)/(2*delta);
u(i) = usave;
end
end
function M=calcMassMat(self, time, u, up)
u=u(:);
up=up(:);
ne = length(u);
M = zeros(ne,ne);
depVarClassType=up;
r0 = calcResidual(self, time, u, up, depVarClassType);
sqrtEps = sqrt(eps);
for i=1:ne
upsave = up(i);
h = sqrtEps * max(upsave, 1);
up(i) = up(i) + h;
rp = calcResidual(self, time, u, up, depVarClassType);
M(:,i) = (rp-r0)/h;
up(i) = upsave;
end
end
function [R,Cxd]=applyConstraints(self, R, Cxd, u,v, vDot,time)
u2=self.femU2FromSysVec(u);
x = self.xmesh;
xl = x(1);
xr = x(end);
%[pl,ql,pr,qr] = self.bcFunc(x(1), u2(:,1),x(end),u2(:,end),time);
[pl,ql,pr,qr] = callVarargFunc(self.bcFunc, ...
{xl, u2(:,1),xr,u2(:,end),time,v, vDot});
self.checkBCSize(pl);
self.checkBCSize(ql);
self.checkBCSize(pr);
self.checkBCSize(qr);
rightDofOff = self.numFEMEqns - self.numDepVars;
m=self.mCoord;
sing = m~=0 && xl==0;
for i=1:self.numDepVars
if ~(sing && self.pdeOpts.automaticBCAtCenter)
if(ql(i) ~= 0)
qli = ql(i);
if ~sing
if (m == 1)
qli = qli/xl;
elseif(m==2)
qli = qli/(xl*xl);
end
end
R(i) = R(i) - pl(i)/qli;
else
% apply dirichlet constraint
R(i) = -pl(i);
Cxd(i) = 0;
end
end
if(qr(i) ~= 0)
qri = qr(i);
if (m == 1)
qri = qri/xr;
elseif(m==2)
qri = qri/(xr*xr);
end
R(i+rightDofOff) = R(i+rightDofOff) + pr(i)/qri;
else
% apply dirichlet constraint using lagrange multiplier
R(i+rightDofOff) = -pr(i);
Cxd(i+rightDofOff) = 0;
end
end
%R = -F + S;
R = -R;
end
function [F, S, Cv, f] = calcFEMEqns(self, t, u,up, v, vDot, depVarClassType)
ndv=self.numDepVars;
nn = self.numNodes;
Cv = zeros(ndv, nn, 'like', depVarClassType);
F = zeros(ndv, nn, 'like', depVarClassType);
S = zeros(ndv, nn, 'like', depVarClassType);
gPts = self.intRule.points;
gWts = self.intRule.wts;
numIntPts = length(gPts);
nen=self.numElemNodes;
N=self.N; %#ok<PROPLC>
dN=self.dN; %#ok<PROPLC>
NN=self.NN; %#ok<PROPLC>
x = self.xmesh(:)';
u2 = self.femU2FromSysVec(u);
up2 = self.femU2FromSysVec(up);
numXpts = size(self.xPts,2);
nenM1=nen-1;
i1=1:nenM1:nn-nen+1;
i2=nen:nenM1:nn;
jac=(x(i2)-x(i1))/2;
dNdx=zeros(nen, length(jac));
uPts = zeros(ndv, numXpts, 'like', depVarClassType);
duPts = zeros(ndv, numXpts, 'like', depVarClassType);
ip = 1:numIntPts:numXpts-numIntPts+1;
for i=1:numIntPts
jj=nenM1;
for j=1:nen
u2j=u2(:,j:nenM1:end-jj);
uPts(:,ip) = uPts(:,ip) + u2j*N(j,i); %#ok<PROPLC>
duPts(:,ip) = duPts(:,ip) + u2j*dN(j,i); %#ok<PROPLC>
jj=jj-1;
end
duPts(:,ip) = duPts(:,ip)./jac;
ip = ip + 1;
end
if(self.vectorized)
% call pde func for all points
[c,f,s] = callVarargFunc(self.pdeFunc, ...
{self.xPts,t,uPts,duPts,v, vDot});
isFullCMat=self.checkCCoefficientMatrixSizeVec(c);
self.checkCoefficientMatrixSizeVec(f, 'f');
self.checkCoefficientMatrixSizeVec(s, 's');
%prtMat(s, 's');
else
f = zeros(ndv, numXpts, 'like', depVarClassType);
s = zeros(ndv, numXpts, 'like', depVarClassType);
for i=1:size(self.xPts,2)
[ci,fi,si] = ...
callVarargFunc(self.pdeFunc, ...
{self.xPts(i),t,uPts(:,i),duPts(:,i),v, vDot});
isFullCMat=self.checkCCoefficientMatrixSize(ci);
self.checkCoefficientMatrixSize(fi, 'f');
self.checkCoefficientMatrixSize(si, 's');
f(:,i) = fi;
s(:,i) = si;
if i==1
if isFullCMat
c = zeros(ndv, ndv, numXpts, 'like', depVarClassType);
else
c = zeros(ndv, numXpts, 'like', depVarClassType);
end
end
if isFullCMat
c(:,:,i) = ci;
else
c(:,i) = ci;
end
end
end
if(all(c==0))
error('pde1d:no_parabolic_eqn', ...
'At least one of the entries in the c-coefficient vector must be non-zero.');
end
% form global vectors
m = self.mCoord;
if(m==1)
xm = self.xPts;
elseif(m==2)
xm = self.xPts.*self.xPts;
end
if m> 0
if ~isFullCMat
c = c.*xm;
else
c = c.*reshape(xm,1,1,[]);
end
f = f.*xm;
s = s.*xm;
end
useDiagMM = self.useDiagMassMat;
ip = 1:numIntPts:numXpts;
for i=1:numIntPts
dNdx = dN(:,i)./jac;
fipJac = f(:, ip).*jac*gWts(i);
sipJac = s(:, ip).*jac*gWts(i);
if isFullCMat
cipJacFull = c(:,:,ip).*(reshape(jac,1,1,[])*gWts(i));
else
cipJac = c(:, ip).*jac*gWts(i);
end
for j=1:nen
ej=j:nenM1:nn-nen+j;
F(:,ej) = F(:,ej) + dNdx(j,:).*fipJac;
S(:,ej) = S(:,ej) + N(j,i)*sipJac;
if(useDiagMM)
Cv(:,ej) = Cv(:,ej) + up2(:,ej).*cipJac/nen;
else
for k=1:nen
ek=k:nenM1:nn-nen+k;
if ~isFullCMat
Cv(:,ej) = Cv(:,ej) + NN(j,k,i)*up2(:,ek).*cipJac;
else
upiek = reshape(NN(j,k,i)*up2(:,ek) ,ndv,1,[]);
Cv(:,ej) = Cv(:,ej) + reshape(multiprod(cipJacFull, upiek),ndv,[]);
end
end
end
end
ip = ip + 1;
end
F=F(:);
S=S(:);
Cv=Cv(:);
end
function printSystemVector(self, v, name, varargin)
if ~self.hasODE
m = reshape(v, self.numDepVars, []);
prtMat(m, name, varargin{:});
else
prtShortVec(v, name, varargin{:});
end
end
function checkCoefficientMatrixSize(self, coeffs, coeffsName)
reqSize=[self.numDepVars, 1];
sc = size(coeffs);
% in non-vec mode, allow row or comumn vector
if length(coeffs) ~= self.numDepVars
fName = func2str(self.pdeFunc);
error('pde1d:coeffSize', ['The \"%s\" coefficient matrix ' ...
'returned from from function \"%s\"\nhas size %d x %d but the ' ...
'expected size is %d x %d.'], coeffsName, fName, ...
sc(1), sc(2), reqSize(1), reqSize(2));
end
end
function isFullCMat=checkCCoefficientMatrixSize(self, c)
ndv = self.numDepVars;
sc = size(c);
if ~ismatrix(c)
fName = func2str(self.pdeFunc);
emsg=sprintf(['The \"c\" coefficient matrix ' ...
'returned from from function \"%s\"\nhas ' ...
'an incorrect number of dimensions'], fName);
error('pde1d:coeffSize', emsg);
end
if isvector(c) && length(c)==ndv
isFullCMat=false;
elseif all(sc == [ndv, ndv])
isFullCMat=true;
else
fName = func2str(self.pdeFunc);
emsg=sprintf(['The \"c\" coefficient matrix ' ...
'returned from from function \"%s\"\nhas '], fName);
emsg=sprintf([emsg 'size %d x %d but the ' ...
'expected size is %d x 1.'], sc(1), sc(2), ndv);
error('pde1d:coeffSize', emsg);
end
end
function checkCoefficientMatrixSizeVec(self, coeffs, coeffsName)
numXpts = size(self.xPts,2);
reqSize=[self.numDepVars, numXpts];
sc = size(coeffs);
if any(sc ~= reqSize)
fName = func2str(self.pdeFunc);
error('pde1d:coeffSize', ['The \"%s\" coefficient matrix ' ...
'returned from from function \"%s\"\nhas size %d x %d but the ' ...
'expected size is %d x %d.'], coeffsName, fName, ...
sc(1), sc(2), reqSize(1), reqSize(2));
end
end
function isFullCMat=checkCCoefficientMatrixSizeVec(self, coeffs)
numXpts = size(self.xPts,2);
ndv = self.numDepVars;
sc = size(coeffs);
dimsc = ndims(coeffs);
if dimsc==2 && all(sc == [ndv, numXpts])
isFullCMat=false;
elseif dimsc==3 && all(sc == [ndv, ndv, numXpts])
isFullCMat=true;
else
fName = func2str(self.pdeFunc);
emsg=sprintf(['The \"c\" coefficient matrix ' ...
'returned from from function \"%s\"\nhas '], fName);
if dimsc == 2
emsg=sprintf([emsg 'size %d x %d but the ' ...
'expected size is %d x %d.'], sc(1), sc(2), ndv, numXpts);
elseif dimsc==3
emsg=sprintf([emsg 'size %d x %d x %d but the ' ...
'expected size is %d x %d x %d.'], sc(1), sc(2), sc(3), ndv, ndv, numXpts);
else
emsg=[emsg 'an incorrect number of dimensions.'];
end
error('pde1d:coeffSize', emsg);
end
end
function J=calcJacPattern(self)
if ~isempty(self.jacobianPattern)
J = self.jacobianPattern;
return;
end
ndv = self.numDepVars;
nn = self.numNodes;
% jacobian is block-tridiagonal with the block size equal numDepVars
J = kron( spdiags(ones(nn,3),[-1 0 1],nn,nn), ones(ndv,ndv));
po = self.pdeOpts.polyOrder;
inds=-po:po;
J = kron( spdiags(ones(nn,length(inds)),inds,nn,nn), ones(ndv,ndv));
% odes may couple with any other vars
if self.hasODE
numODE = self.odeImpl.numODEEquations;
J21 = ones(numODE, self.numFEMEqns);
J22 = ones(numODE, numODE);
J = [J J21'
J21 J22];
end
self.jacobianPattern = J;
end
function [dfdy, dfdyp]=calcSparseJacobian(self, time, u, up)
jPattern = self.calcJacPattern;
if isempty(self.jacobianGroupList)
self.jacobianGroupList=findJacobianGroups(jPattern);
end
f = @(time, u) self.calcResidual(time, u, up, u);
dfdy=numericalJacobian(f, jPattern, u, time, self.jacobianGroupList);
f = @(time, up) self.calcResidual(time, u, up, up);
dfdyp=numericalJacobian(f, jPattern, up, time, self.jacobianGroupList);
end
function v=femUFromSysVec(self,v)
v=v(:);
v = v(1:self.numFEMEqns);
end
function v2=femU2FromSysVec(self,v)
v2 = reshape(v(1:self.numFEMEqns), self.numDepVars, []);
end
function checkBCSize(self, bc)
sb = size(bc);
if any(sb ~= [self.numDepVars 1])
bcName = inputname(2);
bcErrMsg = 'Size of "%s" must be %d X 1';
error('pde1d:bcSize', bcErrMsg, bcName, sb(1));
end
end
function setMesh(self, x)
x=x(:)';
self.xmesh = x;
numXpts = self.numElems*self.numIntegrationPoints;
self.xPts=zeros(1, numXpts);
ip = 1:self.numIntegrationPoints:numXpts;
for i=1:self.numIntegrationPoints
for j=1:self.numElemNodes
jj=self.numElemNodes-j;
self.xPts(ip) = self.xPts(ip) + x(j:self.numElemNodes-1:end-jj)*self.N(j,i);
end
ip = ip + 1;
end
end
function [u,outArgs]=solveMP(self, icf, tspan, y0, yp0, opts)
nfe=self.numFEMEqns;
monitor=3;
mmpde=5;
tau=1;
% FIXME need some rational way to calculate tspan
tspanX=[0 1];
ntOutput=length(tspan);
tne=length(y0);
uOutput=zeros(ntOutput,tne);
xOutput=zeros(ntOutput,nfe);
t=tspan(1);
tfinal=tspan(end);
dt=(tfinal-t)/(self.pdeOpts.numberMPCycles-1);
xIm1=self.xmesh(:);
xtIm1=0*xIm1;
yI=y0;
ypI=yp0;
while(t<tfinal)
% move nodes
y2FEM=self.femU2FromSysVec(yI);
yp2FEM=self.femU2FromSysVec(ypI);
rho = meshDensityFunc(xIm1,y2FEM);
odefun=@(t,x,xt) mmPDEResidual(monitor,mmpde,tau,x,xt,rho);
solX=ode15i(odefun, tspanX, xIm1, xtIm1);
% adjust tspan if necessary
maxDx= max(abs(solX.x(end)-solX.x(end-1)));
[xI,xtI]=deval(solX,solX.x(end));
if 0
else
y2FEMI = interp1(xIm1, y2FEM, xI, 'linear', 'extrap' );
yp2FEMI = interp1(xIm1, yp2FEM, xI, 'linear', 'extrap' );
yI(1:nfe)=y2FEMI(:);
ypI(1:nfe)=yp2FEMI(:);
self.setMesh(xI);
if self.hasODE
self.odeImpl.updateSrcMesh(xI);
end
end
xIm1=xI;
xtIm1=xtI;
% solve the system of pde on new mesh
tNew=t+dt;
tsI=[t tNew];
sol=ode15i(icf, tsI, yI, ypI, opts);
tSol=sol.x(end);
[yI,ypI]=deval(sol,tSol);
if tSol>t
tNew=tSol;
else
error('pde integration failed at t=%g', t);
end
toSave=tspan>=t & tspan<=tNew;
numToSave=sum(toSave);
if numToSave
uOutput(toSave,:)=deval(sol,tspan(toSave))';
xOutput(toSave,:)=repmat(xI', numToSave,1);
end
t=tNew;
end
u=uOutput;
outArgs= {xOutput};
end
end % methods
methods(Access=public, Static)
function shp = shapeLine2( r )
shp = [(1-r)/2 (1+r)/2]';
end
function ds = dShapeLine2( r )
ds = [-.5 .5]'*ones(1,length(r));
end
function shp = shapeLine3( r )
shp = [-r.*(1-r)/2 1-r.^2 r.*(1+r)/2]';
end
function ds = dShapeLine3( r )
r=r(:)';
ds = [-.5+r; -2*r; .5+r];
end
function shp = shapeLine4( r )
r=r(:)';
rp1=r+1;
rm1=r-1;
rp13=r+1/3;
rm13=r-1/3;
shp = [-9/16*(rp13).*(rm13).*(rm1)
27/16*(rp1).*(rm13).*(rm1)
-27/16*(rp1).*(rp13).*(rm1)
9/16*(rp1).*(rp13).*(rm13)
];
end
function ds = dShapeLine4( r )
r=r(:)';
rp1=r+1;
rm1=r-1;
rp13=r+1/3;
rm13=r-1/3;
ds = [
-9/16*(rm13.*rp13 + rm1.*rp13 + rm1.*rm13)
27/16*(rp1.*rm13 + rm1.*rp1 + rm1.*rm13)
-27/16*(rp1.*rp13 + rm1.*rp1 + rm1.*rp13)
9/16*(rp1.*rp13 + rp1.*rm13 + rm13.*rp13)
];
end