//---------------------------------------------------------
void NDG3D::PoissonIPDG3D(CSd& spOP, CSd& spMM)
//---------------------------------------------------------
{
// function [OP,MM] = PoissonIPDG3D()
//
// Purpose: Set up the discrete Poisson matrix directly
// using LDG. The operator is set up in the weak form
DVec faceR("faceR"), faceS("faceS"), faceT("faceT");
DMat V2D; IVec Fm("Fm"); IVec i1_Nfp = Range(1,Nfp);
double opti1=0.0, opti2=0.0; int i=0;
umLOG(1, "\n ==> {OP,MM} assembly: ");
opti1 = timer.read(); // time assembly
// build local face matrices
DMat massEdge[5]; // = zeros(Np,Np,Nfaces);
for (i=1; i<=Nfaces; ++i) {
massEdge[i].resize(Np,Np);
}
// face mass matrix 1
Fm = Fmask(All,1); faceR=r(Fm); faceS=s(Fm);
V2D = Vandermonde2D(N, faceR, faceS);
massEdge[1](Fm,Fm) = inv(V2D*trans(V2D));
// face mass matrix 2
Fm = Fmask(All,2); faceR = r(Fm); faceT = t(Fm);
V2D = Vandermonde2D(N, faceR, faceT);
massEdge[2](Fm,Fm) = inv(V2D*trans(V2D));
// face mass matrix 3
Fm = Fmask(All,3); faceS = s(Fm); faceT = t(Fm);
V2D = Vandermonde2D(N, faceS, faceT);
massEdge[3](Fm,Fm) = inv(V2D*trans(V2D));
// face mass matrix 4
Fm = Fmask(All,4); faceS = s(Fm); faceT = t(Fm);
V2D = Vandermonde2D(N, faceS, faceT);
massEdge[4](Fm,Fm) = inv(V2D*trans(V2D));
// build local volume mass matrix
MassMatrix = trans(invV)*invV;
DMat Dx("Dx"),Dy("Dy"),Dz("Dz"), Dx2("Dx2"),Dy2("Dy2"),Dz2("Dz2");
DMat Dn1("Dn1"),Dn2("Dn2"), mmE("mmE"), OP11("OP11"), OP12("OP12");
DMat mmE_All_Fm1, mmE_Fm1_Fm1, Dn2_Fm2_All;
IMat rows1,cols1,rows2,cols2; int k1=0,f1=0,k2=0,f2=0,id=0;
Index1D entries, entriesMM, idsM; IVec fidM,vidM,Fm1,vidP,Fm2;
double lnx=0.0,lny=0.0,lnz=0.0,lsJ=0.0,hinv=0.0,gtau=0.0;
double N1N1 = double((N+1)*(N+1)); int NpNp = Np*Np;
// build DG derivative matrices
int max_OP = (K*Np*Np*(1+Nfaces));
int max_MM = (K*Np*Np);
// "OP" triplets (i,j,x), extracted to {Ai,Aj,Ax}
IVec OPi(max_OP), OPj(max_OP), Ai,Aj; DVec OPx(max_OP), Ax;
// "MM" triplets (i,j,x)
IVec MMi(max_MM), MMj(max_MM); DVec MMx(max_MM);
IVec OnesNp = Ones(Np);
// global node numbering
entries.reset(1,NpNp); entriesMM.reset(1,NpNp);
OP12.resize(Np,Np);
for (k1=1; k1<=K; ++k1)
{
if (! (k1%250)) { umLOG(1, "%d, ",k1); }
rows1 = outer( Range((k1-1)*Np+1,k1*Np), OnesNp );
cols1 = trans(rows1);
// Build local operators
Dx = rx(1,k1)*Dr + sx(1,k1)*Ds + tx(1,k1)*Dt;
Dy = ry(1,k1)*Dr + sy(1,k1)*Ds + ty(1,k1)*Dt;
Dz = rz(1,k1)*Dr + sz(1,k1)*Ds + tz(1,k1)*Dt;
OP11 = J(1,k1)*(trans(Dx)*MassMatrix*Dx +
trans(Dy)*MassMatrix*Dy +
trans(Dz)*MassMatrix*Dz);
// Build element-to-element parts of operator
for (f1=1; f1<=Nfaces; ++f1) {
k2 = EToE(k1,f1); f2 = EToF(k1,f1);
rows2 = outer( Range((k2-1)*Np+1, k2*Np), OnesNp );
cols2 = trans(rows2);
fidM = (k1-1)*Nfp*Nfaces + (f1-1)*Nfp + i1_Nfp;
vidM = vmapM(fidM); Fm1 = mod(vidM-1,Np)+1;
vidP = vmapP(fidM); Fm2 = mod(vidP-1,Np)+1;
id = 1+(f1-1)*Nfp + (k1-1)*Nfp*Nfaces;
lnx = nx(id); lny = ny(id); lnz = nz(id); lsJ = sJ(id);
hinv = std::max(Fscale(id), Fscale(1+(f2-1)*Nfp, k2));
Dx2 = rx(1,k2)*Dr + sx(1,k2)*Ds + tx(1,k2)*Dt;
Dy2 = ry(1,k2)*Dr + sy(1,k2)*Ds + ty(1,k2)*Dt;
Dz2 = rz(1,k2)*Dr + sz(1,k2)*Ds + tz(1,k2)*Dt;
Dn1 = lnx*Dx + lny*Dy + lnz*Dz;
Dn2 = lnx*Dx2 + lny*Dy2 + lnz*Dz2;
mmE = lsJ*massEdge[f1];
gtau = 2.0 * N1N1 * hinv; // set penalty scaling
if (EToE(k1,f1)==k1) {
OP11 += ( gtau*mmE - mmE*Dn1 - trans(Dn1)*mmE ); // ok
}
else
{
// interior face variational terms
OP11 += 0.5*( gtau*mmE - mmE*Dn1 - trans(Dn1)*mmE );
// extract mapped regions:
mmE_All_Fm1 = mmE(All,Fm1);
mmE_Fm1_Fm1 = mmE(Fm1,Fm1);
Dn2_Fm2_All = Dn2(Fm2,All);
OP12 = 0.0; // reset to zero
OP12(All,Fm2) = -0.5*( gtau*mmE_All_Fm1 );
OP12(Fm1,All) -= 0.5*( mmE_Fm1_Fm1*Dn2_Fm2_All );
//OP12(All,Fm2) -= 0.5*(-trans(Dn1)*mmE_All_Fm1 );
OP12(All,Fm2) += 0.5*( trans(Dn1)*mmE_All_Fm1 );
// load this set of triplets
#if (1)
OPi(entries)=rows1; OPj(entries)=cols2, OPx(entries)=OP12;
entries += (NpNp);
#else
//###########################################################
// load only the lower triangle (after droptol test?)
sk=0; start=entries(1);
for (int i=1; i<=NpNp; ++i) {
eid = start+i;
id=entries(eid); rid=rows1(i); cid=cols2(i);
if (rows1(rid) >= cid) { // take lower triangle
if ( fabs(OP12(id)) > 1e-15) { // drop small entries
++sk; OPi(id)=rid; OPj(id)=cid, OPx(id)=OP12(id);
}
}
}
entries += sk;
//###########################################################
#endif
}
}
OPi(entries )=rows1; OPj(entries )=cols1, OPx(entries )=OP11;
MMi(entriesMM)=rows1; MMj(entriesMM)=cols1; MMx(entriesMM)=J(1,k1)*MassMatrix;
entries += (NpNp); entriesMM += (NpNp);
}
umLOG(1, "\n ==> {OP,MM} to sparse\n");
entries.reset(1, entries.hi()-Np*Np);
// Extract triplets from the large buffers. Note: this
// requires copying each array, and since these arrays
// can be HUGE(!), we force immediate deallocation:
Ai=OPi(entries); OPi.Free();
Aj=OPj(entries); OPj.Free();
Ax=OPx(entries); OPx.Free();
umLOG(1, " ==> triplets ready (OP) nnz = %10d\n", entries.hi());
// adjust triplet indices for 0-based sparse operators
Ai -= 1; Aj -= 1; MMi -= 1; MMj -= 1; int npk=Np*K;
#if defined(NDG_USE_CHOLMOD) || defined(NDG_New_CHOLINC)
// load only the lower triangle tril(OP) free args?
spOP.load(npk,npk, Ai,Aj,Ax, sp_LT, false,1e-15, true); // {LT, false} -> TriL
#else
// select {upper,lower,both} triangles
//spOP.load(npk,npk, Ai,Aj,Ax, sp_LT, true,1e-15,true); // LT -> enforce symmetry
//spOP.load(npk,npk, Ai,Aj,Ax, sp_All,true,1e-15,true); // All-> includes "noise"
//spOP.load(npk,npk, Ai,Aj,Ax, sp_UT, false,1e-15,true); // UT -> triu(OP) only
#endif
Ai.Free(); Aj.Free(); Ax.Free();
umLOG(1, " ==> triplets ready (MM) nnz = %10d\n", entriesMM.hi());
//-------------------------------------------------------
// The mass matrix operator will NOT be factorised,
// Load ALL elements (both upper and lower triangles):
//-------------------------------------------------------
spMM.load(npk,npk, MMi,MMj,MMx, sp_All,false,1.00e-15,true);
MMi.Free(); MMj.Free(); MMx.Free();
opti2 = timer.read(); // time assembly
umLOG(1, " ==> {OP,MM} converted to csc. (%g secs)\n", opti2-opti1);
}
void NDG2D::PoissonIPDGbc2D(
CSd& spOP //[out] sparse operator
)
{
// function [OP] = PoissonIPDGbc2D()
// Purpose: Set up the discrete Poisson matrix directly
// using LDG. The operator is set up in the weak form
// build DG derivative matrices
int max_OP = (K*Np*Np*(1+Nfaces));
//initialize parameters
DVec faceR("faceR"), faceS("faceS");
IVec Fm("Fm"), Fm1("Fm1"), fidM("fidM");
DMat V1D("V1D"); int i=0;
// build local face matrices
DMat massEdge[4]; // = zeros(Np,Np,Nfaces);
for (i=1; i<=Nfaces; ++i) {
massEdge[i].resize(Np,Np);
}
// face mass matrix 1
Fm = Fmask(All,1); faceR = r(Fm);
V1D = Vandermonde1D(N, faceR);
massEdge[1](Fm,Fm) = inv(V1D*trans(V1D));
// face mass matrix 2
Fm = Fmask(All,2); faceR = r(Fm);
V1D = Vandermonde1D(N, faceR);
massEdge[2](Fm,Fm) = inv(V1D*trans(V1D));
// face mass matrix 3
Fm = Fmask(All,3); faceS = s(Fm);
V1D = Vandermonde1D(N, faceS);
massEdge[3](Fm,Fm) = inv(V1D*trans(V1D));
//continue initialize parameters
DMat Dx("Dx"),Dy("Dy"), Dn1("Dn1"), mmE_Fm1("mmE(:,Fm1)");
double lnx=0.0,lny=0.0,lsJ=0.0,hinv=0.0,gtau=0.0;
int k1=0,f1=0,id=0;
IVec i1_Nfp = Range(1,Nfp);
double N1N1 = double((N+1)*(N+1));
// "OP" triplets (i,j,x), extracted to {Ai,Aj,Ax}
IVec OPi(max_OP),OPj(max_OP), Ai,Aj; DVec OPx(max_OP), Ax;
IMat rows1, cols1; Index1D entries; DMat OP11(Np,Nfp, 0.0);
// global node numbering
entries.reset(1,Np*Nfp);
cols1 = outer(Ones(Np), Range(1,Nfp));
umMSG(1, "\n ==> {OP} assembly [bc]: ");
for (k1=1; k1<=K; ++k1)
{
if (! (k1%100)) { umMSG(1, "%d, ",k1); }
rows1 = outer(Range((k1-1)*Np+1,k1*Np), Ones(Nfp));
// Build element-to-element parts of operator
for (f1=1; f1<=Nfaces; ++f1)
{
if (BCType(k1,f1))
{
////////////////////////added by Kevin ///////////////////////////////
Fm1 = Fmask(All,f1);
fidM = (k1-1)*Nfp*Nfaces + (f1-1)*Nfp + i1_Nfp;
id = 1+(f1-1)*Nfp + (k1-1)*Nfp*Nfaces;
lnx = nx(id); lny = ny(id);
lsJ = sJ(id); hinv = Fscale(id);
Dx = rx(1,k1)*Dr + sx(1,k1)*Ds;
Dy = ry(1,k1)*Dr + sy(1,k1)*Ds;
Dn1 = lnx*Dx + lny*Dy;
//mmE = lsJ*massEdge(:,:,f1);
//bc(All,k1) += (gtau*mmE(All,Fm1) - Dn1'*mmE(All,Fm1))*ubc(fidM);
mmE_Fm1 = massEdge[f1](All,Fm1); mmE_Fm1 *= lsJ;
gtau = 10*N1N1*hinv; // set penalty scaling
//bc(All,k1) += (gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1) * ubc(fidM);
switch(BCType(k1,f1)){
case BC_Dirichlet:
OP11 = gtau*mmE_Fm1 - trans(Dn1)*mmE_Fm1;
break;
case BC_Neuman:
OP11 = mmE_Fm1;
break;
default:
std::cout<<"warning: boundary condition is incorrect"<<std::endl;
}
OPi(entries)=rows1; OPj(entries)=cols1; OPx(entries)=OP11;
entries += (Np*Nfp);
}
cols1 += Nfp;
}
}
umMSG(1, "\n ==> {OPbc} to sparse\n");
entries.reset(1, entries.hi()-(Np*Nfp));
// extract triplets from large buffers
Ai=OPi(entries); Aj=OPj(entries); Ax=OPx(entries);
// These arrays can be HUGE, so force deallocation
OPi.Free(); OPj.Free(); OPx.Free();
// return 0-based sparse result
Ai -= 1; Aj -= 1;
//-------------------------------------------------------
// This operator is not symmetric, and will NOT be
// factorised, only used to create reference RHS's:
//
// refrhsbcPR = spOP1 * bcPR;
// refrhsbcUx = spOP2 * bcUx;
// refrhsbcUy = spOP2 * bcUy;
//
// Load ALL elements (both upper and lower triangles):
//-------------------------------------------------------
spOP.load(Np*K, Nfp*Nfaces*K, Ai,Aj,Ax, sp_All,false, 1e-15,true);
Ai.Free(); Aj.Free(); Ax.Free();
umMSG(1, " ==> {OPbc} ready.\n");
#if (1)
// check on original estimates for nnx
umMSG(1, " ==> max_OP: %12d\n", max_OP);
umMSG(1, " ==> nnz_OP: %12d\n", entries.hi());
#endif
}