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minimal-surface.cpp
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minimal-surface.cpp
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// Copyright (c) 2010-2024, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
//
// -------------------------------------------------
// Minimal Surface Miniapp: Compute minimal surfaces
// -------------------------------------------------
//
// This miniapp solves Plateau's problem: the Dirichlet problem for the minimal
// surface equation.
//
// Two problems can be run:
//
// - Problem 0 solves the minimal surface equation of parametric surfaces.
// The surface (-s) option allow the selection of different
// parametrization:
// s=0: Uses the given mesh from command line options
// s=1: Catenoid
// s=2: Helicoid
// s=3: Enneper
// s=4: Hold
// s=5: Costa
// s=6: Shell
// s=7: Scherk
// s=8: FullPeach
// s=9: QuarterPeach
// s=10: SlottedSphere
//
// - Problem 1 solves the minimal surface equation of the form z=f(x,y),
// for the Dirichlet problem, using Picard iterations:
// -div( q grad u) = 0, with q(u) = (1 + |∇u|²)^{-1/2}
//
// Compile with: make minimal-surface
//
// Sample runs: minimal-surface
// minimal-surface -a
// minimal-surface -c
// minimal-surface -c -a
// minimal-surface -no-pa
// minimal-surface -no-pa -a
// minimal-surface -no-pa -a -c
// minimal-surface -p 1
//
// Device sample runs:
// minimal-surface -d debug
// minimal-surface -d debug -a
// minimal-surface -d debug -c
// minimal-surface -d debug -c -a
// minimal-surface -d cuda
// minimal-surface -d cuda -a
// minimal-surface -d cuda -c
// minimal-surface -d cuda -c -a
// minimal-surface -d cuda -no-pa
// minimal-surface -d cuda -no-pa -a
// minimal-surface -d cuda -no-pa -c
// minimal-surface -d cuda -no-pa -c -a
#include "mfem.hpp"
#include "../../general/forall.hpp"
using namespace mfem;
using namespace std;
// Constant variables
constexpr int DIM = 2;
constexpr int SDIM = 3;
constexpr real_t PI = M_PI;
constexpr real_t NRM = 1.e-4;
constexpr real_t EPS = 1.e-14;
constexpr Element::Type QUAD = Element::QUADRILATERAL;
constexpr real_t NL_DMAX = std::numeric_limits<real_t>::max();
// Static variables for GLVis
constexpr int GLVIZ_W = 1024;
constexpr int GLVIZ_H = 1024;
constexpr char vishost[] = "localhost";
// Context/Options for the solver
struct Opt
{
int sz, id;
int pb = 0;
int nx = 6;
int ny = 6;
int order = 3;
int refine = 2;
int niters = 8;
int surface = 5;
// Socket to send visualization data
int visport = 19916;
bool pa = true;
bool vis = true;
bool amr = false;
bool wait = false;
bool print = false;
bool radial = false;
bool by_vdim = false;
bool snapshot = false;
bool vis_mesh = false;
real_t tau = 1.0;
real_t lambda = 0.1;
real_t amr_threshold = 0.6;
const char *keys = "gAm";
const char *device_config = "cpu";
const char *mesh_file = "../../data/mobius-strip.mesh";
void (*Tptr)(const Vector&, Vector&) = nullptr;
};
class Surface: public Mesh
{
protected:
Opt &opt;
Mesh *mesh;
Array<int> bc;
socketstream glvis;
H1_FECollection *fec;
FiniteElementSpace *fes;
public:
// Reading from mesh file
Surface(Opt &opt, const char *file): Mesh(file, true), opt(opt) { }
// Generate 2D empty surface mesh
Surface(Opt &opt, bool): Mesh(), opt(opt) { }
// Generate 2D quad surface mesh
Surface(Opt &opt)
: Mesh(Mesh::MakeCartesian2D(opt.nx, opt.ny, QUAD, true)), opt(opt) { }
// Generate 2D generic surface mesh
Surface(Opt &opt, int nv, int ne, int nbe):
Mesh(DIM, nv, ne, nbe, SDIM), opt(opt) { }
void Create()
{
if (opt.surface > 0)
{
Prefix();
Transform();
}
Postfix();
Refine();
Snap();
fec = new H1_FECollection(opt.order, DIM);
if (opt.amr) { EnsureNCMesh(); }
mesh = new Mesh(*this, true);
fes = new FiniteElementSpace(mesh, fec, opt.by_vdim ? 1 : SDIM);
BoundaryConditions();
}
int Solve()
{
// Initialize GLVis server if 'visualization' is set
if (opt.vis) { opt.vis = glvis.open(vishost, opt.visport) == 0; }
// Send to GLVis the first mesh
if (opt.vis) { Visualize(glvis, opt, mesh, GLVIZ_W, GLVIZ_H); }
// Create and launch the surface solver
if (opt.by_vdim)
{
ByVDim(*this, opt).Solve();
}
else
{
ByNodes(*this, opt).Solve();
}
if (opt.vis && opt.snapshot)
{
opt.keys = "Sq";
Visualize(glvis, opt, mesh, mesh->GetNodes());
}
return 0;
}
~Surface() override
{
if (opt.vis) { glvis.close(); }
delete mesh; delete fec; delete fes;
}
virtual void Prefix()
{
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
}
virtual void Transform() { if (opt.Tptr) { Mesh::Transform(opt.Tptr); } }
virtual void Postfix()
{
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
}
virtual void Refine()
{
for (int l = 0; l < opt.refine; l++)
{
UniformRefinement();
}
}
virtual void Snap()
{
GridFunction &nodes = *GetNodes();
for (int i = 0; i < nodes.Size(); i++)
{
if (std::abs(nodes(i)) < EPS)
{
nodes(i) = 0.0;
}
}
}
void SnapNodesToUnitSphere()
{
Vector node(SDIM);
GridFunction &nodes = *GetNodes();
for (int i = 0; i < nodes.FESpace()->GetNDofs(); i++)
{
for (int d = 0; d < SDIM; d++)
{
node(d) = nodes(nodes.FESpace()->DofToVDof(i, d));
}
node /= node.Norml2();
for (int d = 0; d < SDIM; d++)
{
nodes(nodes.FESpace()->DofToVDof(i, d)) = node(d);
}
}
}
virtual void BoundaryConditions()
{
if (bdr_attributes.Size())
{
Array<int> ess_bdr(bdr_attributes.Max());
ess_bdr = 1;
bc.HostReadWrite();
fes->GetEssentialTrueDofs(ess_bdr, bc);
}
}
// Initialize visualization of some given mesh
static void Visualize(socketstream &glvis,
Opt &opt, const Mesh *mesh,
const int w, const int h,
const GridFunction *sol = nullptr)
{
const GridFunction &solution = sol ? *sol : *mesh->GetNodes();
if (opt.vis_mesh) { glvis << "mesh\n" << *mesh; }
else { glvis << "solution\n" << *mesh << solution; }
glvis.precision(8);
glvis << "window_size " << w << " " << h << "\n";
glvis << "keys " << opt.keys << "\n";
opt.keys = nullptr;
if (opt.wait) { glvis << "pause\n"; }
glvis << std::flush;
}
// Visualize some solution on the given mesh
static void Visualize(socketstream &glvis,
const Opt &opt, const Mesh *mesh,
const GridFunction *sol = nullptr)
{
const GridFunction &solution = sol ? *sol : *mesh->GetNodes();
if (opt.vis_mesh) { glvis << "mesh\n" << *mesh; }
else { glvis << "solution\n" << *mesh << solution; }
if (opt.wait) { glvis << "pause\n"; }
if (opt.snapshot && opt.keys) { glvis << "keys " << opt.keys << "\n"; }
glvis << std::flush;
}
using Mesh::Print;
static void Print(const Opt &opt, Mesh *mesh, const GridFunction *sol)
{
const char *mesh_file = "surface.mesh";
const char *sol_file = "sol.gf";
if (!opt.id)
{
mfem::out << "Printing " << mesh_file << ", " << sol_file << std::endl;
}
std::ofstream mesh_ofs(mesh_file);
mesh_ofs.precision(8);
mesh->Print(mesh_ofs);
mesh_ofs.close();
std::ofstream sol_ofs(sol_file);
sol_ofs.precision(8);
sol->Save(sol_ofs);
sol_ofs.close();
}
// Surface Solver class
class Solver
{
protected:
Opt &opt;
Surface &S;
CGSolver cg;
OperatorPtr A;
BilinearForm a;
GridFunction x, x0, b;
ConstantCoefficient one;
mfem::Solver *M = nullptr;
const int print_iter = -1, max_num_iter = 2000;
const real_t RTOLERANCE = EPS, ATOLERANCE = EPS*EPS;
public:
Solver(Surface &S, Opt &opt): opt(opt), S(S), cg(),
a(S.fes), x(S.fes), x0(S.fes), b(S.fes), one(1.0)
{
cg.SetRelTol(RTOLERANCE);
cg.SetAbsTol(ATOLERANCE);
cg.SetMaxIter(max_num_iter);
cg.SetPrintLevel(print_iter);
}
~Solver() { delete M; }
void Solve()
{
constexpr bool converged = true;
if (opt.pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL);}
for (int i=0; i < opt.niters; ++i)
{
if (opt.amr) { Amr(); }
if (opt.vis) { Surface::Visualize(S.glvis, opt, S.mesh); }
if (!opt.id) { mfem::out << "Iteration " << i << ": "; }
S.mesh->NodesUpdated();
a.Update();
a.Assemble();
if (Step() == converged) { break; }
}
if (opt.print) { Surface::Print(opt, S.mesh, S.mesh->GetNodes()); }
}
virtual bool Step() = 0;
protected:
bool Converged(const real_t rnorm) { return rnorm < NRM; }
bool ParAXeqB()
{
b = 0.0;
Vector X, B;
a.FormLinearSystem(S.bc, x, b, A, X, B);
if (!opt.pa) { M = new GSSmoother((SparseMatrix&)(*A)); }
if (M) { cg.SetPreconditioner(*M); }
cg.SetOperator(*A);
cg.Mult(B, X);
a.RecoverFEMSolution(X, b, x);
const bool by_vdim = opt.by_vdim;
GridFunction *nodes = by_vdim ? &x0 : S.fes->GetMesh()->GetNodes();
x.HostReadWrite();
nodes->HostRead();
real_t rnorm = nodes->DistanceTo(x) / nodes->Norml2();
if (!opt.id) { mfem::out << "rnorm = " << rnorm << std::endl; }
const real_t lambda = opt.lambda;
if (by_vdim)
{
MFEM_VERIFY(!opt.radial,"'VDim solver can't use radial option!");
return Converged(rnorm);
}
if (opt.radial)
{
GridFunction delta(S.fes);
subtract(x, *nodes, delta); // delta = x - nodes
// position and Δ vectors at point i
Vector ni(SDIM), di(SDIM);
for (int i = 0; i < delta.Size()/SDIM; i++)
{
// extract local vectors
const int ndof = S.fes->GetNDofs();
for (int d = 0; d < SDIM; d++)
{
ni(d) = (*nodes)(d*ndof + i);
di(d) = delta(d*ndof + i);
}
// project the delta vector in radial direction
const real_t ndotd = (ni*di) / (ni*ni);
di.Set(ndotd,ni);
// set global vectors
for (int d = 0; d < SDIM; d++) { delta(d*ndof + i) = di(d); }
}
add(*nodes, delta, *nodes);
}
// x = lambda*nodes + (1-lambda)*x
add(lambda, *nodes, (1.0-lambda), x, x);
return Converged(rnorm);
}
void Amr()
{
MFEM_VERIFY(opt.amr_threshold >= 0.0 && opt.amr_threshold <= 1.0, "");
Mesh *smesh = S.mesh;
Array<Refinement> amr;
const int NE = smesh->GetNE();
DenseMatrix Jadjt, Jadj(DIM, SDIM);
for (int e = 0; e < NE; e++)
{
real_t minW = +NL_DMAX;
real_t maxW = -NL_DMAX;
ElementTransformation *eTr = smesh->GetElementTransformation(e);
const Geometry::Type &type =
smesh->GetElement(e)->GetGeometryType();
const IntegrationRule *ir = &IntRules.Get(type, opt.order);
const int NQ = ir->GetNPoints();
for (int q = 0; q < NQ; q++)
{
eTr->SetIntPoint(&ir->IntPoint(q));
const DenseMatrix &J = eTr->Jacobian();
CalcAdjugate(J, Jadj);
Jadjt = Jadj;
Jadjt.Transpose();
const real_t w = Jadjt.Weight();
minW = std::min(minW, w);
maxW = std::max(maxW, w);
}
if (std::abs(maxW) != 0.0)
{
const real_t rho = minW / maxW;
MFEM_VERIFY(rho <= 1.0, "");
if (rho < opt.amr_threshold) { amr.Append(Refinement(e)); }
}
}
if (amr.Size()>0)
{
smesh->GetNodes()->HostReadWrite();
smesh->GeneralRefinement(amr);
S.fes->Update();
x.HostReadWrite();
x.Update();
a.Update();
b.HostReadWrite();
b.Update();
S.BoundaryConditions();
}
}
};
// Surface solver 'by vector'
class ByNodes: public Solver
{
public:
ByNodes(Surface &S, Opt &opt): Solver(S, opt)
{ a.AddDomainIntegrator(new VectorDiffusionIntegrator(one)); }
bool Step() override
{
x = *S.fes->GetMesh()->GetNodes();
bool converge = ParAXeqB();
S.mesh->SetNodes(x);
return converge ? true : false;
}
};
// Surface solver 'by ByVDim'
class ByVDim: public Solver
{
public:
void SetNodes(const GridFunction &Xi, const int c)
{
auto d_Xi = Xi.Read();
auto d_nodes = S.fes->GetMesh()->GetNodes()->Write();
const int ndof = S.fes->GetNDofs();
mfem::forall(ndof, [=] MFEM_HOST_DEVICE (int i)
{
d_nodes[c*ndof + i] = d_Xi[i];
});
}
void GetNodes(GridFunction &Xi, const int c)
{
auto d_Xi = Xi.Write();
const int ndof = S.fes->GetNDofs();
auto d_nodes = S.fes->GetMesh()->GetNodes()->Read();
mfem::forall(ndof, [=] MFEM_HOST_DEVICE (int i)
{
d_Xi[i] = d_nodes[c*ndof + i];
});
}
ByVDim(Surface &S, Opt &opt): Solver(S, opt)
{ a.AddDomainIntegrator(new DiffusionIntegrator(one)); }
bool Step() override
{
bool cvg[SDIM] {false};
for (int c=0; c < SDIM; ++c)
{
GetNodes(x, c);
x0 = x;
cvg[c] = ParAXeqB();
SetNodes(x, c);
}
const bool converged = cvg[0] && cvg[1] && cvg[2];
return converged ? true : false;
}
};
};
// #0: Default surface mesh file
struct MeshFromFile: public Surface
{
MeshFromFile(Opt &opt): Surface(opt, opt.mesh_file) { }
};
// #1: Catenoid surface
struct Catenoid: public Surface
{
Catenoid(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
void Prefix() override
{
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
Array<int> v2v(GetNV());
for (int i = 0; i < v2v.Size(); i++) { v2v[i] = i; }
// identify vertices on vertical lines
for (int j = 0; j <= opt.ny; j++)
{
const int v_old = opt.nx + j * (opt.nx + 1);
const int v_new = j * (opt.nx + 1);
v2v[v_old] = v_new;
}
// renumber elements
for (int i = 0; i < GetNE(); i++)
{
Element *el = GetElement(i);
int *v = el->GetVertices();
const int nv = el->GetNVertices();
for (int j = 0; j < nv; j++)
{
v[j] = v2v[v[j]];
}
}
// renumber boundary elements
for (int i = 0; i < GetNBE(); i++)
{
Element *el = GetBdrElement(i);
int *v = el->GetVertices();
const int nv = el->GetNVertices();
for (int j = 0; j < nv; j++)
{
v[j] = v2v[v[j]];
}
}
RemoveUnusedVertices();
RemoveInternalBoundaries();
}
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(SDIM);
// u in [0,2π] and v in [-π/6,π/6]
const real_t u = 2.0*PI*x[0];
const real_t v = PI*(x[1]-0.5)/3.;
p[0] = cos(u);
p[1] = sin(u);
p[2] = v;
}
};
// #2: Helicoid surface
struct Helicoid: public Surface
{
Helicoid(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(SDIM);
// u in [0,2π] and v in [-2π/3,2π/3]
const real_t u = 2.0*PI*x[0];
const real_t v = 2.0*PI*(2.0*x[1]-1.0)/3.0;
p(0) = sin(u)*v;
p(1) = cos(u)*v;
p(2) = u;
}
};
// #3: Enneper's surface
struct Enneper: public Surface
{
Enneper(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(SDIM);
// (u,v) in [-2, +2]
const real_t u = 4.0*(x[0]-0.5);
const real_t v = 4.0*(x[1]-0.5);
p[0] = +u - u*u*u/3.0 + u*v*v;
p[1] = -v - u*u*v + v*v*v/3.0;
p[2] = u*u - v*v;
}
};
// #4: Hold surface
struct Hold: public Surface
{
Hold(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
void Prefix() override
{
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
Array<int> v2v(GetNV());
for (int i = 0; i < v2v.Size(); i++) { v2v[i] = i; }
// identify vertices on vertical lines
for (int j = 0; j <= opt.ny; j++)
{
const int v_old = opt.nx + j * (opt.nx + 1);
const int v_new = j * (opt.nx + 1);
v2v[v_old] = v_new;
}
// renumber elements
for (int i = 0; i < GetNE(); i++)
{
Element *el = GetElement(i);
int *v = el->GetVertices();
const int nv = el->GetNVertices();
for (int j = 0; j < nv; j++)
{
v[j] = v2v[v[j]];
}
}
// renumber boundary elements
for (int i = 0; i < GetNBE(); i++)
{
Element *el = GetBdrElement(i);
int *v = el->GetVertices();
const int nv = el->GetNVertices();
for (int j = 0; j < nv; j++)
{
v[j] = v2v[v[j]];
}
}
RemoveUnusedVertices();
RemoveInternalBoundaries();
}
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(SDIM);
// u in [0,2π] and v in [0,1]
const real_t u = 2.0*PI*x[0];
const real_t v = x[1];
p[0] = cos(u)*(1.0 + 0.3*sin(3.*u + PI*v));
p[1] = sin(u)*(1.0 + 0.3*sin(3.*u + PI*v));
p[2] = v;
}
};
// #5: Costa minimal surface
#include <complex>
using cdouble = std::complex<real_t>;
#define I cdouble(0.0, 1.0)
// https://dlmf.nist.gov/20.2
cdouble EllipticTheta(const int a, const cdouble u, const cdouble q)
{
cdouble J = 0.0;
real_t delta = std::numeric_limits<real_t>::max();
switch (a)
{
case 1:
for (int n=0; delta > EPS; n+=1)
{
const cdouble j(pow(-real_t(1),real_t(n))*pow(q,
real_t(n*(n+1)))*sin(real_t(2*n+1)*u));
delta = abs(j);
J += j;
}
return cdouble(real_t(2)*pow(q,real_t(0.25))*J);
case 2:
for (int n=0; delta > EPS; n+=1)
{
const cdouble j(pow(q,real_t(n*(n+1)))*cos(real_t(2*n+1)*u));
delta = abs(j);
J += j;
}
return cdouble(real_t(2)*pow(q,real_t(0.25))*J);
case 3:
for (int n=1; delta > EPS; n+=1)
{
const cdouble j(pow(q,real_t(n*n))*cos(real_t(2*n)*u));
delta = abs(j);
J += j;
}
return cdouble(real_t(1) + real_t(2)*J);
case 4:
for (int n=1; delta > EPS; n+=1)
{
const cdouble j(pow(-real_t(1),real_t(n))*pow(q,
real_t(n*n))*cos(real_t(2*n)*u));
delta = abs(j);
J += j;
}
return cdouble(real_t(1) + real_t(2)*J);
}
return J;
}
// https://dlmf.nist.gov/23.6#E5
cdouble WeierstrassP(const cdouble z,
const cdouble w1 = 0.5,
const cdouble w3 = real_t(0.5)*I)
{
const cdouble tau = w3/w1;
const cdouble q = exp(I*PI*tau);
const cdouble e1 = PI*PI/(real_t(12)*w1*w1)*
(pow(EllipticTheta(2,0,q),real_t(4)) +
real_t(2)*pow(EllipticTheta(4,0,q),real_t(4)));
const cdouble u = PI*z / (real_t(2)*w1);
const cdouble P = PI * EllipticTheta(3,0,q)*EllipticTheta(4,0,q) *
EllipticTheta(2,u,q)/(real_t(2)*w1*EllipticTheta(1,u,q));
return P*P + e1;
}
cdouble EllipticTheta1Prime(const int k, const cdouble u, const cdouble q)
{
cdouble J = 0.0;
real_t delta = std::numeric_limits<real_t>::max();
for (int n=0; delta > EPS; n+=1)
{
const real_t alpha = 2.0*n+1.0;
const cdouble Dcosine = pow(alpha,real_t(k))*sin(k*PI/real_t(2) + alpha*u);
const cdouble j(pow(-real_t(1),real_t(n))*pow(q,real_t(n*(n+1)))*Dcosine);
delta = abs(j);
J += j;
}
return cdouble(real_t(2)*pow(q,real_t(0.25))*J);
}
// Logarithmic Derivative of Theta Function 1
cdouble LogEllipticTheta1Prime(const cdouble u, const cdouble q)
{
cdouble J = 0.0;
real_t delta = std::numeric_limits<real_t>::max();
for (int n=1; delta > EPS; n+=1)
{
cdouble q2n = pow(q, real_t(2*n));
if (abs(q2n) < EPS) { q2n = 0.0; }
const cdouble j = q2n/(real_t(1)-q2n)*sin(real_t(2*n)*u);
delta = abs(j);
J += j;
}
return real_t(1)/tan(u) + real_t(4)*J;
}
// https://dlmf.nist.gov/23.6#E13
cdouble WeierstrassZeta(const cdouble z,
const cdouble w1 = 0.5,
const cdouble w3 = real_t(0.5)*I)
{
const cdouble tau = w3/w1;
const cdouble q = exp(I*PI*tau);
const cdouble n1 = -PI*PI/(real_t(12)*w1) *
(EllipticTheta1Prime(3,0,q)/
EllipticTheta1Prime(1,0,q));
const cdouble u = PI*z / (real_t(2)*w1);
return z*n1/w1 + PI/(real_t(2)*w1)*LogEllipticTheta1Prime(u,q);
}
// https://www.mathcurve.com/surfaces.gb/costa/costa.shtml
static real_t ALPHA[4] {0.0};
struct Costa: public Surface
{
Costa(Opt &opt): Surface((opt.Tptr = Parametrization, opt), false) { }
void Prefix() override
{
ALPHA[3] = opt.tau;
const int nx = opt.nx, ny = opt.ny;
MFEM_VERIFY(nx>2 && ny>2, "");
const int nXhalf = (nx%2)==0 ? 4 : 2;
const int nYhalf = (ny%2)==0 ? 4 : 2;
const int nxh = nXhalf + nYhalf;
const int NVert = (nx+1) * (ny+1);
const int NElem = nx*ny - 4 - nxh;
const int NBdrElem = 0;
InitMesh(DIM, SDIM, NVert, NElem, NBdrElem);
// Sets vertices and the corresponding coordinates
for (int j = 0; j <= ny; j++)
{
const real_t cy = ((real_t) j / ny) ;
for (int i = 0; i <= nx; i++)
{
const real_t cx = ((real_t) i / nx);
const real_t coords[SDIM] = {cx, cy, 0.0};
AddVertex(coords);
}
}
// Sets elements and the corresponding indices of vertices
for (int j = 0; j < ny; j++)
{
for (int i = 0; i < nx; i++)
{
if (i == 0 && j == 0) { continue; }
if (i+1 == nx && j == 0) { continue; }
if (i == 0 && j+1 == ny) { continue; }
if (i+1 == nx && j+1 == ny) { continue; }
if ((j == 0 || j+1 == ny) && (abs(nx-(i<<1)-1)<=1)) { continue; }
if ((i == 0 || i+1 == nx) && (abs(ny-(j<<1)-1)<=1)) { continue; }
const int i0 = i + j*(nx+1);
const int i1 = i+1 + j*(nx+1);
const int i2 = i+1 + (j+1)*(nx+1);
const int i3 = i + (j+1)*(nx+1);
const int ind[4] = {i0, i1, i2, i3};
AddQuad(ind);
}
}
RemoveUnusedVertices();
FinalizeQuadMesh(false, 0, true);
FinalizeTopology();
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
}
static void Parametrization(const Vector &X, Vector &p)
{
const real_t tau = ALPHA[3];
Vector x = X;
x -= +0.5;
x *= tau;
x -= -0.5;
p.SetSize(3);
const bool y_top = x[1] > 0.5;
const bool x_top = x[0] > 0.5;
real_t u = x[0];
real_t v = x[1];
if (y_top) { v = 1.0 - x[1]; }
if (x_top) { u = 1.0 - x[0]; }
const cdouble w = u + I*v;
const cdouble w3 = I/real_t(2);
const cdouble w1 = 1./2.;
const cdouble pw = WeierstrassP(w);
const cdouble e1 = WeierstrassP(0.5);
const cdouble zw = WeierstrassZeta(w);
const cdouble dw = WeierstrassZeta(w-w1) - WeierstrassZeta(w-w3);
p[0] = real(PI*(u+PI/(real_t(4)*e1))- zw +PI/(real_t(2)*e1)*(dw));
p[1] = real(PI*(v+PI/(real_t(4)*e1))-I*zw-PI*I/(real_t(2)*e1)*(dw));
p[2] = sqrt(PI/2.)*log(abs((pw-e1)/(pw+e1)));
if (y_top) { p[1] *= -1.0; }
if (x_top) { p[0] *= -1.0; }
const bool nan = std::isnan(p[0]) || std::isnan(p[1]) || std::isnan(p[2]);
MFEM_VERIFY(!nan, "nan");
ALPHA[0] = std::max(p[0], ALPHA[0]);
ALPHA[1] = std::max(p[1], ALPHA[1]);
ALPHA[2] = std::max(p[2], ALPHA[2]);
}
void Snap() override
{
Vector node(SDIM);
MFEM_VERIFY(ALPHA[0] > 0.0,"");
MFEM_VERIFY(ALPHA[1] > 0.0,"");
MFEM_VERIFY(ALPHA[2] > 0.0,"");
GridFunction &nodes = *GetNodes();
const real_t phi = (1.0 + sqrt(5.0)) / 2.0;
for (int i = 0; i < nodes.FESpace()->GetNDofs(); i++)
{
for (int d = 0; d < SDIM; d++)
{
const real_t alpha = d==2 ? phi : 1.0;
const int vdof = nodes.FESpace()->DofToVDof(i, d);
nodes(vdof) /= alpha * ALPHA[d];
}
}
}
};
// #6: Shell surface model
struct Shell: public Surface
{
Shell(Opt &opt): Surface((opt.niters = 1, opt.Tptr = Parametrization, opt)) { }
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(3);
// u in [0,2π] and v in [-15, 6]
const real_t u = 2.0*PI*x[0];
const real_t v = 21.0*x[1]-15.0;
p[0] = +1.0*pow(1.16,v)*cos(v)*(1.0+cos(u));
p[1] = -1.0*pow(1.16,v)*sin(v)*(1.0+cos(u));
p[2] = -2.0*pow(1.16,v)*(1.0+sin(u));
}
};
// #7: Scherk's doubly periodic surface
struct Scherk: public Surface
{
static void Parametrization(const Vector &x, Vector &p)
{
p.SetSize(SDIM);
const real_t alpha = 0.49;
// (u,v) in [-απ, +απ]
const real_t u = alpha*PI*(2.0*x[0]-1.0);
const real_t v = alpha*PI*(2.0*x[1]-1.0);
p[0] = u;
p[1] = v;
p[2] = log(cos(v)/cos(u));
}
Scherk(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
};
// #8: Full Peach street model
struct FullPeach: public Surface
{
static constexpr int NV = 8;
static constexpr int NE = 6;
FullPeach(Opt &opt):
Surface((opt.niters = std::min(4, opt.niters), opt), NV, NE, 0) { }
void Prefix() override
{
const real_t quad_v[NV][SDIM] =
{
{-1, -1, -1}, {+1, -1, -1}, {+1, +1, -1}, {-1, +1, -1},
{-1, -1, +1}, {+1, -1, +1}, {+1, +1, +1}, {-1, +1, +1}
};
const int quad_e[NE][4] =
{
{3, 2, 1, 0}, {0, 1, 5, 4}, {1, 2, 6, 5},
{2, 3, 7, 6}, {3, 0, 4, 7}, {4, 5, 6, 7}
};
for (int j = 0; j < NV; j++) { AddVertex(quad_v[j]); }
for (int j = 0; j < NE; j++) { AddQuad(quad_e[j], j+1); }
FinalizeQuadMesh(false, 0, true);
FinalizeTopology(false);
UniformRefinement();
SetCurvature(opt.order, false, SDIM, Ordering::byNODES);
}
void Snap() override { SnapNodesToUnitSphere(); }
void BoundaryConditions() override
{
Vector X(SDIM);
Array<int> dofs;
Array<int> ess_vdofs, ess_tdofs;
ess_vdofs.SetSize(fes->GetVSize());
MFEM_VERIFY(fes->GetVSize() >= fes->GetTrueVSize(),"");
ess_vdofs = 0;
DenseMatrix PointMat;
mesh->GetNodes()->HostRead();
for (int e = 0; e < fes->GetNE(); e++)
{
fes->GetElementDofs(e, dofs);
const IntegrationRule &ir = fes->GetFE(e)->GetNodes();
ElementTransformation *eTr = mesh->GetElementTransformation(e);
eTr->Transform(ir, PointMat);
Vector one(dofs.Size());
for (int dof = 0; dof < dofs.Size(); dof++)
{
one = 0.0;
one[dof] = 1.0;
const int k = dofs[dof];
MFEM_ASSERT(k >= 0, "");
PointMat.Mult(one, X);
const bool halfX = std::abs(X[0]) < EPS && X[1] <= 0.0;
const bool halfY = std::abs(X[2]) < EPS && X[1] >= 0.0;
const bool is_on_bc = halfX || halfY;
for (int c = 0; c < SDIM; c++)
{ ess_vdofs[fes->DofToVDof(k, c)] = is_on_bc; }
}
}
const SparseMatrix *R = fes->GetRestrictionMatrix();
if (!R)
{
ess_tdofs.MakeRef(ess_vdofs);
}
else
{
R->BooleanMult(ess_vdofs, ess_tdofs);
}
bc.HostReadWrite();
FiniteElementSpace::MarkerToList(ess_tdofs, bc);
}
};
// #9: 1/4th Peach street model
struct QuarterPeach: public Surface
{
QuarterPeach(Opt &opt): Surface((opt.Tptr = Parametrization, opt)) { }
static void Parametrization(const Vector &X, Vector &p)
{
p = X;
const real_t x = 2.0*X[0]-1.0;
const real_t y = X[1];
const real_t r = sqrt(x*x + y*y);