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// MFEM Example 30
//
// Compile with: make ex30
//
// Sample runs: ex30 -m ../data/square-disc.mesh -o 1
// ex30 -m ../data/square-disc.mesh -o 2
// ex30 -m ../data/square-disc.mesh -o 2 -me 1e+4
// ex30 -m ../data/square-disc-nurbs.mesh -o 2
// ex30 -m ../data/star.mesh -o 2 -eo 4
// ex30 -m ../data/fichera.mesh -o 2 -me 1e+5 -e 5e-2
// ex30 -m ../data/disc-nurbs.mesh -o 2
// ex30 -m ../data/ball-nurbs.mesh -o 2 -eo 3 -e 5e-2 -me 1e+5
// ex30 -m ../data/star-surf.mesh -o 2
// ex30 -m ../data/square-disc-surf.mesh -o 2
// ex30 -m ../data/amr-quad.mesh -l 2
//
// Description: This is an example of adaptive mesh refinement preprocessing
// which lowers the data oscillation [1] to a user-defined
// relative threshold. There is no PDE being solved.
//
// MFEM's capability to work with both conforming and
// nonconforming meshes is demonstrated in example 6. In some
// problems, the material data or loading data is not sufficiently
// resolved on the initial mesh. This missing fine scale data
// reduces the accuracy of the solution as well as the accuracy of
// some local error estimators. By preprocessing the mesh before
// solving the PDE, many issues can be avoided.
//
// [1] Morin, P., Nochetto, R. H., & Siebert, K. G. (2000). Data
// oscillation and convergence of adaptive FEM. SIAM Journal
// on Numerical Analysis, 38(2), 466-488.
//
// [2] Mitchell, W. F. (2013). A collection of 2D elliptic
// problems for testing adaptive grid refinement algorithms.
// Applied mathematics and computation, 220, 350-364.
#include "mfem.hpp"
#include <fstream>
#include <iostream>
using namespace std;
using namespace mfem;
// Piecewise-affine function which is sometimes mesh-conforming
real_t affine_function(const Vector &p)
{
real_t x = p(0), y = p(1);
if (x < 0.0)
{
return 1.0 + x + y;
}
else
{
return 1.0;
}
}
// Piecewise-constant function which is never mesh-conforming
real_t jump_function(const Vector &p)
{
if (p.Normlp(2.0) > 0.4 && p.Normlp(2.0) < 0.6)
{
return 1.0;
}
else
{
return 5.0;
}
}
// Singular function derived from the Laplacian of the "steep wavefront" problem
// in [2].
real_t singular_function(const Vector &p)
{
real_t x = p(0), y = p(1);
real_t alpha = 1000.0;
real_t xc = 0.75, yc = 0.5;
real_t r0 = 0.7;
real_t r = sqrt(pow(x - xc,2.0) + pow(y - yc,2.0));
real_t num = - ( alpha - pow(alpha,3) * (pow(r,2) - pow(r0,2)) );
real_t denom = pow(r * ( pow(alpha,2) * pow(r0,2) + pow(alpha,2) * pow(r,2) \
- 2 * pow(alpha,2) * r0 * r + 1.0 ),2);
denom = std::max(denom, (real_t) 1.0e-8);
return num / denom;
}
int main(int argc, char *argv[])
{
// 1. Parse command-line options.
const char *mesh_file = "../data/star.mesh";
int order = 1;
int nc_limit = 1;
int max_elems = 100*1000;
real_t double_max_elems = real_t(max_elems);
bool visualization = true;
real_t osc_threshold = 1e-3;
int enriched_order = 5;
OptionsParser args(argc, argv);
args.AddOption(&mesh_file, "-m", "--mesh",
"Mesh file to use.");
args.AddOption(&order, "-o", "--order",
"Finite element order (polynomial degree).");
args.AddOption(&nc_limit, "-l", "--nc-limit",
"Maximum level of hanging nodes.");
args.AddOption(&double_max_elems, "-me", "--max-elems",
"Stop after reaching this many elements.");
args.AddOption(&osc_threshold, "-e", "--error",
"relative data oscillation threshold.");
args.AddOption(&enriched_order, "-eo", "--enriched_order",
"Enriched quadrature order.");
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
"--no-visualization",
"Enable or disable GLVis visualization.");
args.Parse();
if (!args.Good())
{
args.PrintUsage(cout);
return 1;
}
args.PrintOptions(cout);
max_elems = int(double_max_elems);
Mesh mesh(mesh_file, 1, 1);
// 2. Since a NURBS mesh can currently only be refined uniformly, we need to
// convert it to a piecewise-polynomial curved mesh. First we refine the
// NURBS mesh a bit and then project the curvature to quadratic Nodes.
if (mesh.NURBSext)
{
for (int i = 0; i < 2; i++)
{
mesh.UniformRefinement();
}
mesh.SetCurvature(2);
}
// 3. Define functions and refiner.
FunctionCoefficient affine_coeff(affine_function);
FunctionCoefficient jump_coeff(jump_function);
FunctionCoefficient singular_coeff(singular_function);
CoefficientRefiner coeffrefiner(affine_coeff, order);
// 4. Connect to GLVis.
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock;
if (visualization)
{
sol_sock.open(vishost, visport);
}
// 5. Define custom integration rule (optional).
const IntegrationRule *irs[Geometry::NumGeom];
int order_quad = 2*order + enriched_order;
for (int i = 0; i < Geometry::NumGeom; ++i)
{
irs[i] = &(IntRules.Get(i, order_quad));
}
// 6. Apply custom refiner settings.
coeffrefiner.SetIntRule(irs);
coeffrefiner.SetMaxElements(max_elems);
coeffrefiner.SetThreshold(osc_threshold);
coeffrefiner.SetNCLimit(nc_limit);
coeffrefiner.PrintWarnings();
// 7. Preprocess mesh to control osc (piecewise-affine function). This is
// mostly just a verification check. The oscillation should be zero if the
// function is mesh-conforming and order > 0.
coeffrefiner.PreprocessMesh(mesh);
mfem::out << "\n";
mfem::out << "Function 0 (affine) \n";
mfem::out << "Number of Elements " << mesh.GetNE() << "\n";
mfem::out << "Osc error " << coeffrefiner.GetOsc() << "\n";
// 8. Preprocess mesh to control osc (jump function).
coeffrefiner.ResetCoefficient(jump_coeff);
coeffrefiner.PreprocessMesh(mesh);
mfem::out << "\n";
mfem::out << "Function 1 (discontinuous) \n";
mfem::out << "Number of Elements " << mesh.GetNE() << "\n";
mfem::out << "Osc error " << coeffrefiner.GetOsc() << "\n";
// 9. Preprocess mesh to control osc (singular function).
coeffrefiner.ResetCoefficient(singular_coeff);
coeffrefiner.PreprocessMesh(mesh);
mfem::out << "\n";
mfem::out << "Function 2 (singular) \n";
mfem::out << "Number of Elements " << mesh.GetNE() << "\n";
mfem::out << "Osc error " << coeffrefiner.GetOsc() << "\n";
if (visualization)
{
sol_sock.precision(8);
sol_sock << "mesh\n" << mesh << flush;
}
return 0;
}
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