685 lines
20 KiB
C++
685 lines
20 KiB
C++
// MFEM Example 9
|
|
// Nonlinear Constrained Optimization Modification
|
|
//
|
|
// Compile with: make ex9
|
|
//
|
|
// Sample runs:
|
|
// ex9 -m ../../data/periodic-segment.mesh -r 3 -p 0 -o 2 -dt 0.002 -opt 1
|
|
// ex9 -m ../../data/periodic-segment.mesh -r 3 -p 0 -o 2 -dt 0.002 -opt 2
|
|
//
|
|
// ex9 -m ../../data/periodic-square.mesh -p 0 -r 2 -dt 0.01 -tf 10 -opt 1
|
|
// ex9 -m ../../data/periodic-square.mesh -p 0 -r 2 -dt 0.01 -tf 10 -opt 2
|
|
//
|
|
// ex9 -m ../../data/periodic-square.mesh -p 1 -r 2 -dt 0.005 -tf 9 -opt 1
|
|
// ex9 -m ../../data/periodic-square.mesh -p 1 -r 2 -dt 0.005 -tf 9 -opt 2
|
|
//
|
|
// ex9 -m ../../data/amr-quad.mesh -p 1 -r 1 -dt 0.002 -tf 9 -opt 1
|
|
// ex9 -m ../../data/amr-quad.mesh -p 1 -r 1 -dt 0.002 -tf 9 -opt 2
|
|
//
|
|
// ex9 -m ../../data/disc-nurbs.mesh -p 1 -r 2 -dt 0.005 -tf 9 -opt 1
|
|
// ex9 -m ../../data/disc-nurbs.mesh -p 1 -r 2 -dt 0.005 -tf 9 -opt 2
|
|
//
|
|
// ex9 -m ../../data/disc-nurbs.mesh -p 2 -r 2 -dt 0.01 -tf 9 -opt 1
|
|
// ex9 -m ../../data/disc-nurbs.mesh -p 2 -r 2 -dt 0.01 -tf 9 -opt 2
|
|
//
|
|
// ex9 -m ../../data/periodic-square.mesh -p 3 -r 3 -dt 0.0025 -tf 9 -opt 1
|
|
// ex9 -m ../../data/periodic-square.mesh -p 3 -r 3 -dt 0.0025 -tf 9 -opt 2
|
|
//
|
|
// ex9 -m ../../data/periodic-cube.mesh -p 0 -r 2 -o 2 -dt 0.02 -tf 8 -opt 1
|
|
// ex9 -m ../../data/periodic-cube.mesh -p 0 -r 2 -o 2 -dt 0.02 -tf 8 -opt 2
|
|
|
|
// Description: This example modifies the standard MFEM ex9 by adding nonlinear
|
|
// constrained optimization capabilities through the SLBQP and
|
|
// HIOP solvers. It demonstrates how a user can define a custom
|
|
// class OptimizationProblem that includes linear/nonlinear
|
|
// equality/inequality constraints. This optimization is applied
|
|
// as post-processing to the solution of the transport equation.
|
|
//
|
|
// Description of ex9:
|
|
// This example code solves the time-dependent advection equation
|
|
// du/dt + v.grad(u) = 0, where v is a given fluid velocity, and
|
|
// u0(x)=u(0,x) is a given initial condition.
|
|
//
|
|
// The example demonstrates the use of Discontinuous Galerkin (DG)
|
|
// bilinear forms in MFEM (face integrators), the use of explicit
|
|
// ODE time integrators, the definition of periodic boundary
|
|
// conditions through periodic meshes, as well as the use of GLVis
|
|
// for persistent visualization of a time-evolving solution. The
|
|
// saving of time-dependent data files for external visualization
|
|
// with VisIt (visit.llnl.gov) is also illustrated.
|
|
|
|
#include "mfem.hpp"
|
|
#include <fstream>
|
|
#include <iostream>
|
|
|
|
using namespace std;
|
|
using namespace mfem;
|
|
|
|
// Choice for the problem setup. The fluid velocity, initial condition and
|
|
// inflow boundary condition are chosen based on this parameter.
|
|
int problem;
|
|
|
|
// Nonlinear optimizer.
|
|
int optimizer_type;
|
|
|
|
// Velocity coefficient
|
|
bool invert_velocity = false;
|
|
void velocity_function(const Vector &x, Vector &v);
|
|
|
|
// Initial condition
|
|
double u0_function(const Vector &x);
|
|
|
|
// Inflow boundary condition
|
|
double inflow_function(const Vector &x);
|
|
|
|
// Mesh bounding box
|
|
Vector bb_min, bb_max;
|
|
|
|
/// Computes C(x) = sum w_i x_i, where w is a given Vector.
|
|
class LinearScaleOperator : public Operator
|
|
{
|
|
private:
|
|
const Vector &w;
|
|
mutable DenseMatrix grad;
|
|
|
|
public:
|
|
LinearScaleOperator(const Vector &weight)
|
|
: Operator(1, weight.Size()), w(weight), grad(1, width)
|
|
{
|
|
for (int i = 0; i < width; i++) { grad(0, i) = w(i); }
|
|
}
|
|
|
|
virtual void Mult(const Vector &x, Vector &y) const
|
|
{
|
|
y(0) = w * x;
|
|
}
|
|
|
|
virtual Operator &GetGradient(const Vector &x) const
|
|
{
|
|
return grad;
|
|
}
|
|
};
|
|
|
|
/// Nonlinear monotone bounded operator to test nonlinear ineq constraints.
|
|
/// Computes D(x) = tanh(sum(x_i)).
|
|
class TanhSumOperator : public Operator
|
|
{
|
|
private:
|
|
mutable DenseMatrix grad;
|
|
|
|
public:
|
|
TanhSumOperator(int size) : Operator(1, size), grad(1, width) { }
|
|
|
|
virtual void Mult(const Vector &x, Vector &y) const
|
|
{
|
|
y(0) = std::tanh(x.Sum());
|
|
}
|
|
|
|
virtual Operator &GetGradient(const Vector &x) const
|
|
{
|
|
const double ts = std::tanh(x.Sum());
|
|
const double dtanh = 1.0 - ts * ts;
|
|
for (int i = 0; i < width; i++) { grad(0, i) = dtanh; }
|
|
return grad;
|
|
}
|
|
};
|
|
|
|
/** Monotone and conservative a-posteriori correction for transport solutions:
|
|
* Find x that minimizes 0.5 || x - x_HO ||^2, subject to
|
|
* sum w_i x_i = mass,
|
|
* tanh(sum(x_i_min)) <= tanh(sum(x_i)) <= tanh(sum(x_i_max)),
|
|
* x_i_min <= x_i <= x_i_max,
|
|
*/
|
|
class OptimizedTransportProblem : public OptimizationProblem
|
|
{
|
|
private:
|
|
const Vector &x_HO;
|
|
Vector massvec, d_lo, d_hi;
|
|
const LinearScaleOperator LSoper;
|
|
const TanhSumOperator TSoper;
|
|
|
|
public:
|
|
OptimizedTransportProblem(const Vector &xho, const Vector &w, double mass,
|
|
const Vector &xmin, const Vector &xmax)
|
|
: OptimizationProblem(xho.Size(), NULL, NULL),
|
|
x_HO(xho), massvec(1), d_lo(1), d_hi(1),
|
|
LSoper(w), TSoper(w.Size())
|
|
{
|
|
C = &LSoper;
|
|
massvec(0) = mass;
|
|
SetEqualityConstraint(massvec);
|
|
|
|
D = &TSoper;
|
|
d_lo(0) = std::tanh(xmin.Sum());
|
|
d_hi(0) = std::tanh(xmax.Sum());
|
|
MFEM_ASSERT(d_lo(0) < d_hi(0),
|
|
"The bounds produce an infeasible optimization problem");
|
|
SetInequalityConstraint(d_lo, d_hi);
|
|
|
|
SetSolutionBounds(xmin, xmax);
|
|
}
|
|
|
|
virtual double CalcObjective(const Vector &x) const
|
|
{
|
|
double res = 0.0;
|
|
for (int i = 0; i < input_size; i++)
|
|
{
|
|
const double d = x(i) - x_HO(i);
|
|
res += d * d;
|
|
}
|
|
return 0.5 * res;
|
|
}
|
|
|
|
virtual void CalcObjectiveGrad(const Vector &x, Vector &grad) const
|
|
{
|
|
for (int i = 0; i < input_size; i++) { grad(i) = x(i) - x_HO(i); }
|
|
}
|
|
};
|
|
|
|
|
|
/** A time-dependent operator for the right-hand side of the ODE. The DG weak
|
|
form of du/dt = -v.grad(u) is M du/dt = K u + b, where M and K are the mass
|
|
and advection matrices, and b describes the flow on the boundary. This can
|
|
be written as a general ODE, du/dt = M^{-1} (K u + b), and this class is
|
|
used to evaluate the right-hand side. */
|
|
class FE_Evolution : public TimeDependentOperator
|
|
{
|
|
private:
|
|
SparseMatrix &M, &K;
|
|
const Vector &b;
|
|
DSmoother M_prec;
|
|
CGSolver M_solver;
|
|
|
|
mutable Vector z;
|
|
|
|
double dt;
|
|
BilinearForm &bf;
|
|
Vector &M_rowsums;
|
|
|
|
public:
|
|
FE_Evolution(SparseMatrix &_M, SparseMatrix &_K, const Vector &_b,
|
|
BilinearForm &_bf, Vector &M_rs);
|
|
|
|
void SetTimeStep(double _dt) { dt = _dt; }
|
|
void SetK(SparseMatrix &_K) { K = _K; }
|
|
virtual void Mult(const Vector &x, Vector &y) const;
|
|
|
|
virtual ~FE_Evolution() { }
|
|
};
|
|
|
|
|
|
int main(int argc, char *argv[])
|
|
{
|
|
// 1. Parse command-line options.
|
|
problem = 0;
|
|
optimizer_type = 1;
|
|
const char *mesh_file = "../../data/periodic-hexagon.mesh";
|
|
int ref_levels = 2;
|
|
int order = 3;
|
|
int ode_solver_type = 3;
|
|
double t_final = 1.0;
|
|
double dt = 0.01;
|
|
bool visualization = true;
|
|
bool visit = false;
|
|
bool binary = false;
|
|
int vis_steps = 5;
|
|
|
|
int precision = 8;
|
|
cout.precision(precision);
|
|
|
|
OptionsParser args(argc, argv);
|
|
args.AddOption(&mesh_file, "-m", "--mesh",
|
|
"Mesh file to use.");
|
|
args.AddOption(&problem, "-p", "--problem",
|
|
"Problem setup to use. See options in velocity_function().");
|
|
args.AddOption(&ref_levels, "-r", "--refine",
|
|
"Number of times to refine the mesh uniformly.");
|
|
args.AddOption(&order, "-o", "--order",
|
|
"Order (degree) of the finite elements.");
|
|
args.AddOption(&optimizer_type, "-opt", "--optimizer",
|
|
"Nonlinear optimizer: 1 - SLBQP,\n\t"
|
|
" 2 - HIOP.");
|
|
args.AddOption(&ode_solver_type, "-s", "--ode-solver",
|
|
"ODE solver: 1 - Forward Euler,\n\t"
|
|
" 2 - RK2 SSP, 3 - RK3 SSP, 4 - RK4, 6 - RK6.");
|
|
args.AddOption(&t_final, "-tf", "--t-final",
|
|
"Final time; start time is 0.");
|
|
args.AddOption(&dt, "-dt", "--time-step",
|
|
"Time step.");
|
|
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
|
|
"--no-visualization",
|
|
"Enable or disable GLVis visualization.");
|
|
args.AddOption(&visit, "-visit", "--visit-datafiles", "-no-visit",
|
|
"--no-visit-datafiles",
|
|
"Save data files for VisIt (visit.llnl.gov) visualization.");
|
|
args.AddOption(&binary, "-binary", "--binary-datafiles", "-ascii",
|
|
"--ascii-datafiles",
|
|
"Use binary (Sidre) or ascii format for VisIt data files.");
|
|
args.AddOption(&vis_steps, "-vs", "--visualization-steps",
|
|
"Visualize every n-th timestep.");
|
|
args.Parse();
|
|
if (!args.Good())
|
|
{
|
|
args.PrintUsage(cout);
|
|
return 1;
|
|
}
|
|
args.PrintOptions(cout);
|
|
|
|
// 2. Read the mesh from the given mesh file. We can handle geometrically
|
|
// periodic meshes in this code.
|
|
Mesh *mesh = new Mesh(mesh_file, 1, 1);
|
|
int dim = mesh->Dimension();
|
|
|
|
// 3. Define the ODE solver used for time integration. Several explicit
|
|
// Runge-Kutta methods are available.
|
|
ODESolver *ode_solver = NULL;
|
|
switch (ode_solver_type)
|
|
{
|
|
case 1: ode_solver = new ForwardEulerSolver; break;
|
|
case 2: ode_solver = new RK2Solver(1.0); break;
|
|
case 3: ode_solver = new RK3SSPSolver; break;
|
|
case 4: ode_solver = new RK4Solver; break;
|
|
case 6: ode_solver = new RK6Solver; break;
|
|
default:
|
|
cout << "Unknown ODE solver type: " << ode_solver_type << '\n';
|
|
delete mesh;
|
|
return 3;
|
|
}
|
|
|
|
// 4. Refine the mesh to increase the resolution. In this example we do
|
|
// 'ref_levels' of uniform refinement, where 'ref_levels' is a
|
|
// command-line parameter. If the mesh is of NURBS type, we convert it to
|
|
// a (piecewise-polynomial) high-order mesh.
|
|
for (int lev = 0; lev < ref_levels; lev++)
|
|
{
|
|
mesh->UniformRefinement();
|
|
}
|
|
if (mesh->NURBSext)
|
|
{
|
|
mesh->SetCurvature(max(order, 1));
|
|
}
|
|
mesh->GetBoundingBox(bb_min, bb_max, max(order, 1));
|
|
|
|
// 5. Define the discontinuous DG finite element space of the given
|
|
// polynomial order on the refined mesh.
|
|
DG_FECollection fec(order, dim, BasisType::Positive);
|
|
FiniteElementSpace fes(mesh, &fec);
|
|
|
|
cout << "Number of unknowns: " << fes.GetVSize() << endl;
|
|
|
|
// 6. Set up and assemble the bilinear and linear forms corresponding to the
|
|
// DG discretization. The DGTraceIntegrator involves integrals over mesh
|
|
// interior faces.
|
|
VectorFunctionCoefficient velocity(dim, velocity_function);
|
|
FunctionCoefficient inflow(inflow_function);
|
|
FunctionCoefficient u0(u0_function);
|
|
|
|
BilinearForm m(&fes);
|
|
m.AddDomainIntegrator(new MassIntegrator);
|
|
BilinearForm k(&fes);
|
|
k.AddDomainIntegrator(new ConvectionIntegrator(velocity, -1.0));
|
|
k.AddInteriorFaceIntegrator(
|
|
new TransposeIntegrator(new DGTraceIntegrator(velocity, 1.0, -0.5)));
|
|
k.AddBdrFaceIntegrator(
|
|
new TransposeIntegrator(new DGTraceIntegrator(velocity, 1.0, -0.5)));
|
|
|
|
LinearForm b(&fes);
|
|
b.AddBdrFaceIntegrator(
|
|
new BoundaryFlowIntegrator(inflow, velocity, -1.0, -0.5));
|
|
|
|
m.Assemble();
|
|
m.Finalize();
|
|
int skip_zeros = 0;
|
|
k.Assemble(skip_zeros);
|
|
k.Finalize(skip_zeros);
|
|
b.Assemble();
|
|
|
|
// 7. Define the initial conditions, save the corresponding grid function to
|
|
// a file and (optionally) save data in the VisIt format and initialize
|
|
// GLVis visualization.
|
|
GridFunction u(&fes);
|
|
u.ProjectCoefficient(u0);
|
|
|
|
{
|
|
ofstream omesh("ex9.mesh");
|
|
omesh.precision(precision);
|
|
mesh->Print(omesh);
|
|
ofstream osol("ex9-init.gf");
|
|
osol.precision(precision);
|
|
u.Save(osol);
|
|
}
|
|
|
|
// Create data collection for solution output: either VisItDataCollection for
|
|
// ascii data files, or SidreDataCollection for binary data files.
|
|
DataCollection *dc = NULL;
|
|
if (visit)
|
|
{
|
|
if (binary)
|
|
{
|
|
#ifdef MFEM_USE_SIDRE
|
|
dc = new SidreDataCollection("Example9", mesh);
|
|
#else
|
|
MFEM_ABORT("Must build with MFEM_USE_SIDRE=YES for binary output.");
|
|
#endif
|
|
}
|
|
else
|
|
{
|
|
dc = new VisItDataCollection("Example9", mesh);
|
|
dc->SetPrecision(precision);
|
|
}
|
|
dc->RegisterField("solution", &u);
|
|
dc->SetCycle(0);
|
|
dc->SetTime(0.0);
|
|
dc->Save();
|
|
}
|
|
|
|
socketstream sout;
|
|
if (visualization)
|
|
{
|
|
char vishost[] = "localhost";
|
|
int visport = 19916;
|
|
sout.open(vishost, visport);
|
|
if (!sout)
|
|
{
|
|
cout << "Unable to connect to GLVis server at "
|
|
<< vishost << ':' << visport << endl;
|
|
visualization = false;
|
|
cout << "GLVis visualization disabled.\n";
|
|
}
|
|
else
|
|
{
|
|
sout.precision(precision);
|
|
sout << "solution\n" << *mesh << u;
|
|
sout << "pause\n";
|
|
sout << flush;
|
|
cout << "GLVis visualization paused."
|
|
<< " Press space (in the GLVis window) to resume it.\n";
|
|
}
|
|
}
|
|
|
|
Vector M_rowsums(m.Size());
|
|
m.SpMat().GetRowSums(M_rowsums);
|
|
|
|
// 8. Define the time-dependent evolution operator describing the ODE
|
|
// right-hand side, and perform time-integration (looping over the time
|
|
// iterations, ti, with a time-step dt).
|
|
FE_Evolution adv(m.SpMat(), k.SpMat(), b, k, M_rowsums);
|
|
|
|
double t = 0.0;
|
|
adv.SetTime(t);
|
|
ode_solver->Init(adv);
|
|
|
|
// Compute initial volume.
|
|
const double vol0 = M_rowsums * u;
|
|
|
|
bool done = false;
|
|
for (int ti = 0; !done; )
|
|
{
|
|
double dt_real = min(dt, t_final - t);
|
|
adv.SetTimeStep(dt_real);
|
|
ode_solver->Step(u, t, dt_real);
|
|
ti++;
|
|
|
|
done = (t >= t_final - 1e-8*dt);
|
|
|
|
if (done || ti % vis_steps == 0)
|
|
{
|
|
cout << "time step: " << ti << ", time: " << t << endl;
|
|
|
|
if (visualization)
|
|
{
|
|
sout << "solution\n" << *mesh << u << flush;
|
|
}
|
|
|
|
if (visit)
|
|
{
|
|
dc->SetCycle(ti);
|
|
dc->SetTime(t);
|
|
dc->Save();
|
|
}
|
|
}
|
|
}
|
|
|
|
// Print the error vs exact solution.
|
|
const double max_error = u.ComputeMaxError(u0),
|
|
l1_error = u.ComputeL1Error(u0),
|
|
l2_error = u.ComputeL2Error(u0);
|
|
std::cout << "Linf error = " << max_error << endl
|
|
<< "L1 error = " << l1_error << endl
|
|
<< "L2 error = " << l2_error << endl;
|
|
|
|
// Print error in volume.
|
|
const double vol = M_rowsums * u;
|
|
std::cout << "Vol error = " << vol - vol0 << endl;
|
|
|
|
// 9. Save the final solution. This output can be viewed later using GLVis:
|
|
// "glvis -m ex9.mesh -g ex9-final.gf".
|
|
{
|
|
ofstream osol("ex9-final.gf");
|
|
osol.precision(precision);
|
|
u.Save(osol);
|
|
}
|
|
|
|
// 10. Free the used memory.
|
|
delete ode_solver;
|
|
delete dc;
|
|
delete mesh;
|
|
|
|
return 0;
|
|
}
|
|
|
|
|
|
// Implementation of class FE_Evolution
|
|
FE_Evolution::FE_Evolution(SparseMatrix &_M, SparseMatrix &_K,
|
|
const Vector &_b, BilinearForm &_bf, Vector &M_rs)
|
|
: TimeDependentOperator(_M.Size()),
|
|
M(_M), K(_K), b(_b), M_prec(), M_solver(), z(_M.Size()),
|
|
bf(_bf), M_rowsums(M_rs)
|
|
{
|
|
M_solver.SetPreconditioner(M_prec);
|
|
M_solver.SetOperator(M);
|
|
|
|
M_solver.iterative_mode = false;
|
|
M_solver.SetRelTol(1e-9);
|
|
M_solver.SetAbsTol(0.0);
|
|
M_solver.SetMaxIter(100);
|
|
M_solver.SetPrintLevel(0);
|
|
}
|
|
|
|
void FE_Evolution::Mult(const Vector &x, Vector &y) const
|
|
{
|
|
// Compute bounds y_min, y_max for y from x on the ldofs.
|
|
const int dofs = x.Size();
|
|
Vector y_min(dofs), y_max(dofs);
|
|
const int *In = bf.SpMat().GetI(), *Jn = bf.SpMat().GetJ();
|
|
for (int i = 0, k = 0; i < dofs; i++)
|
|
{
|
|
double x_i_min = +std::numeric_limits<double>::infinity();
|
|
double x_i_max = -std::numeric_limits<double>::infinity();
|
|
for (int end = In[i+1]; k < end; k++)
|
|
{
|
|
const int j = Jn[k];
|
|
if (x(j) > x_i_max) { x_i_max = x(j); }
|
|
if (x(j) < x_i_min) { x_i_min = x(j); }
|
|
}
|
|
y_min(i) = x_i_min;
|
|
y_max(i) = x_i_max;
|
|
}
|
|
for (int i = 0; i < dofs; i++)
|
|
{
|
|
y_min(i) = (y_min(i) - x(i) ) / dt;
|
|
y_max(i) = (y_max(i) - x(i) ) / dt;
|
|
}
|
|
|
|
// Compute the high-order solution y = M^{-1} (K x + b).
|
|
K.Mult(x, z);
|
|
z += b;
|
|
M_solver.Mult(z, y);
|
|
|
|
// The solution y is an increment; it should not introduce new mass.
|
|
const double mass_y = 0.0;
|
|
|
|
// Perform optimization.
|
|
Vector y_out(dofs);
|
|
const int max_iter = 500;
|
|
const double rtol = 1.e-7;
|
|
double atol = 1.e-7;
|
|
|
|
OptimizationSolver *optsolver = NULL;
|
|
if (optimizer_type == 2)
|
|
{
|
|
#ifdef MFEM_USE_HIOP
|
|
HiopNlpOptimizer *tmp_opt_ptr = new HiopNlpOptimizer();
|
|
optsolver = tmp_opt_ptr;
|
|
#else
|
|
MFEM_ABORT("MFEM is not built with HiOp support!");
|
|
#endif
|
|
}
|
|
else
|
|
{
|
|
SLBQPOptimizer *slbqp = new SLBQPOptimizer();
|
|
slbqp->SetBounds(y_min, y_max);
|
|
slbqp->SetLinearConstraint(M_rowsums, mass_y);
|
|
atol = 1.e-15;
|
|
optsolver = slbqp;
|
|
}
|
|
|
|
OptimizedTransportProblem ot_prob(y, M_rowsums, mass_y, y_min, y_max);
|
|
optsolver->SetOptimizationProblem(ot_prob);
|
|
|
|
optsolver->SetMaxIter(max_iter);
|
|
optsolver->SetAbsTol(atol);
|
|
optsolver->SetRelTol(rtol);
|
|
optsolver->SetPrintLevel(0);
|
|
optsolver->Mult(y, y_out);
|
|
|
|
y = y_out;
|
|
|
|
delete optsolver;
|
|
}
|
|
|
|
|
|
// Velocity coefficient
|
|
void velocity_function(const Vector &x, Vector &v)
|
|
{
|
|
int dim = x.Size();
|
|
|
|
// map to the reference [-1,1] domain
|
|
Vector X(dim);
|
|
for (int i = 0; i < dim; i++)
|
|
{
|
|
double center = (bb_min[i] + bb_max[i]) * 0.5;
|
|
X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
|
|
}
|
|
|
|
switch (problem)
|
|
{
|
|
case 0:
|
|
{
|
|
// Translations in 1D, 2D, and 3D
|
|
switch (dim)
|
|
{
|
|
case 1: v(0) = (invert_velocity) ? -1.0 : 1.0; break;
|
|
case 2: v(0) = sqrt(2./3.); v(1) = sqrt(1./3.); break;
|
|
case 3: v(0) = sqrt(3./6.); v(1) = sqrt(2./6.); v(2) = sqrt(1./6.);
|
|
break;
|
|
}
|
|
break;
|
|
}
|
|
case 1:
|
|
case 2:
|
|
{
|
|
// Clockwise rotation in 2D around the origin
|
|
const double w = M_PI/2;
|
|
switch (dim)
|
|
{
|
|
case 1: v(0) = 1.0; break;
|
|
case 2: v(0) = w*X(1); v(1) = -w*X(0); break;
|
|
case 3: v(0) = w*X(1); v(1) = -w*X(0); v(2) = 0.0; break;
|
|
}
|
|
break;
|
|
}
|
|
case 3:
|
|
{
|
|
// Clockwise twisting rotation in 2D around the origin
|
|
const double w = M_PI/2;
|
|
double d = max((X(0)+1.)*(1.-X(0)),0.) * max((X(1)+1.)*(1.-X(1)),0.);
|
|
d = d*d;
|
|
switch (dim)
|
|
{
|
|
case 1: v(0) = 1.0; break;
|
|
case 2: v(0) = d*w*X(1); v(1) = -d*w*X(0); break;
|
|
case 3: v(0) = d*w*X(1); v(1) = -d*w*X(0); v(2) = 0.0; break;
|
|
}
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
|
|
// Initial condition
|
|
double u0_function(const Vector &x)
|
|
{
|
|
int dim = x.Size();
|
|
|
|
// map to the reference [-1,1] domain
|
|
Vector X(dim);
|
|
for (int i = 0; i < dim; i++)
|
|
{
|
|
double center = (bb_min[i] + bb_max[i]) * 0.5;
|
|
X(i) = 2 * (x(i) - center) / (bb_max[i] - bb_min[i]);
|
|
}
|
|
|
|
switch (problem)
|
|
{
|
|
case 0:
|
|
case 1:
|
|
{
|
|
switch (dim)
|
|
{
|
|
case 1:
|
|
return (X(0) > -0.15 && X(0) < 0.15) ? 1.0 : 0.0;
|
|
//return exp(-40.*pow(X(0)-0.0,2));
|
|
case 2:
|
|
case 3:
|
|
{
|
|
double rx = 0.45, ry = 0.25, cx = 0., cy = -0.2, w = 10.;
|
|
if (dim == 3)
|
|
{
|
|
const double s = (1. + 0.25*cos(2*M_PI*X(2)));
|
|
rx *= s;
|
|
ry *= s;
|
|
}
|
|
return ( erfc(w*(X(0)-cx-rx))*erfc(-w*(X(0)-cx+rx)) *
|
|
erfc(w*(X(1)-cy-ry))*erfc(-w*(X(1)-cy+ry)) )/16;
|
|
}
|
|
}
|
|
}
|
|
case 2:
|
|
{
|
|
double x_ = X(0), y_ = X(1), rho, phi;
|
|
rho = hypot(x_, y_);
|
|
phi = atan2(y_, x_);
|
|
return pow(sin(M_PI*rho),2)*sin(3*phi);
|
|
}
|
|
case 3:
|
|
{
|
|
const double f = M_PI;
|
|
return sin(f*X(0))*sin(f*X(1));
|
|
}
|
|
}
|
|
return 0.0;
|
|
}
|
|
|
|
// Inflow boundary condition (zero for the problems considered in this example)
|
|
double inflow_function(const Vector &x)
|
|
{
|
|
switch (problem)
|
|
{
|
|
case 0:
|
|
case 1:
|
|
case 2:
|
|
case 3: return 0.0;
|
|
}
|
|
return 0.0;
|
|
}
|