TEST(solver, sparse_direct_umfpack)
{
// Test CG solver on the system
//
// ( 1 1 ) ( x ) = ( 5 )
// ( 1 0 ) ( y ) = ( 2 )
//
// with the solution x = 2, y = 3.
dealii::SparsityPattern pattern(2, 2);
dealii::SparseMatrix<double> mat;
dealii::Vector<double> rhs(2), sol(2);
pattern.add(0, 1);
pattern.add(1, 0);
pattern.compress();
mat.reinit(pattern);
mat.set(0, 0, 1);
mat.set(0, 1, 1);
mat.set(1, 0, 1);
rhs(0) = 5;
rhs(1) = 2;
dealii::SolverControl solver_control (10, 1e-12);
dealii::SolverCG<> solver (solver_control);
solver.solve (mat, sol, rhs, dealii::PreconditionIdentity());
EXPECT_DOUBLE_EQ( 2, sol(0) );
EXPECT_DOUBLE_EQ( 3, sol(1) );
}
void ISOP2P1::solveStokes()
{
buildStokesSys();
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = 2 * n_dof_v + n_dof_p;
/// 构建系数矩阵和右端项.
/// 这个存放整体的数值解. 没有分割成 u_h[0], u_h[1] 和 p_h.
Vector<double> x(n_total_dof);
/// 将数值解合并一个向量便于边界处理.
for (int i = 0; i < n_dof_v; ++i)
{
x(i) = v_h[0](i);
x(n_dof_v + i) = v_h[1](i);
}
for (int i = 0; i < n_dof_p; ++i)
x(2 * n_dof_v + i) = p_h(i);
rhs.reinit(n_total_dof);
/// 边界条件一起处理了.
boundaryValueStokes();
/// 矩阵求解.
dealii::SolverControl solver_control (4000000, 1e-14, 1);
SolverMinRes<Vector<double> > minres (solver_control);
StokesPreconditioner preconditioner;
/// 预处理矩阵.
SparseMatrix<double> matrix_vxvx(sp_vxvx);
SparseMatrix<double> matrix_vyvy(sp_vyvy);
/// 这里从 Stokes 取是因为加了边界条件.
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
matrix_vxvx.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
matrix_vyvy.global_entry(i) = matrix.global_entry(index_vyvy[i]);
preconditioner.initialize(matrix_vxvx, matrix_vyvy, mat_p_mass);
clock_t t_cost = clock();
minres.solve (matrix, x, rhs, preconditioner);
// minres.solve (matrix, x, rhs, PreconditionIdentity());
t_cost = clock() - t_cost;
std::cout << "time cost: " << (((float)t_cost) / CLOCKS_PER_SEC) << std::endl;
/// 将整体数值解分割成速度和压力.
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) = x(i);
v_h[1](i) = x(i + n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
p_h(i) = x(i + 2 * n_dof_v);
/*
/// 矩阵求解.
SparseMatrix<double> mat_BTx(sp_pvx);
SparseMatrix<double> mat_BTy(sp_pvy);
SparseMatrix<double> mat_Bx(sp_vxp);
SparseMatrix<double> mat_By(sp_vyp);
SparseMatrix<double> mat_Ax(sp_vxvx);
SparseMatrix<double> mat_Ay(sp_vyvy);
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
mat_Ax.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
mat_Ay.global_entry(i) = matrix.global_entry(index_vyvy[i]);
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
mat_BTx.global_entry(i) = matrix.global_entry(index_pvx[i]);
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
mat_BTy.global_entry(i) = matrix.global_entry(index_pvy[i]);
for (int i = 0; i < sp_vxp.n_nonzero_elements(); ++i)
mat_Bx.global_entry(i) = matrix.global_entry(index_vxp[i]);
for (int i = 0; i < sp_vyp.n_nonzero_elements(); ++i)
mat_By.global_entry(i) = matrix.global_entry(index_vyp[i]);
Vector<double> tmp1(n_dof_v);
Vector<double> tmp2(n_dof_v);
Vector<double> rhs_vx(n_dof_v);
Vector<double> rhs_vy(n_dof_v);
Vector<double> rhs_p(n_dof_p);
for (int i = 0; i < n_dof_v; ++i)
{
rhs_vx(i) = rhs(i);
v_h[0](i) = x(i);
rhs_vy(i) = rhs(n_dof_v + i);
v_h[1](i) = x(n_dof_v + i);
}
for (int i = 0; i < n_dof_p; ++i)
{
rhs_p(i) = rhs(2 * n_dof_v + i);
p_h(i) = x(2 * n_dof_v + i);
}
Vector<double> schur_rhs (n_dof_p);
// AMGSolver solverQ(mat_Ax, 1.0e-12, 3, 50, 0.382, 0.25);
AMGSolver solverQ(mat_Ax);
InverseMatrix M(mat_Ax, solverQ);
M.vmult (tmp1, rhs_vx);
M.vmult (tmp2, rhs_vy);
mat_Bx.vmult(schur_rhs, tmp1);
mat_By.vmult_add(schur_rhs, tmp2);
schur_rhs -= rhs_p;
SchurComplement schur_complement(mat_BTx, mat_BTy, mat_Bx, mat_By, M, M);
// SolverControl solver_control_cg (n_dof_p * 2,
// 1e-12*schur_rhs.l2_norm());
SolverControl solver_control_cg (n_dof_p * 20,
1e-15);
// SolverCG<> cg (solver_control_cg);
SolverMinRes<> minres (solver_control_cg);
AMGSolver AQ(mat_p_mass);
ApproxSchurComplement asc(mat_p_mass, AQ);
minres.solve (schur_complement, p_h, schur_rhs, asc);
// cg.solve (schur_complement, p_h, schur_rhs, PreconditionIdentity());
std::cout << solver_control_cg.last_step()
<< " CG Schur complement iterations to obtain convergence."
<< std::endl;
mat_BTx.vmult(tmp1, *dynamic_cast<const Vector<double>* >(&p_h));
mat_BTy.vmult(tmp2, *dynamic_cast<const Vector<double>* >(&p_h));
tmp1 *= -1;
tmp2 *= -1;
tmp1 += rhs_vx;
tmp2 += rhs_vy;
M.vmult(v_h[0], tmp1);
M.vmult(v_h[1], tmp2);
*/
double error;
RealVx accuracy_vx;
error = Functional::L2Error(v_h[0], accuracy_vx, 2);
std::cout << "|| u - u_h ||_L2 = " << error << std::endl;
error = Functional::H1SemiError(v_h[0], accuracy_vx, 1);
std::cout << "|| u - u_h ||_H1 = " << error << std::endl;
RealVy accuracy_vy;
error = Functional::L2Error(v_h[1], accuracy_vy, 2);
std::cout << "|| v - v_h ||_L2 = " << error << std::endl;
error = Functional::H1SemiError(v_h[1], accuracy_vy, 1);
std::cout << "|| v - v_h ||_H1 = " << error << std::endl;
double p_avg = Functional::meanValue(p_h, 2);
RealP accuracy_p(p_avg);
error = Functional::L2Error(p_h, accuracy_p, 2);
std::cout << "|| p - p_h ||_0 = " << error << std::endl;
};
void ISOP2P1::solveStokes()
{
buildStokesSys();
std::cout << "Stokes system builded." << std::endl;
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = 2 * n_dof_v + n_dof_p;
/// 构建系数矩阵和右端项.
/// 这个存放整体的数值解. 没有分割成 u_h[0], u_h[1] 和 p_h.
Vector<double> x(n_total_dof);
/// 将数值解合并一个向量便于边界处理.
for (int i = 0; i < n_dof_v; ++i)
{
x(i) = v_h[0](i);
x(n_dof_v + i) = v_h[1](i);
}
for (int i = 0; i < n_dof_p; ++i)
x(2 * n_dof_v + i) = p_h(i);
rhs.reinit(n_total_dof);
/// 边界条件一起处理了. 修改了matrix, rhs和x.
boundaryValueStokes(x, t);
std::cout << "Stokes boundary applied." << std::endl;
// /// 矩阵求解.
// dealii::SolverControl solver_control (400000, l_tol * rhs.l2_norm(), 1);
// /// 不完全LU分解.
// dealii::SparseILU<double> preconditioner;
// preconditioner.initialize(matrix);
// /// 求解Stokes方程, MinRes要比GMRES求解器要快很多,
// /// 当矩阵规模稍微大点的时候,GMRES会出现不收敛的情况.
// SolverMinRes<Vector<double> > minres (solver_control);
// SolverGMRES<Vector<double> >::AdditionalData para(1000, false, true);
// /// 不用para算不动, 但是均匀网格下可以不用para,算其它的例子也不用para,是不是跟计算区域有关系？
// SolverGMRES<Vector<double> > gmres(solver_control, para);
// // /// 移动网格和时间发展中，这个预处理失效.
// // StokesPreconditioner preconditioner;
// // /// 预处理矩阵.
// // SparseMatrix<double> matrix_vxvx(sp_vxvx);
// // SparseMatrix<double> matrix_vyvy(sp_vyvy);
// // /// 这里从 Stokes 取是因为加了边界条件.
// // for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
// // matrix_vxvx.global_entry(i) = matrix.global_entry(index_vxvx[i]);
// // for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
// // matrix_vyvy.global_entry(i) = matrix.global_entry(index_vyvy[i]);
// // preconditioner.initialize(mat_v_stiff, mat_v_stiff, mat_p_mass);
// clock_t t_cost = clock();
// minres.solve (matrix, x, rhs, PreconditionIdentity());
// // gmres.solve(matrix, x, rhs, preconditioner);
// t_cost = clock() - t_cost;
// std::cout << "time cost: " << (((float)t_cost) / CLOCKS_PER_SEC) << std::endl;
// /// 将整体数值解分割成速度和压力.
// for (int i = 0; i < n_dof_v; ++i)
// {
// v_h[0](i) = x(i);
// v_h[1](i) = x(i + n_dof_v);
// }
// for (int i = 0; i < n_dof_p; ++i)
// p_h(i) = x(i + 2 * n_dof_v);
// /// 计算误差, t为时间参数.
// computeError(t);
// /// debug, 计算惨量的L2 norm。
// Vector<double> res(n_total_dof);
// matrix.vmult(res, x);
// res *= -1;
// res += rhs;
// std::cout << "res_l2norm =" << res.l2_norm() << std::endl;
/// 矩阵求解.
SparseMatrix<double> mat_BTx(sp_pvx);
SparseMatrix<double> mat_BTy(sp_pvy);
SparseMatrix<double> mat_Bx(sp_vxp);
SparseMatrix<double> mat_By(sp_vyp);
SparseMatrix<double> mat_Ax(sp_vxvx);
SparseMatrix<double> mat_Ay(sp_vyvy);
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
mat_Ax.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
mat_Ay.global_entry(i) = matrix.global_entry(index_vyvy[i]);
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
mat_BTx.global_entry(i) = matrix.global_entry(index_pvx[i]);
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
mat_BTy.global_entry(i) = matrix.global_entry(index_pvy[i]);
for (int i = 0; i < sp_vxp.n_nonzero_elements(); ++i)
mat_Bx.global_entry(i) = matrix.global_entry(index_vxp[i]);
for (int i = 0; i < sp_vyp.n_nonzero_elements(); ++i)
mat_By.global_entry(i) = matrix.global_entry(index_vyp[i]);
/// alp对AMGSolver的初始化影响比较大, 如果取得很小，初始化很快.
double alp = dt * viscosity;
AMGSolver solverQ(mat_Ax, 1.0e-12, 3, 100, 0.382, alp);
// AMGSolver solverQ(mat_Ax);
InverseMatrix AInv(mat_Ax, solverQ);
/// 这里没有对速度质量阵进行边界条件处理.
InverseMatrix QInv(mat_v_mass, solverQ);
SchurComplement schur_complement(mat_BTx, mat_BTy, mat_Bx, mat_By, mat_v_mass, QInv, QInv);
AMGSolver solverP(mat_p_mass);
ApproxSchurComplement asc(mat_p_mass, solverQ);
LSCPreconditioner lsc_preconditioner(mat_BTx, mat_BTy, mat_Bx, mat_By, mat_Ax, mat_Ax, mat_v_mass, schur_complement, asc, QInv,
AInv, AInv);
/// 矩阵求解.
dealii::SolverControl solver_control (400000, l_Euler_tol * rhs.l2_norm(), 0);
SolverMinRes<Vector<double> > minres(solver_control);
minres.solve(matrix, x, rhs, lsc_preconditioner);
/// 将整体数值解分割成速度和压力.
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) = x(i);
v_h[1](i) = x(i + n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
p_h(i) = x(i + 2 * n_dof_v);
/// 计算误差, t为时间参数.
computeError(t);
// /// 矩阵求解.
// SparseMatrix<double> mat_BTx(sp_pvx);
// SparseMatrix<double> mat_BTy(sp_pvy);
// SparseMatrix<double> mat_Bx(sp_vxp);
// SparseMatrix<double> mat_By(sp_vyp);
// SparseMatrix<double> mat_Ax(sp_vxvx);
// SparseMatrix<double> mat_Ay(sp_vyvy);
// for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
// mat_Ax.global_entry(i) = matrix.global_entry(index_vxvx[i]);
// for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
// mat_Ay.global_entry(i) = matrix.global_entry(index_vyvy[i]);
// for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
// mat_BTx.global_entry(i) = matrix.global_entry(index_pvx[i]);
// for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
// mat_BTy.global_entry(i) = matrix.global_entry(index_pvy[i]);
// for (int i = 0; i < sp_vxp.n_nonzero_elements(); ++i)
// mat_Bx.global_entry(i) = matrix.global_entry(index_vxp[i]);
// for (int i = 0; i < sp_vyp.n_nonzero_elements(); ++i)
// mat_By.global_entry(i) = matrix.global_entry(index_vyp[i]);
// Vector<double> tmp1(n_dof_v);
// Vector<double> tmp2(n_dof_v);
// Vector<double> rhs_vx(n_dof_v);
// Vector<double> rhs_vy(n_dof_v);
// Vector<double> rhs_p(n_dof_p);
// for (int i = 0; i < n_dof_v; ++i)
// {
// rhs_vx(i) = rhs(i);
// v_h[0](i) = x(i);
// rhs_vy(i) = rhs(n_dof_v + i);
// v_h[1](i) = x(n_dof_v + i);
// }
// for (int i = 0; i < n_dof_p; ++i)
// {
// rhs_p(i) = rhs(2 * n_dof_v + i);
// p_h(i) = x(2 * n_dof_v + i);
// }
// Vector<double> schur_rhs (n_dof_p);
// AMGSolver solverQ(mat_Ax);
// InverseMatrix M(mat_Ax, solverQ);
// M.vmult (tmp1, rhs_vx);
// M.vmult (tmp2, rhs_vy);
// mat_Bx.vmult(schur_rhs, tmp1);
// mat_By.vmult_add(schur_rhs, tmp2);
// schur_rhs -= rhs_p;
// SchurComplement schur_complement(mat_BTx, mat_BTy, mat_Bx, mat_By, mat_v_mass, M, M, dt);
// SolverControl solver_control_cg (n_dof_p * 2,
// 1e-12*schur_rhs.l2_norm());
// SolverCG<> cg (solver_control_cg);
// AMGSolver AQ(mat_p_mass);
// ApproxSchurComplement asc(mat_p_mass, AQ);
// cg.solve (schur_complement, p_h, schur_rhs, asc);
// // cg.solve (schur_complement, p_h, schur_rhs, PreconditionIdentity());
// std::cout << solver_control_cg.last_step()
// << " CG Schur complement iterations to obtain convergence."
// << std::endl;
// mat_BTx.vmult(tmp1, *dynamic_cast<const Vector<double>* >(&p_h));
// mat_BTy.vmult(tmp2, *dynamic_cast<const Vector<double>* >(&p_h));
// tmp1 *= -1;
// tmp2 *= -1;
// tmp1 += rhs_vx;
// tmp2 += rhs_vy;
// M.vmult(v_h[0], tmp1);
// M.vmult(v_h[1], tmp2);
// std::cout << "Stokes system solved." << std::endl;
// /// 计算误差, t为时间参数.
// computeError(t);
// /// debug, 计算惨量的L2 norm。
// Vector<double> res(n_total_dof);
// matrix.vmult(res, x);
// res *= -1;
// res += rhs;
// std::cout << "res_l2norm =" << res.l2_norm() << std::endl;
};
void ISOP2P1::solveNS(int method)
{
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = n_dof_v * 2 + n_dof_p;
/// 开始迭代.
double error_N = 1.0;
int iteration_times = 0;
while (error_N > n_tol)
{
/// Newton 迭代或 Picard 迭代.
/// 先更新和速度场有关的矩阵块.
updateNonlinearMatrix();
/// 构建迭代矩阵.
if (method == 1)
buildNewtonSys4NS();
else if (method == 2)
buildPicardSys4NS();
else if (method == 3)
if (iteration_times < 2)
buildPicardSys4NS();
else
buildNewtonSys4NS();
else
{
std::cout << "Newton: 1, Picard: 2, Hybrid: 3." << std::endl;
exit(1);
}
/// 建立右端项.
rhs.reinit(n_total_dof);
FEMSpace<double, DIM>::ElementIterator the_element_v = fem_space_v.beginElement();
FEMSpace<double, DIM>::ElementIterator end_element_v = fem_space_v.endElement();
FEMSpace<double, DIM>::ElementIterator the_element_p = fem_space_p.beginElement();
FEMSpace<double, DIM>::ElementIterator end_element_p = fem_space_p.endElement();
/// 遍历速度单元, 拼装相关系数矩阵和右端项.
for (the_element_v = fem_space_v.beginElement();
the_element_v != end_element_v; ++the_element_v)
{
/// 当前单元信息.
double volume = the_element_v->templateElement().volume();
/// 积分精度, u 和 p 都是 1 次, 梯度和散度 u 都是常数. 因此矩阵拼
/// 装时积分精度不用超过 1 次. (验证一下!)
const QuadratureInfo<DIM>& quad_info = the_element_v->findQuadratureInfo(4);
std::vector<double> jacobian
= the_element_v->local_to_global_jacobian(quad_info.quadraturePoint());
int n_quadrature_point = quad_info.n_quadraturePoint();
std::vector<Point<DIM> > q_point
= the_element_v->local_to_global(quad_info.quadraturePoint());
/// 速度单元信息.
std::vector<std::vector<std::vector<double> > > basis_gradient_v
= the_element_v->basis_function_gradient(q_point);
std::vector<std::vector<double> > basis_value_v
= the_element_v->basis_function_value(q_point);
const std::vector<int>& element_dof_v = the_element_v->dof();
std::vector<double> fx_value = source_v[0].value(q_point, *the_element_v);
std::vector<double> fy_value = source_v[1].value(q_point, *the_element_v);
int n_element_dof_v = the_element_v->n_dof();
std::vector<double> vx_value = v_h[0].value(q_point, *the_element_v);
std::vector<double> vy_value = v_h[1].value(q_point, *the_element_v);
std::vector<std::vector<double> > vx_gradient = v_h[0].gradient(q_point, *the_element_v);
std::vector<std::vector<double> > vy_gradient = v_h[1].gradient(q_point, *the_element_v);
/// 压力单元信息.
Element<double, DIM> &p_element = fem_space_p.element(index_ele_v2p[the_element_v->index()]);
const std::vector<int>& element_dof_p = p_element.dof();
std::vector<std::vector<std::vector<double> > > basis_gradient_p
= p_element.basis_function_gradient(q_point);
std::vector<std::vector<double> > basis_value_p = p_element.basis_function_value(q_point);
std::vector<double> p_value = p_h.value(q_point, p_element);
int n_element_dof_p = p_element.n_dof();
/// 实际拼装.
for (int l = 0; l < n_quadrature_point; ++l)
{
double Jxw = quad_info.weight(l) * jacobian[l] * volume;
for (int i = 0; i < n_element_dof_v; ++i)
{
double rhs_cont = fx_value[l] * basis_value_v[i][l];
rhs_cont -= (vx_value[l] * vx_gradient[l][0] +
vy_value[l] * vx_gradient[l][1]) * basis_value_v[i][l];
rhs_cont -= viscosity * innerProduct(basis_gradient_v[i][l], vx_gradient[l]);
rhs_cont += p_value[l] * basis_gradient_v[i][l][0];
rhs_cont *= Jxw;
rhs(element_dof_v[i]) += rhs_cont;
rhs_cont = fy_value[l] * basis_value_v[i][l];
rhs_cont -= (vx_value[l] * vy_gradient[l][0] +
vy_value[l] * vy_gradient[l][1]) * basis_value_v[i][l];
rhs_cont -= viscosity * innerProduct(basis_gradient_v[i][l], vy_gradient[l]);
rhs_cont += p_value[l] * basis_gradient_v[i][l][1];
rhs_cont *= Jxw;
rhs(n_dof_v + element_dof_v[i]) += rhs_cont;
}
}
}
/// 遍历压力单元. 拼装矩阵和右端项.
for (the_element_p = fem_space_p.beginElement();
the_element_p != end_element_p; ++the_element_p)
{
const std::vector<int>& element_dof_p = the_element_p->dof();
int n_element_dof_p = the_element_p->n_dof();
for (int i = 0; i < n_element_dof_p; ++i)
{
int idx_p = the_element_p->index();
int n_chi = index_ele_p2v[idx_p].size();
for (int k = 0; k < n_chi; k++)
{
/// 速度单元信息.
Element<double, DIM> &v_element = fem_space_v.element(index_ele_p2v[idx_p][k]);
/// 几何信息.
double volume = v_element.templateElement().volume();
const QuadratureInfo<DIM>& quad_info = v_element.findQuadratureInfo(4);
std::vector<double> jacobian
= v_element.local_to_global_jacobian(quad_info.quadraturePoint());
int n_quadrature_point = quad_info.n_quadraturePoint();
std::vector<Point<DIM> > q_point
= v_element.local_to_global(quad_info.quadraturePoint());
std::vector<std::vector<double> > vx_gradient
= v_h[0].gradient(q_point, v_element);
std::vector<std::vector<double> > vy_gradient
= v_h[1].gradient(q_point, v_element);
std::vector<double> vx_value = v_h[0].value(q_point, v_element);
std::vector<double> vy_value = v_h[1].value(q_point, v_element);
/// 压力单元信息.
std::vector<std::vector<double> > basis_value_p
= the_element_p->basis_function_value(q_point);
/// 具体拼装.
for (int l = 0; l < n_quadrature_point; ++l)
{
double Jxw = quad_info.weight(l) * jacobian[l] * volume;
/// 右端项还是零. 源项和 Neumann 条件.
double rhs_cont = Jxw * basis_value_p[i][l]
* (vx_gradient[l][0] + vy_gradient[l][1]);
rhs(2 * n_dof_v + element_dof_p[i]) += rhs_cont;
}
}
}
}
/// 初始化未知量.
Vector<double> x(n_total_dof);
/// 边界条件处理.
boundaryValueNS(x);
std::cout << "nonlinear res:" << std::endl;
double revx = 0.0;
for (int i = 0; i < n_dof_v; ++i)
revx += rhs(i) * rhs(i);
std::cout << "vx re: " << sqrt(revx) << std::endl;
double revy = 0.0;
for (int i = 0; i < n_dof_v; ++i)
revy += rhs(i + n_dof_v) * rhs(i + n_dof_v);
std::cout << "vy re: " << sqrt(revy) << std::endl;
double rep = 0.0;
for (int i = 0; i < n_dof_p; ++i)
rep += rhs(i + 2 * n_dof_v) * rhs(i + 2 * n_dof_v);
std::cout << "p re: " << sqrt(rep) << std::endl;
double re = revx + revy +rep;
std::cout << "total re: " << sqrt(re) << std::endl;
std::cout << "pause ..." << std::endl;
getchar();
if (sqrt(re) < n_tol)
{
std::cout << "Covergence with residual: " << sqrt(re)
<< " in step " << iteration_times << std::endl;
break;
}
std::cout << "Building precondition matrix ..." << std::endl;
/// 矩阵求解.
SparseMatrix<double> mat_Axx(sp_vxvx);
SparseMatrix<double> mat_Ayy(sp_vyvy);
SparseMatrix<double> mat_Wxy(sp_vyvx);
SparseMatrix<double> mat_Wyx(sp_vxvy);
SparseMatrix<double> mat_BTx(sp_pvx);
SparseMatrix<double> mat_BTy(sp_pvy);
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
mat_Axx.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
mat_Ayy.global_entry(i) = matrix.global_entry(index_vyvy[i]);
for (int i = 0; i < sp_vyvx.n_nonzero_elements(); ++i)
mat_Wxy.global_entry(i) = matrix.global_entry(index_vyvx[i]);
for (int i = 0; i < sp_vxvy.n_nonzero_elements(); ++i)
mat_Wyx.global_entry(i) = matrix.global_entry(index_vxvy[i]);
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
mat_BTx.global_entry(i) = matrix.global_entry(index_pvx[i]);
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
mat_BTy.global_entry(i) = matrix.global_entry(index_pvy[i]);
std::cout << "Precondition matrix builded!" << std::endl;
/// 矩阵求解.
dealii::SolverControl solver_control (4000000, l_tol);
SolverGMRES<Vector<double> >::AdditionalData para(2000, false, true);
SolverGMRES<Vector<double> > gmres (solver_control, para);
std::cout << "Begin to solve linear system ..." << std::endl;
// gmres.solve (matrix, x, rhs, navierstokes_preconditioner);
gmres.solve (matrix, x, rhs, PreconditionIdentity());
/// 调试块: 直接观测真实残量.
// Vector<double> tmp(n_total_dof);
// matrix.vmult(tmp, x);
// tmp -= rhs;
// std::cout << "linear residual: " << tmp.l2_norm() << std::endl;
// getchar();
FEMFunction<double, DIM> res_vx(fem_space_v);
FEMFunction<double, DIM> res_vy(fem_space_v);
FEMFunction<double, DIM> res_p(fem_space_p);
/// 更新数值解.
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) += x(i);
res_vx(i) = x(i);
v_h[1](i) += x(i + n_dof_v);
res_vy(i) = x(i+ n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
{
p_h(i) += x(i + 2 * n_dof_v);
res_p(i) = x(i + 2 * n_dof_v);
}
double r_vx = Functional::L2Norm(res_vx, 1);
double r_vy = Functional::L2Norm(res_vy, 1);
double r_p = Functional::L2Norm(res_p, 1);
/// 这个其实是更新...
error_N = r_vx + r_vy + r_p;
std::cout.setf(std::ios::fixed);
std::cout.precision(20);
std::cout << "updated vx: " << r_vx << std::endl;
std::cout << "updated vy: " << r_vy << std::endl;
std::cout << "updated p: " << r_p << std::endl;
std::cout << "total updated: " << error_N << std::endl;
std::cout << "step " << iteration_times << ", total updated: " << error_N
<< ", GMRES stpes: " << solver_control.last_step() << std::endl;
iteration_times++;
if (iteration_times > 10)
{
std::cout << "Disconvergence at step 10." << std::endl;
break;
}
}
};
void ISOP2P1::run()
{
/// 初始化.
initialize();
/// 构建 Stokes 系统.
buildStokesSys();
/// 边界条件处理.
boundaryValueStokes();
/// 准备预处理.
StokesPreconditioner preconditioner;
/// 预处理矩阵.
SparseMatrix<double> matrix_vxvx(sp_vxvx);
SparseMatrix<double> matrix_vyvy(sp_vyvy);
/// 这里从 Stokes 取是因为加了边界条件.
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
matrix_vxvx.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
matrix_vyvy.global_entry(i) = matrix.global_entry(index_vyvy[i]);
preconditioner.initialize(matrix_vxvx, matrix_vyvy, mat_p_mass);
/// 矩阵求解.
dealii::SolverControl solver_control (4000000, 1e-14, 1);
SolverMinRes<Vector<double> > minres (solver_control);
clock_t t_cost = clock();
minres.solve (matrix, x, rhs, preconditioner);
// minres.solve (matrix, x, rhs, PreconditionIdentity());
t_cost = clock() - t_cost;
std::cout << "time cost: " << (((float)t_cost) / CLOCKS_PER_SEC) << std::endl;
/// 将整体数值解分割成速度和压力.
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = 2 * n_dof_v + n_dof_p;
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) = x(i);
v_h[1](i) = x(i + n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
p_h(i) = x(i + 2 * n_dof_v);
/// 数据输出.
outputTecplotP("P0");
outputTecplot("V0");
/// 误差比较.
double error;
RealVx accuracy_vx;
error = Functional::L2Error(v_h[0], accuracy_vx, 2);
std::cout << "|| u - u_h ||_L2 = " << error << std::endl;
error = Functional::H1SemiError(v_h[0], accuracy_vx, 1);
std::cout << "|| u - u_h ||_H1 = " << error << std::endl;
RealVy accuracy_vy;
error = Functional::L2Error(v_h[1], accuracy_vy, 2);
std::cout << "|| v - v_h ||_L2 = " << error << std::endl;
error = Functional::H1SemiError(v_h[1], accuracy_vy, 1);
std::cout << "|| v - v_h ||_H1 = " << error << std::endl;
double p_avg = Functional::meanValue(p_h, 2);
RealP accuracy_p(p_avg);
error = Functional::L2Error(p_h, accuracy_p, 2);
std::cout << "|| p - p_h ||_0 = " << error << std::endl;
};
void ISOP2P1::updateSolution()
{
/// 更新插值点.
fem_space_p.updateDofInterpPoint();
fem_space_v.updateDofInterpPoint();
/// 备份数值解.
FEMFunction<double, DIM> _u_h(v_h[0]);
FEMFunction<double, DIM> _v_h(v_h[1]);
FEMFunction<double, DIM> _p_h(p_h);
const double& msl = moveStepLength();
/// 因为有限元空间插值点变化, 重新构造矩阵.
buildMatrixStruct();
buildMatrix();
/// 因为网格移动量为小量, 因此时间步长可以相对取的大些.
double _dt = 0.1;
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = DIM * n_dof_v + n_dof_p;
int n_total_dof_v = DIM * n_dof_v;
/// 一步Euler.
for (int m = 1; m > 0; --m)
{
/// 系数矩阵直接使用 Stokes 矩阵结构.
SparseMatrix<double> mat_moving;
mat_moving.reinit(sp_stokes);
/// (0, 0)
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_vxvx[i]) = mat_v_mass.global_entry(i);
/// (1, 1) 这两个对角块仅有质量块.
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_vyvy[i]) = mat_v_mass.global_entry(i);
/// (0, 2) 这个不是方阵. 在矩阵结构定义的时候已经直接排除了对角元优
/// 先.
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_pvx[i]) = (1.0 / m) * mat_pvx_divT.global_entry(i);
/// (1, 2)
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_pvy[i]) = (1.0 / m) * mat_pvy_divT.global_entry(i);
/// (2, 0)
for (int i = 0; i < sp_vxp.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_vxp[i]) = (1.0 / m) * mat_vxp_div.global_entry(i);
/// (2, 1) 这四块直接复制散度矩阵.
for (int i = 0; i < sp_vyp.n_nonzero_elements(); ++i)
mat_moving.global_entry(index_vyp[i]) = (1.0 / m) * mat_vyp_div.global_entry(i);
/// 问题的右端项.
Vector<double> rhs_loc(n_total_dof);
// rhs.reinit(n_total_dof);
FEMSpace<double, DIM>::ElementIterator the_element_v = fem_space_v.beginElement();
FEMSpace<double, DIM>::ElementIterator end_element_v = fem_space_v.endElement();
/// 遍历速度单元, 拼装相关系数矩阵和右端项.
for (the_element_v = fem_space_v.beginElement();
the_element_v != end_element_v; ++the_element_v)
{
/// 当前单元信息.
double volume = the_element_v->templateElement().volume();
/// 积分精度至少为2.
const QuadratureInfo<DIM>& quad_info = the_element_v->findQuadratureInfo(4);
std::vector<double> jacobian = the_element_v->local_to_global_jacobian(quad_info.quadraturePoint());
int n_quadrature_point = quad_info.n_quadraturePoint();
std::vector<Point<DIM> > q_point = the_element_v->local_to_global(quad_info.quadraturePoint());
/// 速度单元信息.
std::vector<std::vector<double> > basis_value_v = the_element_v->basis_function_value(q_point);
std::vector<double> vx_value = _u_h.value(q_point, *the_element_v);
std::vector<double> vy_value = _v_h.value(q_point, *the_element_v);
std::vector<std::vector<double> > vx_gradient = v_h[0].gradient(q_point, *the_element_v);
std::vector<std::vector<double> > vy_gradient = v_h[1].gradient(q_point, *the_element_v);
const std::vector<int>& element_dof_v = the_element_v->dof();
int n_element_dof_v = the_element_v->n_dof();
/// 速度积分点上的移动方向, 注意是速度单元的积分点, 在压力单元上的移动方向.
/// 所以下面一行程序, 仔细算过, 没有问题.
std::vector<std::vector<double> > move_vector = moveDirection(q_point, index_v2p[the_element_v->index()]);
for (int l = 0; l < n_quadrature_point; ++l)
{
double Jxw = quad_info.weight(l) * jacobian[l] * volume;
for (int i = 0; i < n_element_dof_v; ++i)
{
double rhs_cont = (vx_value[l] + (1.0 / m) * msl * innerProduct(move_vector[l], vx_gradient[l])) * basis_value_v[i][l];
rhs_cont *= Jxw;
rhs_loc(element_dof_v[i]) += rhs_cont;
rhs_cont = (vy_value[l] + (1.0 / m) * msl * innerProduct(move_vector[l], vy_gradient[l])) * basis_value_v[i][l];
rhs_cont *= Jxw;
rhs_loc(n_dof_v + element_dof_v[i]) += rhs_cont;
}
}
}
/// 构建系数矩阵和右端项.
Vector<double> x(n_total_dof);
// PoiseuilleVx real_vx (-1.0, 1.0);
// PoiseuilleVy real_vy;
// Operator::L2Project(real_vx, v_h[0], Operator::LOCAL_LEAST_SQUARE, 3);
// Operator::L2Project(real_vy, v_h[1], Operator::LOCAL_LEAST_SQUARE, 3);
// /// 边界处理.
const std::size_t * rowstart = sp_stokes.get_rowstart_indices();
const unsigned int * colnum = sp_stokes.get_column_numbers();
/// 遍历全部维度的速度节点.
for (unsigned int i = 0; i < n_total_dof_v; ++i)
{
/// 边界标志.
int bm = -1;
/// 判断一下是 x 方向还是 y 方向. 分别读取标志.
if (i < n_dof_v)
bm = fem_space_v.dofInfo(i).boundary_mark;
else
bm = fem_space_v.dofInfo(i - n_dof_v).boundary_mark;
if (bm == 0)
continue;
/// 方腔流边界条件.
if (bm == 1 || bm == 2 || bm == 4 || bm == 5)
x(i) = 0.0;
if (bm == 3)
if (i < n_dof_v)
{
/// 不包括顶端的两个端点,称为watertight cavity.
// x(i) = scale * 1.0;
Regularized regularize;
x(i) = scale * regularize.value(fem_space_v.dofInfo(i).interp_point);
}
else
x(i) = 0.0;
// /// poiseuille flow边界条件.
// if (bm == 1 || bm == 2 || bm == 4 || bm == 5)
// if (i < n_dof_v)
// x(i) = scale * real_vx.value(fem_space_v.dofInfo(i).interp_point);
// else
// x(i) = scale * real_vy.value(fem_space_v.dofInfo(i - n_dof_v).interp_point);
/// 右端项这样改, 如果该行和列其余元素均为零, 则在迭代中确
/// 保该数值解和边界一致.
if (bm == 1 || bm == 2 || bm == 3 || bm == 4 || bm == 5)
// if (bm == 1 || bm == 2 || bm == 4 || bm == 5)
{
rhs_loc(i) = mat_moving.diag_element(i) * x(i);
/// 遍历 i 行.
for (unsigned int j = rowstart[i] + 1; j < rowstart[i + 1]; ++j)
{
/// 第 j 个元素消成零(不是第 j 列!). 注意避开了对角元.
mat_moving.global_entry(j) -= mat_moving.global_entry(j);
/// 第 j 个元素是第 k 列.
unsigned int k = colnum[j];
/// 看看第 k 行的 i 列是否为零元.
const unsigned int *p = std::find(&colnum[rowstart[k] + 1],
&colnum[rowstart[k + 1]],
i);
/// 如果是非零元. 则需要将这一项移动到右端项. 因为第 i 个未知量已知.
if (p != &colnum[rowstart[k + 1]])
{
/// 计算 k 行 i 列的存储位置.
unsigned int l = p - &colnum[rowstart[0]];
/// 移动到右端项. 等价于 r(k) = r(k) - x(i) * A(k, i).
rhs_loc(k) -= mat_moving.global_entry(l) * x(i);
/// 移完此项自然是零.
mat_moving.global_entry(l) -= mat_moving.global_entry(l);
}
}
}
}
std::cout << "boundary values for updateSolution OK!" << std::endl;
// /// debug 边界条件处理部分,已经测试过, 边界处理正确.
// RegularMesh<DIM> &mesh_v = irregular_mesh_v->regularMesh();
// for (int i = 0; i < mesh_v.n_geometry(0); ++i)
// {
// Point<DIM> &p= mesh_v.point(i);
// if (fabs(1 - p[1]) < eps || fabs(1 - p[0]) < eps || fabs(-1 - p[1]) < eps || fabs(-1 - p[0]) < eps)
// {
// std::cout << "point(" << i << ") = (" << p[0] << "," << p[1] << ");" << std::endl;
// std::cout << "uh(" << i << ") = " << v_h[0](i) << std::endl;
// std::cout << "vh(" << i << ") = " << v_h[1](i) << std::endl;
// }
// }
// getchar();
// /// debug
clock_t t_cost = clock();
/// 预处理矩阵.
/// 不完全LU分解.
dealii::SparseILU <double> preconditioner;
preconditioner.initialize(mat_moving);
/// 矩阵求解.
dealii::SolverControl solver_control (4000000, l_tol, check);
SolverMinRes<Vector<double> > minres (solver_control);
minres.solve (mat_moving, x, rhs_loc, PreconditionIdentity());
t_cost = clock() - t_cost;
std::cout << "time cost: " << (((float)t_cost) / CLOCKS_PER_SEC) << std::endl;
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) = x(i);
v_h[1](i) = x(i + n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
p_h(i) = x(i + 2 * n_dof_v);
/// debug
std::ofstream mat_deb;
// rowstart = sp_pvx.get_rowstart_indices();
// colnum = sp_pvx.get_column_numbers();
mat_deb.open("mat.m", std::ofstream::out);
mat_deb.setf(std::ios::fixed);
mat_deb.precision(20);
for (int i = 0; i < n_total_dof; ++i)
{
for (int j = rowstart[i]; j < rowstart[i + 1]; ++j)
{
mat_deb << "A(" << i + 1 << ", " << colnum[j] + 1 << ")="
<< mat_moving.global_entry(j) << ";" << std::endl;
}
mat_deb << "x(" << i + 1<< ") = " << x(i) << ";" << std::endl;
mat_deb << "rhs(" << i + 1 << ") = " << rhs_loc(i) << ";" << std::endl;
}
// for(int i = 0; i < n_dof_p; ++i)
// mat_deb << "x(" << i + 1<< ") = " << p_h(i) << ";" << std::endl;
mat_deb.close();
std::cout << "mat output" << std::endl;
Vector<double> res(n_total_dof);
mat_moving.vmult(res, x);
res *= -1;
res += rhs_loc;
std::cout << "res_l2norm =" << res.l2_norm() << std::endl;
// double error;
// error = Functional::L2Error(v_h[0], real_vx, 3);
// std::cout << "|| u - u_h ||_L2 = " << error << std::endl;
// error = Functional::H1SemiError(v_h[0], real_vx, 3);
// std::cout << "|| u - u_h ||_H1 = " << error << std::endl;
/// debug
RegularMesh<DIM> &mesh_p = irregular_mesh_p->regularMesh();
for (int i = 0; i < mesh_p.n_geometry(0); ++i)
{
(*mesh_p.h_geometry<0>(i))[0] += moveDirection(i)[0];
(*mesh_p.h_geometry<0>(i))[1] += moveDirection(i)[1];
}
/// 输出一下.
outputTecplotP("NS_Euler");
getchar();
}
};
void RBEC::stepForward()
{
int i, j, k, l;
int n_dof = fem_space.n_dof();
int n_total_dof = 2 * n_dof;
mat_RBEC.reinit(sp_RBEC);
mat_rere.reinit(sp_rere);
mat_reim.reinit(sp_reim);
mat_imre.reinit(sp_imre);
mat_imim.reinit(sp_imim);
Vector<double> phi(n_total_dof);
FEMFunction <double, DIM> phi_star(fem_space);
Vector<double> rhs(n_total_dof);
Potential V(gamma_x, gamma_y);
/// 准备一个遍历全部单元的迭代器.
FEMSpace<double, DIM>::ElementIterator the_element = fem_space.beginElement();
FEMSpace<double, DIM>::ElementIterator end_element = fem_space.endElement();
/// 循环遍历全部单元, 只是为了统计每一行的非零元个数.
for (; the_element != end_element; ++the_element)
{
/// 当前单元信息.
double volume = the_element->templateElement().volume();
const QuadratureInfo<DIM>& quad_info = the_element->findQuadratureInfo(6);
std::vector<double> jacobian = the_element->local_to_global_jacobian(quad_info.quadraturePoint());
int n_quadrature_point = quad_info.n_quadraturePoint();
std::vector<AFEPack::Point<DIM> > q_point = the_element->local_to_global(quad_info.quadraturePoint());
/// 单元信息.
std::vector<std::vector<std::vector<double> > > basis_gradient = the_element->basis_function_gradient(q_point);
std::vector<std::vector<double> > basis_value = the_element->basis_function_value(q_point);
std::vector<double> phi_re_value = phi_re.value(q_point, *the_element);
std::vector<double> phi_im_value = phi_im.value(q_point, *the_element);
const std::vector<int>& element_dof = the_element->dof();
int n_element_dof = the_element->n_dof();
/// 实际拼装.
for (l = 0; l < n_quadrature_point; ++l)
{
double Jxw = quad_info.weight(l) * jacobian[l] * volume;
for (j = 0; j < n_element_dof; ++j)
{
for (k = 0; k < n_element_dof; ++k)
{
double cont = Jxw * ((1 / dt) * basis_value[j][l] * basis_value[k][l]
+ 0.5 * innerProduct(basis_gradient[j][l], basis_gradient[k][l])
+ V.value(q_point[l]) * basis_value[j][l] * basis_value[k][l]
+ beta * (phi_re_value[l] * phi_re_value[l] + phi_im_value[l] * phi_im_value[l]) * basis_value[j][l] * basis_value[k][l]);
mat_RBEC.add(element_dof[j], element_dof[k], cont);
mat_RBEC.add(element_dof[j] + n_dof, element_dof[k] + n_dof, cont);
}
rhs(element_dof[j]) += Jxw * phi_re_value[l] * basis_value[j][l] / dt;
rhs(element_dof[j] + n_dof) += Jxw * phi_im_value[l] * basis_value[j][l] / dt;
}
}
}
FEMFunction<double, DIM> _phi_re(phi_re);
FEMFunction<double, DIM> _phi_im(phi_im);
boundaryValue(phi, rhs, mat_RBEC);
// AMGSolver solver(mat_RBEC);
// solver.solve(phi, rhs);
dealii::SolverControl solver_control(4000, 1e-15);
SolverGMRES<Vector<double> >::AdditionalData para(500, false, true);
SolverGMRES<Vector<double> > gmres(solver_control, para);
gmres.solve(mat_RBEC, phi, rhs, PreconditionIdentity());
for (int i = 0; i < n_dof; ++i)
{
phi_re(i) = phi(i);
phi_im(i) = phi(n_dof + i);
}
for (int i = 0; i < n_dof; ++i)
phi_star(i) = sqrt(phi_re(i) * phi_re(i) + phi_im(i) * phi_im(i));
double L2Phi = Functional::L2Norm(phi_re, 6);
std::cout << "L2 norm = " << L2Phi << std::endl;
for (int i = 0; i < n_dof; ++i)
{
phi_re(i) /= L2Phi;
phi_im(i) /= L2Phi;
}
double e = energy(phi_re, phi_im, 6);
std::cout << "Energy = " << e << std::endl;
t += dt;
};
void ISOP2P1::stepForwardLinearizedEuler()
{
int n_dof_v = fem_space_v.n_dof();
int n_dof_p = fem_space_p.n_dof();
int n_total_dof = 2 * n_dof_v + n_dof_p;
matrix.reinit(sp_stokes);
/// (0, 0)
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
matrix.global_entry(index_vxvx[i]) = viscosity * mat_v_stiff.global_entry(i)
+ (1.0 / dt) * mat_v_mass.global_entry(i);
/// (1, 1)
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
matrix.global_entry(index_vyvy[i]) = viscosity * mat_v_stiff.global_entry(i)
+ (1.0 / dt) * mat_v_mass.global_entry(i);
// /// (0, 2)
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
matrix.global_entry(index_pvx[i]) = mat_pvx_divT.global_entry(i);
// /// (1, 2)
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
matrix.global_entry(index_pvy[i]) = mat_pvy_divT.global_entry(i);
// /// (2, 0)
for (int i = 0; i < sp_vxp.n_nonzero_elements(); ++i)
matrix.global_entry(index_vxp[i]) = mat_vxp_div.global_entry(i);
// /// (2, 1)
for (int i = 0; i < sp_vyp.n_nonzero_elements(); ++i)
matrix.global_entry(index_vyp[i]) = mat_vyp_div.global_entry(i);
rhs.reinit(n_total_dof);
FEMSpace<double,2>::ElementIterator the_element_v = fem_space_v.beginElement();
FEMSpace<double,2>::ElementIterator end_element_v = fem_space_v.endElement();
FEMSpace<double,2>::ElementIterator the_element_p = fem_space_p.beginElement();
FEMSpace<double,2>::ElementIterator end_element_p = fem_space_p.endElement();
/// 遍历速度单元, 拼装相关系数矩阵和右端项.
for (the_element_v = fem_space_v.beginElement();
the_element_v != end_element_v; ++the_element_v)
{
/// 当前单元信息.
double volume = the_element_v->templateElement().volume();
/// 积分精度, u 和 p 都是 1 次, 梯度和散度 u 都是常数. 因此矩阵拼
/// 装时积分精度不用超过 1 次. (验证一下!)
const QuadratureInfo<2>& quad_info = the_element_v->findQuadratureInfo(4);
std::vector<double> jacobian = the_element_v->local_to_global_jacobian(quad_info.quadraturePoint());
int n_quadrature_point = quad_info.n_quadraturePoint();
std::vector<Point<2> > q_point = the_element_v->local_to_global(quad_info.quadraturePoint());
/// 速度单元信息.
std::vector<std::vector<double> > basis_value_v = the_element_v->basis_function_value(q_point);
std::vector<std::vector<std::vector<double> > > basis_gradient_v = the_element_v->basis_function_gradient(q_point);
std::vector<double> vx_value = v_h[0].value(q_point, *the_element_v);
std::vector<double> vy_value = v_h[1].value(q_point, *the_element_v);
std::vector<double> fx_value = source_v[0].value(q_point, *the_element_v);
std::vector<double> fy_value = source_v[1].value(q_point, *the_element_v);
std::vector<std::vector<double> > vx_gradient = v_h[0].gradient(q_point, *the_element_v);
std::vector<std::vector<double> > vy_gradient = v_h[1].gradient(q_point, *the_element_v);
const std::vector<int>& element_dof_v = the_element_v->dof();
int n_element_dof_v = the_element_v->n_dof();
Element<double, 2> &p_element = fem_space_p.element(index_v2p[the_element_v->index()]);
const std::vector<int>& element_dof_p = p_element.dof();
std::vector<std::vector<std::vector<double> > > basis_gradient_p = p_element.basis_function_gradient(q_point);
std::vector<std::vector<double> > basis_value_p = p_element.basis_function_value(q_point);
int n_element_dof_p = p_element.n_dof();
std::vector<double> p_value = p_h.value(q_point, p_element);
for (int l = 0; l < n_quadrature_point; ++l)
{
double Jxw = quad_info.weight(l) * jacobian[l] * volume;
for (int i = 0; i < n_element_dof_v; ++i)
{
for (int j = 0; j < n_element_dof_v; ++j)
{
double cont = Jxw * (vx_value[l] * basis_gradient_v[j][l][0] +
vy_value[l] * basis_gradient_v[j][l][1]) * basis_value_v[i][l];
matrix.add(element_dof_v[i], element_dof_v[j], cont);
matrix.add(n_dof_v + element_dof_v[i], n_dof_v + element_dof_v[j], cont);
}
/// 右端项. 这里可施加源项和 Neumann 条件.
double rhs_cont = fx_value[l] * basis_value_v[i][l] + (1.0 / dt) * vx_value[l] * basis_value_v[i][l];
rhs_cont *= Jxw;
rhs(element_dof_v[i]) += rhs_cont;
rhs_cont = fy_value[l] * basis_value_v[i][l] + (1.0 / dt ) * vy_value[l] * basis_value_v[i][l];
rhs_cont *= Jxw;
rhs(n_dof_v + element_dof_v[i]) += rhs_cont;
}
}
}
/// 构建系数矩阵和右端项.
/// 这个存放整体的数值解. 没有分割成 u_h[0], u_h[1] 和 p_h.
Vector<double> x(n_total_dof);
for (int i = 0; i < n_dof_v; ++i)
{
x(i) = v_h[0](i);
x(i + n_dof_v) = v_h[1](i);
}
for (int i = 0; i < n_dof_p; ++i)
x(i + 2 * n_dof_v) = p_h(i);
boundaryValueStokes(x);
/// 矩阵求解.
SparseMatrix<double> mat_Axx(sp_vxvx);
SparseMatrix<double> mat_Ayy(sp_vyvy);
SparseMatrix<double> mat_BTx(sp_pvx);
SparseMatrix<double> mat_BTy(sp_pvy);
for (int i = 0; i < sp_vxvx.n_nonzero_elements(); ++i)
mat_Axx.global_entry(i) = matrix.global_entry(index_vxvx[i]);
for (int i = 0; i < sp_vyvy.n_nonzero_elements(); ++i)
mat_Ayy.global_entry(i) = matrix.global_entry(index_vyvy[i]);
for (int i = 0; i < sp_pvx.n_nonzero_elements(); ++i)
mat_BTx.global_entry(i) = matrix.global_entry(index_pvx[i]);
for (int i = 0; i < sp_pvy.n_nonzero_elements(); ++i)
mat_BTy.global_entry(i) = matrix.global_entry(index_pvy[i]);
std::cout << "Precondition matrix builded!" << std::endl;
// NSPreconditioner preconditioner_ns;
// preconditioner_ns.initialize(mat_Axx,
// mat_Ayy,
// mat_BTx,
// mat_BTy,
// mat_p_mass);
clock_t t_cost = clock();
dealii::SolverControl solver_control (4000000, l_tol, check);
SolverGMRES<Vector<double> >::AdditionalData para(2000, false, true);
SolverGMRES<Vector<double> > gmres (solver_control, para);
gmres.solve (matrix, x, rhs, PreconditionIdentity());
t_cost = clock() - t_cost;
std::cout << "time cost: " << (((float)t_cost) / CLOCKS_PER_SEC) << std::endl;
/// 将整体数值解分割成速度和压力.
for (int i = 0; i < n_dof_v; ++i)
{
v_h[0](i) = x(i);
v_h[1](i) = x(i + n_dof_v);
}
for (int i = 0; i < n_dof_p; ++i)
p_h(i) = x(i + 2 * n_dof_v);
};