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::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);
};
