class ParabolicFEMModel(): def __init__(self, pde, mesh, p=1, q=6): from fealpy.functionspace import LagrangeFiniteElementSpace from fealpy.boundarycondition import BoundaryCondition self.space = LagrangeFiniteElementSpace(mesh, p) self.mesh = self.space.mesh self.pde = pde self.ftype = self.mesh.ftype self.itype = self.mesh.itype self.M = self.space.mass_matrix() self.A = self.space.stiff_matrix() def init_solution(self, timeline): NL = timeline.number_of_time_levels() gdof = self.space.number_of_global_dofs() uh = np.zeros((gdof, NL), dtype=self.mesh.ftype) uh[:, 0] = self.space.interpolation(lambda x: self.pde.solution(x, 0.0)) return uh def interpolation(self, u, timeline): NL = timeline.number_of_time_levels() gdof = self.space.number_of_global_dofs() ps = self.space.interpolation_points() uI = np.zeros((gdof, NL), dtype=self.mesh.ftype) times = timeline.all_time_levels() for i, t in enumerate(times): uI[..., i] = u(ps, t) return uI def get_current_left_matrix(self, timeline): dt = timeline.current_time_step_length() return self.M + 0.5 * dt * self.A def get_current_right_vector(self, uh, timeline): i = timeline.current dt = timeline.current_time_step_length() t0 = timeline.current_time_level() t1 = timeline.next_time_level() f = lambda x: self.pde.source(x, t0) + self.pde.source(x, t1) F = self.space.source_vector(f) return self.M @ uh[:, i] - 0.5 * dt * self.A @ uh[:, i] def apply_boundary_condition(self, A, b, timeline): t1 = timeline.next_time_level() bc = BoundaryCondition(self.space, neuamn=lambda x: self.pde.neuman(x, t1)) b = bc.apply_neuman_bc(b) return A, b def solve(self, uh, A, b, solver, timeline): i = timeline.current uh[:, i + 1] = solver(A, b)
def solve_poisson_3d(self, n=2): from fealpy.pde.poisson_3d import CosCosCosData as PDE from fealpy.mesh import MeshFactory from fealpy.functionspace import LagrangeFiniteElementSpace from fealpy.boundarycondition import DirichletBC pde = PDE() mf = MeshFactory() m = 2**n box = [0, 1, 0, 1, 0, 1] mesh = mf.boxmesh3d(box, nx=m, ny=m, nz=m, meshtype='tet') space = LagrangeFiniteElementSpace(mesh, p=1) gdof = space.number_of_global_dofs() NC = mesh.number_of_cells() print('gdof:', gdof, 'NC:', NC) bc = DirichletBC(space, pde.dirichlet) uh = space.function() A = space.stiff_matrix() A = space.parallel_stiff_matrix(q=1) M = space.parallel_mass_matrix(q=2) M = space.mass_matrix() F = space.source_vector(pde.source) A, F = bc.apply(A, F, uh) solver = PETScSolver() solver.solve(A, F, uh) error = space.integralalg.L2_error(pde.solution, uh) print(error)
def test_poisson_fem_2d(): degree = 1 dim = 2 nrefine = 4 maxit = 4 from fealpy.pde.poisson_2d import CosCosData as PDE pde = PDE() mesh = pde.init_mesh(n=nrefine) errorMatrix = np.zeros((2, maxit), dtype=np.float64) NDof = np.zeros(maxit, dtype=np.int64) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, p=degree) NDof[i] = space.number_of_global_dofs() bc = DirichletBC(space, pde.dirichlet) uh = space.function() A = space.stiff_matrix() F = space.source_vector(pde.source) A, F = bc.apply(A, F, uh) uh[:] = spsolve(A, F).reshape(-1) errorMatrix[0, i] = space.integralalg.L2_error(pde.solution, uh) errorMatrix[1, i] = space.integralalg.L2_error(pde.gradient, uh.grad_value) if i < maxit-1: mesh.uniform_refine()
def test_poisson(): p = 1 # 有限元空间次数, 可以增大 p, 看输出结果的变化 n = 4 # 初始网格加密次数 maxit = 4 # 最大迭代次数 pde = PDE() mesh = pde.init_mesh(n=n) errorMatrix = np.zeros((2, maxit), dtype=np.float) NDof = np.zeros(maxit, dtype=np.float) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, p=p) # 建立有限元空间 NDof[i] = space.number_of_global_dofs() # 有限元空间自由度的个数 bc = DirichletBC(space, pde.dirichlet) # DirichletBC 条件 uh = space.function() # 有限元函数 A = space.stiff_matrix() # 刚度矩阵 F = space.source_vector(pde.source) # 载荷向量 A, F = bc.apply(A, F, uh) # 处理边界条件 uh[:] = spsolve(A, F).reshape(-1) # 稀疏矩阵直接解法器 # ml = pyamg.ruge_stuben_solver(A) # 代数多重网格解法器 # uh[:] = ml.solve(F, tol=1e-12, accel='cg').reshape(-1) errorMatrix[0, i] = space.integralalg.L2_error( pde.solution, uh ) # 计算 L2 误差 errorMatrix[1, i] = space.integralalg.L2_error( pde.gradient, uh.grad_value ) # 计算 H1 误差 if i < maxit - 1: mesh.uniform_refine() # 一致加密网格 assert (errorMatrix < 1.0).all()
elif d == 3: from fealpy.pde.poisson_3d import CosCosCosData as PDE #pde = PDE() pde = CosCosData() mesh = pde.init_mesh(n=3) errorType = [ '$|| u - u_h||_{\Omega,0}$', '$||\\nabla u - \\nabla u_h||_{\Omega, 0}$' ] errorMatrix = np.zeros((2, maxit), dtype=np.float) NDof = np.zeros(maxit, dtype=np.float) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, p=p) NDof[i] = space.number_of_global_dofs() uh = space.function() A = space.stiff_matrix() F = space.source_vector(pde.source) bc = NeumannBC(space, pde.neumann) A, F = bc.apply( F, A=A) # Here is the case for pure Neumann bc, we also need modify A # bc.apply(F) # Not pure Neumann bc case uh[:] = spsolve(A, F)[:-1] # we add a addtional dof # Here can not work #ml = pyamg.ruge_stuben_solver(A) #x = ml.solve(F, tol=1e-12, accel='cg').reshape(-1) #uh[:] = x[-1]
p = int(sys.argv[1]) maxit = int(sys.argv[2]) start = timer() solver = MatlabSolver() end = timer() print("The matalb start time:", end - start) pde = CosCosData() mesh = pde.init_mesh(4) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, p=1) gdof = space.number_of_global_dofs() print("The num of dofs:", gdof) A = space.stiff_matrix() b = space.source_vector(pde.source) bc = DirichletBC(space, pde.dirichlet) AD, b = bc.apply(A, b) uh0 = solver.divide(AD, b) start = timer() uh1 = spsolve(AD, b) end = timer() print("The spsolver time:", end - start) print(np.sum(np.abs(uh0 - uh1))) #uh1 = solver.mumps_solver(A, b) #uh2 = solver.ifem_amg_solver(A, b) if i < maxit - 1:
class WaterFloodingModelSolver(): """ Notes ----- """ def __init__(self, model, T=800 * 3600 * 24, NS=32, NT=800 * 24): self.model = model self.mesh = model.space_mesh(n=NS) self.timeline = model.time_mesh(T=T, n=NT) self.GD = model.GD if self.GD == 2: self.vspace = RaviartThomasFiniteElementSpace2d(self.mesh, p=0) # 速度空间 elif self.GD == 3: self.vspace = RaviartThomasFiniteElementSpace3d(self.mesh, p=0) self.pspace = self.vspace.smspace # 压强和饱和度所属的空间, 分片常数 self.cspace = LagrangeFiniteElementSpace(self.mesh, p=1) # 位移空间 # 上一时刻物理量的值 self.v = self.vspace.function() # 速度函数 self.p = self.pspace.function() # 压强函数 self.s = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0 self.u = self.cspace.function(dim=self.GD) # 位移函数 self.phi = self.pspace.function() # 孔隙度函数, 分片常数 # 当前时刻物理量的值, 用于保存临时计算出的值, 模型中系数的计算由当前时刻 # 的物理量的值决定 self.cv = self.vspace.function() # 速度函数 self.cp = self.pspace.function() # 压强函数 self.cs = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0 self.cu = self.cspace.function(dim=self.GD) # 位移函数 self.cphi = self.pspace.function() # 孔隙度函数, 分片常数 # 初值 self.p[:] = model.rock['initial pressure'] # MPa self.phi[:] = model.rock['porosity'] # 初始孔隙度 self.cp[:] = model.rock['initial pressure'] # 初始地层压强 self.cphi[:] = model.rock['porosity'] # 当前孔隙度系数 # 源项, TODO: 注意这里假设用的是结构网格, 换其它的网格需要修改代码 self.fo = self.cspace.function() self.fo[-1] = -self.model.oil['production rate'] # 产出 self.fw = self.cspace.function() self.fw[0] = self.model.water['injection rate'] # 注入 # 一些常数矩阵和向量 # 速度散度矩阵, 速度方程对应的散度矩阵, (\nabla\cdot v, w) self.B = self.vspace.div_matrix() # 压强方程对应的位移散度矩阵, (\nabla\cdot u, w) 位移散度矩阵 # * 注意这里利用了压强空间分片常数, 线性函数导数也是分片常数的事实 c = self.mesh.entity_measure('cell') c *= self.model.rock['biot'] val = self.mesh.grad_lambda() # (NC, TD+1, GD) val *= c[:, None, None] pc2d = self.pspace.cell_to_dof() cc2d = self.cspace.cell_to_dof() pgdof = self.pspace.number_of_global_dofs() cgdof = self.cspace.number_of_global_dofs() I = np.broadcast_to(pc2d, shape=cc2d.shape) J = cc2d self.PU0 = csr_matrix((val[..., 0].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) self.PU1 = csr_matrix((val[..., 1].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) if self.GD == 3: self.PU2 = csr_matrix((val[..., 2].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) # 线弹性矩阵的右端向量 sigma0 = self.pspace.function() sigma0[:] = self.model.rock['initial stress'] self.FU = np.zeros(self.GD * cgdof, dtype=np.float64) self.FU[0 * cgdof:1 * cgdof] -= self.p @ self.PU0 self.FU[1 * cgdof:2 * cgdof] -= self.p @ self.PU1 if self.GD == 3: self.FU[2 * cgdof:3 * cgdof] -= self.p @ self.PU2 # 初始应力和等效应力项 self.FU[0 * cgdof:1 * cgdof] -= sigma0 @ self.PU0 self.FU[1 * cgdof:2 * cgdof] -= sigma0 @ self.PU1 if self.GD == 3: self.FU[2 * cgdof:3 * cgdof] -= sigma0 @ self.PU2 # vtk 文件输出 node, cell, cellType, NC = self.mesh.to_vtk() self.points = vtk.vtkPoints() self.points.SetData(vnp.numpy_to_vtk(node)) self.cells = vtk.vtkCellArray() self.cells.SetCells(NC, vnp.numpy_to_vtkIdTypeArray(cell)) self.cellType = cellType def recover(self, val): """ Notes ----- 给定一个分片常数的量, 恢复为分片连续的量 """ mesh = self.mesh cell = self.mesh.entity('cell') NC = mesh.number_of_cells() NN = mesh.number_of_nodes() w = 1 / self.mesh.entity_measure('cell') # 恢复权重 w = np.broadcast_to(w[:, None], shape=cell.shape) r = np.zeros(NN, dtype=np.float64) d = np.zeros(NN, dtype=np.float64) np.add.at(d, cell, w) np.add.at(r, cell, val[:, None] * w) return r / d def pressure_coefficient(self): """ Notes ----- 计算当前物理量下的压强质量矩阵系数 """ b = self.model.rock['biot'] Ks = self.model.rock['solid grain stiffness'] cw = self.model.water['compressibility'] co = self.model.oil['compressibility'] Sw = self.cs[:].copy() # 当前的水饱和度 (NQ, NC) phi = self.cphi[:].copy() # 当前的孔隙度 val = phi * Sw * cw val += phi * (1 - Sw) * co # 注意这里的 co 是常数, 但在水气混合物条件下应该依赖于压力 val += (b - phi) / Ks return val def saturation_pressure_coefficient(self): """ Notes ----- 计算当前物理量下, 饱和度方程中, 压强项对应的系数 """ b = self.model.rock['biot'] Ks = self.model.rock['solid grain stiffness'] cw = self.model.water['compressibility'] val = self.cs[:].copy() # 当前水饱和度 phi = self.cphi[:].copy() # 当前孔隙度 val *= (b - phi) / Ks + phi * cw return val def flux_coefficient(self): """ Notes ----- 计算**当前**物理量下, 速度方程对应的系数 计算通量的系数, 流体的相对渗透率除以粘性系数得到流动性系数 krg/mu_g 是气体的流动性系数 kro/mu_o 是油的流动性系数 krw/mu_w 是水的流动性系数 这里假设压强的单位是 MPa 1 d = 9.869 233e-13 m^2 = 1000 md 1 cp = 1 mPa s = 1e-9 MPa.s """ muw = self.model.water['viscosity'] # 单位是 1 cp = 1 mPa.s muo = self.model.oil['viscosity'] # 单位是 1 cp = 1 mPa.s # 岩石的绝对渗透率, 这里考虑了量纲的一致性, 压强是 MPa k = self.model.rock['permeability'] * 9.869233e-4 Sw = self.cs[:].copy() # 当前水的饱和度系数 lamw = self.model.water_relative_permeability(Sw) lamw /= muw lamo = self.model.oil_relative_permeability(Sw) lamo /= muo val = 1 / (lamw + lamo) / k # return val def water_fractional_flow_coefficient(self): """ Notes ----- 计算**当前**物理量下, 饱和度方程中, 水的流动性系数 """ Sw = self.cs[:].copy() # 当前水的饱和度系数 lamw = self.model.water_relative_permeability(Sw) lamw /= self.model.water['viscosity'] lamo = self.model.oil_relative_permeability(Sw) lamo /= self.model.oil['viscosity'] val = lamw / (lamw + lamo) return val def get_total_system(self): """ Notes ----- 构造整个系统 二维情形: x = [v, p, s, u0, u1] A = [[ V, VP, None, None, None] [ PV, P, None, PU0, PU1] [None, SP, S, SU0, SU1] [None, UP0, None, U00, U01] [None, UP1, None, U10, U11] F = [FV, FP, FS, FU0, FU1] 三维情形: x = [v, p, s, u0, u1, u2] A = [[ V, VP, None, None, None, None] [ PV, P, None, PU0, PU1, PU2] [None, SP, S, SU0, SU1, SU2] [None, UP0, None, U00, U01, U02] [None, UP1, None, U10, U11, U12] [None, UP2, None, U20, U21, U22]] F = [FV, FP, FS, FU0, FU1, FU2] FS 中考虑的迎风格式 """ GD = self.GD A0, FV, isBdDof0 = self.get_velocity_system(q=2) A1, FP, isBdDof1 = self.get_pressure_system(q=2) A2, FS, isBdDof2 = self.get_saturation_system(q=2) if GD == 2: A3, A4, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A0, A1, A2, A3, A4], format='csr') F = np.r_['0', FV, FP, FS, FU] elif GD == 3: A3, A4, A5, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A0, A1, A2, A3, A4, A5], format='csr') F = np.r_['0', FV, FP, FS, FU] isBdDof = np.r_['0', isBdDof0, isBdDof1, isBdDof2, isBdDof3] return A, F, isBdDof def get_velocity_system(self, q=2): """ Notes ----- 计算速度方程对应的离散系统. if GD == 2: [ V, VP, None, None, None] elif GD == 3: [ V, VP, None, None, None, None] """ GD = self.GD dt = self.timeline.current_time_step_length() cellmeasure = self.mesh.entity_measure('cell') qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() # 速度对应的矩阵 V c = self.flux_coefficient() c *= cellmeasure phi = self.vspace.basis(bcs) V = np.einsum('q, qcin, qcjn, c->cij', ws, phi, phi, c, optimize=True) c2d = self.vspace.cell_to_dof() I = np.broadcast_to(c2d[:, :, None], shape=V.shape) J = np.broadcast_to(c2d[:, None, :], shape=V.shape) gdof = self.vspace.number_of_global_dofs() V = csr_matrix((V.flat, (I.flat, J.flat)), shape=(gdof, gdof)) # 压强矩阵 VP = -self.B # 右端向量, 0 向量 FV = np.zeros(gdof, dtype=np.float64) isBdDof = self.vspace.dof.is_boundary_dof() if GD == 2: return [V, VP, None, None, None], FV, isBdDof elif GD == 3: return [V, VP, None, None, None, None], FV, isBdDof def get_pressure_system(self, q=2): """ Notes ----- 计算压强方程对应的离散系统 这里组装矩阵时, 利用了压强是分片常数的特殊性 [ PV, P, None, PU0, PU1] [ PV, P, None, PU0, PU1, PU2] """ GD = self.GD dt = self.timeline.current_time_step_length() cellmeasure = self.mesh.entity_measure('cell') qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() PV = dt * self.B.T # P 是对角矩阵, 利用分片常数的 c = self.pressure_coefficient() # (NQ, NC) c *= cellmeasure P = diags(c, 0) # 组装压强方程的右端向量 # * 这里利用了压强空间基是分片常数 FP = self.fo.value(bcs) + self.fw.value(bcs) # (NQ, NC) FP *= ws[:, None] FP = np.sum(FP, axis=0) FP *= cellmeasure FP *= dt FP += P @ self.p # 上一步的压强向量 FP += self.PU0 @ self.u[:, 0] FP += self.PU1 @ self.u[:, 1] if GD == 3: FP += self.PU2 @ self.u[:, 2] gdof = self.pspace.number_of_global_dofs() isBdDof = np.zeros(gdof, dtype=np.bool_) if GD == 2: return [PV, P, None, self.PU0, self.PU1], FP, isBdDof elif GD == 3: return [PV, P, None, self.PU0, self.PU1, self.PU2], FP, isBdDof def get_saturation_system(self, q=2): """ Notes ---- 计算饱和度方程对应的离散系统 [ None, SP, S, SU0, SU1] """ GD = self.GD qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() cellmeasure = self.mesh.entity_measure('cell') # SP 是对角矩阵 c = self.saturation_pressure_coefficient() # (NC, ) c *= cellmeasure SP = diags(c, 0) # S 质量矩阵组装, 对角矩阵 c = self.cphi[:] * cellmeasure S = diags(c, 0) # SU0, SU1, 饱和度方程中的位移散度对应的矩阵 val = self.mesh.grad_lambda() # (NC, TD+1, GD) c = self.cs[:] * self.model.rock['biot'] # (NC, ), 注意用当前的水饱和度 c *= cellmeasure val *= c[:, None, None] pgdof = self.pspace.number_of_global_dofs() # 压力空间自由度个数 cgdof = self.cspace.number_of_global_dofs() # 连续空间自由度个数 pc2d = self.pspace.cell_to_dof() cc2d = self.cspace.cell_to_dof() I = np.broadcast_to(pc2d, shape=cc2d.shape) J = cc2d SU0 = csr_matrix((val[..., 0].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) SU1 = csr_matrix((val[..., 1].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) if GD == 3: SU2 = csr_matrix((val[..., 2].flat, (I.flat, J.flat)), shape=(pgdof, cgdof)) # 右端矩阵 dt = self.timeline.current_time_step_length() FS = self.fw.value(bcs) # (NQ, NC) FS *= ws[:, None] FS = np.sum(FS, axis=0) FS *= cellmeasure FS *= dt FS += SP @ self.p # 上一时间步的压强 FS += S @ self.s # 上一时间步的饱和度 FS += SU0 @ self.u[:, 0] # 上一时间步的位移 x 分量 FS += SU1 @ self.u[:, 1] # 上一个时间步的位移 y 分量 if GD == 3: FS += SU2 @ self.u[:, 2] # 上一个时间步的位移 y 分量 # 用当前时刻的速度场, 构造非线性迎风格式 face2cell = self.mesh.ds.face_to_cell() isBdFace = face2cell[:, 0] == face2cell[:, 1] qf = self.mesh.integrator(2, 'face') # 边上的积分公式 bcs, ws = qf.get_quadrature_points_and_weights() facemeasure = self.mesh.entity_measure('face') # 边的定向法线,它是左边单元的外法线,右边单元内法线。 fn = self.mesh.face_unit_normal() val0 = np.einsum('qfm, fm->qf', self.cv.face_value(bcs), fn) # 当前速度和法线的内积 # 水的流动分数, 与水的饱和度有关, 如果饱和度为0, 则为 0 Fw = self.water_fractional_flow_coefficient() val1 = Fw[face2cell[:, 0]] # 边的左边单元的水流动分数 val2 = Fw[face2cell[:, 1]] # 边的右边单元的水流动分数 val2[isBdFace] = 0.0 # 边界的贡献是 0 flag = val0 < 0.0 # 对于左边单元来说,是流出项 # 对于右边单元来说,是流入项 # 左右单元流入流出的绝对量是一样的 val = val0 * val1[None, :] # val0 >= 0.0, 左边单元是流出 # val0 < 0.0, 左边单元是流入 val[flag] = (val0 * val2[None, :])[flag] # val0 >= 0, 右边单元是流入 # val0 < 0, 右边单元是流出 b = np.einsum('q, qf, f->f', ws, val, facemeasure) b *= dt np.subtract.at(FS, face2cell[:, 0], b) isInFace = ~isBdFace # 只处理内部边 np.add.at(FS, face2cell[isInFace, 1], b[isInFace]) isBdDof = np.zeros(pgdof, dtype=np.bool_) if GD == 2: return [None, SP, S, SU0, SU1], FS, isBdDof elif GD == 3: return [None, SP, S, SU0, SU1, SU2], FS, isBdDof def get_dispacement_system(self, q=2): """ Notes ----- 计算位移方程对应的离散系统, 即线弹性方程系统. GD == 2: [None, UP0, None, U00, U01] [None, UP1, None, U10, U11] GD == 3: [None, UP0, None, U00, U01, U02] [None, UP1, None, U10, U11, U12] [None, UP2, None, U20, U21, U22] """ GD = self.GD # 拉梅参数 (lambda, mu) lam, mu = self.model.rock['lame'] U = self.cspace.linear_elasticity_matrix(lam, mu, format='list') isBdDof = self.cspace.dof.is_boundary_dof() if GD == 2: return ([None, -self.PU0.T, None, U[0][0], U[0][1]], [None, -self.PU1.T, None, U[1][0], U[1][1]], self.FU, np.r_['0', isBdDof, isBdDof]) elif GD == 3: return ([None, -self.PU0.T, None, U[0][0], U[0][1], U[0][2] ], [None, -self.PU1.T, None, U[1][0], U[1][1], U[1][2]], [None, -self.PU2.T, None, U[2][0], U[2][1], U[2][2]], self.FU, np.r_['0', isBdDof, isBdDof, isBdDof]) def picard_iteration(self, maxit=10): GD = self.GD e0 = 1.0 k = 0 while e0 > 1e-10: # 构建总系统 A, F, isBdDof = self.get_total_system() # 处理边界条件, 这里是 0 边界 gdof = len(isBdDof) bdIdx = np.zeros(gdof, dtype=np.int_) bdIdx[isBdDof] = 1 Tbd = diags(bdIdx) T = diags(1 - bdIdx) A = T @ A @ T + Tbd F[isBdDof] = 0.0 # 求解 x = spsolve(A, F) vgdof = self.vspace.number_of_global_dofs() pgdof = self.pspace.number_of_global_dofs() cgdof = self.cspace.number_of_global_dofs() e0 = 0.0 start = 0 end = vgdof self.cv[:] = x[start:end] start = end end += pgdof e0 += np.sum((self.cp - x[start:end])**2) self.cp[:] = x[start:end] start = end end += pgdof e0 += np.sum((self.cs - x[start:end])**2) self.cs[:] = x[start:end] e0 = np.sqrt(e0) # 误差的 l2 norm k += 1 for i in range(GD): start = end end += cgdof self.cu[:, i] = x[start:end] print(e0) if k >= maxit: print('picard iteration arrive max iteration with error:', e0) break def update_solution(self): self.v[:] = self.cv self.p[:] = self.cp self.s[:] = self.cs #flag = self.s < 0.0 #self.s[flag] = 0.0 self.u[:] = self.cu def solve(self, step=24): """ Notes ----- 计算所有的时间层。 """ timeline = self.timeline dt = timeline.current_time_step_length() timeline.reset() # 时间置零 n = timeline.current fname = 'test_' + str(n).zfill(10) + '.vtu' self.write_to_vtk(fname) while not timeline.stop(): ct = timeline.current_time_level() / 3600 / 24 # 天为单位 print('当前时刻为第', ct, '天') self.picard_iteration() self.update_solution() timeline.current += 1 if timeline.current % step == 0: n = timeline.current fname = 'test_' + str(n).zfill(10) + '.vtu' self.write_to_vtk(fname) timeline.reset() def write_to_vtk(self, fname): # 重心处的值 mesh = self.mesh GD = self.GD bc = np.array((GD + 1) * [1 / (GD + 1)], dtype=np.float64) ps = self.mesh.bc_to_point(bc) vmesh = vtk.vtkUnstructuredGrid() vmesh.SetPoints(self.points) vmesh.SetCells(self.cellType, self.cells) cdata = vmesh.GetCellData() pdata = vmesh.GetPointData() v = self.v p = self.p s = self.s u = self.u val = v.value(bc) if GD == 2: val = np.concatenate( (val, np.zeros((val.shape[0], 1), dtype=val.dtype)), axis=1) val = vnp.numpy_to_vtk(val) val.SetName('velocity') cdata.AddArray(val) val = self.recover(p[:]) val = vnp.numpy_to_vtk(val) val.SetName('pressure') pdata.AddArray(val) val = self.recover(s[:]) val = vnp.numpy_to_vtk(val) val.SetName('saturation') pdata.AddArray(val) if GD == 2: val = np.concatenate((u[:], np.zeros( (u.shape[0], 1), dtype=u.dtype)), axis=1) val = vnp.numpy_to_vtk(val) val.SetName('displacement') pdata.AddArray(val) writer = vtk.vtkXMLUnstructuredGridWriter() writer.SetFileName(fname) writer.SetInputData(vmesh) writer.Write()
errorMatrix = [[], []] k = 0 while True: mesh = tritree.to_conformmesh() fname = './test-' + str(k) + '.png' mesh.add_plot(plt) plt.savefig(fname) plt.close() space = LagrangeFiniteElementSpace(mesh, p=1) A = space.stiff_matrix(q=2) F = space.source_vector(pde.source) NDof += [space.number_of_global_dofs()] bc = DirichletBC(space, pde.dirichlet) uh = space.function() A, F = bc.apply(A, F, uh) uh[:] = spsolve(A, F) rguh = space.grad_recovery(uh) errorL1 = space.integralalg.error(pde.solution, uh.value) errorH1 = space.integralalg.error(pde.gradient, uh.grad_value) errorMatrix[0] += [errorL1] errorMatrix[1] += [errorH1] if NDof[-1] < maxdof: eta = space.recovery_estimate(uh)
errorType = [ '$|| u - u_h||_{\Omega,0}$', # L2 误差 '$||\\nabla u - \\nabla u_h||_{\Omega, 0}$' # H1 误差 ] errorMatrix = np.zeros((2, maxit), dtype=np.float) NDof = np.zeros(maxit, dtype=np.float) # 初始网格 fig = plt.figure() axes = fig.gca() mesh.add_plot(axes) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, p=p) # 建立有限元空间 NDof[i] = space.number_of_global_dofs() # 有限元空间自由度的个数 bc = DirichletBC(space, pde.dirichlet) # DirichletBC 条件 uh = space.function() # 有限元函数 A = space.stiff_matrix() # 刚度矩阵 F = space.source_vector(pde.source) # 载荷向量 A, F = bc.apply(A, F, uh) # 处理边界条件 uh[:] = spsolve(A, F).reshape(-1) # 稀疏矩阵直接解法器 #ml = pyamg.ruge_stuben_solver(A) # 代数多重网格解法器 #uh[:] = ml.solve(F, tol=1e-12, accel='cg').reshape(-1) errorMatrix[0, i] = space.integralalg.L2_error(pde.solution, uh) # 计算 L2 误差
class ParallelTwoFluidsWithGeostressSimulator(): """ Notes ----- 这是一个并行模拟两种流体和地质力学耦合的程序, 这里只是线性系统是并行的 * S_0: 流体 0 的饱和度,一般是水 * S_1: 流体 1 的饱和度,可以为气或油 * v: 总速度 * p: 压力 * u: 岩石位移 饱和度 P0 间断元,单元边界用迎风格式 速度 RT0 元 压强 P0 元 岩石位移 P1 连续元离散 目前, 模型 * 忽略了毛细管压强和重力作用 * 饱和度用分片线性间断元求解, 非线性的迎风格式 渐近解决方案: 1. Picard 迭代 2. 气的可压性随着压强的变化而变化 3. 考虑渗透率随着孔隙度的变化而变化 4. 考虑裂缝,裂缝用自适应网格加密,设设置更大的渗透率实现 体积模量: K = E/3/(1 - 2*nu) = lambda + 2/3* mu Develop ------ """ def __init__(self, mesh, args, ctx): self.ctx = ctx self.args = args # 模拟相关参数 NT = int((args.T1 - args.T0)/args.DT) self.timeline = UniformTimeLine(args.T0, args.T1, NT) self.mesh = mesh if self.ctx.myid == 0: self.GD = mesh.geo_dimension() if self.GD == 2: self.vspace = RaviartThomasFiniteElementSpace2d(self.mesh, p=0) # 速度空间 elif self.GD == 3: self.vspace = RaviartThomasFiniteElementSpace3d(self.mesh, p=0) self.pspace = self.vspace.smspace # 压力和饱和度所属的空间, 分片常数 self.cspace = LagrangeFiniteElementSpace(self.mesh, p=1) # 位移空间 # 上一时刻物理量的值 self.v = self.vspace.function() # 速度函数 self.p = self.pspace.function() # 压力函数 self.s = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0 self.u = self.cspace.function(dim=self.GD) # 位移函数 self.phi = self.pspace.function() # 孔隙度函数, 分片常数 # 当前时刻物理量的值, 用于保存临时计算出的值, 模型中系数的计算由当前时刻 # 的物理量的值决定 self.cv = self.vspace.function() # 速度函数 self.cp = self.pspace.function() # 压力函数 self.cs = self.pspace.function() # 水的饱和度函数 默认为0, 初始时刻区域内水的饱和度为0 self.cu = self.cspace.function(dim=self.GD) # 位移函数 self.cphi = self.pspace.function() # 孔隙度函数, 分片常数 # 初值 self.s[:] = self.mesh.celldata['fluid_0'] self.p[:] = self.mesh.celldata['pressure_0'] # MPa self.phi[:] = self.mesh.celldata['porosity_0'] # 初始孔隙度 self.s[:] = self.mesh.celldata['fluid_0'] self.cp[:] = self.p # 初始地层压力 self.cphi[:] = self.phi # 当前孔隙度系数 # 源汇项 self.f0 = self.cspace.function() self.f1 = self.cspace.function() self.f0[:] = self.mesh.nodedata['source_0'] # 流体 0 的源汇项 self.f1[:] = self.mesh.nodedata['source_1'] # 流体 1 的源汇项 # 一些常数矩阵和向量 # 速度散度矩阵, 速度方程对应的散度矩阵, (\nabla\cdot v, w) self.B = self.vspace.div_matrix() # 压力方程对应的位移散度矩阵, (\nabla\cdot u, w) 位移散度矩阵 # * 注意这里利用了压力空间分片常数, 线性函数导数也是分片常数的事实 c = self.mesh.entity_measure('cell') c *= self.mesh.celldata['biot'] val = self.mesh.grad_lambda() # (NC, TD+1, GD) val *= c[:, None, None] pc2d = self.pspace.cell_to_dof() cc2d = self.cspace.cell_to_dof() pgdof = self.pspace.number_of_global_dofs() cgdof = self.cspace.number_of_global_dofs() I = np.broadcast_to(pc2d, shape=cc2d.shape) J = cc2d self.PU0 = csr_matrix( (val[..., 0].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) self.PU1 = csr_matrix( (val[..., 1].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) if self.GD == 3: self.PU2 = csr_matrix( (val[..., 2].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) # 线弹性矩阵的右端向量 self.FU = np.zeros(self.GD*cgdof, dtype=np.float64) self.FU[0*cgdof:1*cgdof] -= [email protected] self.FU[1*cgdof:2*cgdof] -= [email protected] if self.GD == 3: self.FU[2*cgdof:3*cgdof] -= [email protected] # 初始应力和等效应力项 sigma = self.mesh.celldata['stress_0'] + self.mesh.celldata['stress_eff']# 初始应力和等效应力之和 self.FU[0*cgdof:1*cgdof] -= [email protected] self.FU[1*cgdof:2*cgdof] -= [email protected] if self.GD == 3: self.FU[2*cgdof:3*cgdof] -= [email protected] def recover(self, val): """ Notes ----- 给定一个分片常数的量, 恢复为分片连续的量 """ mesh = self.mesh cell = self.mesh.entity('cell') NC = mesh.number_of_cells() NN = mesh.number_of_nodes() w = 1/self.mesh.entity_measure('cell') # 恢复权重 w = np.broadcast_to(w[:, None], shape=cell.shape) r = np.zeros(NN, dtype=np.float64) d = np.zeros(NN, dtype=np.float64) np.add.at(d, cell, w) np.add.at(r, cell, val[:, None]*w) return r/d def add_time(self, n): """ Notes ---- 增加 n 步的计算时间 """ if self.ctx.myid == 0: self.timeline.add_time(n) def pressure_coefficient(self): """ Notes ----- 计算当前物理量下的压强质量矩阵系数 """ b = self.mesh.celldata['biot'] K = self.mesh.celldata['K'] c0 = self.mesh.meshdata['fluid_0']['compressibility'] c1 = self.mesh.meshdata['fluid_1']['compressibility'] s = self.cs # 流体 0 当前的饱和度 (NQ, NC) phi = self.cphi # 当前的孔隙度 val = phi*s*c0 val += phi*(1 - s)*c1 # 注意这里的 co 是常数, 但在水气混合物条件下应该依赖于压力 val += (b - phi)/K return val def saturation_pressure_coefficient(self): """ Notes ----- 计算当前物理量下, 饱和度方程中, 压强项对应的系数 """ b = self.mesh.celldata['biot'] K = self.mesh.celldata['K'] c0 = self.mesh.meshdata['fluid_0']['compressibility'] s = self.cs # 当前流体 0 的饱和度 phi = self.cphi # 当前孔隙度 val = (b - phi)/K val += phi*c0 val *= s return val def flux_coefficient(self): """ Notes ----- 计算**当前**物理量下, 速度方程对应的系数 计算通量的系数, 流体的相对渗透率除以粘性系数得到流动性系数 k_0/mu_0 是气体的流动性系数 k_1/mu_1 是油的流动性系数 这里假设压强的单位是 MPa 1 d = 9.869 233e-13 m^2 = 1000 md 1 cp = 1 mPa s = 1e-9 MPa*s = 1e-3 Pa*s """ mu0 = self.mesh.meshdata['fluid_0']['viscosity'] # 单位是 1 cp = 1 mPa.s mu1 = self.mesh.meshdata['fluid_1']['viscosity'] # 单位是 1 cp = 1 mPa.s # 岩石的绝对渗透率, 这里考虑了量纲的一致性, 压强单位是 Pa k = self.mesh.celldata['permeability']*9.869233e-4 s = self.cs # 流体 0 当前的饱和度系数 lam0 = self.mesh.fluid_relative_permeability_0(s) # TODO: 考虑更复杂的饱和度和渗透的关系 lam0 /= mu0 lam1 = self.mesh.fluid_relative_permeability_1(s) # TODO: 考虑更复杂的饱和度和渗透的关系 lam1 /= mu1 val = 1/(lam0 + lam1) val /=k return val def fluid_fractional_flow_coefficient_0(self): """ Notes ----- 计算**当前**物理量下, 饱和度方程中, 主流体的的流动性系数 """ s = self.cs # 流体 0 当前的饱和度系数 mu0 = self.mesh.meshdata['fluid_0']['viscosity'] # 单位是 1 cp = 1 mPa.s lam0 = self.mesh.fluid_relative_permeability_0(s) lam0 /= mu0 lam1 = self.mesh.fluid_relative_permeability_1(s) mu1 = self.mesh.meshdata['fluid_1']['viscosity'] # 单位是 1 cp = 1 mPa.s lam1 /= mu1 val = lam0/(lam0 + lam1) return val def get_total_system(self): """ Notes ----- 构造整个系统 二维情形: x = [s, v, p, u0, u1] A = [[ S, None, SP, SU0, SU1] [None, V, VP, None, None] [None, PV, P, PU0, PU1] [None, None, UP0, U00, U01] [None, None, UP1, U10, U11]] F = [FS, FV, FP, FU0, FU1] 三维情形: x = [s, v, p, u0, u1, u2] A = [[ S, None, SP, SU0, SU1, SU2] [None, V, VP, None, None, None] [None, PV, P, PU0, PU1, PU2] [None, None, UP0, U00, U01, U02] [None, None, UP1, U10, U11, U12] [None, None, UP2, U20, U21, U22]] F = [FS, FV, FP, FU0, FU1, FU2] """ GD = self.GD A0, FS, isBdDof0 = self.get_saturation_system(q=2) A1, FV, isBdDof1 = self.get_velocity_system(q=2) A2, FP, isBdDof2 = self.get_pressure_system(q=2) if GD == 2: A3, A4, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A0, A1, A2, A3, A4], format='csr') F = np.r_['0', FS, FV, FP, FU] elif GD == 3: A3, A4, A5, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A0, A1, A2, A3, A4, A5], format='csr') F = np.r_['0', FS, FV, FP, FU] isBdDof = np.r_['0', isBdDof0, isBdDof1, isBdDof2, isBdDof3] return A, F, isBdDof def get_saturation_system(self, q=2): """ Notes ---- 计算饱和度方程对应的离散系统 [ S, None, SP, SU0, SU1] [ S, None, SP, SU0, SU1, SU2] """ GD = self.GD qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() cellmeasure = self.mesh.entity_measure('cell') # SP 是对角矩阵 c = self.saturation_pressure_coefficient() # (NC, ) c *= cellmeasure SP = diags(c, 0) # S 质量矩阵组装, 对角矩阵 c = self.cphi[:]*cellmeasure S = diags(c, 0) # SU0, SU1, 饱和度方程中的位移散度对应的矩阵 b = self.mesh.celldata['biot'] val = self.mesh.grad_lambda() # (NC, TD+1, GD) c = self.cs[:]*b # (NC, ), 注意用当前的水饱和度 c *= cellmeasure val *= c[:, None, None] pgdof = self.pspace.number_of_global_dofs() # 压力空间自由度个数 cgdof = self.cspace.number_of_global_dofs() # 连续空间自由度个数 pc2d = self.pspace.cell_to_dof() cc2d = self.cspace.cell_to_dof() I = np.broadcast_to(pc2d, shape=cc2d.shape) J = cc2d SU0 = csr_matrix( (val[..., 0].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) SU1 = csr_matrix( (val[..., 1].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) if GD == 3: SU2 = csr_matrix( (val[..., 2].flat, (I.flat, J.flat)), shape=(pgdof, cgdof) ) # 右端矩阵 dt = self.timeline.current_time_step_length() FS = self.f0.value(bcs) # (NQ, NC) FS *= ws[:, None] FS = np.sum(FS, axis=0) FS *= cellmeasure FS *= dt FS += [email protected] # 上一时间步的压强 FS += [email protected] # 上一时间步的饱和度 FS += [email protected][:, 0] # 上一时间步的位移 x 分量 FS += [email protected][:, 1] # 上一个时间步的位移 y 分量 if GD == 3: FS += [email protected][:, 2] # 上一个时间步的位移 z 分量 # 用当前时刻的速度场, 构造迎风格式 face2cell = self.mesh.ds.face_to_cell() isBdFace = face2cell[:, 0] == face2cell[:, 1] qf = self.mesh.integrator(2, 'face') # 边或面上的积分公式 bcs, ws = qf.get_quadrature_points_and_weights() facemeasure = self.mesh.entity_measure('face') # 边的定向法线,它是左边单元的外法线,右边单元内法线。 fn = self.mesh.face_unit_normal() val0 = np.einsum('qfm, fm->qf', self.cv.face_value(bcs), fn) # 当前速度和法线的内积 # 流体 0 的流动分数, 与水的饱和度有关, 如果饱和度为0, 则为 0 F0 = self.fluid_fractional_flow_coefficient_0() val1 = F0[face2cell[:, 0]] # 边的左边单元的水流动分数 val2 = F0[face2cell[:, 1]] # 边的右边单元的水流动分数 val2[isBdFace] = 0.0 # 边界的贡献是 0,没有流入和流出 flag = val0 < 0.0 # 对于左边单元来说,是流入项 # 对于右边单元来说,是流出项 # 左右单元流入流出的绝对量是一样的 val = val0*val1[None, :] # val0 >= 0.0, 左边单元是流出 # val0 < 0.0, 左边单元是流入 val[flag] = (val0*val2[None, :])[flag] # val0 >= 0, 右边单元是流入 # val0 < 0, 右边单元是流出 b = np.einsum('q, qf, f->f', ws, val, facemeasure) b *= dt np.subtract.at(FS, face2cell[:, 0], b) isInFace = ~isBdFace # 只处理内部边 np.add.at(FS, face2cell[isInFace, 1], b[isInFace]) isBdDof = np.zeros(pgdof, dtype=np.bool_) if GD == 2: return [ S, None, SP, SU0, SU1], FS, isBdDof elif GD == 3: return [ S, None, SP, SU0, SU1, SU2], FS, isBdDof def get_velocity_system(self, q=2): """ Notes ----- 计算速度方程对应的离散系统. [None, V, VP, None, None] [None, V, VP, None, None, None] """ GD = self.GD dt = self.timeline.current_time_step_length() cellmeasure = self.mesh.entity_measure('cell') qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() # 速度对应的矩阵 V c = self.flux_coefficient() c *= cellmeasure phi = self.vspace.basis(bcs) V = np.einsum('q, qcin, qcjn, c->cij', ws, phi, phi, c, optimize=True) c2d = self.vspace.cell_to_dof() I = np.broadcast_to(c2d[:, :, None], shape=V.shape) J = np.broadcast_to(c2d[:, None, :], shape=V.shape) gdof = self.vspace.number_of_global_dofs() V = csr_matrix( (V.flat, (I.flat, J.flat)), shape=(gdof, gdof) ) # 压强矩阵 VP = -self.B # 右端向量, 0 向量 FV = np.zeros(gdof, dtype=np.float64) isBdDof = self.vspace.dof.is_boundary_dof() if GD == 2: return [None, V, VP, None, None], FV, isBdDof elif GD == 3: return [None, V, VP, None, None, None], FV, isBdDof def get_pressure_system(self, q=2): """ Notes ----- 计算压强方程对应的离散系统 这里组装矩阵时, 利用了压强是分片常数的特殊性 [ Noe, PV, P, PU0, PU1] [ None, PV, P, PU0, PU1, PU2] """ GD = self.GD dt = self.timeline.current_time_step_length() cellmeasure = self.mesh.entity_measure('cell') qf = self.mesh.integrator(q, etype='cell') bcs, ws = qf.get_quadrature_points_and_weights() PV = dt*self.B.T # P 是对角矩阵, 利用分片常数的 c = self.pressure_coefficient() # (NQ, NC) c *= cellmeasure P = diags(c, 0) # 组装压力方程的右端向量 # * 这里利用了压力空间基是分片常数 FP = self.f1.value(bcs) + self.f0.value(bcs) # (NQ, NC) FP *= ws[:, None] FP = np.sum(FP, axis=0) FP *= cellmeasure FP *= dt FP += [email protected] # 上一步的压力向量 FP += [email protected][:, 0] FP += [email protected][:, 1] if GD == 3: FP += [email protected][:, 2] gdof = self.pspace.number_of_global_dofs() isBdDof = np.zeros(gdof, dtype=np.bool_) if GD == 2: return [None, PV, P, self.PU0, self.PU1], FP, isBdDof elif GD == 3: return [None, PV, P, self.PU0, self.PU1, self.PU2], FP, isBdDof def get_dispacement_system(self, q=2): """ Notes ----- 计算位移方程对应的离散系统, 即线弹性方程系统. GD == 2: [None, None, UP0, U00, U01] [None, None, UP1, U10, U11] GD == 3: [None, None, UP0, U00, U01, U02] [None, None, UP1, U10, U11, U12] [None, None, UP2, U20, U21, U22] """ GD = self.GD # 拉梅参数 (lambda, mu) lam = self.mesh.celldata['lambda'] mu = self.mesh.celldata['mu'] U = self.linear_elasticity_matrix(lam, mu, format='list') isBdDof = self.cspace.dof.is_boundary_dof() if GD == 2: return ( [None, None, -self.PU0.T, U[0][0], U[0][1]], [None, None, -self.PU1.T, U[1][0], U[1][1]], self.FU, np.r_['0', isBdDof, isBdDof] ) elif GD == 3: return ( [None, None, -self.PU0.T, U[0][0], U[0][1], U[0][2]], [None, None, -self.PU1.T, U[1][0], U[1][1], U[1][2]], [None, None, -self.PU2.T, U[2][0], U[2][1], U[2][2]], self.FU, np.r_['0', isBdDof, isBdDof, isBdDof] ) def linear_elasticity_matrix(self, lam, mu, format='csr', q=None): """ Notes ----- 注意这里的拉梅常数是单元分片常数 lam.shape == (NC, ) # MPa mu.shape == (NC, ) # MPa """ GD = self.GD if GD == 2: idx = [(0, 0), (0, 1), (1, 1)] imap = {(0, 0):0, (0, 1):1, (1, 1):2} elif GD == 3: idx = [(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2)] imap = {(0, 0):0, (0, 1):1, (0, 2):2, (1, 1):3, (1, 2):4, (2, 2):5} A = [] qf = self.cspace.integrator if q is None else self.mesh.integrator(q, 'cell') bcs, ws = qf.get_quadrature_points_and_weights() grad = self.cspace.grad_basis(bcs) # (NQ, NC, ldof, GD) # 分块组装矩阵 gdof = self.cspace.number_of_global_dofs() cellmeasure = self.cspace.cellmeasure for k, (i, j) in enumerate(idx): Aij = np.einsum('i, ijm, ijn, j->jmn', ws, grad[..., i], grad[..., j], cellmeasure) A.append(Aij) if GD == 2: C = [[None, None], [None, None]] D = mu[:, None, None]*(A[0] + A[2]) elif GD == 3: C = [[None, None, None], [None, None, None], [None, None, None]] D = mu[:, None, None]*(A[0] + A[3] + A[5]) cell2dof = self.cspace.cell_to_dof() # (NC, ldof) ldof = self.cspace.number_of_local_dofs() NC = self.mesh.number_of_cells() shape = (NC, ldof, ldof) I = np.broadcast_to(cell2dof[:, :, None], shape=shape) J = np.broadcast_to(cell2dof[:, None, :], shape=shape) for i in range(GD): Aii = D + (mu + lam)[:, None, None]*A[imap[(i, i)]] C[i][i] = csr_matrix((Aii.flat, (I.flat, J.flat)), shape=(gdof, gdof)) for j in range(i+1, GD): Aij = lam[:, None, None]*A[imap[(i, j)]] + mu[:, None, None]*A[imap[(i, j)]].swapaxes(-1, -2) C[i][j] = csr_matrix((Aij.flat, (I.flat, J.flat)), shape=(gdof, gdof)) C[j][i] = C[i][j].T if format == 'csr': return bmat(C, format='csr') # format = bsr ?? elif format == 'bsr': return bmat(C, format='bsr') elif format == 'list': return C def solve_linear_system_0(self): # 构建总系统 if self.ctx.myid == 0: A, F, isBdDof = self.get_total_system() # 处理边界条件, 这里是 0 边界 gdof = len(isBdDof) bdIdx = np.zeros(gdof, dtype=np.int_) bdIdx[isBdDof] = 1 Tbd = diags(bdIdx) T = diags(1-bdIdx) A = T@A@T + Tbd F[isBdDof] = 0.0 # 求解 self.ctx.set_centralized_sparse(A) x = F.copy() self.ctx.set_rhs(x) # Modified in place self.ctx.run(job=6) # Analysis + Factorization + Solve if self.ctx.myid == 0: vgdof = self.vspace.number_of_global_dofs() pgdof = self.pspace.number_of_global_dofs() cgdof = self.cspace.number_of_global_dofs() start = 0 end = pgdof s = x[start:end] start = end end += vgdof v = x[start:end] start = end end += pgdof p = x[start:end] start = end u = x[start:] return s, v, p, u def solve_linear_system_1(self): """ Notes ----- 构造整个系统 二维情形: x = [s, v, p, u0, u1] A = [[ S, None, SP, SU0, SU1] [None, V, VP, None, None] [None, PV, P, PU0, PU1] [None, None, UP0, U00, U01] [None, None, UP1, U10, U11]] F = [FS, FV, FP, FU0, FU1] 三维情形: x = [s, v, p, u0, u1, u2] A = [[ S, None, SP, SU0, SU1, SU2] [None, V, VP, None, None, None] [None, PV, P, PU0, PU1, PU2] [None, None, UP0, U00, U01, U02] [None, None, UP1, U10, U11, U12] [None, None, UP2, U20, U21, U22]] F = [FS, FV, FP, FU0, FU1, FU2] """ if self.ctx.myid == 0: vgdof = self.vspace.number_of_global_dofs() pgdof = self.pspace.number_of_global_dofs() cgdof = self.cspace.number_of_global_dofs() GD = self.GD A0, FS, isBdDof0 = self.get_saturation_system(q=2) A1, FV, isBdDof1 = self.get_velocity_system(q=2) A2, FP, isBdDof2 = self.get_pressure_system(q=2) if GD == 2: A3, A4, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A1[1:], A2[1:], A3[1:], A4[1:]], format='csr') F = np.r_['0', FV, FP, FU] elif GD == 3: A3, A4, A5, FU, isBdDof3 = self.get_dispacement_system(q=2) A = bmat([A1[1:], A2[1:], A3[1:], A4[1:], A5[1:]], format='csr') F = np.r_['0', FV, FP, FU] isBdDof = np.r_['0', isBdDof1, isBdDof2, isBdDof3] gdof = len(isBdDof) bdIdx = np.zeros(gdof, dtype=np.int_) bdIdx[isBdDof] = 1 Tbd = diags(bdIdx) T = diags(1-bdIdx) A = T@A@T + Tbd F[isBdDof] = 0.0 #[ S, None, SP, SU0, SU1, SU2] self.ctx.set_centralized_sparse(A) x = F.copy() self.ctx.set_rhs(x) # Modified in place self.ctx.run(job=6) # Analysis + Factorization + Solve if self.ctx.myid == 0: v = x[0:vgdof] p = x[vgdof:vgdof+pgdof] u = x[vgdof+pgdof:] s = FS s -= A0[2]@p s -= A0[3]@u[0*cgdof:1*cgdof] s -= A0[4]@u[1*cgdof:2*cgdof] if GD == 3: s -= A0[5]@u[2*cgdof:3*cgdof] s /= A0[0].diagonal() return s, v, p, u else: return None, None, None, None def picard_iteration(self): """ """ e0 = 1.0 k = 0 while e0 > 1e-10: s, v, p, u = self.solve_linear_system_1() if self.ctx.myid == 0: e0 = 0.0 e0 += np.sum((self.cs - s)**2) self.cs[:] = s e0 += np.sum((self.cp - p)**2) self.cp[:] = p self.cv[:] = v self.cu.T.flat[:] = u e0 = np.sqrt(e0) # 误差的 l2 norm print(e0) e0 = self.ctx.comm.bcast(e0, root=0) k += 1 if k >= self.args.npicard: if self.ctx.myid == 0: print('picard iteration arrive max iteration with error:', e0) break if self.ctx.myid == 0: # 更新解 self.s[:] = self.cs self.v[:] = self.cv self.p[:] = self.cp self.u[:] = self.cu def update_mesh_data(self): """ Notes ----- 更新 mesh 中的数据 """ GD = self.GD bc = np.array((GD+1)*[1/(GD+1)], dtype=np.float64) mesh = self.mesh v = self.v p = self.p s = self.s u = self.u # 单元中心的流体速度 val = v.value(bc) if GD == 2: val = np.concatenate((val, np.zeros((val.shape[0], 1), dtype=val.dtype)), axis=1) mesh.celldata['velocity'] = val # 分片常数的压强 val = self.recover(p[:]) mesh.nodedata['pressure'] = val # 分片常数的饱和度 val = self.recover(s[:]) mesh.nodedata['fluid_0'] = val val = self.recover(1 - s) mesh.nodedata['fluid_1'] = val # 节点处的位移 if GD == 2: u = np.concatenate((u[:], np.zeros((u.shape[0], 1), dtype=u.dtype)), axis=1) mesh.nodedata['displacement'] = u[:] # 增加应变和应力的计算 NC = self.mesh.number_of_cells() s0 = np.zeros((NC, 3, 3), dtype=np.float64) s1 = np.zeros((NC, 3, 3), dtype=np.float64) s = u.grad_value(bc) # (NC, GD, GD) s0[:, 0:GD, 0:GD] = s + s.swapaxes(-1, -2) s0[:, 0:GD, 0:GD] /= 2 lam = self.mesh.celldata['lambda'] mu = self.mesh.celldata['mu'] s1[:, 0:GD, 0:GD] = s0[:, 0:GD, 0:GD] s1[:, 0:GD, 0:GD] *= mu[:, None, None] s1[:, 0:GD, 0:GD] *= 2 s1[:, range(GD), range(GD)] += (lam*s0.trace(axis1=-1, axis2=-2))[:, None] s1[:, range(GD), range(GD)] += mesh.celldata['stress_0'][:, None] s1[:, range(GD), range(GD)] -= (mesh.celldata['biot']*(p - mesh.celldata['pressure_0']))[:, None] mesh.celldata['strain'] = s0.reshape(NC, -1) mesh.celldata['stress'] = s1.reshape(NC, -1) def run(self, writer=None): """ Notes ----- 计算所有时间层物理量。 """ args = self.args timeline = self.timeline dt = timeline.current_time_step_length() if (self.ctx.myid == 0) and (writer is not None): n = timeline.current fname = args.output + str(n).zfill(10) + '.vtu' self.update_mesh_data() writer(fname, self.mesh) while not timeline.stop(): if self.ctx.myid == 0: ct = timeline.current_time_level()/3600/24 # 天为单位 print('当前时刻为第', ct, '天') self.picard_iteration() timeline.current += 1 if timeline.current%args.step == 0: if (self.ctx.myid == 0) and (writer is not None): n = timeline.current fname = args.output + str(n).zfill(10) + '.vtu' self.update_mesh_data() writer(fname, self.mesh) if (self.ctx.myid == 0) and (writer is not None): n = timeline.current fname = args.output + str(n).zfill(10) + '.vtu' self.update_mesh_data() writer(fname, self.mesh)
def alg_3_4(self, maxit=None): """ 1. 最粗网格上求解最小特征特征值问题,得到最小特征值 d_H 和特征向量 u_H 2. 自适应求解 - \Delta u_h = u_H * u_H 插值到下一层网格上做为新 u_H 3. 最新网格层上求出的 uh 做为一个基函数,加入到最粗网格的有限元空间中, 求解最小特征值问题。 自适应 maxit, picard:2 """ print("算法 3.4") if maxit is None: maxit = self.maxit start = timer() if self.step == 0: idx = [] else: idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1] mesh = self.pde.init_mesh(n=self.numrefine) integrator = mesh.integrator(self.q) # 1. 粗网格上求解最小特征值问题 space = LagrangeFiniteElementSpace(mesh, 1) AH = self.get_stiff_matrix(space) MH = self.get_mass_matrix(space) isFreeHDof = ~(space.boundary_dof()) gdof = space.number_of_global_dofs() print("initial mesh :", gdof) uH = np.zeros(gdof, dtype=np.float) A = AH[isFreeHDof, :][:, isFreeHDof].tocsr() M = MH[isFreeHDof, :][:, isFreeHDof].tocsr() if self.matlab is False: uH[isFreeHDof], d = self.eig(A, M) else: uH[isFreeHDof], d = self.meigs(A, M) uh = space.function() uh[:] = uH GD = mesh.geo_dimension() if (self.step > 0) and (0 in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_4_0_' + str(NN) + '.pdf') plt.close() self.savemesh(mesh, self.resultdir + 'mesh_3_4_0_' + str(NN) + '.mat') # 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H I = eye(gdof) final = 0 for i in range(maxit): eta = self.residual_estimate(uh) markedCell = mark(eta, self.theta) IM = mesh.bisect(markedCell, returnim=True) print(i + 1, "refine :", mesh.number_of_nodes()) if (self.step > 0) and (i in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_4_' + str(i + 1) + '_' + str(NN) + '.mat') I = IM @ I uH = IM @ uH space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) isFreeDof = ~(space.boundary_dof()) b = M @ uH uh = space.function() if self.matlab is False: uh[isFreeDof] = self.psolve( A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof], M[isFreeDof, :][:, isFreeDof].tocsr()) else: uh[isFreeDof] = self.msolve( A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof]) final = i + 1 if gdof > self.maxdof: break if self.step > 0: NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_4_' + str(final) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_4_' + str(final) + '_' + str(NN) + '.mat') # 3. 把 uh 加入粗网格空间, 组装刚度和质量矩阵 w0 = uh @ A w1 = w0 @ uh w2 = w0 @ I AA = bmat([[AH, w2.reshape(-1, 1)], [w2, w1]], format='csr') w0 = uh @ M w1 = w0 @ uh w2 = w0 @ I MM = bmat([[MH, w2.reshape(-1, 1)], [w2, w1]], format='csr') isFreeDof = np.r_[isFreeHDof, True] u = np.zeros(len(isFreeDof)) ## 求解特征值 A = AA[isFreeDof, :][:, isFreeDof].tocsr() M = MM[isFreeDof, :][:, isFreeDof].tocsr() if self.multieigs is True: self.A = A self.M = M self.ml = pyamg.ruge_stuben_solver(self.A) self.eigs() else: if self.matlab is False: u[isFreeDof], d = self.eig(A, M) else: u[isFreeDof], d = self.meigs(A, M) print("smallest eigns:", d) uh *= u[-1] uh += I @ u[:-1] uh /= np.max(np.abs(uh)) uh = space.function(array=uh) return uh end = timer() print("with time: ", end - start)
def alg_3_3(self, maxit=None): """ 1. 最粗网格上求解最小特征特征值问题,得到最小特征值 d_H 和特征向量 u_H 2. 自适应求解 - \Delta u_h = u_H * u_H 插值到下一层网格上做为新 u_H 3. 在最细网格上求解一次最小特征值问题 自适应 maxit, picard:2 """ print("算法 3.3") if maxit is None: maxit = self.maxit start = timer() if self.step == 0: idx = [] else: idx = list(range(0, self.maxit, self.step)) mesh = self.pde.init_mesh(n=self.numrefine) # 1. 粗网格上求解最小特征值问题 space = LagrangeFiniteElementSpace(mesh, 1) AH = self.get_stiff_matrix(space) MH = self.get_mass_matrix(space) isFreeHDof = ~(space.boundary_dof()) gdof = space.number_of_global_dofs() uH = np.zeros(gdof, dtype=mesh.ftype) print("initial mesh :", gdof) A = AH[isFreeHDof, :][:, isFreeHDof].tocsr() M = MH[isFreeHDof, :][:, isFreeHDof].tocsr() if self.matlab is False: uH[isFreeHDof], d = self.eig(A, M) else: uH[isFreeHDof], d = self.meigs(A, M) uh = space.function() uh[:] = uH GD = mesh.geo_dimension() if (self.step > 0) and (0 in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_3_0_' + str(NN) + '.pdf') plt.close() self.savemesh(mesh, self.resultdir + 'mesh_3_3_0_' + str(NN) + '.mat') # 2. 以 u_H 为右端项自适应求解 -\Deta u = u_H I = eye(gdof) final = 0 for i in range(maxit - 1): eta = self.residual_estimate(uh) markedCell = mark(eta, self.theta) IM = mesh.bisect(markedCell, returnim=True) NN = mesh.number_of_nodes() print(i + 1, "refine : ", NN) if (self.step > 0) and (i in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_3_' + str(i + 1) + '_' + str(NN) + '.mat') final = i + 1 if NN > self.maxdof: break I = IM @ I uH = IM @ uH space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) isFreeDof = ~(space.boundary_dof()) b = M @ uH uh = space.function() if self.matlab is False: uh[isFreeDof] = self.psolve( A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof], M[isFreeDof, :][:, isFreeDof].tocsr()) else: uh[isFreeDof] = self.msolve( A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof]) # 3. 在最细网格上求解一次最小特征值问题 if self.step > 0: NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_3_' + str(final) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_3_' + str(final) + '_' + str(NN) + '.mat') space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) isFreeDof = ~(space.boundary_dof()) uh = space.function(array=uh) A = A[isFreeDof, :][:, isFreeDof].tocsr() M = M[isFreeDof, :][:, isFreeDof].tocsr() uh = space.function() if self.multieigs is True: self.A = A self.M = M self.ml = pyamg.ruge_stuben_solver(self.A) self.eigs() else: if self.matlab is False: uh[isFreeDof], d = self.eig(A, M) else: uh[isFreeDof], d = self.meigs(A, M) print("smallest eigns:", d) return uh end = timer() print("with time: ", end - start)
def alg_3_2(self, maxit=None): """ 1. 自适应求解 -\Delta u = 1。 1. 在最细网格上求最小特征值和特征向量。 refine maxit, picard: 1 """ print("算法 3.2") if maxit is None: maxit = self.maxit start = timer() if self.step == 0: idx = [] else: idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1] mesh = self.pde.init_mesh(n=self.numrefine) GD = mesh.geo_dimension() if (self.step > 0) and (0 in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_2_0_' + str(NN) + '.pdf') plt.close() self.savemesh(mesh, self.resultdir + 'mesh_3_2_0_' + str(NN) + '.mat') final = 0 integrator = mesh.integrator(self.q) for i in range(maxit): space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) b = M @ np.ones(gdof) isFreeDof = ~(space.boundary_dof()) A = A[isFreeDof, :][:, isFreeDof].tocsr() M = M[isFreeDof, :][:, isFreeDof].tocsr() uh = space.function() if self.matlab is False: uh[isFreeDof] = self.psolve(A, b[isFreeDof], M) else: uh[isFreeDof] = self.msolve(A, b[isFreeDof]) eta = self.residual_estimate(uh) markedCell = mark(eta, self.theta) IM = mesh.bisect(markedCell, returnim=True) NN = mesh.number_of_nodes() print(i + 1, "refine :", NN) if (self.step > 0) and (i in idx): fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_2_' + str(i + 1) + '_' + str(NN) + '.mat') final = i + 1 if NN > self.maxdof: break if self.step > 0: NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_2_' + str(final) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_2_' + str(final) + '_' + str(NN) + '.mat') space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) isFreeDof = ~(space.boundary_dof()) A = A[isFreeDof, :][:, isFreeDof].tocsr() M = M[isFreeDof, :][:, isFreeDof].tocsr() if self.multieigs is True: self.A = A self.M = M self.ml = pyamg.ruge_stuben_solver(self.A) self.eigs() else: uh = IM @ uh if self.matlab is False: uh[isFreeDof], d = self.eig(A, M) else: uh[isFreeDof], d = self.meigs(A, M) print("smallest eigns:", d) return space.function(array=uh) end = timer() print("with time: ", end - start)
def alg_3_1(self, maxit=None): """ 1. 自适应在每层网格上求解最小特征值问题 refine: maxit, picard: maxit + 1 """ print("算法 3.1") if maxit is None: maxit = self.maxit start = timer() if self.step == 0: idx = [] else: idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1] mesh = self.pde.init_mesh(n=self.numrefine) GD = mesh.geo_dimension() if (self.step > 0) and (0 in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_1_0_' + str(NN) + '.pdf') plt.close() self.savemesh(mesh, self.resultdir + 'mesh_3_1_0_' + str(NN) + '.mat') space = LagrangeFiniteElementSpace(mesh, 1) isFreeDof = ~(space.boundary_dof()) gdof = space.number_of_global_dofs() uh = np.ones(gdof, dtype=mesh.ftype) uh[~isFreeDof] = 0 IM = eye(gdof) for i in range(maxit + 1): area = mesh.entity_measure('cell') A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) uh = IM @ uh A = A[isFreeDof, :][:, isFreeDof].tocsr() M = M[isFreeDof, :][:, isFreeDof].tocsr() if self.matlab is False: uh[isFreeDof], d = self.eig(A, M) else: uh[isFreeDof], d = self.meigs(A, M) if i < maxit: uh = space.function(array=uh) eta = self.residual_estimate(uh) markedCell = mark(eta, self.theta) IM = mesh.bisect(markedCell, returnim=True) print(i + 1, "refine: ", mesh.number_of_nodes()) if (self.step > 0) and (i in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' + str(NN) + '.mat') space = LagrangeFiniteElementSpace(mesh, 1) isFreeDof = ~(space.boundary_dof()) gdof = space.number_of_global_dofs() if gdof > self.maxdof: break if self.step > 0: NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_3_1_' + str(i + 1) + '_' + str(NN) + '.mat') if self.multieigs is True: self.A = A self.M = M self.ml = pyamg.ruge_stuben_solver(self.A) self.eigs() end = timer() print("smallest eigns:", d, "with time: ", end - start) uh = space.function(array=uh) return uh
def alg_0(self, maxit=None): """ 1. 最粗网格上求解最小特征特征值问题,得到最小特征值 d_H 和特征向量 u_H 2. 自适应求解 - \Delta u_h = d_H*u_H * 每层网格上求出的 u_h,插值到下一层网格上做为 u_H * 并更新 d_H = u_h@A@u_h/u_h@M@u_h, 其中 A 是当前网格层上的刚度矩 阵,M 为当前网格层的质量矩阵。 自适应 maxit, picard: 1 """ print("算法 0") if maxit is None: maxit = self.maxit start = timer() if self.step == 0: idx = [] else: idx = list(range(0, self.maxit, self.step)) + [self.maxit - 1] mesh = self.pde.init_mesh(n=self.numrefine) # 1. 粗网格上求解最小特征值问题 space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() print("initial mesh:", gdof) uh = np.zeros(gdof, dtype=np.float) AH = self.get_stiff_matrix(space) MH = self.get_mass_matrix(space) isFreeHDof = ~(space.boundary_dof()) A = AH[isFreeHDof, :][:, isFreeHDof].tocsr() M = MH[isFreeHDof, :][:, isFreeHDof].tocsr() if self.picard is True: uh[isFreeHDof], d = picard(A, M, np.ones(sum(isFreeHDof)), sigma=self.sigma) else: uh[isFreeHDof], d = self.eig(A, M) GD = mesh.geo_dimension() if (self.step > 0) and (0 in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_0_0_' + str(NN) + '.pdf') plt.close() self.savemesh(mesh, self.resultdir + 'mesh_0_0_' + str(NN) + '.mat') # 2. 以 u_h 为右端项自适应求解 -\Deta u = d*u_h I = eye(gdof) for i in range(maxit): uh = space.function(array=uh) eta = self.residual_estimate(uh) markedCell = mark(eta, self.theta) IM = mesh.bisect(markedCell, returnim=True) print(i + 1, "refine: ", mesh.number_of_nodes()) if (self.step > 0) and (i in idx): NN = mesh.number_of_nodes() fig = plt.figure() fig.set_facecolor('white') if GD == 2: axes = fig.gca() else: axes = Axes3D(fig) mesh.add_plot(axes, cellcolor='w') fig.savefig(self.resultdir + 'mesh_0_' + str(i + 1) + '_' + str(NN) + '.pdf') plt.close() self.savemesh( mesh, self.resultdir + 'mesh_0_' + str(i + 1) + '_' + str(NN) + '.mat') I = IM @ I uh = IM @ uh space = LagrangeFiniteElementSpace(mesh, 1) gdof = space.number_of_global_dofs() A = self.get_stiff_matrix(space) M = self.get_mass_matrix(space) isFreeDof = ~(space.boundary_dof()) b = d * M @ uh if self.sigma is None: ml = pyamg.ruge_stuben_solver( A[isFreeDof, :][:, isFreeDof].tocsr()) uh[isFreeDof] = ml.solve(b[isFreeDof], x0=uh[isFreeDof], tol=1e-12, accel='cg').reshape((-1, )) else: K = A[isFreeDof, :][:, isFreeDof].tocsr( ) + self.sigma * M[isFreeDof, :][:, isFreeDof].tocsr() b += self.sigma * M @ uh ml = pyamg.ruge_stuben_solver(K) uh[isFreeDof] = ml.solve(b[isFreeDof], x0=uh[isFreeDof], tol=1e-12, accel='cg').reshape(-1) # uh[isFreeDof] = spsolve(A[isFreeDof, :][:, isFreeDof].tocsr(), b[isFreeDof]) d = uh @ A @ uh / (uh @ M @ uh) if gdof > self.maxdof: break if self.multieigs is True: self.A = A[isFreeDof, :][:, isFreeDof].tocsr() self.M = M[isFreeDof, :][:, isFreeDof].tocsr() self.ml = pyamg.ruge_stuben_solver(self.A) self.eigs() end = timer() print("smallest eigns:", d, "with time: ", end - start) uh = space.function(array=uh) return uh
err = np.sqrt(np.sum(eta**2)) print('errrefine', err) if err < tol: break else: # 加密并插值 NN0 = smesh.number_of_nodes() edge = smesh.entity('edge') isMarkedCell = smesh.refine_marker(eta, rtheta, method='L2') smesh.refine_triangle_rg(isMarkedCell) i += 1 smesh.add_plot(plt) plt.savefig('./test-' + str(i + 1) + '.png') plt.close() space = LagrangeFiniteElementSpace(smesh, p=1) print('refinedof', space.number_of_global_dofs()) uh00 = space.function() nn2e = smesh.newnode2edge uh00[:NN0] = uh0 uh00[NN0:] = np.average(uh0[edge[nn2e]], axis=-1) uh0 = space.function() uh0[:] = uh00 # 粗化网格并插值 isMarkedCell = smesh.refine_marker(eta, ctheta, 'COARSEN') smesh.coarsen_triangle_rg(isMarkedCell) i += 1 smesh.add_plot(plt) plt.savefig('./test-' + str(i + 1) + '.png') plt.close() space = LagrangeFiniteElementSpace(smesh, p=1) print('coarsendof', space.number_of_global_dofs())