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_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_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_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