errorType = [
'$|| u - u_h||_{\Omega,0}$', '$||\\nabla u - \\nabla u_h||_{\Omega, 0}$',
'$|| u - u_I||_{\Omega,0}$', '$||\\nabla u - \\nabla u_I||_{\Omega, 0}$',
'$|| u_I - u_h ||_{\Omega, \infty}$'
]
errorMatrix = np.zeros((len(errorType), maxit), dtype=np.float64)
NDof = np.zeros(maxit, dtype=np.int_)
m = 4
for i in range(maxit):
print("The {}-th computation:".format(i))
space = ParametricLagrangeFiniteElementSpace(mesh, p=sdegree)
NDof[i] = space.number_of_global_dofs()
uI = space.interpolation(pde.solution)
A = space.stiff_matrix()
C = space.integral_basis()
F = space.source_vector(pde.source)
NN = mesh.number_of_corner_nodes()
NC = mesh.number_of_cells()
delta = (A @ uI - F)
A = bmat([[A, C.reshape(-1, 1)], [C, None]], format='csr')
F = np.r_[F, 0]
uh = space.function()
class ModelSover():
def __init__(self, model, p, NS=0, NT=100):
self.model = model
self.mesh0 = model.space_mesh(NS=NS, p=p) # 初始网格
self.mesh1 = model.space_mesh(NS=NS, p=p) # 计算网格
self.timeline = model.time_mesh(NT=NT)
self.cspace = ParametricLagrangeFiniteElementSpace(self.mesh1,
p=p,
spacetype='C')
self.dspace = ParametricLagrangeFiniteElementSpace(self.mesh1,
p=p - 1,
spacetype='D')
self.integrator = self.cspace.integralalg.integrator
bc = self.mesh0.entity_barycenter('cell')
self.rho = model.init_rho(bc) # (NC, )
self.gamma = model.adiabatic_index(bc) #(NC, )
self.MV = self.cspace.mass_matrix(c=self.rho)
self.ME = self.dspace.mass_matrix(c=self.rho)
GD = self.mesh1.geo_dimension()
self.x = self.cspace.function(dim=GD) # 位置
self.v = self.cspace.function(dim=GD) # 速度
self.e = self.dspace.function() # 能量
self.x[:] = self.mesh1.entity('node') # 初始化自由度位置
model.init_velocity(self.v) # 初始化速度
model.init_energe(self.e) # 初始化能量
self.cx = self.cspace.function(dim=GD) # 保存临时的解
self.cv = self.cspace.function(dim=GD)
self.ce = self.dspace.function()
self.cx[:] = self.mesh1.entity('node')
model.init_velocity(self.cv)
model.init_energe(self.ce)
self.mesh1.celldata['rho'] = self.rho
self.mesh1.celldata['gamma'] = self.gamma
self.mesh1.nodedata['velocity'] = self.cv
def get_force_matrix(self, q=None):
# 积分公式
qf = self.integrator if q is None else self.mesh1.integrator(
q, etype='cell')
# bcs : (NQ, n)
# ws : (NQ, )
bcs, ws = qf.get_quadrature_points_and_weights()
rm = self.mesh1.reference_cell_measure() # 参考单元测度
d = self.mesh1.first_fundamental_form(bcs)
d = np.sqrt(np.linalg.det(d))
d *= self.model.stress(bcs, self.ce, self.rho, self.gamma, self.mesh0,
self.mesh1)
gphi = self.cspace.grad_basis(bcs) # (NQ, NC, ldof, GD)
phi = self.dspace.basis(bcs)
M0 = np.einsum('q, qci, qcj, qc->cij', ws * rm, gphi[..., 0], phi,
d) # (NC, ldof0, ldof1)
M1 = np.einsum('q, qci, qcj, qc->cij', ws * rm, gphi[..., 1], phi,
d) # (NC, ldof0, ldof1)
c2d0 = self.cspace.cell_to_dof()
c2d1 = self.dspace.cell_to_dof()
I = np.broadcast_to(c2d0[:, :, None], shape=M0.shape)
J = np.broadcast_to(c2d1[:, None, :], shape=M0.shape)
gdof0 = self.cspace.number_of_global_dofs()
gdof1 = self.dspace.number_of_global_dofs()
M0 = csr_matrix((M0.flat, (I.flat, J.flat)), shape=(gdof0, gdof1))
M1 = csr_matrix((M1.flat, (I.flat, J.flat)), shape=(gdof0, gdof1))
return M0, M1
def solve_one_step(self):
dt = self.timeline.current_time_step_length()
M0, M1 = self.get_force_matrix()
one = np.ones(M0.shape[1])
F0 = spsolve(self.MV, M0 @ one)
F1 = spsolve(self.MV, M1 @ one)
self.cv[:, 0] = self.v[:, 0] - dt / 2 * F0
self.cv[:, 1] = self.v[:, 1] - dt / 2 * F1
# 边界条件处理
edge2cell = self.mesh1.ds.edge_to_cell()
isBdEdge = edge2cell[:, 0] == edge2cell[:, 1]
edge2dof = self.cspace.edge_to_dof()[isBdEdge]
en = self.mesh1.edge_unit_normal(index=isBdEdge) # (NE, GD)
# (NE, ldof, GD) * (NE, 1, GD) = (NE, ldof, GD) --> (NE, ldof)
l = np.sum(self.cv[edge2dof, :] * en[:, None, :], axis=-1)
val = l[..., None] * en[:, None, :] # (NE, ldof, 1) * (NE, 1, GD)-->
np.subtract.at(self.cv, (edge2dof, np.s_[:]), val)
F = spsolve(self.ME, self.cv[:, 0] @ M0 + self.cv[:, 1] @ M1)
self.ce[:] = self.e + dt / 2 * F
self.cx[:] = self.x + dt / 2 * self.cv
self.mesh1.node[:] = self.cx
M0, M1 = self.get_force_matrix()
F0 = spsolve(self.MV, M0 @ one)
F1 = spsolve(self.MV, M1 @ one)
self.cv[:, 0] = self.v[:, 0] - dt * F0
self.cv[:, 1] = self.v[:, 1] - dt * F1
l = np.sum(self.cv[edge2dof, :] * en[:, None, :],
axis=-1) # (NE, ldof)
val = l[..., None] * en[:, None, :] # (NE, ldof, GD)
np.subtract.at(self.cv, (edge2dof, np.s_[:]), val)
v = (self.cv + self.v) / 2
F = spsolve(self.ME, v[:, 0] @ M0 + v[:, 1] @ M1)
self.ce[:] = self.e + dt * F
self.cx[:] = self.x + dt * v
self.x[:] = self.cx
self.v[:] = self.cv
self.e[:] = self.ce
self.mesh1.node[:] = self.x
self.mesh1.nodedata['velocity'] = self.v
def solve(self, step=1):
"""
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.mesh1.to_vtk(fname=fname)
while not timeline.stop():
self.solve_one_step()
timeline.current += 1
if timeline.current % step == 0:
n = timeline.current
fname = 'test_' + str(n).zfill(10) + '.vtu'
self.mesh1.to_vtk(fname=fname)
timeline.reset()
class LagrangianHydrodynamicsSimulator():
def __init__(self, model, p, NS=0, NT=100):
self.model = model # 物理模型
self.mesh0 = model.space_mesh(NS=NS, p=p) # 初始网格
self.mesh1 = model.space_mesh(NS=NS, p=p) # 计算网格
self.timeline = model.time_mesh(NT=NT) # 时间离散网格
# 运动学空间
self.cspace = ParametricLagrangeFiniteElementSpace(self.mesh1,
p=p,
spacetype='C')
# 热力学空间
self.dspace = ParametricLagrangeFiniteElementSpace(self.mesh1,
p=p - 1,
spacetype='D')
self.integrator = self.cspace.integralalg.integrator
bc = self.mesh0.entity_barycenter('cell')
# 物质密度
self.rho = model.init_rho(bc) # (NC, )
# 绝热指数
self.gamma = model.adiabatic_index(bc) #(NC, )
self.MV = self.cspace.mass_matrix(c=self.rho) # 运动空间的质量矩阵
self.ME = self.dspace.mass_matrix(c=self.rho) # 热力学空间的质量矩阵
GD = self.mesh1.geo_dimension()
self.x = self.cspace.function(dim=GD) # 位置
self.v = self.cspace.function(dim=GD) # 速度
self.e = self.dspace.function() # 能量
self.x[:] = self.mesh1.entity('node') # 初始化自由度位置
model.init_velocity(self.v) # 初始化速度
model.init_energe(self.e) # 初始化能量
self.cx = self.cspace.function(dim=GD) # 保存临时的解
self.cv = self.cspace.function(dim=GD)
self.ce = self.dspace.function()
self.cx[:] = self.mesh1.entity('node')
model.init_velocity(self.cv)
model.init_energe(self.ce)
self.mesh1.celldata['rho'] = self.rho
self.mesh1.celldata['gamma'] = self.gamma
self.mesh1.nodedata['velocity'] = self.cv
def get_force_matrix(self, q=None):
# 积分公式
qf = self.integrator if q is None else self.mesh1.integrator(
q, etype='cell')
# bcs : (NQ, n)
# ws : (NQ, )
bcs, ws = qf.get_quadrature_points_and_weights()
rm = self.mesh1.reference_cell_measure() # 参考单元测度
d = self.mesh1.first_fundamental_form(bcs)
d = np.sqrt(np.linalg.det(d))
d *= self.model.stress(bcs, self.ce, self.rho, self.gamma, self.mesh0,
self.mesh1)
gphi = self.cspace.grad_basis(bcs) # (NQ, NC, ldof, GD)
phi = self.dspace.basis(bcs)
M0 = np.einsum('q, qci, qcj, qc->cij', ws * rm, gphi[..., 0], phi,
d) # (NC, ldof0, ldof1)
M1 = np.einsum('q, qci, qcj, qc->cij', ws * rm, gphi[..., 1], phi,
d) # (NC, ldof0, ldof1)
c2d0 = self.cspace.cell_to_dof()
c2d1 = self.dspace.cell_to_dof()
I = np.broadcast_to(c2d0[:, :, None], shape=M0.shape)
J = np.broadcast_to(c2d1[:, None, :], shape=M0.shape)
gdof0 = self.cspace.number_of_global_dofs()
gdof1 = self.dspace.number_of_global_dofs()
M0 = csr_matrix((M0.flat, (I.flat, J.flat)), shape=(gdof0, gdof1))
M1 = csr_matrix((M1.flat, (I.flat, J.flat)), shape=(gdof0, gdof1))
return M0, M1
def solve_one_step(self):
dt = self.timeline.current_time_step_length()
M0, M1 = self.get_force_matrix()
one = np.ones(M0.shape[1])
F0 = spsolve(self.MV, M0 @ one)
F1 = spsolve(self.MV, M1 @ one)
self.cv[:, 0] = self.v[:, 0] - dt / 2 * F0
self.cv[:, 1] = self.v[:, 1] - dt / 2 * F1
# 网格节点自由度
# dof == 0: 表示固定点
# dof == 1: 表示边界上的点
# dof == 2: 区域内部点
dof = self.mesh0.nodedata['dof']
# 边界条件处理
self.cv[dof == 0] = 0.0
en = self.mesh0.meshdata['bd_normal']
vv = self.cv[dof == 1]
l = np.sum(vv * en, axis=-1) # (NE, )
self.cv[dof == 1] -= l[:, None] * en
F = spsolve(self.ME, self.cv[:, 0] @ M0 + self.cv[:, 1] @ M1)
self.ce[:] = self.e + dt / 2 * F
self.cx[:] = self.x + dt / 2 * self.cv
self.mesh1.node[:] = self.cx
M0, M1 = self.get_force_matrix()
F0 = spsolve(self.MV, M0 @ one)
F1 = spsolve(self.MV, M1 @ one)
self.cv[:, 0] = self.v[:, 0] - dt * F0
self.cv[:, 1] = self.v[:, 1] - dt * F1
# 边界条件处理
self.cv[dof == 0] = 0.0
vv = self.cv[dof == 1]
l = np.sum(vv * en, axis=-1) # (NE, )
self.cv[dof == 1] -= l[:, None] * en
v = (self.cv + self.v) / 2
F = spsolve(self.ME, v[:, 0] @ M0 + v[:, 1] @ M1)
self.ce[:] = self.e + dt * F
self.cx[:] = self.x + dt * v
self.x[:] = self.cx
self.v[:] = self.cv
self.e[:] = self.ce
self.mesh1.node[:] = self.x
self.mesh1.nodedata['velocity'] = self.v
def solve(self, step=1):
"""
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.mesh1.to_vtk(fname=fname)
while not timeline.stop():
self.solve_one_step()
timeline.current += 1
if timeline.current % step == 0:
n = timeline.current
fname = 'test_' + str(n).zfill(10) + '.vtu'
self.mesh1.to_vtk(fname=fname)
timeline.reset()