class SurfaceTriangleMesh():
def __init__(self, mesh, surface, p=1, scale=None):
"""
Initial a object of Surface Triangle Mesh.
Parameters
----------
self :
Surface Triangle Mesh Object
mesh :
mesh object, represents a triangulation with flat triangle faces.
surface :
The continuous surface which was represented as a level set
function.
p : int
The degree of the Lagrange space
Returns
-------
See Also
--------
Notes
-----
"""
self.mesh = mesh
self.p = p
self.space = LagrangeFiniteElementSpace(mesh, p)
self.surface = surface
self.ds = mesh.ds
self.scale = scale
if scale is not None:
self.mesh.node *= scale
if scale is None:
self.node, d = self.surface.project(
self.space.interpolation_points())
else:
self.node, d = self.surface.project(
self.space.interpolation_points() / scale)
self.node *= scale
self.meshtype = 'stri'
self.ftype = mesh.ftype
self.itype = mesh.itype
self.nodedata = {}
self.celldata = {}
def vtk_cell_type(self):
return 69
def project(self, p):
if self.scale is None:
return self.surface.project(p)
else:
p, d = self.surface.project(p / self.scale)
return p * self.scale, d * self.scale
def integrator(self, k, etype='cell'):
return TriangleQuadrature(k)
def entity(self, etype=2):
if etype in {'cell', 2}:
return self.ds.cell
elif etype in {'edge', 1}:
return self.ds.edge
elif etype in {'node', 0}:
return self.mesh.node
else:
raise ValueError("`entitytype` is wrong!")
def entity_measure(self, etype=2):
p = self.p
if etype in {'cell', 2}:
return self.area(p + 1)
elif etype in {'edge', 'face', 1}:
return self.mesh.entity_measure('edge')
else:
raise ValueError("`entitytype` is wrong!")
def entity_barycenter(self, etype=2):
p = self.p
return self.mesh.entity_barycenter(etype=etype)
def number_of_nodes(self):
return self.node.shape[0]
def number_of_edges(self):
return self.mesh.ds.NE
def number_of_cells(self):
return self.mesh.ds.NC
def geo_dimension(self):
return self.node.shape[1]
def top_dimension(self):
return 2
def jacobi_matrix(self, bc, index=np.s_[:]):
mesh = self.mesh
cell2dof = self.space.dof.cell2dof
grad = self.space.grad_basis(bc, index=index)
# the tranpose of the jacobi matrix between S_h and K
Jh = mesh.jacobi_matrix(index=index)
# the tranpose of the jacobi matrix between S_p and S_h
Jph = np.einsum('ijm, ...ijk->...imk', self.node[cell2dof[index], :],
grad)
# the transpose of the jacobi matrix between S_p and K
Jp = np.einsum('...ijk, imk->...imj', Jph, Jh)
grad = np.einsum('ijk, ...imk->...imj', Jh, grad)
return Jp, grad
def normal(self, bc, index=None):
Js, _, ps = self.surface_jacobi_matrix(bc, index=index)
n = np.cross(Js[..., 0, :], Js[..., 1, :], axis=-1)
return n, ps
def surface_jacobi_matrix(self, bc, index=None):
Jp, grad = self.jacobi_matrix(bc, index=index)
ps = self.bc_to_point(bc, index=index)
Jsp = self.surface.jacobi_matrix(ps)
Js = np.einsum('...ijk, ...imk->...imj', Jsp, Jp)
return Js, grad, ps
def bc_to_point(self, bc, index=None):
phi = self.space.basis(bc)
cell2dof = self.space.dof.cell2dof
if index is None:
bcp = np.einsum('...ij, ijk->...ik', phi, self.node[cell2dof, :])
else:
bcp = np.einsum('...ij, ijk->...ik', phi,
self.node[cell2dof[index], :])
bcp, _ = self.project(bcp)
return bcp
def area(self, q=3):
integrator = self.integrator(q)
bcs, ws = integrator.quadpts, integrator.weights
Jp, _ = self.jacobi_matrix(bcs)
n = np.cross(Jp[..., 0, :], Jp[..., 1, :], axis=-1)
l = np.sqrt(np.sum(n**2, axis=-1))
a = np.einsum('i, ij->j', ws, l) / 2.0
return a
def add_plot(self,
plot,
nodecolor='w',
edgecolor='k',
cellcolor=[0.5, 0.9, 0.45],
aspect='equal',
linewidths=1,
markersize=50,
showaxis=False,
showcolorbar=False,
cmap='rainbow'):
if isinstance(plot, ModuleType):
fig = plot.figure()
fig.set_facecolor('white')
axes = fig.gca()
else:
axes = plot
return show_mesh_2d(axes,
self.mesh,
nodecolor=nodecolor,
edgecolor=edgecolor,
cellcolor=cellcolor,
aspect=aspect,
linewidths=linewidths,
markersize=markersize,
showaxis=showaxis,
showcolorbar=showcolorbar,
cmap=cmap)
def find_node(self,
axes,
node=None,
index=None,
showindex=False,
color='r',
markersize=100,
fontsize=24,
fontcolor='k'):
if node is None:
node = self.node
if (index is None) and (showindex is True):
index = np.array(range(node.shape[0]))
find_node(axes,
node,
index=index,
showindex=showindex,
color=color,
markersize=markersize,
fontsize=fontsize,
fontcolor=fontcolor)
def find_edge(self,
axes,
index=None,
showindex=False,
color='g',
markersize=150,
fontsize=24,
fontcolor='k'):
find_entity(axes,
self.mesh,
entity='edge',
index=index,
showindex=showindex,
color=color,
markersize=markersize,
fontsize=fontsize,
fontcolor=fontcolor)
def find_cell(self,
axes,
index=None,
showindex=False,
color='y',
markersize=200,
fontsize=24,
fontcolor='k'):
find_entity(axes,
self.mesh,
entity='cell',
index=index,
showindex=showindex,
color=color,
markersize=markersize,
fontsize=fontsize,
fontcolor=fontcolor)
class HuZhangFiniteElementSpace():
"""
Hu-Zhang Mixed Finite Element Space.
"""
def __init__(self, mesh, p):
self.space = LagrangeFiniteElementSpace(mesh, p) # the scalar space
self.mesh = mesh
self.p = p
self.dof = self.space.dof
self.dim = self.space.GD
self.init_orth_matrices()
self.init_cell_to_dof()
def init_orth_matrices(self):
"""
Initialize the othogonal symetric matrix basis.
"""
mesh = self.mesh
gdim = self.geo_dimension()
NE = mesh.number_of_edges()
if gdim == 2:
idx = np.array([(0, 0), (1, 1), (0, 1)])
self.TE = np.zeros((NE, 3, 3), dtype=np.float)
self.T = np.array([[(1, 0), (0, 0)], [(0, 0), (0, 1)],
[(0, 1), (1, 0)]])
elif gdim == 3:
idx = np.array([(0, 0), (1, 1), (2, 2), (1, 2), (0, 2), (0, 1)])
self.TE = np.zeros((NE, 6, 6), dtype=np.float)
self.T = np.array([[(1, 0, 0), (0, 0, 0), (0, 0, 0)],
[(0, 0, 0), (0, 1, 0), (0, 0, 0)],
[(0, 0, 0), (0, 0, 0), (0, 0, 1)],
[(0, 0, 0), (0, 0, 1), (0, 1, 0)],
[(0, 0, 1), (0, 0, 0), (1, 0, 0)],
[(0, 1, 0), (1, 0, 0), (0, 0, 0)]])
t = mesh.edge_unit_tagent()
_, _, frame = np.linalg.svd(
t[:, np.newaxis, :]) # get the axis frame on the edge by svd
frame[:, 0, :] = t
for i, (j, k) in enumerate(idx):
self.TE[:,
i] = (frame[:, j, idx[:, 0]] * frame[:, k, idx[:, 1]] +
frame[:, j, idx[:, 1]] * frame[:, k, idx[:, 0]]) / 2
self.TE[:, gdim:] *= np.sqrt(2)
if gdim == 3:
NF = mesh.number_of_faces()
self.TF = np.zeros((NF, 6, 6), dtype=np.float)
n = mesh.face_unit_normal()
_, _, frame = np.linalg.svd(
n[:, np.newaxis, :]) # get the axis frame on the edge by svd
frame[:, 0, :] = n
for i, (j, k) in enumerate(idx):
self.TF[:, i] = (
frame[:, j, idx[:, 0]] * frame[:, k, idx[:, 1]] +
frame[:, j, idx[:, 1]] * frame[:, k, idx[:, 0]]) / 2
self.TF[:, gdim:] *= np.sqrt(2)
def __str__(self):
return "Hu-Zhang mixed finite element space!"
def number_of_global_dofs(self):
"""
"""
p = self.p
gdim = self.geo_dimension()
tdim = self.tensor_dimension()
mesh = self.mesh
NC = mesh.number_of_cells()
NN = mesh.number_of_nodes()
gdof = tdim * NN
if p > 1:
edof = p - 1
NE = mesh.number_of_edges()
gdof += (tdim - 1) * edof * NE # 边内部连续自由度的个数
E = mesh.number_of_edges_of_cells() # 单元边的个数
gdof += NC * E * edof # 边内部不连续自由度的个数
if p > 2:
fdof = (p + 1) * (p + 2) // 2 - 3 * p # 面内部自由度的个数
if gdim == 2:
gdof += tdim * fdof * NC
elif gdim == 3:
NF = mesh.number_of_faces()
gdof += 3 * fdof * NF # 面内部连续自由度的个数
F = mesh.number_of_faces_of_cells() # 每个单元面的个数
gdof += 3 * F * fdof * NC # 面内部不连续自由度的个数
if (p > 3) and (gdim == 3):
ldof = self.dof.number_of_local_dofs()
V = mesh.number_of_nodes_of_cells() # 单元顶点的个数
cdof = ldof - E * edof - F * fdof - V
gdof += tdim * cdof * NC
return gdof
def number_of_local_dofs(self):
tdim = self.tensor_dimension()
ldof = self.dof.number_of_local_dofs()
return tdim * ldof
def cell_to_dof(self):
return self.cell2dof
def init_cell_to_dof(self):
"""
构建局部自由度到全局自由度的映射矩阵
Returns
-------
cell2dof : ndarray with shape (NC, ldof*tdim)
NC: 单元个数
ldof: p 次标量空间局部自由度的个数
tdim: 对称张量的维数
"""
mesh = self.mesh
NN = mesh.number_of_nodes()
NE = mesh.number_of_edges()
NC = mesh.number_of_cells()
gdim = self.geo_dimension()
tdim = self.tensor_dimension() # 张量维数
p = self.p
dof = self.dof # 标量空间自由度管理对象
c2d = dof.cell2dof[..., np.newaxis]
ldof = dof.number_of_local_dofs() # ldof : 标量空间单元上自由度个数
cell2dof = np.zeros((NC, ldof, tdim),
dtype=np.int) # 每个标量自由度变成 tdim 个自由度
dofFlags = self.dof_flags_1() # 把不同类型的自由度区分开来
idx, = np.nonzero(dofFlags[0]) # 局部顶点自由度的编号
cell2dof[:, idx, :] = tdim * c2d[:, idx] + np.arange(tdim)
base0 = 0
base1 = 0
idx, = np.nonzero(dofFlags[1]) # 边内部自由度的编号
if len(idx) > 0:
base0 += NN # 这是标量编号的新起点
base1 += tdim * NN # 这是张量自由度编号的新起点
# 0号局部自由度对应的是切向不连续的自由度, 留到后面重新编号
cell2dof[:, idx,
1:] = base1 + (tdim - 1) * (c2d[:, idx] -
base0) + np.arange(tdim - 1)
idx, = np.nonzero(dofFlags[2])
if len(idx) > 0:
edof = p - 1
base0 += edof * NE
base1 += (tdim - 1) * edof * NE
if gdim == 2:
cell2dof[:, idx, :] = base1 + tdim * (c2d[:, idx] -
base0) + np.arange(tdim)
elif gdim == 3:
# 1, 2, 3 号局部自由度对应切向不连续的张量自由度, 留到后面重新编号
# TODO: check it is right
cell2dof[:, idx.reshape(-1, 1),
np.array([0, 4, 5])] = base1 + (tdim - 3) * (
c2d[:, idx] - base0) + np.arange(tdim - 3)
fdof = (p + 1) * (p + 2) // 2 - 3 * p # 边内部自由度
if gdim == 3:
idx, = np.nonzero(dofFlags[3])
if len(idx) > 0:
NF = mesh.number_of_faces()
base0 += fdof * NF
base1 += (tdim - 3) * fdof * NF
cell2dof[:, idx, :] = base1 + tdim * (c2d[:, idx] -
base0) + np.arange(tdim)
cdof = ldof - 4 * fdof - 6 * edof - 4 # 单元内部自由度
else:
cdof = fdof
idx, = np.nonzero(dofFlags[1])
if len(idx) > 0:
base1 += tdim * cdof * NC
cell2dof[:, idx, 0] = base1 + np.arange(NC * len(idx)).reshape(
NC, len(idx))
if gdim == 3:
base1 += NC * len(idx)
idx, = np.nonzero(dofFlags[2])
print(idx)
if len(idx) > 0:
cell2dof[:, idx.reshape(-1, 1),
np.array([1, 2, 3])] = base1 + np.arange(
NC * len(idx) * 3).reshape(NC, len(idx), 3)
self.cell2dof = cell2dof.reshape(NC, -1)
def geo_dimension(self):
return self.dim
def tensor_dimension(self):
dim = self.dim
return dim * (dim - 1) // 2 + dim
def interpolation_points(self):
return self.dof.interpolation_points()
def dof_flags(self):
""" 对标量空间中的自由度进行分类, 分为边内部自由度, 面内部自由度(如果是三维空间的话)及其它自由度
Returns
-------
isOtherDof : ndarray, (ldof,)
除了边内部和面内部自由度的其它自由度
isEdgeDof : ndarray, (ldof, 3) or (ldof, 6)
每个边内部的自由度
isFaceDof : ndarray, (ldof, 4)
每个面内部的自由度
-------
"""
dim = self.geo_dimension()
dof = self.dof
isPointDof = dof.is_on_node_local_dof()
isEdgeDof = dof.is_on_edge_local_dof()
isEdgeDof[isPointDof] = False
isEdgeDof0 = np.sum(isEdgeDof, axis=-1) > 0 #
isOtherDof = (~isEdgeDof0) # 除了边内部自由度之外的其它自由度
# dim = 2: 包括点和面内部自由度
# dim = 3: 包括点, 面内部和体内部自由度
if dim == 2:
return isOtherDof, isEdgeDof
elif dim == 3:
isFaceDof = dof.is_on_face_local_dof()
isFaceDof[isPointDof, :] = False
isFaceDof[isEdgeDof0, :] = False
isFaceDof0 = np.sum(isFaceDof, axis=-1) > 0
isOtherDof = isOtherDof & (~isFaceDof0) # 三维情形下, 从其它自由度中除去面内部自由度
return isOtherDof, isEdgeDof, isFaceDof
else:
raise ValueError('`dim` should be 2 or 3!')
def dof_flags_1(self):
"""
对标量空间中的自由度进行分类, 分为:
点上的自由由度
边内部的自由度
面内部的自由度
体内部的自由度
Returns
-------
"""
gdim = self.geo_dimension() # the geometry space dimension
dof = self.dof
isPointDof = dof.is_on_node_local_dof()
isEdgeDof = dof.is_on_edge_local_dof()
isEdgeDof[isPointDof] = False
isEdgeDof0 = np.sum(isEdgeDof, axis=-1) > 0
if gdim == 2:
return isPointDof, isEdgeDof0, ~(isPointDof | isEdgeDof0)
elif gdim == 3:
isFaceDof = dof.is_on_face_local_dof()
isFaceDof[isPointDof, :] = False
isFaceDof[isEdgeDof0, :] = False
isFaceDof0 = np.sum(isFaceDof, axis=-1) > 0
return isPointDof, isEdgeDof0, isFaceDof0, ~(
isPointDof | isEdgeDof0 | isFaceDof0)
else:
raise ValueError('`dim` should be 2 or 3!')
def basis(self, bc, cellidx=None):
"""
Parameters
----------
bc : ndarray with shape (NQ, dim+1)
bc[i, :] is i-th quad point
cellidx : ndarray
有时我我们只需要计算部分单元上的基函数
Returns
-------
phi : ndarray with shape (NQ, NC, ldof*tdim, 3 or 6)
NQ: 积分点个数
NC: 单元个数
ldof: 标量空间的单元自由度个数
tdim: 对称张量的维数
"""
mesh = self.mesh
gdim = self.geo_dimension()
tdim = self.tensor_dimension()
if cellidx is None:
NC = mesh.number_of_cells()
cell2edge = mesh.ds.cell_to_edge()
else:
NC = len(cellidx)
cell2edge = mesh.ds.cell_to_edge()[cellidx]
phi0 = self.space.basis(bc) # the shape of phi0 is (NQ, ldof)
shape = list(phi0.shape)
shape.insert(-1, NC)
shape += [tdim, tdim]
# The shape of `phi` is (NQ, NC, ldof, tdim, tdim), where
# NQ : the number of quadrature points
# NC : the number of cells
# ldof : the number of dofs in each cell
# tdim : the dimension of symmetric tensor matrix
phi = np.zeros(shape, dtype=np.float)
dofFlag = self.dof_flags()
# the dof on the vertex and the interior of the cell
isOtherDof = dofFlag[0]
idx, = np.nonzero(isOtherDof)
if len(idx) > 0:
phi[..., idx[..., np.newaxis],
range(tdim),
range(tdim)] = phi0[..., np.newaxis, idx, np.newaxis]
isEdgeDof = dofFlag[1]
for i, isDof in enumerate(isEdgeDof.T):
phi[..., isDof, :, :] = np.einsum('...j, imn->...ijmn',
phi0[..., isDof],
self.TE[cell2edge[:, i]])
if gdim == 3:
if cellidx is None:
cell2face = mesh.ds.cell_to_face()
else:
cell2face = mesh.ds.cell_to_face()[cellidx]
isFaceDof = dofFlag[2]
for i, isDof in enumerate(isFaceDof.T):
phi[..., isDof, :, :] = np.einsum('...j, imn->...ijmn',
phi0[..., isDof],
self.TF[cell2face[:, i]])
# The shape of `phi` should be (NQ, NC, ldof*tdim, tdim)?
shape = phi.shape[:-3] + (-1, tdim)
return phi.reshape(shape)
def div_basis(self, bc, cellidx=None):
mesh = self.mesh
gdim = self.geo_dimension()
tdim = self.tensor_dimension()
# the shape of `gphi` is (NQ, NC, ldof, gdim)
gphi = self.space.grad_basis(bc, cellidx=cellidx)
shape = list(gphi.shape)
shape.insert(-1, tdim)
# the shape of `dphi` is (NQ, NC, ldof, tdim, gdim)
dphi = np.zeros(shape, dtype=np.float)
dofFlag = self.dof_flags()
# the dof on the vertex and the interior of the cell
isOtherDof = dofFlag[0]
dphi[..., isOtherDof, :, :] = np.einsum('...ijm, kmn->...ijkn',
gphi[...,
isOtherDof, :], self.T)
if cellidx is None:
cell2edge = mesh.ds.cell_to_edge()
else:
cell2edge = mesh.ds.cell_to_edge()[cellidx]
isEdgeDof = dofFlag[1]
for i, isDof in enumerate(isEdgeDof.T):
VAL = np.einsum('ijk, kmn->ijmn', self.TE[cell2edge[:, i]], self.T)
dphi[..., isDof, :, :] = np.einsum('...ikm, ijmn->...ikjn',
gphi[..., isDof, :], VAL)
if gdim == 3:
if cellidx is None:
cell2face = mesh.ds.cell_to_face()
else:
cell2face = mesh.ds.cell_to_face()[cellidx]
isFaceDof = dofFlag[2]
for i, isDof in enumerate(isFaceDof.T):
VAL = np.einsum('ijk, kmn->ijmn', self.TF[cell2face[:, i]],
self.T)
dphi[..., isDof, :, :] = np.einsum('...ikm, ijmn->...ikjn',
gphi[..., isDof, :], VAL)
# The new shape of `dphi` is `(NQ, NC, ldof*tdim, gdim)`, where
shape = dphi.shape[:-3] + (-1, gdim)
return dphi.reshape(shape)
def value(self, uh, bc, cellidx=None):
phi = self.basis(bc, cellidx=cellidx)
cell2dof = self.cell_to_dof()
tdim = self.tensor_dimension()
if cellidx is None:
uh = uh[cell2dof]
else:
uh = uh[cell2dof[cellidx]]
phi = np.einsum('...jk, kmn->...jmn', phi, self.T)
val = np.einsum('...ijmn, ij->...imn', phi, uh)
return val
def div_value(self, uh, bc, cellidx=None):
dphi = self.div_basis(bc, cellidx=cellidx)
cell2dof = self.cell_to_dof()
tdim = self.tensor_dimension()
if cellidx is None:
uh = uh[cell2dof]
else:
uh = uh[cell2dof[cellidx]]
val = np.einsum('...ijm, ij->...im', dphi, uh)
return val
def interpolation(self, u):
mesh = self.mesh
gdim = self.geo_dimension()
tdim = self.tensor_dimension()
if gdim == 2:
idx = np.array([(0, 0), (1, 1), (0, 1)])
elif gdim == 3:
idx = np.array([(0, 0), (1, 1), (2, 2), (1, 2), (0, 2), (0, 1)])
ipoint = self.dof.interpolation_points()
c2d = self.dof.cell2dof
val = u(ipoint)[c2d]
ldof = self.dof.number_of_local_dofs()
cell2dof = self.cell2dof.reshape(-1, ldof, tdim)
uI = Function(self)
dofFlag = self.dof_flags()
isOtherDof = dofFlag[0]
idx0, = np.nonzero(isOtherDof)
uI[cell2dof[:, idx0, :]] = val[:, idx0][..., idx[:, 0], idx[:, 1]]
isEdgeDof = dofFlag[1]
cell2edge = self.mesh.ds.cell_to_edge()
for i, isDof in enumerate(isEdgeDof.T):
TE = np.einsum('ijk, kmn->ijmn', self.TE[cell2edge[:, i]], self.T)
uI[cell2dof[:, isDof, :]] = np.einsum('ikmn, ijmn->ikj',
val[:, isDof, :, :], TE)
if gdim == 3:
cell2face = mesh.ds.cell_to_face()
isFaceDof = dofFlag[2]
for i, isDof in enumerate(isFaceDof.T):
TF = np.einsum('ijk, kmn->ijmn', self.TF[cell2face[:, i]],
self.T)
uI[cell2dof[:, isDof, :]] = np.einsum('ikmn, ijmn->ikj',
val[...,
isDof, :, :], TF)
return uI
def function(self, dim=None):
f = Function(self)
return f
def array(self, dim=None):
gdof = self.number_of_global_dofs()
return np.zeros(gdof, dtype=np.float)
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)
from fealpy.mesh import MeshFactory as MF
from fealpy.mesh import TriangleMesh
from fealpy.mesh.core import multi_index_matrix2d
from fealpy.functionspace import LagrangeFiniteElementSpace
mesh = MF.one_triangle_mesh(meshtype='equ')
space = LagrangeFiniteElementSpace(mesh, p=5)
n = 50
bcs = multi_index_matrix2d(n) / n # ldof = (n+1)(n+2)/2
ps = mesh.bc_to_point(bcs).reshape(-1, 2)
val = space.basis(bcs) # (NQ, 1, ldof)
val = space.grad_basis(bcs) # (NQ, NC, ldof, GD)
val = val[:, 0, 0]
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
mesh.find_node(axes, node=ps, markersize=20)
fig = plt.figure()
axes = fig.add_subplot(projection='3d')
axes.plot_trisurf(ps[:, 0],
ps[:, 1],
val,
linewidth=0.2,
antialiased=True,
from fealpy.tools.show import showmultirate, show_error_table
n = 10
p = 1
mf = MeshFactory()
mesh = mf.boxmesh2d([0,1,0,1], nx=n, ny=n,meshtype='tri')
pde = PDE()
space = LagrangeFiniteElementSpace(mesh, p=p)
gdof = space.number_of_global_dofs()
cell2dof = space.cell_to_dof()
cellmeasure = mesh.entity_measure('cell')#计算每个单元的面积
qf = mesh.integrator(1, 'cell')#积分点
bcs, ws = qf.get_quadrature_points_and_weights()
gphi = space.grad_basis(bcs)#(1,2,3,2)一个积分点，2个单元，3个基函数，每个基函数梯度两个分量
a = np.array([(10.0,-1.0),(-1.0,2.0)],dtype=np.float64)
dgphi = np.einsum('mn,ijkn->ijkm',a,gphi)#(2,3,2)
A = np.einsum('i, ijkl, ijml, j->jkm', ws,dgphi, gphi, cellmeasure)#(2,3,3)
I = np.broadcast_to(cell2dof[:,:,None], shape=A.shape)
J = np.broadcast_to(cell2dof[:,None,:],shape=A.shape)
A = csr_matrix((A.flat,(I.flat,J.flat)),shape = (gdof,gdof))#(gdof,gdof)刚度矩阵
# A = space.stiff_matrix(c=a)
# print(A.toarray())
# #质量矩阵组装
# phi = space.basis(bcs)#(1,1,3)一个积分点，每个单元上的重心坐标
@cartesian
def r(p):
x = p[...,0]
y = p[...,1]
NC = mesh.number_of_cells()
print("网格中节点、边和单元的个数分别为：", NN, NE, NC)
print('创建拉格朗日有限元空间...')
space = LagrangeFiniteElementSpace(mesh, p=p)
ldof = space.number_of_local_dofs()
gdof = space.number_of_global_dofs()
print('拉格朗日空间的次数为：', p)
print('每个单元上的局部自由度个数：', ldof)
print('每个单元上的全局自由度个数', gdof)
print('计算空间基函数在每个单元重心坐标点处的值...')
bc = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=np.float64) # (3, 3)
ps = mesh.bc_to_point(bc) # (NQ, NC, 2)
phi = space.basis(bc)
gphi = space.grad_basis(bc)
print('重心坐标数组：', bc)
print('bc.shape:', bc.shape)
print('phi.shape:', phi.shape)
print('gphi.shape:', gphi.shape)
fig = plt.figure()
axes = fig.gca()
mesh.add_plot(axes)
mesh.find_node(axes, showindex=True)
mesh.find_cell(axes, showindex=True)
plt.show()
