def get_costate_current_right_vector(self, sq, nt):
'''
sq: (NE, 1) n 到 N 时间层的累加
nt: 正序的 n-1 个时间层的值, 则逆序所对应的时间层为 NL - nt -1
NL: 时间方向剖分的总的层数
'''
dt = self.timeline.current_time_step_length()
NL = self.timeline.number_of_time_levels()
# f1 = -\sum_{i=1}^n-1\Delta t (q^i, v)
f1 = -sq
# f2 = (\tilde p^n_h - \tilde p^n_d, v)
b = self.uspace.source_vector(
cartesian(lambda x: self.pde.tpd(x, (NL - nt - 1) * dt)))
f2 = self.A @ self.tph[:, NL - nt - 1] - b
# f3 = \Delta t(y^n_h - y^n_d, w) + (z^n_h, w)
bb0 = self.pspace.source_vector(
cartesian(lambda x: self.pde.yd(x, (NL - nt - 1) * dt)))
bb1 = self.M @ self.zh[:, NL - nt - 1]
bb2 = dt * self.M @ self.yh[:, NL - nt - 1] - dt * bb0
f3 = bb1 + bb2
return np.r_[f1, f2, f3]
def __init__(self, pde, mesh, timeline):
self.pde = pde
self.mesh = mesh
self.uspace = RaviartThomasFiniteElementSpace2d(mesh, p=0)
self.pspace = self.uspace.smspace
self.timeline = timeline
NL = timeline.number_of_time_levels()
# state variable
self.yh = self.pspace.function(dim=NL)
self.uh = self.pspace.function(dim=NL)
self.tph = self.uspace.function(dim=NL)
self.ph = self.uspace.function()
f = cartesian(lambda p: pde.y_solution(p, 0))
self.yh[:, 0] = self.pspace.local_projection(f)
# costate variable
self.zh = self.pspace.function(dim=NL)
self.tqh = self.uspace.function(dim=NL)
self.qh = self.uspace.function()
self.A = self.uspace.stiff_matrix() # RT 质量矩阵
self.D = self.uspace.div_matrix() # (p, \div v)
self.M = self.pspace.mass_matrix()
def __init__(self, pde, timeline, n=1):
self.pde = pde
box = pde.domain()
mf = MeshFactory()
self.mesh = mf.boxmesh2d(box, nx=n, ny=n, meshtype='tri')
self.uspace = RaviartThomasFiniteElementSpace2d(self.mesh, p=0)
self.pspace = self.uspace.smspace
self.timeline = timeline
NL = timeline.number_of_time_levels()
# state variable
self.yh = self.pspace.function(dim=NL)
self.uh = self.pspace.function(dim=NL)
self.tph = self.uspace.function(dim=NL)
self.ph = self.uspace.function()
bc = self.mesh.entity_barycenter('cell')
f = cartesian(lambda p: pde.y_solution(p, 0))
self.yh[:, 0] = self.pspace.local_projection(f)
# costate variable
self.zh = self.pspace.function(dim=NL)
self.tqh = self.uspace.function(dim=NL)
self.qh = self.uspace.function()
self.A = self.uspace.stiff_matrix() # RT 质量矩阵
self.D = self.uspace.div_matrix() # (p, \div v)
self.M = self.pspace.mass_matrix()
def L2error(self):
dt = self.timeline.current_time_step_length()
NL = self.timeline.number_of_time_levels()
T = dt * (NL - 1)
pL2error = self.uspace.integralalg.error(cartesian(lambda x: \
self.pde.p_solution(x, T)), barycentric(self.ph))
print('pL2error', pL2error)
tpL2error = self.uspace.integralalg.error(cartesian(lambda x: \
self.pde.tp_solution(x, T)), \
barycentric(self.uspace.function(array=self.tph[:, NL-1])))
print('tpL2error', tpL2error)
uL2error = self.pspace.integralalg.error(
cartesian(lambda x: self.pde.u_solution(x, T)),
cartesian(self.pspace.function(array=self.uh[:, NL - 1])))
print('uerror', uL2error)
yL2error = self.pspace.integralalg.error(
cartesian(lambda x: self.pde.y_solution(x, T)),
cartesian(self.pspace.function(array=self.yh[:, NL - 1])))
print('yerror', yL2error)
qL2error = self.uspace.integralalg.error(cartesian(lambda x: \
self.pde.q_solution(x, T)), barycentric(self.qh))
print('qL2error', qL2error)
tqL2error = self.uspace.integralalg.error(cartesian(lambda x: \
self.pde.tq(x, T, T)), \
barycentric(self.uspace.function(array=self.tqh[:, NL-1])))
print('tqL2error', tqL2error)
zL2error = self.pspace.integralalg.error(
cartesian(lambda x: self.pde.z_solution(x, T)),
cartesian(self.pspace.function(array=self.zh[:, NL - 1])))
print('zerror', zL2error)
return pL2error
def get_state_current_right_vector(self, sp, nt):
'''
sp: (NE, 1), 前 n-1 层的累加
nt: 表示 n-1 时间层,要求的是 n 个时间层的方程
'''
dt = self.timeline.current_time_step_length()
u = self.uh[:, nt + 1]
NE = self.mesh.number_of_edges()
# f1 = -\sum_{i=1}^n-1\Delta t (p^i, v^i)
f1 = -sp
# f2 = 0
f2 = np.zeros(NE, dtype=self.mesh.ftype)
# f3 = \Delta t(f^n + u^n, w_h) + (y^{n-1}, w_h)
b = self.pspace.source_vector(
cartesian(lambda x: self.pde.source(x, (nt + 1) * dt)))
f3 = dt * b + dt * self.M @ u + self.M @ self.yh[:, nt]
return np.r_[f1, f2, f3]
def costate_solve(self):
timeline = self.timeline
dt = timeline.current_time_step_length()
NL = timeline.number_of_time_levels()
NE = self.mesh.number_of_edges()
cellmeasure = self.mesh.entity_measure('cell')
sq = np.zeros(NE, dtype=self.mesh.ftype)
zh1 = self.pspace.function()
timeline.reset()
while not timeline.stop():
QZ = self.costate_one_step_solve(timeline.current, sq)
self.tqh[:, NL - timeline.current - 2] = QZ[:NE]
self.qh[:] = QZ[NE:2 * NE]
self.zh[:, NL - timeline.current - 2] = QZ[2 * NE:]
zh1[:] = self.zh[:, NL - timeline.current - 2]
e = self.pspace.integralalg.cell_integral(cartesian(zh1))
self.uh[:, NL-timeline.current-1] = max(0, np.sum(e)/np.sum(cellmeasure)) \
- self.zh[:, NL - timeline.current-2]
timeline.current += 1
sq = sq + dt * self.A @ self.qh
timeline.reset()
return sq