def runTest(self):
# solve Poisson by standard P1 method
# and P1(dg) interior penalty method.
#
# this will test interior facet assembly and normal
# vectors.
from spfem.mesh import MeshTri
from spfem.asm import AssemblerElement
from spfem.element import ElementTriP1, ElementTriDG
from spfem.utils import direct
import numpy as np
m = MeshTri()
m.refine(3)
e = ElementTriDG(ElementTriP1())
e1 = ElementTriP1()
a = AssemblerElement(m, e)
a1 = AssemblerElement(m, e1)
b1 = AssemblerElement(m, e1, e)
A = a.iasm(lambda du, dv: du[0]*dv[0] + du[1]*dv[1])
M = a.iasm(lambda u, v: u*v)
b = a.iasm(lambda v, x: np.sin(2.0*np.pi*x[0])*v)
B1 = a.fasm(lambda u1,u2,v1,v2,h: 1/h*(u1-u2)*(v1-v2), interior=True)
B2 = a.fasm(lambda u,v,h: 1/h*u*v)
B=B1+B2
C1 = a.fasm(lambda du1,du2,v1,v2,n: 0.5*(du1[0]*n[0]*v1 - du2[0]*n[0]*v2
+ du1[1]*n[1]*v1 - du2[1]*n[1]*v2), interior=True)
C2 = a.fasm(lambda du,v,n: (du[0]*n[0]+du[1]*n[1])*v)
C=C1+C2
D = C+C.T
eps = 1e-3
eta = 1e5
x = direct(A + eta*B - 0*D, b)
K = a1.iasm(lambda du, dv: du[0]*dv[0] + du[1]*dv[1])
f = a1.iasm(lambda v, x: np.sin(2.0*np.pi*x[0])*v)
y = direct(K, f, I=m.interior_nodes())
P = b1.iasm(lambda u,v: u*v)
self.assertAlmostEqual(np.linalg.norm(x-direct(M,P*y)),0,places=4)
def Poisson_triP1():
mesh = MeshTri()
mesh.refine(1)
# mesh.draw()
# mesh.draw_debug()
# mesh.draw_nodes(6)
# mesh.plot(z=[0, 0, 0, 0, 0, 0, 1, 0, 0])
# mesh.plot3(z=[0, 0, 0, 0, 0, 0, 1, 0, 0])
# plt.show()
# print(mesh._neighbors()) # TODO 弄懂邻居
# boundary and interior node sets
D1 = np.nonzero(mesh.p[0, :] == 0)[0] # x坐标为0的点
D2 = np.nonzero(mesh.p[1, :] == 0)[0] # y坐标为0的点
D3 = np.nonzero(mesh.p[0, :] == 1)[0] # x坐标为1的点
D4 = np.nonzero(mesh.p[1, :] == 1)[0] # y坐标为1的点
D = np.union1d(D1, D2)
D = np.union1d(D, D3)
D = np.union1d(D, D4)
I = np.setdiff1d(np.arange(0, mesh.p.shape[1]), D) # 位于内部的点的编号，这里暗示了节点编号必须从0开始，依次递增
bilin = lambda u, v, du, dv, x, h: du[0] * dv[0] + du[1] * dv[1]
lin = lambda v, dv, x, h: 1 * v
a = AssemblerElement(mesh, ElementTriP1()) # 用这个例子好好理解 mesh._build_mappings(), FIXME 重要
# 只在编号为1的单元计算单刚
A_ = a.iasm(bilin, tind=[1])
f_ = a.iasm(lin, tind=[1])
A = a.iasm(bilin)
f = a.iasm(lin)
x = np.zeros(A.shape[0])
I = I
x[I] = spsolve(A[np.ix_(I, I)], f[I])
# assert np.round(np.max(x) - 0.073614737354524146, 8) == 0 # 需要将初始网格加密5次
print(np.max(x))
def contact_mesh(h, nref=0):
points = [
(-1.0, 0.0),
(-1.0, -1.0),
(1.0, -1.0),
(1.0, 0.0), # lower block
(-scale, 0.0),
(-scale, 1.0),
(scale, 1.0),
(scale, 0.0)
] # upper block
facets = [(0, 1), (1, 2), (2, 3), (3, 0), (4, 5), (5, 6), (6, 7), (7, 4)]
g = GeometryMeshPyTriangle(points, facets)
m = g.mesh(h)
if nref > 0:
m.refine(nref)
my = .3333 * np.sum(m.p[1, m.t], axis=0)
ixlower = np.nonzero(my < 0.0)[0]
ixupper = np.nonzero(my > 0.0)[0]
mupper = MeshTri(*m.remove_elements(ixlower))
mlower = MeshTri(*m.remove_elements(ixupper))
return mlower, mupper
def runTest(self):
import numpy as np
from spfem.mesh import MeshTri
from spfem.asm import AssemblerAbstract, AssemblerElement
from spfem.element import AbstractElementTriPp, ElementTriP1
from spfem.utils import direct
m = MeshTri()
m.refine(3)
e = AbstractElementTriPp(1)
e1 = ElementTriP1()
a = AssemblerAbstract(m,e)
b = AssemblerElement(m,e1)
A=a.iasm(lambda du,dv: du[0]*dv[0]+du[1]*dv[1])
f=a.iasm(lambda v: 1.0*v)
x=direct(A,f,I=m.boundary_nodes())
K=a.inorm(lambda u: u[0], {0:x})
self.assertAlmostEqual(np.sqrt(np.sum(K)),b.L2error(x,lambda X:0*X[0]))
def example():
mesh = MeshLine()
mesh.refine(2)
print(mesh.boundary_nodes())
print(mesh.interior_nodes())
# u = np.array([1, 5, 3, 2, 4]) # 在网格节点上的函数值，有顺序
# mesh.plot(u)
# plt.show()
mapping = mesh.mapping() # 对一维没什么内容，重点看二维，三维
tri = MeshTri(initmesh="reftri")
tri.refine(1)
tri.draw()
mapping = tri.mapping() # CONTINUE
pass
def TriP2Test():
"""Test triangular h-refinement.
Also test facet assembly."""
def U(x):
return 1+x[0]-x[0]**2*x[1]**2
def dUdx(x):
return 1-2*x[0]*x[1]**2
def dUdy(x):
return -2*x[0]**2*x[1]
def dudv(du,dv):
return du[0]*dv[0]+du[1]*dv[1]
def uv(u,v):
return u*v
def F(x,y):
return 2*x**2+2*y**2
def fv(v,x):
return F(x[0],x[1])*v
def G(x,y):
return (x==1)*(3-3*y**2)+\
(x==0)*(0)+\
(y==1)*(1+x-3*x**2)+\
(y==0)*(1+x)
def gv(v,x):
return G(x[0],x[1])*v
dexact={}
dexact[0]=dUdx
dexact[1]=dUdy
mesh=MeshTri()
mesh.draw()
mesh.refine(1)
mesh.draw()
hs=np.array([])
H1errs=np.array([])
L2errs=np.array([])
for itr in range(2):
mesh.refine()
a=AssemblerElement(mesh,ElementTriP2())
A=a.iasm(dudv)
f=a.iasm(fv)
B=a.fasm(uv)
g=a.fasm(gv)
u=np.zeros(a.dofnum_u.N)
u=spsolve(A+B,f+g)
hs=np.append(hs,mesh.param())
L2errs=np.append(L2errs,a.L2error(u,U))
H1errs=np.append(H1errs,a.H1error(u,dexact))
# https://docs.scipy.org/doc/numpy-1.15.0/reference/generated/numpy.polyfit.html
pfit=np.polyfit(np.log10(hs),np.log10(H1errs),1)
assert (pfit[0] >= 0.95*2)
