BB = np.kron(B.T,np.identity(dpn))
matDT = constitutive(material, dpn)
k += gauss.wgt[i] * j * np.dot(np.dot(BB.T, matDT), BB)
# assemble global K matrix
K[eft[:, np.newaxis], eft] += k
# derive body force vector f
bodyf = grule(p+3)
for i, xi in enumerate(bodyf.xi):
phi, dphi = np.array(shapes(xi,p))
Xxi = np.dot(X.T,[phi[0],phi[1]])
#Xxi = mapping(xi,e)
j = h/2
matDT = constitutive(material, dpn)
f += bodyf.wgt[i] * j * phi * BodyF(matDT, Xxi)
# assemble global body force vector
F[eft] += f
#-- Applying boundary conditions...
zero = bcs[0]
F -= K[:, zero] * bcs[1]
K[:, zero] = 0;
K[zero, :] = 0;
K[zero, zero] = 1
F[zero] = bcs[1]
# apply loads
F[load[0]] += load[1]
# assemble global K matrix
K[eft_0[:, np.newaxis], eft_0] += k_0
K[eft_er[:, np.newaxis], eft_er] += k_er
K[eft_0[:, np.newaxis], eft_er] += k_0_er
K[eft_er[:, np.newaxis], eft_0] += k_er_0
print(K.toarray())
# derive body force vector f
bodyf = grule(p + 3)
for i, xi in enumerate(bodyf.xi):
N_0, dN_0 = fns_l2(xi, X)
x = np.dot(X.T, [N_0[0], N_0[1]])
j = h / 2
matDT = 1
f_0 += bodyf.wgt[i] * j * N_0 * BodyF(matDT, x, 1)
f_er += bodyf.wgt[i] * j * N_er * BodyF(matDT, x, 1)
# assemble global body force vector
F[eft_0] += f_0
F[eft_er] += f_er
print('F:', F)
K_0 = K[0:nnodes, 0:nnodes]
K_er = K[nnodes:, nnodes:]
K_0_er = K[0:nnodes, nnodes:]
K_er_0 = K[nnodes:, 0:nnodes]
print(K_0.toarray())
print(K_er.toarray())
print(K_0_er.toarray())
print(K_er_0.toarray())
Jinv, j = jacobian(X, dN)
B = np.dot(dN, Jinv)
BB = np.kron(B.T, np.identity(dpn))
matDT = constitutive(material, dpn)
k += gauss.wgt[i] * j * np.dot(np.dot(BB.T, matDT), BB)
print('k', k)
# assemble global K matrix
K[eft[:, np.newaxis], eft] += k
bodyforce = quadrature(etype, 4)
for i, xi in enumerate(bodyforce.xi):
N, dN = eval('fns_{}'.format(etype))(xi, X)
Jinv, j = jacobian(X, dN)
matDT = constitutive(material, dpn)
f += bodyforce.wgt[i] * j * N * BodyF(matDT, xi)
# assemble global body force vector
F[eft] += f
print('F', F)
print('K', K.toarray())
print("-- Applying boundary conditions...")
zero = bcs[0] # array of rows/columns which are to be zeroed out
F -= K[:, zero] * bcs[1] # modify right hand side with prescribed values
K[:, zero] = 0
K[zero, :] = 0
# zero-out rows/columns
K[zero, zero] = 1 # add 1 in the diagonal
F[zero] = bcs[1] # prescribed values
Jinv, j = jacobian(X, dN)
B = np.dot(dN, Jinv)
BB = np.kron(B.T, np.identity(dpn))
matDT = constitutive(material, dpn)
k += gauss.wgt[i] * j * np.dot(np.dot(BB.T, matDT), BB)
# assemble global K matrix
K[eft[:, np.newaxis], eft] += k
bodyf = quadrature(etype, 3)
for i, xi in enumerate(bodyf.xi):
N, dN = eval('fns_{}'.format(etype))(xi, X)
Xxi = np.dot(X.T,N) #map xi back to X and then sustitude into the bodyforce!!!!
Jinv, j = jacobian(X, dN)
matDT = constitutive(material, dpn)
f += bodyf.wgt[i] * j * N * BodyF(matDT,Xxi,1)
# assemble global body force vector
F[eft] += f
#print('F', F)
print('K', K.toarray())
print("-- Applying boundary conditions...")
zero = bcs[0] # array of rows/columns which are to be zeroed out
F -= K[:, zero] * bcs[1] # modify right hand side with prescribed values
K[:, zero] = 0;
K[zero, :] = 0; # zero-out rows/columns
K[zero, zero] = 1 # add 1 in the diagonal
F[zero] = bcs[1] # prescribed values
def pFEMsol2(p, case):
h = 0.2
nodes = np.array([[0.0], [0.2], [0.4], [0.6], [0.8], [1.0]])
elems = np.array([[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]])
bcs = [[0, 4 * p + 1], [0.0, 0.0]]
load = [[0, 4 * p + 1], [0, 0]]
nnodes, dpn = nodes.shape # node count and dofs per node
dofs = dpn * nnodes + 5 * (p - 1) # total number of dofs
material = tuple([1, 1])
gauss = grule(p + 1)
K = sp.lil_matrix((dofs, dofs))
F = np.zeros(dofs)
#-- Assembling stiffness matrix and force vector...
for e, conn in enumerate(elems):
# coordinate array for the element
X = nodes[conn]
ldofs = dpn * len(conn) + (p - 1)
k = np.zeros((ldofs, ldofs))
f = np.zeros(ldofs)
# element degree of freedom
if p == 1:
eft = np.array([dpn * n + i for n in conn for i in range(dpn)])
if p > 1:
if e == 0:
eft = np.array([dpn * n for n in range(ldofs)])
elif e > 0:
eft_0 = np.array([1 + p * (e - 1), 1 + p * e])
eft_1 = np.array([1 + p * e + (n + 1) for n in range(p - 1)])
eft = np.append(eft_0, eft_1)
# derive element k matrix
for i, xi in enumerate(gauss.xi):
phi, dphi = shapes(xi, p)
j = h / 2
Jinv = 1 / j
B = np.dot(dphi, Jinv)
BB = np.kron(B.T, np.identity(dpn))
matDT = constitutive(material, dpn)
k += gauss.wgt[i] * j * np.dot(np.dot(BB.T, matDT), BB)
# assemble global K matrix
K[eft[:, np.newaxis], eft] += k
# derive body force vector f
bodyf = grule(p + 3)
for i, xi in enumerate(bodyf.xi):
phi, dphi = np.array(shapes(xi, p))
Xxi = np.dot(X.T, [phi[0], phi[1]])
j = h / 2
matDT = constitutive(material, dpn)
f += bodyf.wgt[i] * j * phi * BodyF(matDT, Xxi, 2)
# assemble global body force vector
F[eft] += f
#print('F', F)
#-- Applying boundary conditions...
zero = bcs[0]
F -= K[:, zero] * bcs[1]
K[:, zero] = 0
K[zero, :] = 0
K[zero, zero] = 1
F[zero] = bcs[1]
# apply loads
F[load[0]] += load[1]
#-- Solving system of equations...
u = spsolve(K.tocsr(), F)
#-- Calculating strain energy...
U = 0.5 * np.dot((u.T * K), u)
return dofs, U