Lee’s Frame (Continuum Elements)
Contents
Lee’s Frame (Continuum Elements)#
The problem provided in this example is a L frame with pinned-pinned boundary conditions subjected to a point load at the center on top of the frame. The frame is modeled using continuum elements with a compressible neo-Hookean material constitutive law.
1from dolfin import *
2import numpy as np
3import matplotlib.pyplot as plt
4import os
5from arc_length.force_control_solver import force_control # import force control formulation of arc-length solver
6
7
8parameters["form_compiler"]["cpp_optimize"] = True
9ffc_options = {"optimize": True, \
10 "eliminate_zeros": True, \
11 "precompute_basis_const": True, \
12 "precompute_ip_const": True}
Import Mesh and define function space#
1# Create mesh and define function space
2h = 10
3L = 1200
4l1 = L / 5
5E = 7200
6nu = 0
7
8mesh = Mesh('mesh/lees-frame-2d.xml')
9V = VectorFunctionSpace(mesh, "Lagrange", 1)
10
11# Define functions for the arc length method
12du = TrialFunction(V) # Incremental displacement
13v = TestFunction(V) # Test function
14u = Function(V) # Solution
15
16plot(mesh)
[<matplotlib.lines.Line2D at 0x7f8498e9a2c0>,
<matplotlib.lines.Line2D at 0x7f8498e9a590>]
Define Dirichlet Boundary Conditions#
1def Hinge1(x, on_boundary):
2 return near(x[0], 2*h, 1e-6) and near(x[1], 0.0, 1e-6)
3
4def Hinge2(x, on_boundary):
5 return near(x[0], L+h, 1e-6) and near(x[1], L-h, 1e-6)
6
7def force(x,on_boundary):
8 return between(x[0],[l1-l1/25,l1+l1/25]) #and near(x[1],L-h,1e-6)
9
10facets = MeshFunction("size_t", mesh, mesh.topology().dim()-1)
11facets.set_all(0)
12AutoSubDomain(force).mark(facets,1)
13
14bc1 = DirichletBC(V, Constant((0.0, 0.0)), Hinge1, method='pointwise')
15bc2 = DirichletBC(V, Constant((0.0, 0.0)), Hinge2, method='pointwise')
16bcs = [bc1, bc2]
Define Function for Point Load#
Note that this is an approximation of a point load since the FEniCS UserExpression will be projected into a discontinuous space. To approach a point load the area near the point load will need to have fine mesh. In our case it is not neccessary since the point load is just a perturbation.
1# Function for Point Load
2class PointLoad(UserExpression):
3 def __init__(self, x0, f, tol,**kwargs):
4 super().__init__(**kwargs)
5 self.x0 = x0
6 self.f = f
7 self.tol = tol
8 def eval(self, values, x):
9 if near (x[0], self.x0[0],self.tol) and near(x[1], self.x0[1],self.tol):
10 values[0] = self.f[0]
11 values[1] = self.f[1]
12 else:
13 values[0] = 0
14 values[1] = 0
15 def value_shape(self):
16 return (2,)
Kinematics and Weak Form#
1# Kinematics
2I = Identity(mesh.topology().dim()) # Identity tensor
3F = I + grad(u) # Deformation gradient
4C = F.T*F # Right Cauchy-Green tensor
5
6# Invariants of deformation tensors
7Ic = tr(C)
8J = det(F)
9
10# Elasticity parameters
11mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))
12
13# Stored strain energy density (compressible neo-Hookean model)
14psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2
1# Applied Traction and Body Force
2load = Expression("-t", t=0, degree = 0)
3T = Constant((0,1)) # Traction
4B = Constant((0,0)) # Body Force
5P = PointLoad(x0 = [l1,L+h], f = [0,1], tol = 1e-4, degree = 5, element=V.ufl_element()) # Point load
6
7# define sub domains
8ds = Measure('ds', domain=mesh, subdomain_data=facets)
9
10# Solve Variational Problem - Arc Length Method
11F_int = derivative(psi*dx, u, v)
12F_ext = derivative(dot(B, u)*dx + load*dot(P,u)*ds , u, v)
13residual = F_int-F_ext
14J = derivative(residual, u, du)
15
16# Plot the point load function
17plt.figure(figsize = (7,7))
18plot(project(P,V), mode = 'displacement', edgecolor = 'k', cmap = 'Reds', linewidth = 0.1)
<matplotlib.collections.PolyCollection at 0x7f8490cf00d0>
Solver#
To use our solver we first have to define the type of solver (i.e. displacement control or force control) and solver parameters before using the solver. Note that the correct type of solver has to first be imported (see first cell).
Solver parameters#
Here the parameters for both types of solvers:
psi: the scalar arc-length parameter. When psi = 1, the method becomes the spherical arc-length method and when psi = 0 the method becomes the cylindrical arc-length method
abs_tol(optional) : absolute residual tolerance for the linear solver (default value: 1e-10)
rel_tol(optional) : relative residual tolerance for solver; the relative residual is defined as the ration between the current residual and initial residual (default value: DOLFIN_EPS)
lmbda0: the initial load parameter
max_iter: maximum number of iterations for the linear solver
solver(optional): type of linear solver for the FEniCS linear solve function – default FEniCS linear solver is used if no argument is used.
Aside from these solver parameters, the arguments need to solve the FEA problem must also be passed into the solver:
u: the solution function
F_int: First variation of strain energy (internal nodal forces)
F_ext: Externally applied load (external applied force)
J: The Jacobian of the residual with respect to the deformation (tangential stiffness matrix)
displacement_factor: The incremental displacement factor
The solver can be called by:
solver = force_control(psi,abs_tol,rel_tol,lmbda0,max_iter,u,F_int,F_ext,bcs,J,displacement_factor,solver)
Using the solver#
Initialize the solver by calling solver.initialize()
Iteratively call solver.solve() until desired stopping condition
1# Solver Parameters
2psi = 1.0
3abs_tol = 1.0e-6
4lmbda0 = 4.0
5max_iter = 30
6
7# Set up arc-length solver
8solver = force_control(psi=psi, abs_tol=abs_tol, lmbda0=lmbda0, max_iter=max_iter, u=u,
9 F_int=F_int, F_ext=F_ext, bcs=bcs, J=J, load_factor=load)
1disp = [u.vector()[:]]
2lmbda = [0]
3
4for ii in range(0,38):
5 solver.solve()
6 if solver.converged:
7 disp.append(u.vector()[:])
8 lmbda.append(load.t)
Initializing solver parameters...
Starting initial Force Control with Newton Method:
Iteration 0: |
Absolute Residual: 2.8284e+01| Relative Residual: 1.0000e+00
Iteration 1: |
Absolute Residual: 1.2595e+04| Relative Residual: 4.4530e+02
Iteration 2: |
Absolute Residual: 1.3625e+02| Relative Residual: 4.8172e+00
Iteration 3: |
Absolute Residual: 4.5732e+03| Relative Residual: 1.6169e+02
Iteration 4: |
Absolute Residual: 1.5026e+01| Relative Residual: 5.3124e-01
Iteration 5: |
Absolute Residual: 3.6557e+02| Relative Residual: 1.2925e+01
Iteration 6: |
Absolute Residual: 1.0960e-01| Relative Residual: 3.8749e-03
Iteration 7: |
Absolute Residual: 1.1267e-01| Relative Residual: 3.9836e-03
Iteration 8: |
Absolute Residual: 8.9173e-09| Relative Residual: 3.1528e-10
Arc-Length Step 1 :
Iteration: 0
|Total Norm: 8.9295e-09 |Residual: 8.9173e-09 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Arc-Length Step 2 :
Iteration: 0
|Total Norm: 4.6279e+04 |Residual: 4.6279e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.8857e+05 |Residual: 2.5306e+03 |A: 3.8856e+05| Relative Norm : 8.3962e+00
Iteration: 2
|Total Norm: 3.3018e+05 |Residual: 4.5160e+03 |A: 3.3015e+05| Relative Norm : 7.1344e+00
Iteration: 3
|Total Norm: 1.6699e+04 |Residual: 6.2596e+02 |A: 1.6688e+04| Relative Norm : 3.6084e-01
Iteration: 4
|Total Norm: 1.9658e+05 |Residual: 2.8620e+03 |A: 1.9656e+05| Relative Norm : 4.2478e+00
Iteration: 5
|Total Norm: 6.9296e+03 |Residual: 1.1737e+02 |A: 6.9286e+03| Relative Norm : 1.4973e-01
Iteration: 6
|Total Norm: 2.7110e+04 |Residual: 3.0103e+02 |A: 2.7108e+04| Relative Norm : 5.8579e-01
Iteration: 7
|Total Norm: 2.3607e+02 |Residual: 2.9648e+00 |A: 2.3606e+02| Relative Norm : 5.1011e-03
Iteration: 8
|Total Norm: 9.1752e+01 |Residual: 9.5373e-01 |A: 9.1747e+01| Relative Norm : 1.9826e-03
Iteration: 9
|Total Norm: 3.6673e-03 |Residual: 4.8428e-05 |A: 3.6669e-03| Relative Norm : 7.9242e-08
Iteration: 10
|Total Norm: 2.7570e-08 |Residual: 2.2956e-09 |A: 2.7474e-08| Relative Norm : 5.9572e-13
Arc-Length Step 3 :
Iteration: 0
|Total Norm: 3.8695e+04 |Residual: 3.8695e+04 |A: 2.7474e-08| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.1289e+05 |Residual: 1.2919e+03 |A: 2.1288e+05| Relative Norm : 5.5018e+00
Iteration: 2
|Total Norm: 4.4531e+04 |Residual: 6.9012e+02 |A: 4.4526e+04| Relative Norm : 1.1508e+00
Iteration: 3
|Total Norm: 4.6822e+03 |Residual: 1.2545e+02 |A: 4.6805e+03| Relative Norm : 1.2100e-01
Iteration: 4
|Total Norm: 7.2786e+03 |Residual: 8.4133e+01 |A: 7.2782e+03| Relative Norm : 1.8810e-01
Iteration: 5
|Total Norm: 4.6482e+02 |Residual: 6.5173e+00 |A: 4.6477e+02| Relative Norm : 1.2012e-02
Iteration: 6
|Total Norm: 4.5428e+01 |Residual: 5.2681e-01 |A: 4.5425e+01| Relative Norm : 1.1740e-03
Iteration: 7
|Total Norm: 2.3996e-02 |Residual: 2.9607e-04 |A: 2.3994e-02| Relative Norm : 6.2013e-07
Iteration: 8
|Total Norm: 8.9482e-08 |Residual: 3.6537e-09 |A: 8.9407e-08| Relative Norm : 2.3125e-12
Arc-Length Step 4 :
Iteration: 0
|Total Norm: 3.1291e+04 |Residual: 3.1291e+04 |A: 8.9407e-08| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.2123e+05 |Residual: 8.8470e+02 |A: 1.2123e+05| Relative Norm : 3.8743e+00
Iteration: 2
|Total Norm: 6.4124e+03 |Residual: 1.0960e+02 |A: 6.4115e+03| Relative Norm : 2.0493e-01
Iteration: 3
|Total Norm: 3.0028e+03 |Residual: 5.5386e+01 |A: 3.0023e+03| Relative Norm : 9.5961e-02
Iteration: 4
|Total Norm: 1.2052e+02 |Residual: 1.5574e+00 |A: 1.2051e+02| Relative Norm : 3.8516e-03
Iteration: 5
|Total Norm: 5.2743e+00 |Residual: 6.1432e-02 |A: 5.2740e+00| Relative Norm : 1.6855e-04
Iteration: 6
|Total Norm: 2.1733e-04 |Residual: 2.4746e-06 |A: 2.1731e-04| Relative Norm : 6.9453e-09
Iteration: 7
|Total Norm: 4.2888e-09 |Residual: 4.2888e-09 |A: 0.0000e+00| Relative Norm : 1.3706e-13
Arc-Length Step 5 :
Iteration: 0
|Total Norm: 2.6358e+04 |Residual: 2.6358e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 8.8896e+04 |Residual: 7.2061e+02 |A: 8.8893e+04| Relative Norm : 3.3726e+00
Iteration: 2
|Total Norm: 6.0449e+03 |Residual: 1.2585e+02 |A: 6.0435e+03| Relative Norm : 2.2934e-01
Iteration: 3
|Total Norm: 1.8833e+03 |Residual: 3.2735e+01 |A: 1.8830e+03| Relative Norm : 7.1450e-02
Iteration: 4
|Total Norm: 4.3943e+02 |Residual: 4.7284e+00 |A: 4.3941e+02| Relative Norm : 1.6672e-02
Iteration: 5
|Total Norm: 8.0747e+00 |Residual: 8.6750e-02 |A: 8.0743e+00| Relative Norm : 3.0635e-04
Iteration: 6
|Total Norm: 3.6395e-03 |Residual: 3.7576e-05 |A: 3.6393e-03| Relative Norm : 1.3808e-07
Iteration: 7
|Total Norm: 4.8955e-09 |Residual: 4.8733e-09 |A: -4.6566e-10| Relative Norm : 1.8573e-13
Arc-Length Step 6 :
Iteration: 0
|Total Norm: 2.3328e+04 |Residual: 2.3328e+04 |A: -9.3132e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 9.3070e+04 |Residual: 6.6958e+02 |A: 9.3068e+04| Relative Norm : 3.9897e+00
Iteration: 2
|Total Norm: 4.0511e+03 |Residual: 9.3121e+01 |A: 4.0501e+03| Relative Norm : 1.7366e-01
Iteration: 3
|Total Norm: 1.0910e+03 |Residual: 2.0626e+01 |A: 1.0908e+03| Relative Norm : 4.6769e-02
Iteration: 4
|Total Norm: 6.3259e+01 |Residual: 6.3480e-01 |A: 6.3256e+01| Relative Norm : 2.7118e-03
Iteration: 5
|Total Norm: 4.8411e-01 |Residual: 5.0766e-03 |A: 4.8408e-01| Relative Norm : 2.0753e-05
Iteration: 6
|Total Norm: 4.4585e-06 |Residual: 4.4010e-08 |A: 4.4582e-06| Relative Norm : 1.9112e-10
Iteration: 7
|Total Norm: 5.5275e-09 |Residual: 5.4485e-09 |A: -9.3132e-10| Relative Norm : 2.3695e-13
Arc-Length Step 7 :
Iteration: 0
|Total Norm: 2.0633e+04 |Residual: 2.0633e+04 |A: -9.3132e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.7032e+05 |Residual: 2.0131e+03 |A: 1.7031e+05| Relative Norm : 8.2547e+00
Iteration: 2
|Total Norm: 1.5018e+04 |Residual: 2.1754e+02 |A: 1.5016e+04| Relative Norm : 7.2785e-01
Iteration: 3
|Total Norm: 2.7075e+04 |Residual: 3.2949e+02 |A: 2.7073e+04| Relative Norm : 1.3122e+00
Iteration: 4
|Total Norm: 1.3125e+02 |Residual: 9.0033e-01 |A: 1.3125e+02| Relative Norm : 6.3614e-03
Iteration: 5
|Total Norm: 5.3725e+01 |Residual: 4.7797e-01 |A: 5.3723e+01| Relative Norm : 2.6039e-03
Iteration: 6
|Total Norm: 3.9028e-04 |Residual: 1.9599e-06 |A: 3.9027e-04| Relative Norm : 1.8915e-08
Iteration: 7
|Total Norm: 7.8770e-09 |Residual: 7.8633e-09 |A: -4.6566e-10| Relative Norm : 3.8177e-13
Arc-Length Step 8 :
Iteration: 0
|Total Norm: 1.7928e+04 |Residual: 1.7928e+04 |A: -4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.5112e+04 |Residual: 3.3668e+02 |A: 3.5111e+04| Relative Norm : 1.9586e+00
Iteration: 2
|Total Norm: 4.7559e+03 |Residual: 7.6024e+01 |A: 4.7553e+03| Relative Norm : 2.6528e-01
Iteration: 3
|Total Norm: 1.4531e+03 |Residual: 1.1969e+01 |A: 1.4530e+03| Relative Norm : 8.1052e-02
Iteration: 4
|Total Norm: 2.0507e+01 |Residual: 1.8418e-01 |A: 2.0506e+01| Relative Norm : 1.1439e-03
Iteration: 5
|Total Norm: 2.5959e-02 |Residual: 2.1549e-04 |A: 2.5958e-02| Relative Norm : 1.4480e-06
Iteration: 6
|Total Norm: 1.1075e-08 |Residual: 8.5946e-09 |A: 6.9849e-09| Relative Norm : 6.1777e-13
Arc-Length Step 9 :
Iteration: 0
|Total Norm: 1.6219e+04 |Residual: 1.6219e+04 |A: 6.5193e-09| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.9509e+04 |Residual: 3.6790e+02 |A: 1.9505e+04| Relative Norm : 1.2028e+00
Iteration: 2
|Total Norm: 9.8209e+03 |Residual: 1.3267e+02 |A: 9.8200e+03| Relative Norm : 6.0551e-01
Iteration: 3
|Total Norm: 3.3106e+03 |Residual: 3.3224e+01 |A: 3.3104e+03| Relative Norm : 2.0412e-01
Iteration: 4
|Total Norm: 2.2870e+02 |Residual: 2.1682e+00 |A: 2.2869e+02| Relative Norm : 1.4100e-02
Iteration: 5
|Total Norm: 1.7736e+00 |Residual: 1.5767e-02 |A: 1.7735e+00| Relative Norm : 1.0935e-04
Iteration: 6
|Total Norm: 5.5722e-05 |Residual: 4.8990e-07 |A: 5.5720e-05| Relative Norm : 3.4356e-09
Iteration: 7
|Total Norm: 8.5548e-09 |Residual: 8.5421e-09 |A: 4.6566e-10| Relative Norm : 5.2745e-13
Arc-Length Step 10 :
Iteration: 0
|Total Norm: 1.5839e+04 |Residual: 1.5839e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.2634e+04 |Residual: 4.1333e+02 |A: 2.2630e+04| Relative Norm : 1.4290e+00
Iteration: 2
|Total Norm: 1.6271e+04 |Residual: 1.9949e+02 |A: 1.6270e+04| Relative Norm : 1.0272e+00
Iteration: 3
|Total Norm: 1.1131e+03 |Residual: 1.6208e+01 |A: 1.1130e+03| Relative Norm : 7.0273e-02
Iteration: 4
|Total Norm: 1.3356e+02 |Residual: 1.4270e+00 |A: 1.3355e+02| Relative Norm : 8.4320e-03
Iteration: 5
|Total Norm: 1.2732e-01 |Residual: 1.2897e-03 |A: 1.2731e-01| Relative Norm : 8.0382e-06
Iteration: 6
|Total Norm: 1.2150e-06 |Residual: 1.5430e-08 |A: 1.2149e-06| Relative Norm : 7.6708e-11
Iteration: 7
|Total Norm: 1.0172e-08 |Residual: 1.0162e-08 |A: -4.6566e-10| Relative Norm : 6.4223e-13
Arc-Length Step 11 :
Iteration: 0
|Total Norm: 1.6273e+04 |Residual: 1.6273e+04 |A: -4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.7116e+04 |Residual: 4.7038e+02 |A: 2.7112e+04| Relative Norm : 1.6663e+00
Iteration: 2
|Total Norm: 4.3349e+04 |Residual: 4.6506e+02 |A: 4.3346e+04| Relative Norm : 2.6639e+00
Iteration: 3
|Total Norm: 1.1793e+03 |Residual: 2.8243e+01 |A: 1.1789e+03| Relative Norm : 7.2469e-02
Iteration: 4
|Total Norm: 5.9373e+01 |Residual: 9.7497e-01 |A: 5.9365e+01| Relative Norm : 3.6486e-03
Iteration: 5
|Total Norm: 4.2291e-01 |Residual: 4.6703e-03 |A: 4.2288e-01| Relative Norm : 2.5989e-05
Iteration: 6
|Total Norm: 5.5849e-07 |Residual: 1.3396e-08 |A: 5.5833e-07| Relative Norm : 3.4320e-11
Arc-Length Step 12 :
Iteration: 0
|Total Norm: 1.7185e+04 |Residual: 1.7185e+04 |A: 5.5879e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.5182e+04 |Residual: 6.2641e+02 |A: 3.5177e+04| Relative Norm : 2.0472e+00
Iteration: 2
|Total Norm: 9.5370e+05 |Residual: 8.9702e+03 |A: 9.5366e+05| Relative Norm : 5.5495e+01
Iteration: 3
|Total Norm: nan |Residual: nan |A: 2.9767e+09| Relative Norm : nan
Arc-Length Step 12 :
Iteration: 0
|Total Norm: 6.5161e+03 |Residual: 6.5161e+03 |A: 1.3970e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.7284e+03 |Residual: 4.5992e+01 |A: 3.7281e+03| Relative Norm : 5.7218e-01
Iteration: 2
|Total Norm: 4.3400e+03 |Residual: 4.6610e+01 |A: 4.3398e+03| Relative Norm : 6.6605e-01
Iteration: 3
|Total Norm: 1.8961e+01 |Residual: 1.5799e-01 |A: 1.8961e+01| Relative Norm : 2.9100e-03
Iteration: 4
|Total Norm: 3.6750e-02 |Residual: 3.8852e-04 |A: 3.6748e-02| Relative Norm : 5.6399e-06
Iteration: 5
|Total Norm: 1.2820e-08 |Residual: 1.2074e-08 |A: 4.3074e-09| Relative Norm : 1.9674e-12
Arc-Length Step 13 :
Iteration: 0
|Total Norm: 4.6268e+03 |Residual: 4.6268e+03 |A: 4.3074e-09| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.8116e+03 |Residual: 2.0941e+01 |A: 1.8115e+03| Relative Norm : 3.9155e-01
Iteration: 2
|Total Norm: 2.4525e+03 |Residual: 2.5966e+01 |A: 2.4524e+03| Relative Norm : 5.3006e-01
Iteration: 3
|Total Norm: 5.7014e+00 |Residual: 4.4027e-02 |A: 5.7013e+00| Relative Norm : 1.2323e-03
Iteration: 4
|Total Norm: 3.1425e-03 |Residual: 3.5973e-05 |A: 3.1422e-03| Relative Norm : 6.7918e-07
Iteration: 5
|Total Norm: 1.1578e-08 |Residual: 1.1576e-08 |A: 2.3283e-10| Relative Norm : 2.5024e-12
Arc-Length Step 14 :
Iteration: 0
|Total Norm: 1.4510e+04 |Residual: 1.4510e+04 |A: 9.3132e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.9439e+04 |Residual: 4.7535e+02 |A: 2.9435e+04| Relative Norm : 2.0289e+00
Iteration: 2
|Total Norm: 5.0892e+07 |Residual: 3.5587e+05 |A: 5.0891e+07| Relative Norm : 3.5075e+03
Iteration: 3
|Total Norm: 4.5917e+07 |Residual: 1.9459e+05 |A: 4.5917e+07| Relative Norm : 3.1646e+03
Iteration: 4
|Total Norm: 1.9823e+07 |Residual: 2.8255e+05 |A: 1.9821e+07| Relative Norm : 1.3662e+03
Iteration: 5
|Total Norm: 1.2212e+08 |Residual: 3.3796e+05 |A: 1.2212e+08| Relative Norm : 8.4164e+03
Iteration: 6
|Total Norm: nan |Residual: nan |A: 1.7034e+08| Relative Norm : nan
Arc-Length Step 14 :
Iteration: 0
|Total Norm: 4.8884e+03 |Residual: 4.8884e+03 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.1889e+03 |Residual: 2.6376e+01 |A: 2.1888e+03| Relative Norm : 4.4778e-01
Iteration: 2
|Total Norm: 4.3286e+03 |Residual: 4.6887e+01 |A: 4.3284e+03| Relative Norm : 8.8550e-01
Iteration: 3
|Total Norm: 2.1227e+01 |Residual: 1.7320e-01 |A: 2.1227e+01| Relative Norm : 4.3424e-03
Iteration: 4
|Total Norm: 2.0632e-02 |Residual: 2.7602e-04 |A: 2.0631e-02| Relative Norm : 4.2207e-06
Iteration: 5
|Total Norm: 1.3508e-08 |Residual: 1.3394e-08 |A: 1.7462e-09| Relative Norm : 2.7632e-12
Arc-Length Step 15 :
Iteration: 0
|Total Norm: 5.2545e+03 |Residual: 5.2545e+03 |A: 1.5134e-09| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.9512e+03 |Residual: 3.6518e+01 |A: 2.9510e+03| Relative Norm : 5.6166e-01
Iteration: 2
|Total Norm: 8.3161e+03 |Residual: 9.1486e+01 |A: 8.3156e+03| Relative Norm : 1.5827e+00
Iteration: 3
|Total Norm: 1.0052e+02 |Residual: 8.8760e-01 |A: 1.0052e+02| Relative Norm : 1.9130e-02
Iteration: 4
|Total Norm: 1.7992e-01 |Residual: 3.1852e-03 |A: 1.7989e-01| Relative Norm : 3.4241e-05
Iteration: 5
|Total Norm: 1.3123e-07 |Residual: 1.2657e-08 |A: 1.3062e-07| Relative Norm : 2.4975e-11
Arc-Length Step 16 :
Iteration: 0
|Total Norm: 1.7133e+04 |Residual: 1.7133e+04 |A: 5.2201e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 6.2450e+04 |Residual: 8.8482e+02 |A: 6.2444e+04| Relative Norm : 3.6450e+00
Iteration: 2
|Total Norm: 8.1605e+04 |Residual: 8.8383e+02 |A: 8.1601e+04| Relative Norm : 4.7631e+00
Iteration: 3
|Total Norm: 9.3255e+04 |Residual: 1.0126e+03 |A: 9.3250e+04| Relative Norm : 5.4430e+00
Iteration: 4
|Total Norm: 1.3381e+05 |Residual: 1.3315e+03 |A: 1.3380e+05| Relative Norm : 7.8098e+00
Iteration: 5
|Total Norm: 1.7512e+07 |Residual: 1.6210e+05 |A: 1.7511e+07| Relative Norm : 1.0221e+03
Iteration: 6
|Total Norm: 4.8909e+07 |Residual: 5.4697e+04 |A: 4.8909e+07| Relative Norm : 2.8547e+03
Iteration: 7
|Total Norm: 1.9413e+07 |Residual: 2.0253e+05 |A: 1.9412e+07| Relative Norm : 1.1331e+03
Iteration: 8
|Total Norm: 1.3814e+09 |Residual: 3.7140e+06 |A: 1.3814e+09| Relative Norm : 8.0629e+04
Iteration: 9
|Total Norm: 8.1738e+08 |Residual: 2.2081e+06 |A: 8.1738e+08| Relative Norm : 4.7708e+04
Iteration: 10
|Total Norm: nan |Residual: nan |A: 2.3962e+08| Relative Norm : nan
Arc-Length Step 16 :
Iteration: 0
|Total Norm: 5.7875e+03 |Residual: 5.7875e+03 |A: 1.3073e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 4.5741e+03 |Residual: 5.5447e+01 |A: 4.5737e+03| Relative Norm : 7.9034e-01
Iteration: 2
|Total Norm: 1.7537e+04 |Residual: 1.9386e+02 |A: 1.7536e+04| Relative Norm : 3.0301e+00
Iteration: 3
|Total Norm: 6.4707e+02 |Residual: 6.1981e+00 |A: 6.4704e+02| Relative Norm : 1.1180e-01
Iteration: 4
|Total Norm: 3.8358e+00 |Residual: 7.7444e-02 |A: 3.8350e+00| Relative Norm : 6.6277e-04
Iteration: 5
|Total Norm: 1.2665e-04 |Residual: 2.2863e-06 |A: 1.2663e-04| Relative Norm : 2.1884e-08
Iteration: 6
|Total Norm: 1.2848e-08 |Residual: 1.2848e-08 |A: -1.1642e-10| Relative Norm : 2.2200e-12
Arc-Length Step 17 :
Iteration: 0
|Total Norm: 6.5933e+03 |Residual: 6.5933e+03 |A: -1.1642e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 8.3034e+03 |Residual: 9.3315e+01 |A: 8.3029e+03| Relative Norm : 1.2594e+00
Iteration: 2
|Total Norm: 4.5656e+04 |Residual: 5.0220e+02 |A: 4.5653e+04| Relative Norm : 6.9246e+00
Iteration: 3
|Total Norm: 1.0906e+04 |Residual: 1.0844e+02 |A: 1.0905e+04| Relative Norm : 1.6540e+00
Iteration: 4
|Total Norm: 1.4770e+03 |Residual: 1.4947e+01 |A: 1.4769e+03| Relative Norm : 2.2401e-01
Iteration: 5
|Total Norm: 8.9655e+00 |Residual: 1.3307e-01 |A: 8.9645e+00| Relative Norm : 1.3598e-03
Iteration: 6
|Total Norm: 3.1738e-03 |Residual: 2.9769e-05 |A: 3.1737e-03| Relative Norm : 4.8137e-07
Iteration: 7
|Total Norm: 1.2942e-08 |Residual: 1.2941e-08 |A: 1.1642e-10| Relative Norm : 1.9629e-12
Arc-Length Step 18 :
Iteration: 0
|Total Norm: 2.3105e+04 |Residual: 2.3105e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.2265e+05 |Residual: 2.4105e+03 |A: 2.2264e+05| Relative Norm : 9.6364e+00
Iteration: 2
|Total Norm: 2.3044e+04 |Residual: 5.9228e+02 |A: 2.3036e+04| Relative Norm : 9.9735e-01
Iteration: 3
|Total Norm: 3.4929e+04 |Residual: 4.0310e+02 |A: 3.4926e+04| Relative Norm : 1.5117e+00
Iteration: 4
|Total Norm: 1.4648e+05 |Residual: 1.6359e+03 |A: 1.4647e+05| Relative Norm : 6.3396e+00
Iteration: 5
|Total Norm: 1.4033e+07 |Residual: 1.2209e+05 |A: 1.4032e+07| Relative Norm : 6.0733e+02
Iteration: 6
|Total Norm: 3.4112e+07 |Residual: 1.2620e+05 |A: 3.4112e+07| Relative Norm : 1.4764e+03
Iteration: 7
|Total Norm: 1.7729e+07 |Residual: 3.2642e+05 |A: 1.7726e+07| Relative Norm : 7.6734e+02
Iteration: 8
|Total Norm: nan |Residual: nan |A: 1.3379e+10| Relative Norm : nan
Arc-Length Step 18 :
Iteration: 0
|Total Norm: 7.8406e+03 |Residual: 7.8406e+03 |A: 1.1642e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.7713e+04 |Residual: 1.7786e+02 |A: 1.7712e+04| Relative Norm : 2.2592e+00
Iteration: 2
|Total Norm: 2.7663e+05 |Residual: 3.0211e+03 |A: 2.7661e+05| Relative Norm : 3.5281e+01
Iteration: 3
|Total Norm: 3.3224e+04 |Residual: 4.3476e+02 |A: 3.3221e+04| Relative Norm : 4.2375e+00
Iteration: 4
|Total Norm: 3.4438e+05 |Residual: 2.7505e+03 |A: 3.4437e+05| Relative Norm : 4.3923e+01
Iteration: 5
|Total Norm: 8.2087e+04 |Residual: 1.0439e+03 |A: 8.2080e+04| Relative Norm : 1.0469e+01
Iteration: 6
|Total Norm: 8.5872e+03 |Residual: 1.2083e+02 |A: 8.5864e+03| Relative Norm : 1.0952e+00
Iteration: 7
|Total Norm: 1.1322e+05 |Residual: 1.3013e+03 |A: 1.1321e+05| Relative Norm : 1.4440e+01
Iteration: 8
|Total Norm: 4.7764e+04 |Residual: 3.7823e+02 |A: 4.7763e+04| Relative Norm : 6.0919e+00
Iteration: 9
|Total Norm: 9.3625e+05 |Residual: 8.2386e+03 |A: 9.3621e+05| Relative Norm : 1.1941e+02
Iteration: 10
|Total Norm: 7.0541e+05 |Residual: 7.5036e+03 |A: 7.0537e+05| Relative Norm : 8.9969e+01
Iteration: 11
|Total Norm: 1.2695e+05 |Residual: 8.2069e+02 |A: 1.2695e+05| Relative Norm : 1.6192e+01
Iteration: 12
|Total Norm: 1.0583e+06 |Residual: 9.9214e+03 |A: 1.0583e+06| Relative Norm : 1.3498e+02
Iteration: 13
|Total Norm: 2.5486e+05 |Residual: 1.7170e+03 |A: 2.5486e+05| Relative Norm : 3.2506e+01
Iteration: 14
|Total Norm: 3.7569e+04 |Residual: 4.5904e+02 |A: 3.7566e+04| Relative Norm : 4.7916e+00
Iteration: 15
|Total Norm: 3.0911e+04 |Residual: 4.5085e+02 |A: 3.0907e+04| Relative Norm : 3.9424e+00
Iteration: 16
|Total Norm: 3.3119e+04 |Residual: 3.2167e+02 |A: 3.3118e+04| Relative Norm : 4.2241e+00
Iteration: 17
|Total Norm: 2.7000e+03 |Residual: 2.5507e+01 |A: 2.6998e+03| Relative Norm : 3.4436e-01
Iteration: 18
|Total Norm: 8.3665e+02 |Residual: 7.8830e+00 |A: 8.3661e+02| Relative Norm : 1.0671e-01
Iteration: 19
|Total Norm: 4.9741e-01 |Residual: 2.9090e-03 |A: 4.9740e-01| Relative Norm : 6.3440e-05
Iteration: 20
|Total Norm: 1.2324e-04 |Residual: 1.3652e-06 |A: 1.2323e-04| Relative Norm : 1.5718e-08
Iteration: 21
|Total Norm: 1.2550e-08 |Residual: 1.2545e-08 |A: -3.4925e-10| Relative Norm : 1.6006e-12
Arc-Length Step 19 :
Iteration: 0
|Total Norm: 9.7302e+03 |Residual: 9.7302e+03 |A: -1.1642e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 4.1256e+04 |Residual: 3.6961e+02 |A: 4.1254e+04| Relative Norm : 4.2399e+00
Iteration: 2
|Total Norm: 9.1756e+05 |Residual: 1.0655e+04 |A: 9.1749e+05| Relative Norm : 9.4299e+01
Iteration: 3
|Total Norm: 1.8798e+06 |Residual: 2.8710e+04 |A: 1.8796e+06| Relative Norm : 1.9319e+02
Iteration: 4
|Total Norm: 9.0113e+06 |Residual: 7.8267e+04 |A: 9.0109e+06| Relative Norm : 9.2611e+02
Iteration: 5
|Total Norm: 4.7525e+06 |Residual: 6.9160e+04 |A: 4.7520e+06| Relative Norm : 4.8842e+02
Iteration: 6
|Total Norm: 4.2332e+06 |Residual: 1.5435e+05 |A: 4.2304e+06| Relative Norm : 4.3506e+02
Iteration: 7
|Total Norm: 8.0982e+06 |Residual: 2.0830e+05 |A: 8.0955e+06| Relative Norm : 8.3227e+02
Iteration: 8
|Total Norm: 7.8557e+06 |Residual: 2.3198e+05 |A: 7.8523e+06| Relative Norm : 8.0735e+02
Iteration: 9
|Total Norm: 7.7839e+06 |Residual: 1.6025e+05 |A: 7.7822e+06| Relative Norm : 7.9997e+02
Iteration: 10
|Total Norm: 1.0799e+07 |Residual: 8.1717e+04 |A: 1.0799e+07| Relative Norm : 1.1099e+03
Iteration: 11
|Total Norm: 1.0266e+07 |Residual: 1.8161e+05 |A: 1.0265e+07| Relative Norm : 1.0551e+03
Iteration: 12
|Total Norm: nan |Residual: nan |A: 1.2119e+11| Relative Norm : nan
Arc-Length Step 19 :
Iteration: 0
|Total Norm: 3.6745e+03 |Residual: 3.6745e+03 |A: -8.7311e-11| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 4.3968e+03 |Residual: 3.6821e+01 |A: 4.3967e+03| Relative Norm : 1.1966e+00
Iteration: 2
|Total Norm: 7.5690e+03 |Residual: 8.3220e+01 |A: 7.5685e+03| Relative Norm : 2.0599e+00
Iteration: 3
|Total Norm: 2.0725e+02 |Residual: 2.4798e+00 |A: 2.0723e+02| Relative Norm : 5.6401e-02
Iteration: 4
|Total Norm: 1.1846e+00 |Residual: 2.0617e-02 |A: 1.1844e+00| Relative Norm : 3.2238e-04
Iteration: 5
|Total Norm: 4.8261e-05 |Residual: 7.2364e-07 |A: 4.8255e-05| Relative Norm : 1.3134e-08
Iteration: 6
|Total Norm: 1.1992e-08 |Residual: 1.1991e-08 |A: 1.1642e-10| Relative Norm : 3.2636e-12
Arc-Length Step 20 :
Iteration: 0
|Total Norm: 2.9231e+03 |Residual: 2.9231e+03 |A: 1.4552e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.0452e+03 |Residual: 2.2992e+01 |A: 3.0451e+03| Relative Norm : 1.0418e+00
Iteration: 2
|Total Norm: 2.8823e+03 |Residual: 3.0791e+01 |A: 2.8822e+03| Relative Norm : 9.8604e-01
Iteration: 3
|Total Norm: 3.0354e+01 |Residual: 3.9760e-01 |A: 3.0351e+01| Relative Norm : 1.0384e-02
Iteration: 4
|Total Norm: 3.3009e-02 |Residual: 5.3524e-04 |A: 3.3005e-02| Relative Norm : 1.1292e-05
Iteration: 5
|Total Norm: 6.3794e-08 |Residual: 1.2421e-08 |A: 6.2573e-08| Relative Norm : 2.1824e-11
Arc-Length Step 21 :
Iteration: 0
|Total Norm: 9.6917e+03 |Residual: 9.6917e+03 |A: 2.5029e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 4.4145e+04 |Residual: 3.4715e+02 |A: 4.4144e+04| Relative Norm : 4.5549e+00
Iteration: 2
|Total Norm: 1.8273e+06 |Residual: 2.0695e+04 |A: 1.8272e+06| Relative Norm : 1.8854e+02
Iteration: 3
|Total Norm: 1.9482e+06 |Residual: 2.1617e+04 |A: 1.9481e+06| Relative Norm : 2.0102e+02
Iteration: 4
|Total Norm: 6.5545e+06 |Residual: 5.7443e+04 |A: 6.5543e+06| Relative Norm : 6.7630e+02
Iteration: 5
|Total Norm: 1.5769e+07 |Residual: 2.2359e+05 |A: 1.5767e+07| Relative Norm : 1.6270e+03
Iteration: 6
|Total Norm: 1.1710e+07 |Residual: 2.2068e+05 |A: 1.1708e+07| Relative Norm : 1.2083e+03
Iteration: 7
|Total Norm: 6.0905e+06 |Residual: 1.7860e+05 |A: 6.0879e+06| Relative Norm : 6.2843e+02
Iteration: 8
|Total Norm: nan |Residual: nan |A: 1.0641e+09| Relative Norm : nan
Arc-Length Step 21 :
Iteration: 0
|Total Norm: 3.2546e+03 |Residual: 3.2546e+03 |A: 6.2573e-08| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.8821e+03 |Residual: 2.7695e+01 |A: 3.8820e+03| Relative Norm : 1.1928e+00
Iteration: 2
|Total Norm: 2.1860e+03 |Residual: 2.3637e+01 |A: 2.1859e+03| Relative Norm : 6.7167e-01
Iteration: 3
|Total Norm: 2.3875e+01 |Residual: 3.2589e-01 |A: 2.3873e+01| Relative Norm : 7.3358e-03
Iteration: 4
|Total Norm: 1.7689e-02 |Residual: 2.7091e-04 |A: 1.7687e-02| Relative Norm : 5.4353e-06
Iteration: 5
|Total Norm: 2.9300e-08 |Residual: 1.2837e-08 |A: 2.6339e-08| Relative Norm : 9.0029e-12
Arc-Length Step 22 :
Iteration: 0
|Total Norm: 3.5279e+03 |Residual: 3.5279e+03 |A: 2.6281e-08| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 4.1142e+03 |Residual: 2.7079e+01 |A: 4.1141e+03| Relative Norm : 1.1662e+00
Iteration: 2
|Total Norm: 7.7908e+02 |Residual: 9.2653e+00 |A: 7.7903e+02| Relative Norm : 2.2084e-01
Iteration: 3
|Total Norm: 4.9716e+00 |Residual: 6.8680e-02 |A: 4.9711e+00| Relative Norm : 1.4092e-03
Iteration: 4
|Total Norm: 3.7196e-04 |Residual: 5.5751e-06 |A: 3.7192e-04| Relative Norm : 1.0543e-07
Iteration: 5
|Total Norm: 1.3147e-08 |Residual: 1.3147e-08 |A: 2.9104e-11| Relative Norm : 3.7267e-12
Arc-Length Step 23 :
Iteration: 0
|Total Norm: 1.0831e+04 |Residual: 1.0831e+04 |A: -2.3283e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.5994e+04 |Residual: 2.4203e+02 |A: 3.5994e+04| Relative Norm : 3.3232e+00
Iteration: 2
|Total Norm: 6.4908e+03 |Residual: 1.1371e+02 |A: 6.4898e+03| Relative Norm : 5.9926e-01
Iteration: 3
|Total Norm: 9.2306e+02 |Residual: 1.0378e+01 |A: 9.2300e+02| Relative Norm : 8.5221e-02
Iteration: 4
|Total Norm: 2.8341e+02 |Residual: 3.3541e+00 |A: 2.8339e+02| Relative Norm : 2.6166e-02
Iteration: 5
|Total Norm: 8.1679e-01 |Residual: 7.6468e-03 |A: 8.1675e-01| Relative Norm : 7.5409e-05
Iteration: 6
|Total Norm: 2.2184e-04 |Residual: 2.5540e-06 |A: 2.2183e-04| Relative Norm : 2.0481e-08
Iteration: 7
|Total Norm: 1.2876e-08 |Residual: 1.2876e-08 |A: -1.1642e-10| Relative Norm : 1.1888e-12
Arc-Length Step 24 :
Iteration: 0
|Total Norm: 3.8080e+04 |Residual: 3.8080e+04 |A: -9.3132e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.8182e+05 |Residual: 2.7098e+03 |A: 3.8181e+05| Relative Norm : 1.0027e+01
Iteration: 2
|Total Norm: 4.4581e+06 |Residual: 8.1859e+04 |A: 4.4573e+06| Relative Norm : 1.1707e+02
Iteration: 3
|Total Norm: 1.1244e+06 |Residual: 1.4979e+04 |A: 1.1243e+06| Relative Norm : 2.9528e+01
Iteration: 4
|Total Norm: 5.6965e+05 |Residual: 2.2196e+04 |A: 5.6922e+05| Relative Norm : 1.4959e+01
Iteration: 5
|Total Norm: 1.4352e+05 |Residual: 2.5714e+03 |A: 1.4350e+05| Relative Norm : 3.7689e+00
Iteration: 6
|Total Norm: 6.5277e+06 |Residual: 6.6777e+04 |A: 6.5274e+06| Relative Norm : 1.7142e+02
Iteration: 7
|Total Norm: 1.4348e+06 |Residual: 9.0614e+03 |A: 1.4348e+06| Relative Norm : 3.7678e+01
Iteration: 8
|Total Norm: 2.2885e+05 |Residual: 1.6219e+03 |A: 2.2884e+05| Relative Norm : 6.0097e+00
Iteration: 9
|Total Norm: 1.0099e+05 |Residual: 3.0348e+03 |A: 1.0094e+05| Relative Norm : 2.6521e+00
Iteration: 10
|Total Norm: 3.6759e+03 |Residual: 9.6082e+01 |A: 3.6747e+03| Relative Norm : 9.6533e-02
Iteration: 11
|Total Norm: 3.6155e+05 |Residual: 6.0626e+03 |A: 3.6150e+05| Relative Norm : 9.4946e+00
Iteration: 12
|Total Norm: 1.6533e+04 |Residual: 9.8012e+01 |A: 1.6533e+04| Relative Norm : 4.3417e-01
Iteration: 13
|Total Norm: 3.0244e+03 |Residual: 5.3247e+01 |A: 3.0240e+03| Relative Norm : 7.9424e-02
Iteration: 14
|Total Norm: 3.2828e+00 |Residual: 4.3550e-02 |A: 3.2825e+00| Relative Norm : 8.6209e-05
Iteration: 15
|Total Norm: 4.1418e-03 |Residual: 9.7772e-05 |A: 4.1406e-03| Relative Norm : 1.0877e-07
Iteration: 16
|Total Norm: 1.3319e-08 |Residual: 1.3286e-08 |A: 9.3132e-10| Relative Norm : 3.4977e-13
Arc-Length Step 25 :
Iteration: 0
|Total Norm: 3.1426e+04 |Residual: 3.1426e+04 |A: 9.3132e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 2.2799e+05 |Residual: 1.2024e+03 |A: 2.2798e+05| Relative Norm : 7.2547e+00
Iteration: 2
|Total Norm: 3.4383e+05 |Residual: 6.7427e+03 |A: 3.4377e+05| Relative Norm : 1.0941e+01
Iteration: 3
|Total Norm: 1.3346e+04 |Residual: 2.2112e+02 |A: 1.3344e+04| Relative Norm : 4.2468e-01
Iteration: 4
|Total Norm: 6.8367e+04 |Residual: 1.7128e+03 |A: 6.8346e+04| Relative Norm : 2.1755e+00
Iteration: 5
|Total Norm: 1.1967e+03 |Residual: 9.6134e+00 |A: 1.1967e+03| Relative Norm : 3.8079e-02
Iteration: 6
|Total Norm: 4.0802e+03 |Residual: 4.7701e+01 |A: 4.0799e+03| Relative Norm : 1.2983e-01
Iteration: 7
|Total Norm: 1.0422e+01 |Residual: 7.8938e-02 |A: 1.0422e+01| Relative Norm : 3.3164e-04
Iteration: 8
|Total Norm: 4.3546e-01 |Residual: 4.8218e-03 |A: 4.3544e-01| Relative Norm : 1.3857e-05
Iteration: 9
|Total Norm: 1.6305e-07 |Residual: 1.3197e-08 |A: 1.6252e-07| Relative Norm : 5.1884e-12
Arc-Length Step 26 :
Iteration: 0
|Total Norm: 2.3965e+04 |Residual: 2.3965e+04 |A: 1.6252e-07| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.1688e+05 |Residual: 1.1421e+03 |A: 1.1687e+05| Relative Norm : 4.8771e+00
Iteration: 2
|Total Norm: 4.4668e+03 |Residual: 1.9675e+02 |A: 4.4625e+03| Relative Norm : 1.8639e-01
Iteration: 3
|Total Norm: 3.4277e+03 |Residual: 3.6778e+01 |A: 3.4275e+03| Relative Norm : 1.4303e-01
Iteration: 4
|Total Norm: 3.7230e+02 |Residual: 4.2702e+00 |A: 3.7228e+02| Relative Norm : 1.5535e-02
Iteration: 5
|Total Norm: 1.4047e+01 |Residual: 1.4129e-01 |A: 1.4046e+01| Relative Norm : 5.8614e-04
Iteration: 6
|Total Norm: 6.6511e-03 |Residual: 6.6720e-05 |A: 6.6508e-03| Relative Norm : 2.7753e-07
Iteration: 7
|Total Norm: 1.5214e-08 |Residual: 1.4484e-08 |A: 4.6566e-09| Relative Norm : 6.3484e-13
Arc-Length Step 27 :
Iteration: 0
|Total Norm: 2.1730e+04 |Residual: 2.1730e+04 |A: 5.5879e-09| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 8.6210e+04 |Residual: 6.4378e+02 |A: 8.6208e+04| Relative Norm : 3.9673e+00
Iteration: 2
|Total Norm: 5.7578e+03 |Residual: 9.2190e+01 |A: 5.7571e+03| Relative Norm : 2.6497e-01
Iteration: 3
|Total Norm: 1.2540e+03 |Residual: 1.3372e+01 |A: 1.2539e+03| Relative Norm : 5.7705e-02
Iteration: 4
|Total Norm: 1.4192e+02 |Residual: 1.3767e+00 |A: 1.4191e+02| Relative Norm : 6.5310e-03
Iteration: 5
|Total Norm: 5.3616e-01 |Residual: 5.0869e-03 |A: 5.3613e-01| Relative Norm : 2.4673e-05
Iteration: 6
|Total Norm: 2.3287e-05 |Residual: 2.2229e-07 |A: 2.3286e-05| Relative Norm : 1.0717e-09
Iteration: 7
|Total Norm: 1.4588e-08 |Residual: 1.4588e-08 |A: 0.0000e+00| Relative Norm : 6.7130e-13
Arc-Length Step 28 :
Iteration: 0
|Total Norm: 1.9967e+04 |Residual: 1.9967e+04 |A: 0.0000e+00| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 7.4364e+04 |Residual: 5.2152e+02 |A: 7.4362e+04| Relative Norm : 3.7243e+00
Iteration: 2
|Total Norm: 1.2985e+04 |Residual: 1.6675e+02 |A: 1.2984e+04| Relative Norm : 6.5031e-01
Iteration: 3
|Total Norm: 1.5391e+02 |Residual: 4.3768e+00 |A: 1.5385e+02| Relative Norm : 7.7082e-03
Iteration: 4
|Total Norm: 6.0569e+00 |Residual: 6.7935e-02 |A: 6.0566e+00| Relative Norm : 3.0334e-04
Iteration: 5
|Total Norm: 4.0643e-04 |Residual: 4.1460e-06 |A: 4.0641e-04| Relative Norm : 2.0355e-08
Iteration: 6
|Total Norm: 1.5499e-08 |Residual: 1.5499e-08 |A: 0.0000e+00| Relative Norm : 7.7622e-13
Arc-Length Step 29 :
Iteration: 0
|Total Norm: 1.9503e+04 |Residual: 1.9503e+04 |A: 0.0000e+00| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 6.7342e+04 |Residual: 4.0182e+02 |A: 6.7341e+04| Relative Norm : 3.4528e+00
Iteration: 2
|Total Norm: 8.8934e+03 |Residual: 1.0985e+02 |A: 8.8927e+03| Relative Norm : 4.5599e-01
Iteration: 3
|Total Norm: 8.6688e+01 |Residual: 5.4285e+00 |A: 8.6518e+01| Relative Norm : 4.4447e-03
Iteration: 4
|Total Norm: 3.5514e-01 |Residual: 1.6536e-02 |A: 3.5476e-01| Relative Norm : 1.8209e-05
Iteration: 5
|Total Norm: 1.3422e-05 |Residual: 7.7838e-07 |A: 1.3399e-05| Relative Norm : 6.8816e-10
Iteration: 6
|Total Norm: 1.5585e-08 |Residual: 1.5585e-08 |A: 0.0000e+00| Relative Norm : 7.9908e-13
Arc-Length Step 30 :
Iteration: 0
|Total Norm: 2.0136e+04 |Residual: 2.0136e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 6.3741e+04 |Residual: 3.7528e+02 |A: 6.3740e+04| Relative Norm : 3.1656e+00
Iteration: 2
|Total Norm: 1.1461e+04 |Residual: 7.1536e+01 |A: 1.1461e+04| Relative Norm : 5.6920e-01
Iteration: 3
|Total Norm: 4.5811e+02 |Residual: 9.3043e+00 |A: 4.5802e+02| Relative Norm : 2.2751e-02
Iteration: 4
|Total Norm: 1.0607e+01 |Residual: 1.2099e-01 |A: 1.0606e+01| Relative Norm : 5.2677e-04
Iteration: 5
|Total Norm: 4.9916e-03 |Residual: 5.2681e-05 |A: 4.9913e-03| Relative Norm : 2.4790e-07
Iteration: 6
|Total Norm: 1.5827e-08 |Residual: 1.5820e-08 |A: 4.6566e-10| Relative Norm : 7.8599e-13
Arc-Length Step 31 :
Iteration: 0
|Total Norm: 2.1848e+04 |Residual: 2.1848e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 1.0711e+05 |Residual: 7.4886e+02 |A: 1.0711e+05| Relative Norm : 4.9028e+00
Iteration: 2
|Total Norm: 6.0557e+04 |Residual: 1.2371e+02 |A: 6.0557e+04| Relative Norm : 2.7718e+00
Iteration: 3
|Total Norm: 5.3200e+03 |Residual: 9.0501e+01 |A: 5.3192e+03| Relative Norm : 2.4350e-01
Iteration: 4
|Total Norm: 1.1583e+02 |Residual: 1.2889e+00 |A: 1.1583e+02| Relative Norm : 5.3019e-03
Iteration: 5
|Total Norm: 4.6912e+00 |Residual: 5.4836e-02 |A: 4.6909e+00| Relative Norm : 2.1472e-04
Iteration: 6
|Total Norm: 7.1440e-05 |Residual: 8.2681e-07 |A: 7.1436e-05| Relative Norm : 3.2699e-09
Iteration: 7
|Total Norm: 1.6554e-08 |Residual: 1.6547e-08 |A: 4.6566e-10| Relative Norm : 7.5770e-13
Arc-Length Step 32 :
Iteration: 0
|Total Norm: 2.6281e+04 |Residual: 2.6281e+04 |A: 4.6566e-10| Relative Norm : 1.0000e+00
Iteration: 1
|Total Norm: 3.7475e+06 |Residual: 1.2820e+04 |A: 3.7475e+06| Relative Norm : 1.4259e+02
Iteration: 2
|Total Norm: 1.0457e+06 |Residual: 2.4465e+03 |A: 1.0457e+06| Relative Norm : 3.9788e+01
Iteration: 3
|Total Norm: 3.3021e+05 |Residual: 3.2819e+02 |A: 3.3021e+05| Relative Norm : 1.2564e+01
Iteration: 4
|Total Norm: 4.1320e+04 |Residual: 6.1620e+02 |A: 4.1315e+04| Relative Norm : 1.5722e+00
Iteration: 5
|Total Norm: 4.4405e+03 |Residual: 5.9815e+01 |A: 4.4401e+03| Relative Norm : 1.6896e-01
Iteration: 6
|Total Norm: 2.2128e+03 |Residual: 2.5250e+01 |A: 2.2126e+03| Relative Norm : 8.4196e-02
Iteration: 7
|Total Norm: 3.2994e+01 |Residual: 3.9106e-01 |A: 3.2991e+01| Relative Norm : 1.2554e-03
Iteration: 8
|Total Norm: 9.5014e-02 |Residual: 1.0361e-03 |A: 9.5008e-02| Relative Norm : 3.6153e-06
Iteration: 9
|Total Norm: 7.7840e-08 |Residual: 1.9191e-08 |A: 7.5437e-08| Relative Norm : 2.9618e-12
Post Processing#
Here we plot the final deformed shape and the equilibrium path.
1# Plot final deformation
2plt.figure(figsize=(7,7))
3plot(u, mode = 'displacement', cmap = 'Reds', edgecolor= 'k', linewidth = 0.1)
<matplotlib.collections.PolyCollection at 0x7f8490b54430>
1# get force dof
2y_dofs = V.sub(1).dofmap().dofs()
3dof_coords = V.tabulate_dof_coordinates().reshape((-1, 2))
4
5eps = 1e-5
6for kk in y_dofs:
7 if dof_coords[kk,1] >= L+h:
8 if l1-eps < dof_coords[kk,0] < l1+eps:
9 force_node = kk
10
11force_disp = []
12for ii in range(0,len(disp)):
13 force_disp.append(-disp[ii][force_node])
14
15plt.figure(figsize=(7,7))
16
17plt.plot(force_disp,lmbda, marker = 'o', color = 'k')
18plt.xlabel('Displacement')
19plt.ylabel('Load Factor')
Text(0, 0.5, 'Load Factor')
Optional: Creating an animation from solution snapshots#
1from matplotlib import animation, rc
2
3plt.rcParams["animation.html"] = "jshtml"
4
5u_plot = Function(V)
6
7fig = plt.figure(figsize=(7,7))
8ax = fig.gca()
9
10ax.set_xlim([-L/5,L+L/5])
11ax.set_ylim([-L/5,L+L/5])
12
13def drawframe(n):
14 fig.clf()
15 ax = fig.gca()
16 ax.set_xlim([0,L])
17 ax.set_xlim([-L/5,L+L/3])
18 ax.set_ylim([-L/3,L+L/10])
19 u_plot = Function(V)
20 u_plot.vector()[:] = disp[n][:]
21 p = plot(u_plot, mode = 'displacement', cmap = 'Reds', edgecolor= 'k', linewidth = 0.1)
22 return p,
23# blit=True re-draws only the parts that have changed.
24anim = animation.FuncAnimation(fig, drawframe, frames=len(lmbda), interval=40, blit=True)
25anim
