Lee’s Frame#

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 with 2D geometrically exact beams (i.e. Simo-Reissner Beams). For more information on beams see here.

../../../../_images/lee_frame.png 
 1%matplotlib inline
 2from dolfin import *
 3import numpy as np
 4import matplotlib.pyplot as plt
 5import os
 6from arc_length.force_control_solver import force_control # import force control formulation of arc-length solver
 7# Dealing with ufl legacy
 8try:
 9    from ufl import diag, Jacobian, shape
10except:
11    from ufl_legacy import diag, Jacobian, shape
12
13parameters["form_compiler"]["cpp_optimize"] = True
14parameters["form_compiler"]["quadrature_degree"] = 1
15parameters['reorder_dofs_serial'] = False
16
17
18ffc_options = {"optimize": True, \
19               "eliminate_zeros": True, \
20               "precompute_basis_const": True, \
21               "precompute_ip_const": True}

Import Mesh and define function spaces#

In the case of 2D beams we also define the rotation matrix about the \(z\) axis and directional derivative with respect to the beam centerline.

 1mesh = Mesh()
 2with XDMFFile('mesh/lee_frame.xdmf') as infile:
 3    infile.read(mesh)
 4    
 5
 6Ue = VectorElement("CG", mesh.ufl_cell(), 1, dim=2) # displacement
 7Te = FiniteElement("CG", mesh.ufl_cell(), 1) # rotation
 8V = FunctionSpace(mesh, MixedElement([Ue, Te]))   
 9
10v_ = TestFunction(V)
11u_, theta_ = split(v_)
12dv = TrialFunction(V)
13v = Function(V, name="Generalized displacement")
14u, theta = split(v)
15
16VR = TensorFunctionSpace(mesh, "DG", 0, shape=(2, 2))
17
18V0 = FunctionSpace(mesh, "DG", 0)
19
20
21Vu = V.sub(0).collapse()
22disp = Function(Vu)
23
24Jac = Jacobian(mesh)
25gdim = mesh.geometry().dim()
26Jac = as_vector([Jac[i, 0] for i in range(gdim)])
27g01 = Jac/sqrt(dot(Jac, Jac))
28g02 = as_vector([-g01[1],g01[0]])
29
30r01 = outer(g01,as_vector([1,0]))
31r02 = outer(g02, as_vector([0,1]))
32
33R0 = r01+r02
34
35#-----------------------Define Functions for beams-----------------------------------#
36def tgrad(u): # directional derivative w.r.t. beam centerline
37    return dot(grad(u), g01)
38
39def rotation_matrix(theta): # 2D rotation matrix -- there is no need to do rotation parametrization for 2D beams
40    return as_tensor([[cos(theta), -sin(theta)],[sin(theta), cos(theta)]])
41Rot = rotation_matrix(theta)

Define Dirichlet Boundary Conditions#

1# Beam Geometry
2Lx = 120
3Ly = 120
4Lf = 24

 1def Pinned_Bottom(x, on_boundary):
 2    return near(x[0], 0, 1e-6) and near(x[1], 0, 1e-6)
 3    
 4def Pinned_Top(x, on_boundary):
 5    return near(x[0], Lx, 1e-6) and near(x[1],Ly, 1e-6)
 6
 7def load(x, on_boundary):
 8    return near(x[0],Lf, 1e-6)
 9
10facets = MeshFunction("size_t", mesh, 0)
11facets.set_all(0)
12AutoSubDomain(load).mark(facets,1)
13
14Pinned_bot = DirichletBC(V.sub(0), Constant((0.0, 0.0)), Pinned_Bottom, method='pointwise') # displacement + rotation
15Pinned_top = DirichletBC(V.sub(0), Constant((0.0, 0.0)), Pinned_Top, method='pointwise') # displacement + rotation
16bcs = [Pinned_bot,Pinned_top]

Kinematics and Weak form#

1# 2D beam Kinematics
2defo = dot(R0.T,dot(Rot.T, g01 + tgrad(u)) - g01)
3curv =  tgrad(theta)

 1# Geometrical properties
 2S = 6.0 # cross-sectional area
 3I = 2.0 # Area moment
 4E = 720.0 # Elastic modulus
 5nu = 0.3 # Poisson's Ratio
 6
 7G = E/(2*(1+nu)) # Shear Modulus
 8kappa = 1.0 # Shear correction
 9
10
11
12# Stiffness moduli
13ES = E*S
14GS = G*S
15GS_2 = G*S*kappa
16GS_3 = G*S*kappa
17EI = E*I

 1# Constitutive Equations
 2C_N = diag(as_vector([ES, GS_2]))
 3
 4# Applied Load:
 5F_max = Constant((0.0,-1.0))
 6M_max = Constant(0.0)
 7load = Expression("t", t=0, degree = 0)
 8
 9dS = Measure("dS", domain=mesh, subdomain_data=facets) # Note its dS not ds -- interior facet
10dx = Measure("dx", domain=mesh)
11
12elastic_energy = 0.5 * (dot(defo, dot(C_N, defo)) + (EI*curv**2))*dx
13
14F_int = derivative(elastic_energy, v, v_)
15F_ext = avg(load * (-M_max*theta_ + dot(F_max, u_))) * dS(1) # avg function is used to integrte over disconticuous interior facets
16residual = F_int - F_ext
17tangent_form = derivative(residual, v, dv)

 1# Get dof coordinates:
 2x_dofs = V.sub(0).sub(0).dofmap().dofs()
 3y_dofs = V.sub(0).sub(1).dofmap().dofs()
 4theta_dofs = V.sub(1).dofmap().dofs()
 5dofs = V.tabulate_dof_coordinates()
 6dof_coords = dofs.reshape((-1, 2))
 7
 8eps = 1e-5
 9dof = []
10# Identify y-dof at the center for force application
11for ii in y_dofs:
12    if abs(dof_coords[ii,1]-Ly) <= eps:
13        if abs(dof_coords[ii,0]-Lf) <=eps:
14            force_dof = ii

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)

  • load_factor : The incremental load 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,load_factor,solver)

Using the solver#

  1. Initialize the solver by calling solver.initialize()

  2. Iteratively call solver.solve() until desired stopping condition

 1# Solver Parameters
 2psi = 1.0
 3abs_tol = 1.0e-6
 4lmbda0 = 0.5
 5max_iter = 10
 6solver = 'mumps' # optional
 7
 8# Set up arc-length solver
 9solver = force_control(psi=psi, abs_tol=abs_tol, lmbda0=lmbda0, max_iter=max_iter, u=v,
10                       F_int=F_int, F_ext=F_ext, bcs=bcs, J=tangent_form, load_factor=load, solver=solver)

1disp = [v.vector()[:]]
2lmbda = [0]
3
4for ii in range(47):
5    solver.solve()
6    if solver.converged: # We only want to save the solution step if the solver coverges       
7        disp.append(v.vector()[:])
8        lmbda.append(load.t)

Initializing solver parameters...
Starting initial Force Control with Newton Method:
Iteration 0: | 
Absolute Residual: 5.0000e-01| Relative Residual: 1.0000e+00
Iteration 1: | 
Absolute Residual: 5.4001e+01| Relative Residual: 1.0800e+02
Iteration 2: | 
Absolute Residual: 4.5229e-01| Relative Residual: 9.0457e-01
Iteration 3: | 
Absolute Residual: 1.9289e+00| Relative Residual: 3.8578e+00
Iteration 4: | 
Absolute Residual: 3.9445e-03| Relative Residual: 7.8891e-03
Iteration 5: | 
Absolute Residual: 4.0046e-04| Relative Residual: 8.0091e-04
Iteration 6: | 
Absolute Residual: 1.9783e-10| Relative Residual: 3.9567e-10

Arc-Length Step 1 :
Iteration: 0 
|Total Norm: 1.9783e-10 |Residual: 1.9783e-10 |A: 2.8422e-14| Relative Norm : 1.0000e+00

Arc-Length Step 2 :
Iteration: 0 
|Total Norm: 1.7172e+02 |Residual: 1.7172e+02 |A: 2.8422e-14| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 9.9491e+00 |Residual: 3.4885e+00 |A: 9.3175e+00| Relative Norm : 5.7938e-02
Iteration: 2 
|Total Norm: 3.9705e+00 |Residual: 1.3821e+00 |A: 3.7222e+00| Relative Norm : 2.3122e-02
Iteration: 3 
|Total Norm: 2.1698e-02 |Residual: 9.0788e-03 |A: 1.9707e-02| Relative Norm : 1.2636e-04
Iteration: 4 
|Total Norm: 3.0068e-04 |Residual: 1.2656e-04 |A: 2.7275e-04| Relative Norm : 1.7510e-06
Iteration: 5 
|Total Norm: 2.1397e-09 |Residual: 9.2762e-10 |A: 1.9282e-09| Relative Norm : 1.2460e-11

Arc-Length Step 3 :
Iteration: 0 
|Total Norm: 1.4570e+02 |Residual: 1.4570e+02 |A: 1.9282e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 8.0083e+00 |Residual: 2.4449e+00 |A: 7.6259e+00| Relative Norm : 5.4966e-02
Iteration: 2 
|Total Norm: 1.2841e+00 |Residual: 5.0455e-01 |A: 1.1808e+00| Relative Norm : 8.8137e-03
Iteration: 3 
|Total Norm: 1.3094e-02 |Residual: 4.3318e-03 |A: 1.2356e-02| Relative Norm : 8.9869e-05
Iteration: 4 
|Total Norm: 7.4796e-05 |Residual: 1.5359e-05 |A: 7.3202e-05| Relative Norm : 5.1337e-07
Iteration: 5 
|Total Norm: 5.3913e-11 |Residual: 1.4603e-11 |A: 5.1898e-11| Relative Norm : 3.7004e-13

Arc-Length Step 4 :
Iteration: 0 
|Total Norm: 1.1935e+02 |Residual: 1.1935e+02 |A: 5.1955e-11| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 6.3104e+00 |Residual: 2.8690e+00 |A: 5.6205e+00| Relative Norm : 5.2873e-02
Iteration: 2 
|Total Norm: 3.7789e-01 |Residual: 1.8299e-01 |A: 3.3063e-01| Relative Norm : 3.1663e-03
Iteration: 3 
|Total Norm: 3.7002e-03 |Residual: 9.9497e-04 |A: 3.5639e-03| Relative Norm : 3.1003e-05
Iteration: 4 
|Total Norm: 8.4051e-07 |Residual: 1.7424e-07 |A: 8.2225e-07| Relative Norm : 7.0424e-09

Arc-Length Step 5 :
Iteration: 0 
|Total Norm: 9.7272e+01 |Residual: 9.7272e+01 |A: 8.2225e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 4.7515e+00 |Residual: 2.5372e+00 |A: 4.0174e+00| Relative Norm : 4.8848e-02
Iteration: 2 
|Total Norm: 1.8428e-01 |Residual: 8.6827e-02 |A: 1.6254e-01| Relative Norm : 1.8945e-03
Iteration: 3 
|Total Norm: 3.4694e-04 |Residual: 2.9875e-04 |A: 1.7639e-04| Relative Norm : 3.5667e-06
Iteration: 4 
|Total Norm: 3.4552e-08 |Residual: 1.8096e-08 |A: 2.9435e-08| Relative Norm : 3.5521e-10

Arc-Length Step 6 :
Iteration: 0 
|Total Norm: 8.0188e+01 |Residual: 8.0188e+01 |A: 2.9435e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 3.6319e+00 |Residual: 1.9844e+00 |A: 3.0418e+00| Relative Norm : 4.5292e-02
Iteration: 2 
|Total Norm: 1.8237e-01 |Residual: 8.2015e-02 |A: 1.6289e-01| Relative Norm : 2.2743e-03
Iteration: 3 
|Total Norm: 1.6881e-03 |Residual: 6.0209e-04 |A: 1.5770e-03| Relative Norm : 2.1051e-05
Iteration: 4 
|Total Norm: 8.5257e-07 |Residual: 2.5856e-07 |A: 8.1241e-07| Relative Norm : 1.0632e-08

Arc-Length Step 7 :
Iteration: 0 
|Total Norm: 6.7515e+01 |Residual: 6.7515e+01 |A: 8.1241e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.9540e+00 |Residual: 1.4898e+00 |A: 2.5508e+00| Relative Norm : 4.3753e-02
Iteration: 2 
|Total Norm: 1.9761e-01 |Residual: 9.2417e-02 |A: 1.7467e-01| Relative Norm : 2.9269e-03
Iteration: 3 
|Total Norm: 2.0827e-03 |Residual: 6.1166e-04 |A: 1.9909e-03| Relative Norm : 3.0849e-05
Iteration: 4 
|Total Norm: 1.5844e-06 |Residual: 4.3482e-07 |A: 1.5235e-06| Relative Norm : 2.3467e-08
Iteration: 5 
|Total Norm: 8.3858e-12 |Residual: 8.3828e-12 |A: 2.2737e-13| Relative Norm : 1.2421e-13

Arc-Length Step 8 :
Iteration: 0 
|Total Norm: 5.8122e+01 |Residual: 5.8122e+01 |A: 1.1369e-13| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.6429e+00 |Residual: 1.1033e+00 |A: 2.4016e+00| Relative Norm : 4.5471e-02
Iteration: 2 
|Total Norm: 1.6677e-01 |Residual: 7.7366e-02 |A: 1.4774e-01| Relative Norm : 2.8693e-03
Iteration: 3 
|Total Norm: 1.7292e-03 |Residual: 4.6856e-04 |A: 1.6645e-03| Relative Norm : 2.9751e-05
Iteration: 4 
|Total Norm: 9.2442e-07 |Residual: 2.4134e-07 |A: 8.9236e-07| Relative Norm : 1.5905e-08

Arc-Length Step 9 :
Iteration: 0 
|Total Norm: 5.0857e+01 |Residual: 5.0857e+01 |A: 8.9236e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.6901e+00 |Residual: 8.2523e-01 |A: 2.5604e+00| Relative Norm : 5.2896e-02
Iteration: 2 
|Total Norm: 8.7612e-02 |Residual: 4.0419e-02 |A: 7.7731e-02| Relative Norm : 1.7227e-03
Iteration: 3 
|Total Norm: 8.6740e-04 |Residual: 2.1917e-04 |A: 8.3926e-04| Relative Norm : 1.7056e-05
Iteration: 4 
|Total Norm: 6.5059e-08 |Residual: 1.6095e-08 |A: 6.3037e-08| Relative Norm : 1.2793e-09

Arc-Length Step 10 :
Iteration: 0 
|Total Norm: 4.4807e+01 |Residual: 4.4807e+01 |A: 6.3037e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 3.2031e+00 |Residual: 7.6482e-01 |A: 3.1105e+00| Relative Norm : 7.1488e-02
Iteration: 2 
|Total Norm: 5.1422e-02 |Residual: 3.4089e-02 |A: 3.8499e-02| Relative Norm : 1.1476e-03
Iteration: 3 
|Total Norm: 7.7724e-04 |Residual: 2.0268e-04 |A: 7.5034e-04| Relative Norm : 1.7346e-05
Iteration: 4 
|Total Norm: 3.2559e-09 |Residual: 1.1364e-09 |A: 3.0512e-09| Relative Norm : 7.2666e-11

Arc-Length Step 11 :
Iteration: 0 
|Total Norm: 3.9416e+01 |Residual: 3.9416e+01 |A: 3.0513e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 4.0474e+00 |Residual: 1.0748e+00 |A: 3.9021e+00| Relative Norm : 1.0268e-01
Iteration: 2 
|Total Norm: 1.2316e-01 |Residual: 7.5335e-02 |A: 9.7430e-02| Relative Norm : 3.1245e-03
Iteration: 3 
|Total Norm: 2.9434e-02 |Residual: 7.4632e-03 |A: 2.8472e-02| Relative Norm : 7.4674e-04
Iteration: 4 
|Total Norm: 1.3513e-05 |Residual: 3.7606e-06 |A: 1.2979e-05| Relative Norm : 3.4282e-07
Iteration: 5 
|Total Norm: 1.8388e-10 |Residual: 4.3743e-11 |A: 1.7860e-10| Relative Norm : 4.6651e-12

Arc-Length Step 12 :
Iteration: 0 
|Total Norm: 3.4508e+01 |Residual: 3.4508e+01 |A: 1.7852e-10| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 3.7348e+00 |Residual: 1.1806e+00 |A: 3.5433e+00| Relative Norm : 1.0823e-01
Iteration: 2 
|Total Norm: 1.4186e-01 |Residual: 8.9161e-02 |A: 1.1033e-01| Relative Norm : 4.1109e-03
Iteration: 3 
|Total Norm: 3.5039e-02 |Residual: 8.8288e-03 |A: 3.3908e-02| Relative Norm : 1.0154e-03
Iteration: 4 
|Total Norm: 2.6747e-05 |Residual: 7.1268e-06 |A: 2.5780e-05| Relative Norm : 7.7511e-07
Iteration: 5 
|Total Norm: 5.0711e-10 |Residual: 1.1615e-10 |A: 4.9363e-10| Relative Norm : 1.4696e-11

Arc-Length Step 13 :
Iteration: 0 
|Total Norm: 3.0193e+01 |Residual: 3.0193e+01 |A: 4.9371e-10| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.1007e+00 |Residual: 8.7456e-01 |A: 1.9100e+00| Relative Norm : 6.9574e-02
Iteration: 2 
|Total Norm: 4.2850e-02 |Residual: 2.8108e-02 |A: 3.2343e-02| Relative Norm : 1.4192e-03
Iteration: 3 
|Total Norm: 1.1161e-03 |Residual: 3.2199e-04 |A: 1.0687e-03| Relative Norm : 3.6967e-05
Iteration: 4 
|Total Norm: 3.2761e-08 |Residual: 8.7873e-09 |A: 3.1560e-08| Relative Norm : 1.0850e-09

Arc-Length Step 14 :
Iteration: 0 
|Total Norm: 2.6719e+01 |Residual: 2.6719e+01 |A: 3.1560e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.0998e+00 |Residual: 6.2790e-01 |A: 9.0297e-01| Relative Norm : 4.1163e-02
Iteration: 2 
|Total Norm: 1.0731e-01 |Residual: 3.4330e-02 |A: 1.0167e-01| Relative Norm : 4.0162e-03
Iteration: 3 
|Total Norm: 3.3027e-04 |Residual: 6.5629e-05 |A: 3.2369e-04| Relative Norm : 1.2361e-05
Iteration: 4 
|Total Norm: 1.0264e-08 |Residual: 2.1176e-09 |A: 1.0043e-08| Relative Norm : 3.8416e-10

Arc-Length Step 15 :
Iteration: 0 
|Total Norm: 2.4291e+01 |Residual: 2.4291e+01 |A: 1.0043e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 7.2487e-01 |Residual: 4.6312e-01 |A: 5.5764e-01| Relative Norm : 2.9841e-02
Iteration: 2 
|Total Norm: 2.2705e-01 |Residual: 5.3670e-02 |A: 2.2062e-01| Relative Norm : 9.3470e-03
Iteration: 3 
|Total Norm: 3.8771e-04 |Residual: 8.3976e-05 |A: 3.7851e-04| Relative Norm : 1.5961e-05
Iteration: 4 
|Total Norm: 6.6281e-08 |Residual: 1.4041e-08 |A: 6.4777e-08| Relative Norm : 2.7286e-09

Arc-Length Step 16 :
Iteration: 0 
|Total Norm: 2.2947e+01 |Residual: 2.2947e+01 |A: 6.4777e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.7709e-01 |Residual: 3.4485e-01 |A: 4.6272e-01| Relative Norm : 2.5149e-02
Iteration: 2 
|Total Norm: 2.7838e-01 |Residual: 6.4852e-02 |A: 2.7072e-01| Relative Norm : 1.2131e-02
Iteration: 3 
|Total Norm: 2.9435e-04 |Residual: 6.7775e-05 |A: 2.8644e-04| Relative Norm : 1.2827e-05
Iteration: 4 
|Total Norm: 4.8967e-08 |Residual: 1.0869e-08 |A: 4.7746e-08| Relative Norm : 2.1339e-09

Arc-Length Step 17 :
Iteration: 0 
|Total Norm: 2.2538e+01 |Residual: 2.2538e+01 |A: 4.7746e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.1977e-01 |Residual: 2.6388e-01 |A: 4.4780e-01| Relative Norm : 2.3062e-02
Iteration: 2 
|Total Norm: 2.8430e-01 |Residual: 6.7904e-02 |A: 2.7607e-01| Relative Norm : 1.2614e-02
Iteration: 3 
|Total Norm: 2.1609e-04 |Residual: 5.1260e-05 |A: 2.0993e-04| Relative Norm : 9.5882e-06
Iteration: 4 
|Total Norm: 2.0855e-08 |Residual: 4.9361e-09 |A: 2.0262e-08| Relative Norm : 9.2534e-10

Arc-Length Step 18 :
Iteration: 0 
|Total Norm: 2.2827e+01 |Residual: 2.2827e+01 |A: 2.0262e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.0838e-01 |Residual: 2.1776e-01 |A: 4.5939e-01| Relative Norm : 2.2271e-02
Iteration: 2 
|Total Norm: 2.8299e-01 |Residual: 6.9817e-02 |A: 2.7425e-01| Relative Norm : 1.2397e-02
Iteration: 3 
|Total Norm: 1.6497e-04 |Residual: 3.9151e-05 |A: 1.6026e-04| Relative Norm : 7.2272e-06
Iteration: 4 
|Total Norm: 6.8555e-09 |Residual: 1.8092e-09 |A: 6.6124e-09| Relative Norm : 3.0033e-10

Arc-Length Step 19 :
Iteration: 0 
|Total Norm: 2.3614e+01 |Residual: 2.3614e+01 |A: 6.6125e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.2222e-01 |Residual: 2.0173e-01 |A: 4.8168e-01| Relative Norm : 2.2115e-02
Iteration: 2 
|Total Norm: 2.9978e-01 |Residual: 7.6448e-02 |A: 2.8986e-01| Relative Norm : 1.2695e-02
Iteration: 3 
|Total Norm: 1.4709e-04 |Residual: 3.2847e-05 |A: 1.4337e-04| Relative Norm : 6.2290e-06
Iteration: 4 
|Total Norm: 1.4457e-09 |Residual: 5.3421e-10 |A: 1.3434e-09| Relative Norm : 6.1223e-11

Arc-Length Step 20 :
Iteration: 0 
|Total Norm: 2.4800e+01 |Residual: 2.4800e+01 |A: 1.3433e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.5188e-01 |Residual: 2.0614e-01 |A: 5.1193e-01| Relative Norm : 2.2253e-02
Iteration: 2 
|Total Norm: 3.5770e-01 |Residual: 9.4105e-02 |A: 3.4510e-01| Relative Norm : 1.4423e-02
Iteration: 3 
|Total Norm: 1.7489e-04 |Residual: 3.2944e-05 |A: 1.7176e-04| Relative Norm : 7.0520e-06
Iteration: 4 
|Total Norm: 1.0321e-09 |Residual: 4.9133e-10 |A: 9.0762e-10| Relative Norm : 4.1616e-11

Arc-Length Step 21 :
Iteration: 0 
|Total Norm: 2.6412e+01 |Residual: 2.6412e+01 |A: 9.0745e-10| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 5.9775e-01 |Residual: 2.2273e-01 |A: 5.5471e-01| Relative Norm : 2.2632e-02
Iteration: 2 
|Total Norm: 4.9543e-01 |Residual: 1.3417e-01 |A: 4.7691e-01| Relative Norm : 1.8758e-02
Iteration: 3 
|Total Norm: 3.0268e-04 |Residual: 4.2580e-05 |A: 2.9967e-04| Relative Norm : 1.1460e-05
Iteration: 4 
|Total Norm: 1.9572e-08 |Residual: 5.0251e-09 |A: 1.8916e-08| Relative Norm : 7.4102e-10

Arc-Length Step 22 :
Iteration: 0 
|Total Norm: 2.8615e+01 |Residual: 2.8615e+01 |A: 1.8916e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 6.7385e-01 |Residual: 2.5232e-01 |A: 6.2483e-01| Relative Norm : 2.3548e-02
Iteration: 2 
|Total Norm: 7.9647e-01 |Residual: 2.2191e-01 |A: 7.6493e-01| Relative Norm : 2.7834e-02
Iteration: 3 
|Total Norm: 7.6818e-04 |Residual: 9.4129e-05 |A: 7.6239e-04| Relative Norm : 2.6845e-05
Iteration: 4 
|Total Norm: 2.6787e-07 |Residual: 6.8015e-08 |A: 2.5909e-07| Relative Norm : 9.3610e-09

Arc-Length Step 23 :
Iteration: 0 
|Total Norm: 3.1760e+01 |Residual: 3.1760e+01 |A: 2.5909e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 8.2066e-01 |Residual: 3.0928e-01 |A: 7.6016e-01| Relative Norm : 2.5839e-02
Iteration: 2 
|Total Norm: 1.4467e+00 |Residual: 4.1409e-01 |A: 1.3861e+00| Relative Norm : 4.5549e-02
Iteration: 3 
|Total Norm: 2.7856e-03 |Residual: 5.3614e-04 |A: 2.7335e-03| Relative Norm : 8.7708e-05
Iteration: 4 
|Total Norm: 4.2288e-06 |Residual: 1.1408e-06 |A: 4.0720e-06| Relative Norm : 1.3315e-07
Iteration: 5 
|Total Norm: 1.6777e-11 |Residual: 1.6776e-11 |A: 1.4211e-13| Relative Norm : 5.2823e-13

Arc-Length Step 24 :
Iteration: 0 
|Total Norm: 3.6487e+01 |Residual: 3.6487e+01 |A: 5.4001e-13| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.1364e+00 |Residual: 4.2602e-01 |A: 1.0536e+00| Relative Norm : 3.1146e-02
Iteration: 2 
|Total Norm: 2.8588e+00 |Residual: 8.3538e-01 |A: 2.7340e+00| Relative Norm : 7.8351e-02
Iteration: 3 
|Total Norm: 1.3990e-02 |Residual: 4.0674e-03 |A: 1.3386e-02| Relative Norm : 3.8343e-04
Iteration: 4 
|Total Norm: 7.9985e-05 |Residual: 2.3911e-05 |A: 7.6327e-05| Relative Norm : 2.1921e-06
Iteration: 5 
|Total Norm: 2.3932e-11 |Residual: 1.8756e-11 |A: 1.4865e-11| Relative Norm : 6.5590e-13

Arc-Length Step 25 :
Iteration: 0 
|Total Norm: 4.3946e+01 |Residual: 4.3946e+01 |A: 1.4893e-11| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.8667e+00 |Residual: 6.7816e-01 |A: 1.7391e+00| Relative Norm : 4.2476e-02
Iteration: 2 
|Total Norm: 6.0522e+00 |Residual: 1.7889e+00 |A: 5.7818e+00| Relative Norm : 1.3772e-01
Iteration: 3 
|Total Norm: 9.3134e-02 |Residual: 3.2826e-02 |A: 8.7157e-02| Relative Norm : 2.1193e-03
Iteration: 4 
|Total Norm: 1.7913e-03 |Residual: 6.2033e-04 |A: 1.6804e-03| Relative Norm : 4.0760e-05
Iteration: 5 
|Total Norm: 6.5001e-09 |Residual: 2.6232e-09 |A: 5.9472e-09| Relative Norm : 1.4791e-10

Arc-Length Step 26 :
Iteration: 0 
|Total Norm: 5.6232e+01 |Residual: 5.6232e+01 |A: 5.9472e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 3.7113e+00 |Residual: 1.2862e+00 |A: 3.4813e+00| Relative Norm : 6.6000e-02
Iteration: 2 
|Total Norm: 1.4772e+01 |Residual: 4.4069e+00 |A: 1.4099e+01| Relative Norm : 2.6270e-01
Iteration: 3 
|Total Norm: 8.7608e-01 |Residual: 3.3603e-01 |A: 8.0908e-01| Relative Norm : 1.5580e-02
Iteration: 4 
|Total Norm: 6.1306e-02 |Residual: 2.4407e-02 |A: 5.6238e-02| Relative Norm : 1.0902e-03
Iteration: 5 
|Total Norm: 1.2700e-04 |Residual: 3.7163e-05 |A: 1.2144e-04| Relative Norm : 2.2585e-06
Iteration: 6 
|Total Norm: 6.4767e-10 |Residual: 1.6832e-10 |A: 6.2542e-10| Relative Norm : 1.1518e-11

Arc-Length Step 27 :
Iteration: 0 
|Total Norm: 7.7012e+01 |Residual: 7.7012e+01 |A: 6.2542e-10| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 8.6888e+00 |Residual: 2.8358e+00 |A: 8.2129e+00| Relative Norm : 1.1282e-01
Iteration: 2 
|Total Norm: 5.7627e+01 |Residual: 1.7967e+01 |A: 5.4754e+01| Relative Norm : 7.4829e-01
Iteration: 3 
|Total Norm: 1.1496e+01 |Residual: 5.0569e+00 |A: 1.0324e+01| Relative Norm : 1.4928e-01
Iteration: 4 
|Total Norm: 2.4594e+01 |Residual: 6.2715e+00 |A: 2.3781e+01| Relative Norm : 3.1936e-01
Iteration: 5 
|Total Norm: 2.0221e+01 |Residual: 6.6243e+00 |A: 1.9105e+01| Relative Norm : 2.6257e-01
Iteration: 6 
|Total Norm: 1.9236e+00 |Residual: 1.1017e+00 |A: 1.5768e+00| Relative Norm : 2.4978e-02
Iteration: 7 
|Total Norm: 6.1535e-01 |Residual: 2.2650e-01 |A: 5.7215e-01| Relative Norm : 7.9904e-03
Iteration: 8 
|Total Norm: 1.8305e-02 |Residual: 5.8862e-03 |A: 1.7333e-02| Relative Norm : 2.3769e-04
Iteration: 9 
|Total Norm: 8.6351e-07 |Residual: 3.2907e-07 |A: 7.9835e-07| Relative Norm : 1.1213e-08

Arc-Length Step 28 :
Iteration: 0 
|Total Norm: 1.1043e+02 |Residual: 1.1043e+02 |A: 7.9835e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.9767e+01 |Residual: 6.0153e+00 |A: 1.8830e+01| Relative Norm : 1.7900e-01
Iteration: 2 
|Total Norm: 3.1631e+02 |Residual: 1.2203e+02 |A: 2.9182e+02| Relative Norm : 2.8643e+00
Iteration: 3 
|Total Norm: 3.3971e+02 |Residual: 1.0003e+02 |A: 3.2465e+02| Relative Norm : 3.0762e+00
Iteration: 4 
|Total Norm: 9.4795e+04 |Residual: 7.0406e+04 |A: 6.3475e+04| Relative Norm : 8.5841e+02
Iteration: 5 
|Total Norm: 4.6885e+05 |Residual: 4.5323e+05 |A: 1.2003e+05| Relative Norm : 4.2456e+03
Iteration: 6 
|Total Norm: 3.5039e+05 |Residual: 3.4755e+05 |A: 4.4498e+04| Relative Norm : 3.1729e+03
Iteration: 7 
|Total Norm: 1.2888e+06 |Residual: 1.2749e+06 |A: 1.8867e+05| Relative Norm : 1.1671e+04
Iteration: 8 
|Total Norm: 7.1081e+05 |Residual: 6.9366e+05 |A: 1.5523e+05| Relative Norm : 6.4367e+03
Iteration: 9 
|Total Norm: 4.3856e+06 |Residual: 4.3645e+06 |A: 4.2910e+05| Relative Norm : 3.9713e+04

Arc-Length Step 28 :
Iteration: 0 
|Total Norm: 3.5098e+01 |Residual: 3.5098e+01 |A: 1.9959e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.2498e+00 |Residual: 6.8928e-01 |A: 2.1416e+00| Relative Norm : 6.4100e-02
Iteration: 2 
|Total Norm: 2.3191e+00 |Residual: 6.9806e-01 |A: 2.2115e+00| Relative Norm : 6.6074e-02
Iteration: 3 
|Total Norm: 4.2223e-02 |Residual: 1.6638e-02 |A: 3.8807e-02| Relative Norm : 1.2030e-03
Iteration: 4 
|Total Norm: 9.0311e-04 |Residual: 2.5078e-04 |A: 8.6760e-04| Relative Norm : 2.5731e-05
Iteration: 5 
|Total Norm: 1.9077e-08 |Residual: 7.2195e-09 |A: 1.7658e-08| Relative Norm : 5.4353e-10

Arc-Length Step 29 :
Iteration: 0 
|Total Norm: 2.7338e+01 |Residual: 2.7338e+01 |A: 1.7658e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.3327e+00 |Residual: 3.9577e-01 |A: 1.2725e+00| Relative Norm : 4.8746e-02
Iteration: 2 
|Total Norm: 4.7094e-01 |Residual: 1.5135e-01 |A: 4.4596e-01| Relative Norm : 1.7226e-02
Iteration: 3 
|Total Norm: 1.9027e-03 |Residual: 8.0536e-04 |A: 1.7239e-03| Relative Norm : 6.9599e-05
Iteration: 4 
|Total Norm: 1.4582e-06 |Residual: 3.9491e-07 |A: 1.4037e-06| Relative Norm : 5.3337e-08
Iteration: 5 
|Total Norm: 1.5099e-11 |Residual: 1.5098e-11 |A: 1.1369e-13| Relative Norm : 5.5229e-13

Arc-Length Step 30 :
Iteration: 0 
|Total Norm: 1.0778e+02 |Residual: 1.0778e+02 |A: 8.5265e-13| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.3614e+01 |Residual: 3.8573e+00 |A: 1.3056e+01| Relative Norm : 1.2631e-01
Iteration: 2 
|Total Norm: 1.3680e+00 |Residual: 7.6185e-01 |A: 1.1362e+00| Relative Norm : 1.2692e-02
Iteration: 3 
|Total Norm: 4.6927e-02 |Residual: 2.3621e-02 |A: 4.0549e-02| Relative Norm : 4.3540e-04
Iteration: 4 
|Total Norm: 6.6833e-04 |Residual: 2.3533e-04 |A: 6.2553e-04| Relative Norm : 6.2009e-06
Iteration: 5 
|Total Norm: 7.0171e-09 |Residual: 2.6323e-09 |A: 6.5047e-09| Relative Norm : 6.5106e-11

Arc-Length Step 31 :
Iteration: 0 
|Total Norm: 1.5287e+02 |Residual: 1.5287e+02 |A: 6.5048e-09| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.6986e+01 |Residual: 4.4984e+00 |A: 1.6379e+01| Relative Norm : 1.1111e-01
Iteration: 2 
|Total Norm: 6.5886e+00 |Residual: 2.3800e+00 |A: 6.1437e+00| Relative Norm : 4.3100e-02
Iteration: 3 
|Total Norm: 1.2248e-01 |Residual: 3.8303e-02 |A: 1.1634e-01| Relative Norm : 8.0121e-04
Iteration: 4 
|Total Norm: 1.1423e-01 |Residual: 2.8862e-02 |A: 1.1053e-01| Relative Norm : 7.4727e-04
Iteration: 5 
|Total Norm: 4.8126e-05 |Residual: 2.2376e-05 |A: 4.2608e-05| Relative Norm : 3.1482e-07
Iteration: 6 
|Total Norm: 5.0744e-08 |Residual: 1.2960e-08 |A: 4.9061e-08| Relative Norm : 3.3195e-10

Arc-Length Step 32 :
Iteration: 0 
|Total Norm: 1.0515e+02 |Residual: 1.0515e+02 |A: 4.9061e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 7.8999e+00 |Residual: 2.0631e+00 |A: 7.6258e+00| Relative Norm : 7.5130e-02
Iteration: 2 
|Total Norm: 7.5349e+00 |Residual: 2.2180e+00 |A: 7.2010e+00| Relative Norm : 7.1658e-02
Iteration: 3 
|Total Norm: 1.1834e-01 |Residual: 4.0741e-02 |A: 1.1111e-01| Relative Norm : 1.1254e-03
Iteration: 4 
|Total Norm: 1.7440e-01 |Residual: 4.5086e-02 |A: 1.6848e-01| Relative Norm : 1.6586e-03
Iteration: 5 
|Total Norm: 9.4193e-05 |Residual: 4.2804e-05 |A: 8.3906e-05| Relative Norm : 8.9579e-07
Iteration: 6 
|Total Norm: 1.8443e-07 |Residual: 4.8595e-08 |A: 1.7792e-07| Relative Norm : 1.7540e-09

Arc-Length Step 33 :
Iteration: 0 
|Total Norm: 6.4867e+01 |Residual: 6.4867e+01 |A: 1.7792e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 3.7454e+00 |Residual: 1.2589e+00 |A: 3.5275e+00| Relative Norm : 5.7740e-02
Iteration: 2 
|Total Norm: 2.0078e+00 |Residual: 8.2296e-01 |A: 1.8313e+00| Relative Norm : 3.0952e-02
Iteration: 3 
|Total Norm: 2.8810e-02 |Residual: 9.5637e-03 |A: 2.7176e-02| Relative Norm : 4.4414e-04
Iteration: 4 
|Total Norm: 5.3415e-03 |Residual: 1.4188e-03 |A: 5.1496e-03| Relative Norm : 8.2346e-05
Iteration: 5 
|Total Norm: 2.0724e-07 |Residual: 7.4164e-08 |A: 1.9351e-07| Relative Norm : 3.1949e-09

Arc-Length Step 34 :
Iteration: 0 
|Total Norm: 5.0424e+01 |Residual: 5.0424e+01 |A: 1.9352e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.7954e+00 |Residual: 7.7547e-01 |A: 2.6857e+00| Relative Norm : 5.5439e-02
Iteration: 2 
|Total Norm: 3.2297e-01 |Residual: 2.5084e-01 |A: 2.0344e-01| Relative Norm : 6.4050e-03
Iteration: 3 
|Total Norm: 9.3013e-04 |Residual: 5.0181e-04 |A: 7.8315e-04| Relative Norm : 1.8446e-05
Iteration: 4 
|Total Norm: 3.3345e-06 |Residual: 1.3256e-06 |A: 3.0597e-06| Relative Norm : 6.6130e-08
Iteration: 5 
|Total Norm: 1.8932e-11 |Residual: 1.8931e-11 |A: 8.5265e-14| Relative Norm : 3.7545e-13

Arc-Length Step 35 :
Iteration: 0 
|Total Norm: 4.6135e+01 |Residual: 4.6135e+01 |A: -1.9895e-13| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.5442e+00 |Residual: 5.3314e-01 |A: 2.4877e+00| Relative Norm : 5.5146e-02
Iteration: 2 
|Total Norm: 1.3451e-01 |Residual: 8.6866e-02 |A: 1.0269e-01| Relative Norm : 2.9155e-03
Iteration: 3 
|Total Norm: 1.6000e-03 |Residual: 3.3330e-04 |A: 1.5649e-03| Relative Norm : 3.4681e-05
Iteration: 4 
|Total Norm: 1.3114e-07 |Residual: 2.8454e-08 |A: 1.2801e-07| Relative Norm : 2.8425e-09

Arc-Length Step 36 :
Iteration: 0 
|Total Norm: 4.3147e+01 |Residual: 4.3147e+01 |A: 1.2801e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.2654e+00 |Residual: 4.4314e-01 |A: 2.2216e+00| Relative Norm : 5.2503e-02
Iteration: 2 
|Total Norm: 2.2249e-01 |Residual: 6.5269e-02 |A: 2.1270e-01| Relative Norm : 5.1565e-03
Iteration: 3 
|Total Norm: 2.4201e-03 |Residual: 4.8789e-04 |A: 2.3704e-03| Relative Norm : 5.6090e-05
Iteration: 4 
|Total Norm: 1.2231e-06 |Residual: 2.4926e-07 |A: 1.1974e-06| Relative Norm : 2.8347e-08
Iteration: 5 
|Total Norm: 2.1730e-11 |Residual: 2.1730e-11 |A: 5.6843e-14| Relative Norm : 5.0363e-13

Arc-Length Step 37 :
Iteration: 0 
|Total Norm: 4.0005e+01 |Residual: 4.0005e+01 |A: 1.4211e-13| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.0067e+00 |Residual: 4.2313e-01 |A: 1.9615e+00| Relative Norm : 5.0160e-02
Iteration: 2 
|Total Norm: 2.7051e-01 |Residual: 7.7172e-02 |A: 2.5927e-01| Relative Norm : 6.7619e-03
Iteration: 3 
|Total Norm: 1.6779e-03 |Residual: 3.5865e-04 |A: 1.6391e-03| Relative Norm : 4.1942e-05
Iteration: 4 
|Total Norm: 1.3023e-06 |Residual: 2.6985e-07 |A: 1.2740e-06| Relative Norm : 3.2553e-08
Iteration: 5 
|Total Norm: 1.6854e-11 |Residual: 1.6854e-11 |A: 5.6843e-14| Relative Norm : 4.2130e-13

Arc-Length Step 38 :
Iteration: 0 
|Total Norm: 3.7320e+01 |Residual: 3.7320e+01 |A: 8.5265e-14| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.8443e+00 |Residual: 4.8955e-01 |A: 1.7781e+00| Relative Norm : 4.9419e-02
Iteration: 2 
|Total Norm: 2.5136e-01 |Residual: 7.9652e-02 |A: 2.3841e-01| Relative Norm : 6.7354e-03
Iteration: 3 
|Total Norm: 9.0916e-04 |Residual: 2.0962e-04 |A: 8.8467e-04| Relative Norm : 2.4361e-05
Iteration: 4 
|Total Norm: 6.0277e-07 |Residual: 1.2701e-07 |A: 5.8924e-07| Relative Norm : 1.6152e-08

Arc-Length Step 39 :
Iteration: 0 
|Total Norm: 3.5613e+01 |Residual: 3.5613e+01 |A: 5.8924e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.7504e+00 |Residual: 5.8489e-01 |A: 1.6498e+00| Relative Norm : 4.9151e-02
Iteration: 2 
|Total Norm: 2.0346e-01 |Residual: 7.1220e-02 |A: 1.9059e-01| Relative Norm : 5.7132e-03
Iteration: 3 
|Total Norm: 5.0394e-04 |Residual: 1.2631e-04 |A: 4.8786e-04| Relative Norm : 1.4151e-05
Iteration: 4 
|Total Norm: 2.2691e-07 |Residual: 4.8870e-08 |A: 2.2158e-07| Relative Norm : 6.3715e-09

Arc-Length Step 40 :
Iteration: 0 
|Total Norm: 3.4983e+01 |Residual: 3.4983e+01 |A: 2.2158e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.6980e+00 |Residual: 6.9680e-01 |A: 1.5485e+00| Relative Norm : 4.8538e-02
Iteration: 2 
|Total Norm: 1.5627e-01 |Residual: 6.0959e-02 |A: 1.4389e-01| Relative Norm : 4.4669e-03
Iteration: 3 
|Total Norm: 3.1927e-04 |Residual: 8.6744e-05 |A: 3.0726e-04| Relative Norm : 9.1262e-06
Iteration: 4 
|Total Norm: 9.1883e-08 |Residual: 2.0408e-08 |A: 8.9588e-08| Relative Norm : 2.6265e-09

Arc-Length Step 41 :
Iteration: 0 
|Total Norm: 3.5319e+01 |Residual: 3.5319e+01 |A: 8.9588e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.6815e+00 |Residual: 8.3932e-01 |A: 1.4570e+00| Relative Norm : 4.7609e-02
Iteration: 2 
|Total Norm: 1.2257e-01 |Residual: 5.3570e-02 |A: 1.1025e-01| Relative Norm : 3.4705e-03
Iteration: 3 
|Total Norm: 2.5327e-04 |Residual: 7.3731e-05 |A: 2.4230e-04| Relative Norm : 7.1710e-06
Iteration: 4 
|Total Norm: 4.8776e-08 |Residual: 1.1298e-08 |A: 4.7450e-08| Relative Norm : 1.3810e-09

Arc-Length Step 42 :
Iteration: 0 
|Total Norm: 3.6534e+01 |Residual: 3.6534e+01 |A: 4.7450e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.7215e+00 |Residual: 1.0450e+00 |A: 1.3680e+00| Relative Norm : 4.7120e-02
Iteration: 2 
|Total Norm: 1.0838e-01 |Residual: 5.1387e-02 |A: 9.5424e-02| Relative Norm : 2.9666e-03
Iteration: 3 
|Total Norm: 2.8926e-04 |Residual: 8.8884e-05 |A: 2.7526e-04| Relative Norm : 7.9176e-06
Iteration: 4 
|Total Norm: 4.3875e-08 |Residual: 1.0720e-08 |A: 4.2545e-08| Relative Norm : 1.2009e-09

Arc-Length Step 43 :
Iteration: 0 
|Total Norm: 3.8726e+01 |Residual: 3.8726e+01 |A: 4.2546e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 1.8943e+00 |Residual: 1.3849e+00 |A: 1.2924e+00| Relative Norm : 4.8915e-02
Iteration: 2 
|Total Norm: 1.2622e-01 |Residual: 5.8019e-02 |A: 1.1210e-01| Relative Norm : 3.2593e-03
Iteration: 3 
|Total Norm: 5.8250e-04 |Residual: 1.8682e-04 |A: 5.5173e-04| Relative Norm : 1.5042e-05
Iteration: 4 
|Total Norm: 9.6052e-08 |Residual: 2.5052e-08 |A: 9.2728e-08| Relative Norm : 2.4803e-09

Arc-Length Step 44 :
Iteration: 0 
|Total Norm: 4.2335e+01 |Residual: 4.2335e+01 |A: 9.2728e-08| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.4522e+00 |Residual: 2.0487e+00 |A: 1.3477e+00| Relative Norm : 5.7923e-02
Iteration: 2 
|Total Norm: 2.3920e-01 |Residual: 8.4841e-02 |A: 2.2365e-01| Relative Norm : 5.6501e-03
Iteration: 3 
|Total Norm: 2.6305e-03 |Residual: 8.8207e-04 |A: 2.4782e-03| Relative Norm : 6.2134e-05
Iteration: 4 
|Total Norm: 7.9455e-07 |Residual: 2.2530e-07 |A: 7.6194e-07| Relative Norm : 1.8768e-08

Arc-Length Step 45 :
Iteration: 0 
|Total Norm: 4.8568e+01 |Residual: 4.8568e+01 |A: 7.6194e-07| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 4.5336e+00 |Residual: 3.7242e+00 |A: 2.5854e+00| Relative Norm : 9.3345e-02
Iteration: 2 
|Total Norm: 9.2459e-01 |Residual: 1.7722e-01 |A: 9.0744e-01| Relative Norm : 1.9037e-02
Iteration: 3 
|Total Norm: 3.2711e-02 |Residual: 1.1802e-02 |A: 3.0508e-02| Relative Norm : 6.7350e-04
Iteration: 4 
|Total Norm: 3.5362e-05 |Residual: 1.1494e-05 |A: 3.3441e-05| Relative Norm : 7.2808e-07
Iteration: 5 
|Total Norm: 3.6598e-10 |Residual: 1.0682e-10 |A: 3.5004e-10| Relative Norm : 7.5354e-12

Arc-Length Step 46 :
Iteration: 0 
|Total Norm: 6.1121e+01 |Residual: 6.1121e+01 |A: 3.5013e-10| Relative Norm : 1.0000e+00
Iteration: 1 
|Total Norm: 2.4311e+01 |Residual: 1.0453e+01 |A: 2.1948e+01| Relative Norm : 3.9774e-01
Iteration: 2 
|Total Norm: 1.0069e+01 |Residual: 6.2584e-01 |A: 1.0050e+01| Relative Norm : 1.6475e-01
Iteration: 3 
|Total Norm: 1.0705e+00 |Residual: 4.6269e-01 |A: 9.6537e-01| Relative Norm : 1.7515e-02
Iteration: 4 
|Total Norm: 6.5728e-03 |Residual: 3.0212e-03 |A: 5.8373e-03| Relative Norm : 1.0754e-04
Iteration: 5 
|Total Norm: 4.9659e-05 |Residual: 1.4534e-05 |A: 4.7485e-05| Relative Norm : 8.1247e-07
Iteration: 6 
|Total Norm: 3.1662e-11 |Residual: 2.7925e-11 |A: 1.4921e-11| Relative Norm : 5.1802e-13

Post Processing#

Here we plot the final deformed shape and the equilibrium path. To verify or solution, we also compare our solutions to ones found in literature.

Solutions are obtained from the paper: A simple extrapolated predictor for overcoming the starting and tracking issues in the arc-length method for nonlinear structural mechanics

 1x_nodal_coord = dof_coords[x_dofs][:,0]
 2y_nodal_coord = dof_coords[y_dofs][:,1]
 3# Get nodal values 
 4
 5# Plot displacement field
 6disp_x = x_nodal_coord + disp[-1][x_dofs]
 7disp_y = y_nodal_coord + disp[-1][y_dofs]
 8
 9
10
11plt.scatter(x_nodal_coord,y_nodal_coord)
12plt.scatter(disp_x, disp_y)
13
14plt.xlabel('x-coordinates')
15plt.ylabel('y-coordinates')
16plt.axis('equal')
17plt.show()
18
19print(x_nodal_coord.shape)
../../../../_images/2d2954adea2e88e870bba1d6486fcc08a878377ff053a9706b1d596668a90ced.png 
(21,)

1# get paper solution
2import scipy.io
3paper_soln = scipy.io.loadmat('lit_soln/solution_lee.mat')

 1plt.figure(figsize=(7,7))
 2plt.scatter(paper_soln['soln'][:,0],paper_soln['soln'][:,1], label = 'Paper implementation', facecolors = 'None', edgecolors = 'r', marker = 's')
 3plt.scatter(disp_x, disp_y, marker = '.', label = 'Our implementation')
 4plt.scatter(x_nodal_coord,y_nodal_coord,label = 'Undeformed', color = 'k')
 5
 6plt.xlabel('x-coordinates')
 7plt.ylabel('y-coordinates')
 8plt.title('Beam Deformation')
 9#plt.axis('equal')
10plt.legend()
11plt.show()
../../../../_images/5122e4251487ec59aad404e323df3a5651477ed9355223daf71075dfc6123786.png 
1# get paper equilibrium path
2paper_eq = np.loadtxt('lit_soln/path_lee.dat')
3
4# Get displacment at force application
5force_disp = []
6for ii in range(len(disp)):
7    force_disp.append(-disp[ii][force_dof])
8    

1plt.figure(figsize=(7,7))
2plt.scatter(-paper_eq[:,2], paper_eq[:,0], label = 'Paper implementation', facecolors = 'None', edgecolors = 'r', marker = 's')
3
4plt.scatter(force_disp, lmbda, marker = '.', label = 'Our Implementation')
5plt.title('Equilibrium Path')
6plt.ylabel('Normalized Load')
7plt.xlabel('Normalized Displacement')
8plt.legend()

<matplotlib.legend.Legend at 0x7fb9aade94b0>
../../../../_images/db610c2fc4d59df39cf4f1b663b7a1fb8d168537c6678b8335d590f1114590e7.png

Optional: Creating an animation from solution snapshots#

 1from matplotlib import animation, rc
 2
 3plt.rcParams["animation.html"] = "jshtml"
 4
 5fig = plt.figure(figsize=(7,7))
 6ax = fig.add_subplot(111)
 7
 8ax.set_xlim([-25,150])
 9ax.set_ylim([-25,150])
10
11deformed, = ax.plot([], [], lw = 7, c = 'r', label = 'Deformed Configuration', ls = 'None', marker = 'o')
12init, = ax.plot(x_nodal_coord, y_nodal_coord, c='k', lw = 5, ls = 'None', label = 'Initial Configuration', marker = 'o')
13ax.legend(loc = 'lower right')
14
15def drawframe(n):
16    disp_x = x_nodal_coord + disp[n][x_dofs]
17    disp_y = y_nodal_coord + disp[n][y_dofs]
18    
19    deformed.set_data(disp_x,disp_y)
20    return deformed,
21
22plt.close()
23# blit=True re-draws only the parts that have changed.
24anim = animation.FuncAnimation(fig, drawframe, frames=len(lmbda), interval=40, blit=True)
25
26anim