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LinearElasticity1#

A cantilever beam undergoing bending deformation.

Note that this simulation is also performed in examples/Elastic/Elas1.py.

  • $uy$
  • LinearElasticity1
 15 import matplotlib.pyplot as plt
 16 import numpy as np
 17
 18 from EasyFEA import Display, ElemType, Models, Simulations
 19 from EasyFEA.FEM import Field, BiLinearForm, FeArray, Sym_Grad, Trace
 20 from EasyFEA.Geoms import Domain
 21
 22 if __name__ == "__main__":
 23     Display.Clear()
 24
 25     # ----------------------------------------------
 26     # Configuration
 27     # ----------------------------------------------
 28
 29     dim = 2
 30
 31     # geom
 32     L = 120  # mm
 33     h = 13
 34
 35     # load
 36     F = -800  # N
 37
 38     # model
 39     E = 210000  # MPa
 40     v = 0.3
 41
 42     elastic = Models.Elastic.Isotropic(dim, E, v, planeStress=True, thickness=h)
 43     lmbda = elastic.get_lambda()
 44     mu = elastic.get_mu()
 45
 46     # ----------------------------------------------
 47     # Mesh
 48     # ----------------------------------------------
 49
 50     contour = Domain((0, 0), (L, h), h / 3)
 51
 52     if dim == 2:
 53         mesh = contour.Mesh_2D([], ElemType.QUAD9, isOrganised=True)
 54     else:
 55         mesh = contour.Mesh_Extrude(
 56             [], [0, 0, h], [3], ElemType.HEXA27, isOrganised=True
 57         )
 58
 59     nodesX0 = mesh.Nodes_Conditions(lambda x, y, z: x == 0)
 60     nodesXL = mesh.Nodes_Conditions(lambda x, y, z: x == L)
 61
 62     # ----------------------------------------------
 63     # Formulations
 64     # ----------------------------------------------
 65
 66     field = Field(mesh.groupElem, dim)
 67
 68     def S(u: Field) -> FeArray:
 69         Eps = Sym_Grad(u)
 70         return 2 * mu * Eps + lmbda * Trace(Eps) * np.eye(dim)
 71
 72     @BiLinearForm
 73     def ComputeK(u: Field, v: Field):
 74         Sig = S(u)
 75         Eps = Sym_Grad(v)
 76         return Sig.ddot(Eps)
 77
 78     weakForms = Models.WeakForms(field, ComputeK)
 79
 80     # ----------------------------------------------
 81     # Simulations
 82     # ----------------------------------------------
 83
 84     simu = Simulations.WeakForms(mesh, weakForms)
 85
 86     simu.add_dirichlet(nodesX0, [0] * dim, simu.Get_unknowns())
 87     simu.add_surfLoad(nodesXL, [F / h**2], ["y"])
 88
 89     simu.Solve()
 90
 91     # ----------------------------------------------
 92     # Results
 93     # ----------------------------------------------
 94
 95     Display.Plot_Result(simu, "uy", 10, plotMesh=True)
 96
 97     def von_mises_stress(u):
 98         Stress = S(u)
 99         if dim == 2:
100             xx = Stress[..., 0, 0]
101             yy = Stress[..., 1, 1]
102             xy = Stress[..., 0, 1]
103             return np.sqrt(xx**2 + yy**2 - xx * yy + 3 * xy**2)
104         else:
105             Stress_d = Stress - 1 / 3 * Trace(Stress) * np.eye(3)
106             return np.sqrt(3 / 2 * Stress_d.ddot(Stress_d))
107
108     Svm_e = field.Evaluate_e(von_mises_stress, simu.u)
109
110     Display.Plot_Result(simu, Svm_e, plotMesh=True, ncolors=11)
111
112     plt.show()

Total running time of the script: (0 minutes 0.222 seconds)

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