Benchmark problem: Wedged Plate - finite deformations

Features

• Using PatchMesher to model the plate

• nodal boundary conditions using location-based search

• face loads using location-based search

• finite deformation Triangle6 and Quad9 elements

• history plot feature

import numpy as np

from femedu.examples import Example

from femedu.domain import System
# from femedu.solver import NewtonRaphsonSolver
from femedu.solver import NewtonRaphsonSolverSparse as NewtonRaphsonSolver
from femedu.elements.finite import Quad9, Triangle6
from femedu.materials import PlaneStress
from femedu.mesher import *


class Example21_Benchmark01(Example):

    def problem(self):
        # ========== setting mesh parameters ==============

        Nx = 8  # number of elements in the mesh
        Ny = 4  # number of elements in the mesh
        L1 = 48.0
        L2 = 44.0
        L3 = 16.0

        # ========== setting material parameters ==============

        params = dict(
            E=1000.,  # Young's modulus
            nu=0.3,  # Poisson's ratio
            t=1.00  # thickness of the plate
        )

        # ========== setting load parameters ==============

        px = 0.0  # uniform load normal to x=Lx
        pxy = 100.0 / L3  # uniform shear load on x=L1

        # ========== setting analysis parameters ==============

        target_load_level = 5.00  # reference load
        max_steps = 25  # number of load steps: 2 -> [0.0, 1.0]

        # define a list of target load levels
        load_levels = np.linspace(0, target_load_level, max_steps+1)

        #
        # ==== Build the system model ====
        #

        model = System()
        model.setSolver(NewtonRaphsonSolver())

        # create nodes

        mesher = PatchMesher(model, (0., 0.), (L1, L2), (L1, L2 + L3), (0., L2))
        nodes, elements = mesher.quadMesh(Nx, Ny, Quad9, PlaneStress(params))

        mesher.shift(1.25 * L1, 0.0)
        nodes2, elements2 = mesher.triangleMesh(Nx, Ny, Triangle6, PlaneStress(params))

        nodes += nodes2
        elements += elements2

        # ==== Apply boundary conditions ====

        #
        # the left model
        #

        ## fix left side
        for node, _ in model.findNodesAlongLine((0.0, 0.0), (0.0, 1.0)):
            node.fixDOF('ux', 'uy')

        ## define loads ...
        for _, face in model.findFacesAlongLine((L1, 0.0), (0.0, 1.0), orientation=+1):
            face.setLoad(px, pxy)

        ## locate the node at the centerline
        nodeA, dist = model.findNodesAt((L1, L2 + L3))[0]

        #
        # the right model
        #

        ## fix left side
        for node, _ in model.findNodesAlongLine((1.25 * L1, 0.0), (0.0, 1.0)):
            node.fixDOF('ux', 'uy')

        ## define loads ...
        for _, face in model.findFacesAlongLine((2.25 * L1, 0.0), (0.0, 1.0), orientation=+1):
            face.setLoad(px, pxy)

        ## locate the node at the centerline
        nodeB, dist = model.findNodesAt((2.25 * L1, L2 + L3))[0]

        # model.report()

        # set up a recorder
        model.initRecorder(variables=['ux', 'uy'], nodes=[nodeA, nodeB])
        model.startRecorder()

        model.plot(factor=0, title="undeformed system", filename="benchmark01_undeformed.png", show_bc=1, show_loads=1)

        for lf in load_levels:
            model.setLoadFactor(lf)
            model.solve(verbose=True)

            # model.report()

        model.plot(factor=1., show_bc=1, show_loads=1, show_reactions=1)

        model.valuePlot('sxx', show_mesh=1)
        model.valuePlot('syy', show_mesh=1)
        model.valuePlot('sxy', show_mesh=1)

        # create a history plot for the end node

        # model.historyPlot('lam', ['ux','uy'], nodes=[nodeA,nodeA])
        # model.historyPlot('lam', ['ux','uy'], nodes=[nodeB,nodeB])
        model.historyPlot('lam', ['ux', 'uy', 'ux', 'uy'], nodes=[nodeA, nodeA, nodeB, nodeB])

Run the example by creating an instance of the problem and executing it by calling Example.run()

if __name__ == "__main__":
    ex = Example21_Benchmark01()
    ex.run()

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norm of the out-of-balance force:   1.1562e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   4.4288e+00
norm of the out-of-balance force:   1.8504e-03
norm of the out-of-balance force:   2.1912e-06
norm of the out-of-balance force:   4.3933e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   4.2997e+00
norm of the out-of-balance force:   2.1512e-03
norm of the out-of-balance force:   4.2592e-06
norm of the out-of-balance force:   4.6466e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   4.1752e+00
norm of the out-of-balance force:   2.6218e-03
norm of the out-of-balance force:   6.1816e-06
norm of the out-of-balance force:   3.6921e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   4.0559e+00
norm of the out-of-balance force:   3.1923e-03
norm of the out-of-balance force:   7.9418e-06
norm of the out-of-balance force:   4.5753e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.9424e+00
norm of the out-of-balance force:   3.8177e-03
norm of the out-of-balance force:   9.5289e-06
norm of the out-of-balance force:   4.2242e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.8351e+00
norm of the out-of-balance force:   4.4715e-03
norm of the out-of-balance force:   1.0937e-05
norm of the out-of-balance force:   4.3992e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.7342e+00
norm of the out-of-balance force:   5.1378e-03
norm of the out-of-balance force:   1.2164e-05
norm of the out-of-balance force:   4.2938e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.6402e+00
norm of the out-of-balance force:   5.8059e-03
norm of the out-of-balance force:   1.3214e-05
norm of the out-of-balance force:   4.4627e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.5530e+00
norm of the out-of-balance force:   6.4686e-03
norm of the out-of-balance force:   1.4090e-05
norm of the out-of-balance force:   3.7944e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.4730e+00
norm of the out-of-balance force:   7.1204e-03
norm of the out-of-balance force:   1.4801e-05
norm of the out-of-balance force:   5.1315e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.4000e+00
norm of the out-of-balance force:   7.7573e-03
norm of the out-of-balance force:   1.5358e-05
norm of the out-of-balance force:   4.1534e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.3341e+00
norm of the out-of-balance force:   8.3763e-03
norm of the out-of-balance force:   1.5770e-05
norm of the out-of-balance force:   4.7822e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.2752e+00
norm of the out-of-balance force:   8.9750e-03
norm of the out-of-balance force:   1.6049e-05
norm of the out-of-balance force:   4.4194e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.2230e+00
norm of the out-of-balance force:   9.5518e-03
norm of the out-of-balance force:   1.6209e-05
norm of the out-of-balance force:   4.3191e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.1773e+00
norm of the out-of-balance force:   1.0105e-02
norm of the out-of-balance force:   1.6262e-05
norm of the out-of-balance force:   5.1873e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.1379e+00
norm of the out-of-balance force:   1.0635e-02
norm of the out-of-balance force:   1.6219e-05
norm of the out-of-balance force:   4.6965e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.1044e+00
norm of the out-of-balance force:   1.1140e-02
norm of the out-of-balance force:   1.6093e-05
norm of the out-of-balance force:   4.7504e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0764e+00
norm of the out-of-balance force:   1.1620e-02
norm of the out-of-balance force:   1.5895e-05
norm of the out-of-balance force:   4.4994e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0536e+00
norm of the out-of-balance force:   1.2076e-02
norm of the out-of-balance force:   1.5635e-05
norm of the out-of-balance force:   4.6813e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0355e+00
norm of the out-of-balance force:   1.2506e-02
norm of the out-of-balance force:   1.5324e-05
norm of the out-of-balance force:   4.6317e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0218e+00
norm of the out-of-balance force:   1.2913e-02
norm of the out-of-balance force:   1.4970e-05
norm of the out-of-balance force:   4.2574e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0121e+00
norm of the out-of-balance force:   1.3296e-02
norm of the out-of-balance force:   1.4582e-05
norm of the out-of-balance force:   4.9222e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0059e+00
norm of the out-of-balance force:   1.3655e-02
norm of the out-of-balance force:   1.4167e-05
norm of the out-of-balance force:   4.4703e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0029e+00
norm of the out-of-balance force:   1.3992e-02
norm of the out-of-balance force:   1.3731e-05
norm of the out-of-balance force:   4.3146e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.0408e+01
norm of the out-of-balance force:   3.0028e+00
norm of the out-of-balance force:   1.4307e-02
norm of the out-of-balance force:   1.3280e-05
norm of the out-of-balance force:   4.6276e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+