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 Triangle and Quad 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.elements.finite import Quad, Triangle
from femedu.materials import PlaneStress
from femedu.mesher import *


class Example20_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 = 26                # 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)

        #
        # ==== 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, Quad, PlaneStress(params))

        mesher.shift(1.25*L1, 0.0)
        nodes2, elements2 = mesher.triangleMesh(Nx, Ny, Triangle, 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 np.linspace(0.0, target_load_level, max_steps):

            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=True)
        model.valuePlot('syy', show_mesh=True)
        model.valuePlot('sxy', show_mesh=True)

        # 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 = Example20_Benchmark01()
    ex.run()

$Contours of '$\sigma_{xx}$'$
$Contours of '$\sigma_{yy}$'$
$Contours of '$\sigma_{xy}$'$

+ Quad_6854: nodes ( Node_4321 Node_4322 Node_4331 Node_4330 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6854: nodes ( Node_4321 Node_4322 Node_4331 Node_4330 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6862: nodes ( Node_4330 Node_4331 Node_4340 Node_4339 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6862: nodes ( Node_4330 Node_4331 Node_4340 Node_4339 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6870: nodes ( Node_4339 Node_4340 Node_4349 Node_4348 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6870: nodes ( Node_4339 Node_4340 Node_4349 Node_4348 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6878: nodes ( Node_4348 Node_4349 Node_4358 Node_4357 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Quad_6878: nodes ( Node_4348 Node_4349 Node_4358 Node_4357 )
    material: list
    strain (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (0): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (1): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (2): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
    strain (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress (3): xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6893: nodes ( Node_4366 Node_4367 Node_4375 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6894: nodes ( Node_4376 Node_4375 Node_4367 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6894: nodes ( Node_4376 Node_4375 Node_4367 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6909: nodes ( Node_4375 Node_4376 Node_4384 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6910: nodes ( Node_4385 Node_4384 Node_4376 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6910: nodes ( Node_4385 Node_4384 Node_4376 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6925: nodes ( Node_4384 Node_4385 Node_4393 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6926: nodes ( Node_4394 Node_4393 Node_4385 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6926: nodes ( Node_4394 Node_4393 Node_4385 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6941: nodes ( Node_4393 Node_4394 Node_4402 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6942: nodes ( Node_4403 Node_4402 Node_4394 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
+ Triangle_6942: nodes ( Node_4403 Node_4402 Node_4394 )
    material: PlaneStress
    strain: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=-0.000e+00
    stress: xx=0.000e+00 yy=0.000e+00 xy=0.000e+00 zz=0.000e+00
norm of the out-of-balance force:   2.7688e-11
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.7967e+00
norm of the out-of-balance force:   8.7339e-04
norm of the out-of-balance force:   1.3869e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.6992e+00
norm of the out-of-balance force:   8.4504e-04
norm of the out-of-balance force:   1.3341e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.6018e+00
norm of the out-of-balance force:   8.1663e-04
norm of the out-of-balance force:   1.2800e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.5050e+00
norm of the out-of-balance force:   7.8823e-04
norm of the out-of-balance force:   1.2278e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.4088e+00
norm of the out-of-balance force:   7.5995e-04
norm of the out-of-balance force:   1.1697e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.3136e+00
norm of the out-of-balance force:   7.3186e-04
norm of the out-of-balance force:   1.1009e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.2193e+00
norm of the out-of-balance force:   7.0406e-04
norm of the out-of-balance force:   1.0257e-09
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.1263e+00
norm of the out-of-balance force:   6.7662e-04
norm of the out-of-balance force:   9.6203e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   3.0346e+00
norm of the out-of-balance force:   6.4960e-04
norm of the out-of-balance force:   9.0858e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.9445e+00
norm of the out-of-balance force:   6.2307e-04
norm of the out-of-balance force:   8.5029e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.8559e+00
norm of the out-of-balance force:   5.9709e-04
norm of the out-of-balance force:   7.7695e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.7691e+00
norm of the out-of-balance force:   5.7170e-04
norm of the out-of-balance force:   7.2541e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.6841e+00
norm of the out-of-balance force:   5.4694e-04
norm of the out-of-balance force:   6.5840e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.6009e+00
norm of the out-of-balance force:   5.2286e-04
norm of the out-of-balance force:   6.1889e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.5197e+00
norm of the out-of-balance force:   4.9947e-04
norm of the out-of-balance force:   5.4816e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.4404e+00
norm of the out-of-balance force:   4.7681e-04
norm of the out-of-balance force:   5.0640e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.3631e+00
norm of the out-of-balance force:   4.5488e-04
norm of the out-of-balance force:   4.6202e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.2879e+00
norm of the out-of-balance force:   4.3371e-04
norm of the out-of-balance force:   4.2264e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.2147e+00
norm of the out-of-balance force:   4.1330e-04
norm of the out-of-balance force:   3.8200e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.1436e+00
norm of the out-of-balance force:   3.9364e-04
norm of the out-of-balance force:   3.5541e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.0745e+00
norm of the out-of-balance force:   3.7475e-04
norm of the out-of-balance force:   3.1965e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   2.0075e+00
norm of the out-of-balance force:   3.5661e-04
norm of the out-of-balance force:   2.8849e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   1.9426e+00
norm of the out-of-balance force:   3.3921e-04
norm of the out-of-balance force:   2.5962e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   1.8796e+00
norm of the out-of-balance force:   3.2255e-04
norm of the out-of-balance force:   2.5385e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+
norm of the out-of-balance force:   1.3229e+01
norm of the out-of-balance force:   1.8187e+00
norm of the out-of-balance force:   3.0661e-04
norm of the out-of-balance force:   2.1568e-10
Recorder.addData: 'stability' not initialized by the recorder: ignored
+

Total running time of the script: (0 minutes 23.394 seconds)

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