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Patch test for quadrilateral plate elements under in-plane loading
The patch test is an empirical minimum test which every finite element has to pass to ensure convergence with mesh refinement.
It consists of a problem for which a known homogeneous solution exists. For plates, we commonly use a rectangular plate subject to homogeneous edge loading, e.g., constant tension in the x-direction, or constant shear, etc.
The mesh must contain distorted elements and at least one element not attached to any node on the boundary.
Author: Peter Mackenzie-Helnwein
import numpy as np
from femedu.examples import Example
from femedu.domain import System, Node
from femedu.solver import NewtonRaphsonSolver
from femedu.elements.linear import Quad
from femedu.materials import PlaneStress
class ExamplePlate09(Example):
def problem(self):
# ========== setting mesh parameters ==============
N = 8 # number of elements in the mesh
Lx = 100.0 # length of plate in the x-direction
Ly = 80.0 # length of plate in the y-direction
# ========== setting material parameters ==============
params = dict(
E = 20000., # Young's modulus
nu = 0.250, # Poisson's ratio
t = 1.00 # thickness of the plate
)
# ========== setting load parameters ==============
px = 10.0 # uniform load normal to x=const
py = 0.0 # uniform load normal to y=const
pxy = 0.0 # uniform shear load on x=const and y=const
# ========== setting analysis parameters ==============
target_load_level = 1.00 # reference load
max_steps = 2 # 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
nodes = (
Node(0.0*Lx, 0.0*Ly), # nd 0
Node(0.2*Lx, 0.0*Ly), # nd 1
Node(0.5*Lx, 0.0*Ly), # nd 2
Node(0.7*Lx, 0.0*Ly), # nd 3
Node(1.0*Lx, 0.0*Ly), # nd 4
#
Node(0.0*Lx, 0.2*Ly), # nd 5
Node(0.15*Lx,0.3*Ly), # nd 6
Node(0.5*Lx, 0.2*Ly), # nd 7
Node(0.8*Lx, 0.3*Ly), # nd 8
Node(1.0*Lx, 0.2*Ly), # nd 9
#
Node(0.0*Lx, 0.6*Ly), # nd 10
Node(0.2*Lx, 0.5*Ly), # nd 11
Node(0.7*Lx, 0.7*Ly), # nd 12
Node(0.9*Lx, 0.6*Ly), # nd 13
Node(1.0*Lx, 0.7*Ly), # nd 14
#
Node(0.0*Lx, 1.0*Ly), # nd 15
Node(0.3*Lx, 1.0*Ly), # nd 16
Node(0.55*Lx,1.0*Ly), # nd 17
Node(0.8*Lx, 1.0*Ly), # nd 18
Node(1.0*Lx, 1.0*Ly), # nd 19
)
elements = (
Quad(nodes[0],nodes[1],nodes[6],nodes[5],PlaneStress(params)), # elem 0
Quad(nodes[1],nodes[2],nodes[7],nodes[6],PlaneStress(params)), # elem 1
Quad(nodes[2],nodes[3],nodes[8],nodes[7],PlaneStress(params)), # elem 2
Quad(nodes[3],nodes[4],nodes[9],nodes[8],PlaneStress(params)), # elem 3
#
Quad(nodes[5],nodes[6],nodes[11],nodes[10],PlaneStress(params)), # elem 4
Quad(nodes[6],nodes[7],nodes[12],nodes[11],PlaneStress(params)), # elem 5
Quad(nodes[7],nodes[8],nodes[13],nodes[12],PlaneStress(params)), # elem 6
Quad(nodes[8],nodes[9],nodes[14],nodes[13],PlaneStress(params)), # elem 7
#
Quad(nodes[10],nodes[11],nodes[16],nodes[15],PlaneStress(params)), # elem 8
Quad(nodes[11],nodes[12],nodes[17],nodes[16],PlaneStress(params)), # elem 9
Quad(nodes[12],nodes[13],nodes[18],nodes[17],PlaneStress(params)), # elem 10
Quad(nodes[13],nodes[14],nodes[19],nodes[18],PlaneStress(params)), # elem 11
#
)
model.addNode(*nodes)
model.addElement(*elements)
# define support(s)
fix_x = (0,)
fix_y = (0,4)
for idx in fix_x:
nodes[idx].fixDOF('ux') # horizontal support left end
for idx in fix_y:
nodes[idx].fixDOF('uy') # vertical support right end
# ==== complete the reference load ====
# surface loads on the left side
elements[0].setSurfaceLoad(3,px)
elements[4].setSurfaceLoad(3,px)
elements[8].setSurfaceLoad(3,px)
# surface loads on the right side
elements[ 3].setSurfaceLoad(1,px)
elements[ 7].setSurfaceLoad(1,px)
elements[11].setSurfaceLoad(1,px)
# these are only nodal forces as part of the reference load
# .. load only the upper node
model.plot(factor=0., title="undeformed system", show_bc=1)
model.setLoadFactor(0.0)
model.solve()
# for k in range(25):
# name = f"plate08_mode{k:2d}.png"
# model.plotBucklingMode(mode=k,filename=name,factor=25)
model.setLoadFactor(10.0)
model.solve()
model.solver.showKt()
model.report()
model.plot(factor=25.)
model.valuePlot('sxx', show_mesh=1)
model.valuePlot('syy', show_mesh=1)
model.valuePlot('sxy', show_mesh=1)
Run the example by creating an instance of the problem and executing it by calling Example.run()
if __name__ == "__main__":
ex = ExamplePlate09()
ex.run()
+
+
System Analysis Report
=======================
Nodes:
---------------------
Node_675:
x: [0.000 0.000]
fix: ['ux', 'uy']
u: [0.000 0.000]
Node_676:
x: [20.000 0.000]
u: [0.100 -0.000]
Node_677:
x: [50.000 0.000]
u: [0.250 -0.000]
Node_678:
x: [70.000 0.000]
u: [0.350 -0.000]
Node_679:
x: [100.000 0.000]
fix: ['uy']
u: [0.500 0.000]
Node_680:
x: [0.000 16.000]
u: [-0.000 -0.020]
Node_681:
x: [15.000 24.000]
u: [0.075 -0.030]
Node_682:
x: [50.000 16.000]
u: [0.250 -0.020]
Node_683:
x: [80.000 24.000]
u: [0.400 -0.030]
Node_684:
x: [100.000 16.000]
u: [0.500 -0.020]
Node_685:
x: [0.000 48.000]
u: [-0.000 -0.060]
Node_686:
x: [20.000 40.000]
u: [0.100 -0.050]
Node_687:
x: [70.000 56.000]
u: [0.350 -0.070]
Node_688:
x: [90.000 48.000]
u: [0.450 -0.060]
Node_689:
x: [100.000 56.000]
u: [0.500 -0.070]
Node_690:
x: [0.000 80.000]
u: [-0.000 -0.100]
Node_691:
x: [30.000 80.000]
u: [0.150 -0.100]
Node_692:
x: [55.000 80.000]
u: [0.275 -0.100]
Node_693:
x: [80.000 80.000]
u: [0.400 -0.100]
Node_694:
x: [100.000 80.000]
u: [0.500 -0.100]
Elements:
---------------------
Quad_873: nodes ( Node_675 Node_676 Node_681 Node_680 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-2.428e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=2.366e-12 xy=-1.942e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-2.272e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=-1.183e-12 xy=-1.818e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-2.176e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=3.553e-12 xy=-1.741e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=1.514e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=1.187e-12 xy=1.211e-12 zz=0.000e+00
Quad_874: nodes ( Node_676 Node_677 Node_682 Node_681 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=1.748e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=2.370e-12 xy=1.399e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-1.490e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=3.553e-12 xy=-1.192e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-2.020e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-4.739e-12 xy=-1.616e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-4.152e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=0.000e+00 xy=-3.321e-12 zz=0.000e+00
Quad_875: nodes ( Node_677 Node_678 Node_683 Node_682 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-4.098e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=-1.183e-12 xy=-3.278e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-2.583e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=3.553e-12 xy=-2.067e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-4.831e-17 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=0.000e+00 xy=-3.865e-13 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-7.157e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=2.366e-12 xy=-5.726e-12 zz=0.000e+00
Quad_876: nodes ( Node_678 Node_679 Node_684 Node_683 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-3.055e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=0.000e+00 xy=-2.444e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-1.864e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=1.187e-12 xy=-1.492e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-6.040e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-2.370e-12 xy=-4.832e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=1.562e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=7.105e-12 xy=1.250e-12 zz=0.000e+00
Quad_877: nodes ( Node_680 Node_681 Node_686 Node_685 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-2.174e-17 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=-2.370e-12 xy=-1.739e-13 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=1.507e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=3.553e-12 xy=1.205e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-1.436e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-1.183e-12 xy=-1.149e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-4.615e-17 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=-1.183e-12 xy=-3.692e-13 zz=0.000e+00
Quad_878: nodes ( Node_681 Node_682 Node_687 Node_686 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=1.752e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=2.366e-12 xy=1.402e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-1.708e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=-4.739e-12 xy=-1.366e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=5.468e-17 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-8.288e-12 xy=4.374e-13 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-6.084e-17 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=-1.183e-12 xy=-4.867e-13 zz=0.000e+00
Quad_879: nodes ( Node_682 Node_683 Node_688 Node_687 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=1.674e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=-2.370e-12 xy=1.339e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=5.848e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=1.187e-12 xy=4.678e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=3.364e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-2.366e-12 xy=2.691e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=8.505e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=0.000e+00 xy=6.804e-12 zz=0.000e+00
Quad_880: nodes ( Node_683 Node_684 Node_689 Node_688 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-3.858e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=5.919e-12 xy=-3.086e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-2.609e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=-1.187e-12 xy=-2.087e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-1.943e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=1.183e-12 xy=-1.554e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-3.523e-17 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=2.370e-12 xy=-2.819e-13 zz=0.000e+00
Quad_881: nodes ( Node_685 Node_686 Node_691 Node_690 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=2.093e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=3.553e-12 xy=1.675e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=1.524e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=4.736e-12 xy=1.219e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=4.599e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=-4.739e-12 xy=3.679e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=3.502e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=0.000e+00 xy=2.802e-12 zz=0.000e+00
Quad_882: nodes ( Node_686 Node_687 Node_692 Node_691 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=-1.144e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=2.366e-12 xy=-9.150e-13 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=2.635e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=-2.370e-12 xy=2.108e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-9.626e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=9.475e-12 xy=-7.701e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=-3.084e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=1.187e-12 xy=-2.467e-12 zz=0.000e+00
Quad_883: nodes ( Node_687 Node_688 Node_693 Node_692 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=1.882e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=4.739e-12 xy=1.506e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=-1.989e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=-1.066e-11 xy=-1.591e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=-4.238e-16 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=0.000e+00 xy=-3.390e-12 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=6.587e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=0.000e+00 xy=5.269e-12 zz=0.000e+00
Quad_884: nodes ( Node_688 Node_689 Node_694 Node_693 )
material: list
strain (0): xx=5.000e-03 yy=-1.250e-03 xy=6.956e-16 zz=-9.375e-04
stress (0): xx=1.000e+02 yy=-8.288e-12 xy=5.565e-12 zz=0.000e+00
strain (1): xx=5.000e-03 yy=-1.250e-03 xy=2.904e-16 zz=-9.375e-04
stress (1): xx=1.000e+02 yy=2.370e-12 xy=2.323e-12 zz=0.000e+00
strain (2): xx=5.000e-03 yy=-1.250e-03 xy=1.619e-15 zz=-9.375e-04
stress (2): xx=1.000e+02 yy=7.105e-12 xy=1.296e-11 zz=0.000e+00
strain (3): xx=5.000e-03 yy=-1.250e-03 xy=7.029e-16 zz=-9.375e-04
stress (3): xx=1.000e+02 yy=1.183e-12 xy=5.623e-12 zz=0.000e+00
Total running time of the script: (0 minutes 0.134 seconds)