You can view and download this file on Github: geometricallyExactBeamTest.py
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# This is an EXUDYN example
#
# Details: Test models for GeometricallyExactBeam (2-node shear deformable beam,
# Lie group formulation for work of elastic forces);
# test models: cantilever beam with tip force and torque
#
# Author: Johannes Gerstmayr
# Date: 2023-04-05
#
# Copyright:This file is part of Exudyn. Exudyn is free software. You can redistribute it and/or modify it under the terms of the Exudyn license. See 'LICENSE.txt' for more details.
#
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
import exudyn as exu
from exudyn.utilities import * #includes itemInterface and rigidBodyUtilities
import exudyn.graphics as graphics #only import if it does not conflict
import numpy as np
useGraphics = True #without test
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
#you can erase the following lines and all exudynTestGlobals related operations if this is not intended to be used as TestModel:
try: #only if called from test suite
from modelUnitTests import exudynTestGlobals #for globally storing test results
useGraphics = exudynTestGlobals.useGraphics
except:
class ExudynTestGlobals:
pass
exudynTestGlobals = ExudynTestGlobals()
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SC = exu.SystemContainer()
mbs = SC.AddSystem()
compute2D = False
compute3D = True
#test examples
#2011 MUBO, Nachbagauer Pechstein Irschik Gerstmayr (2D)
#2013 CND, Nachbagauer Gruber Gerstmayr (static, 3D); "Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples"
### not yet: 2013 CND, Nachbagauer Gerstmayr (dynamic, 3D)
cases = ['CantileverLinear2011', 'Cantilever2011', 'GeneralBending2013', 'PrincetonBeamF2', 'PrincetonBeamF3', 'Eigenmodes2013']
nElementsList = [1,2,4,8,16,32,64,128,256,512,1024]
# nElementsList = [8,32, 128]
nElements = 8
betaList = [0,15,30,45,60,75,90]
betaDegree = 45
caseList = [0,1,2,3,4] #case 0 not working for Geometrically exact beam
#case=2
useGraphics = False
verbose = 1*0
useEP = True #for geometrically exact beam node
if useEP:
NodeClass = NodeRigidBodyEP
initialRotationsGE = eulerParameters0
else: #does not work for static case, as static solver currently (2023-04) cannot solve for Lie group nodes
NodeClass = NodeRigidBodyRotVecLG
initialRotationsGE = [0,0,0]
bodyFixedLoad = False
testErrorSum = 0
case=4
printCase = True
#for nElements in nElementsList:
#for betaDegree in betaList:
for case in caseList:
#if True:
if printCase:
printCase=False
exu.Print('case=', case, cases[case])
mbs.Reset()
computeEigenmodes = False
csFact = 1
sectionData = exu.BeamSection()
fTip = 0
MxTip = 0
MyTip = 0
ks1=1 #shear correction, torsion
ks2=1 #shear correction, bending
ks3=1 #shear correction, bending
ff=1 #drawing factor
if case == 0 or case == 1:
caseName = cases[case]
L = 2 #length of beam
w = 0.1 #width of beam
h = 0.5 #height Y
fTip = 5e5*h**3
if case == 1:
fTip *= 1000
Em = 2.07e11
rho = 1e2
A=h*w
nu = 0.3 # Poisson ratio
ks2= 10*(1+nu)/(12+11*nu)
ks3=ks2
elif case == 2:
L = 2 #length of beam
h = 0.2 #height Y
w = 0.4 #width Z of beam
Em = 2.07e11
rho = 1e2
A=h*w
nu = 0.3 # Poisson ratio
ks1= 0.5768 #torsion correction factor if J=Jyy+Jzz
ks2= 0.8331
ks3= 0.7961
MxTip = 0.5e6
MyTip = 2e6
csFact = 10
elif case == 3 or case == 4: #Princeton beam example
L = 0.508 #length of beam
h = 12.3777e-3 #height Y; 12.3777e-3 with Obrezkov's paper
w = 3.2024e-3 #width Z of beam
Em = 71.7e9
ks1=0.198
nu = 0.31
ks2=1
ks3=1
# ks2=0.9
# ks3=0.9
rho = 1e2 #unused
A=h*w
MxTip = 0
MyTip = 0
if case == 3:
fTip = 8.896 #F2
elif case == 4:
fTip = 13.345 #F3
#if kk==0: exu.Print('load=', fTip)
beta = betaDegree/180*pi #beta=0 => negative y-axis
bodyFixedLoad = False
csFact = 10
Gm = Em/(2*(1+nu)) # Shear modulus
# Cross-section properties
Iyy = h*w**3/12 # Second moment of area of the beam cross-section
Izz = w*h**3/12 # Second moment of area of the beam cross-section
J = (Iyy+Izz) # approximation; Polar moment of area of the beam cross-section
sectionData.stiffnessMatrix = np.diag([Em*A, Gm*A*ks2, Gm*A*ks3, Gm*J*ks1, Em*Iyy, Em*Izz])
rhoA = rho*A
if False:
#linear solution:
uzTip = fTip*L**3/(3*Em*Iyy)
exu.Print('uz linear=',uzTip)
uyTip = fTip*L**3/(3*Em*Izz)
exu.Print('uy linear=',uyTip)
sectionData.inertia= rho*J*np.eye(3)
sectionData.massPerLength = rhoA
sectionGeometry = exu.BeamSectionGeometry()
#points, in positive rotation sense viewing in x-direction, points in [Y,Z]-plane
#points do not need to be closed!
lp = exu.Vector2DList()
if True:
lp.Append([h*ff,-w*ff])
lp.Append([h*ff,w*ff])
lp.Append([-h*ff,w*ff])
lp.Append([-h*ff,-w*ff])
sectionGeometry.polygonalPoints = lp
#exu.Print('HERE\n',sectionGeometry.polygonalPoints)
nGround = mbs.AddNode(NodePointGround(referenceCoordinates=[0,0,0])) #ground node for coordinate constraint
mnGround = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nGround, coordinate=0))
eY=[0,1,0]
eZ=[0,0,1]
lElem = L/nElements
useGeometricallyExact = True
if compute3D:
if useGeometricallyExact:
n0 = mbs.AddNode(NodeClass(referenceCoordinates=[0,0,0]+initialRotationsGE))
else:
initialRotations = eY+eZ
n0 = mbs.AddNode(NodePointSlope23(referenceCoordinates=[0,0,0]+initialRotations))
nInit = n0
for k in range(nElements):
if useGeometricallyExact:
n1 = mbs.AddNode(NodeClass(referenceCoordinates=[lElem*(k+1),0,0]+initialRotationsGE))
oBeam = mbs.AddObject(ObjectBeamGeometricallyExact(nodeNumbers=[n0,n1], physicsLength = lElem,
sectionData = sectionData,
visualization=VBeam3D(sectionGeometry=sectionGeometry)))
else:
n1 = mbs.AddNode(NodePointSlope23(referenceCoordinates=[lElem*(k+1),0,0]+initialRotations))
oBeam = mbs.AddObject(ObjectANCFBeam(nodeNumbers=[n0,n1], physicsLength = lElem,
#testBeamRectangularSize = [h,w],
sectionData = sectionData,
crossSectionPenaltyFactor = [csFact,csFact,csFact],
visualization=VANCFBeam(sectionGeometry=sectionGeometry)))
n0 = n1
mTip = mbs.AddMarker(MarkerNodeRigid(nodeNumber = n1))
if fTip != 0:
if case < 3:
mbs.AddLoad(Force(markerNumber=mTip, loadVector = [0,fTip,0], bodyFixed = bodyFixedLoad))
elif case >= 3:
mbs.AddLoad(Force(markerNumber=mTip, loadVector = [0,-fTip*cos(beta),fTip*sin(beta)], bodyFixed = bodyFixedLoad))
if MxTip != 0 or MyTip != 0:
mbs.AddLoad(Torque(markerNumber=mTip, loadVector = [MxTip, MyTip,0]))#, bodyFixed = True ))
if useGeometricallyExact:
nm0 = mbs.AddMarker(MarkerNodeRigid(nodeNumber=nInit))
nmGround = mbs.AddMarker(MarkerNodeRigid(nodeNumber=nGround))
mbs.AddObject(GenericJoint(markerNumbers=[nmGround, nm0]))
else:
for i in range(9):
#if i != 4 and i != 8: #exclude constraining the slope lengths
if True:
nm0 = mbs.AddMarker(MarkerNodeCoordinate(nodeNumber=nInit, coordinate=i))
mbs.AddObject(CoordinateConstraint(markerNumbers=[mnGround, nm0]))
# exu.Print(mbs)
mbs.Assemble()
tEnd = 100 #end time of simulation
stepSize = 0.5*0.01*0.1 #step size; leads to 1000 steps
simulationSettings = exu.SimulationSettings()
simulationSettings.solutionSettings.solutionWritePeriod = 2e-2 #output interval general
simulationSettings.solutionSettings.sensorsWritePeriod = 1e-1 #output interval of sensors
simulationSettings.timeIntegration.numberOfSteps = int(tEnd/stepSize) #must be integer
simulationSettings.timeIntegration.endTime = tEnd
#simulationSettings.solutionSettings.solutionInformation = "This is the info\nNew line\n and another new line \n"
simulationSettings.timeIntegration.generalizedAlpha.spectralRadius = 0.5
#simulationSettings.timeIntegration.simulateInRealtime=True
#simulationSettings.timeIntegration.realtimeFactor=0.1
simulationSettings.timeIntegration.verboseMode = verbose
simulationSettings.staticSolver.verboseMode = verbose
#simulationSettings.parallel.numberOfThreads = 4
simulationSettings.timeIntegration.newton.useModifiedNewton = True
#simulationSettings.timeIntegration.newton.numericalDifferentiation.minimumCoordinateSize = 1e0
#simulationSettings.timeIntegration.newton.numericalDifferentiation.relativeEpsilon = 1e-4
simulationSettings.timeIntegration.newton.relativeTolerance = 1e-6
# simulationSettings.displayComputationTime = True
simulationSettings.linearSolverType = exu.LinearSolverType.EigenSparse
# simulationSettings.parallel.numberOfThreads = 4
#simulationSettings.staticSolver.newton.numericalDifferentiation.relativeEpsilon = 5e-5
#simulationSettings.staticSolver.newton.numericalDifferentiation.forODE2 = True
#simulationSettings.staticSolver.newton.relativeTolerance = 1e-6
# simulationSettings.staticSolver.newton.numericalDifferentiation.relativeEpsilon = 1e-4
simulationSettings.staticSolver.numberOfLoadSteps = 5
simulationSettings.staticSolver.adaptiveStep = True
#simulationSettings.staticSolver.stabilizerODE2term = 100
if useGeometricallyExact:
# simulationSettings.staticSolver.newton.numericalDifferentiation.forODE2 = True
# simulationSettings.staticSolver.newton.numericalDifferentiation.relativeEpsilon = 1e-5
# simulationSettings.staticSolver.newton.relativeTolerance = 1e-5
# simulationSettings.staticSolver.newton.absoluteTolerance = 1e-5
if case == 0:
simulationSettings.staticSolver.newton.relativeTolerance = 1e-4
simulationSettings.staticSolver.newton.absoluteTolerance = 1e-5
simulationSettings.staticSolver.numberOfLoadSteps = 1 #otherwise makes problems
if nElements > 32 and case==0: #change tolerance, because otherwise no convergence
simulationSettings.staticSolver.newton.relativeTolerance = 1e-6
if case == 1: #tolerance changed from 1e-8 to 5e-10 to achieve values of paper (1024 has difference at last digit in paper)
simulationSettings.staticSolver.newton.relativeTolerance = 0.5e-9
#add some drawing parameters for this example
SC.visualizationSettings.nodes.drawNodesAsPoint=False
SC.visualizationSettings.nodes.defaultSize=0.01
SC.visualizationSettings.bodies.beams.axialTiling = 50
SC.visualizationSettings.general.drawWorldBasis = True
SC.visualizationSettings.general.worldBasisSize = 0.1
SC.visualizationSettings.openGL.multiSampling = 4
# [M, K, D] = exu.solver.ComputeLinearizedSystem(mbs, simulationSettings, useSparseSolver=True)
# exu.Print('M=',M.round(1))
if useGraphics:
exu.StartRenderer()
mbs.WaitForUserToContinue()
# if computeEigenmodes:
# nModes = 3*(1+int(compute3D))
# nRigidModes = 3*(1+int(compute3D))
# if compute2D:
# constrainedCoordinates=[0,1,mbs.systemData.ODE2Size()-2]
# else:
# constrainedCoordinates=[0,1,2,5,mbs.systemData.ODE2Size()-8,mbs.systemData.ODE2Size()-7]
# # constrainedCoordinates=[]
# compeig=mbs.ComputeODE2Eigenvalues(simulationSettings, useSparseSolver=False,
# numberOfEigenvalues= nRigidModes+nModes,
# constrainedCoordinates=constrainedCoordinates,
# convert2Frequencies= False)
# exu.Print('eigvalues=',np.sqrt(compeig[0][nRigidModes:]))
# if False: #show modes:
# for i in range(nModes):
# iMode = nRigidModes+i
# mbs.systemData.SetODE2Coordinates(5*compeig[1][:,iMode], exudyn.ConfigurationType.Visualization)
# mbs.systemData.SetTime(np.sqrt(compeig[0][iMode]), exudyn.ConfigurationType.Visualization)
# mbs.SendRedrawSignal()
# mbs.WaitForUserToContinue()
# else:
mbs.SolveStatic(simulationSettings)
# mbs.SolveDynamic(simulationSettings)
#mbs.SolveDynamic(simulationSettings, solverType = exu.DynamicSolverType.RK44)
#check jacobian
if False:
#%%+++++++++++++++++++++++++++++++++++
solver=mbs.sys['staticSolver']
solver.InitializeSolver(mbs, simulationSettings)
solver.ComputeJacobianODE2RHS(mbs)
J=solver.GetSystemJacobian()
print((1e-6*J[:14,:7]).round(3))
print((1e-6*J[:14,7:14]).round(3))
#%%+++++++++++++++++++++++++++++++++++
if useGraphics:
SC.WaitForRenderEngineStopFlag()
exu.StopRenderer() #safely close rendering window!
##evaluate final (=current) output values
uTip = mbs.GetNodeOutput(n1, exu.OutputVariableType.Displacement)
errorFact = 1
if case != 1:
errorFact *= 100
testErrorSum += np.linalg.norm(uTip)
if case < 2:
pTip = mbs.GetNodeOutput(n1, exu.OutputVariableType.Position)
exu.Print('ne=',nElements, ', ux=',L-pTip[0], ', uy=',pTip[1])
elif case == 2:
rotTip = mbs.GetNodeOutput(n1, exu.OutputVariableType.Rotation)
exu.Print('ne=',nElements, ', u=',list(uTip))
# exu.Print('ne=',nElements, ', rot=',rotTip)
elif case == 3 or case == 4:
exu.Print('ne=', nElements, ', beta=', round(beta*180/pi,1), ', u=',uTip.round(7))
exu.Print('Solution of geometricallyExactBeamTest=', testErrorSum)
exudynTestGlobals.testError = testErrorSum - (1.012822053539261)
exudynTestGlobals.testResult = testErrorSum
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
#all results are taken from ANCFBeam (shear deformable 2-node 3D beam):
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# case= 0/CantileverLinear2011
#NachbagauerPechsteinIrschikGerstmayrMUBO2011 (2D):
# ne=1, 9.12273046e–8, 6.16666566e–4, 0.000193
# ne=2, 1.61293091e–7, 7.61594059e–4, 4.831e–5
# ne=4, 1.81763233e–7, 7.97825954e–4, 1.208e–5
# ne=256, 1.88847418e–7, 8.09900305e–4, 2.945e–9
#Exudyn: ksFact=1
# ne= 1 , ux= 9.122730637578513e-08 , uy= 0.0006166665660910789
# ne= 2 , ux= 1.612930911054633e-07 , uy= 0.0007615940599560586
# ne= 4 , ux= 1.8176323512975046e-07 , uy= 0.0007978259537503566
# ne= 8 , ux= 1.8706537496804287e-07 , uy= 0.0008068839288072378
# ne= 16 , ux= 1.8840244964124508e-07 , uy= 0.0008091484226773518
# ne= 32 , ux= 1.887374359021976e-07 , uy= 0.0008097145461515286
# ne= 64 , ux= 1.888212299849812e-07 , uy= 0.0008098560770202866
# ne= 128 , ux= 1.8884218011550047e-07 , uy= 0.000809891459736643
# ne= 256 , ux= 1.8884741770364144e-07 , uy= 0.0008099003054122335
# case= 1/Cantilever2011
#NachbagauerPechsteinIrschikGerstmayrMUBO2011 (2D):
# ne=1, 0.07140274, 0.54225823, 0.168310
# ne=2, 0.12379212, 0.65687111, 0.053697
# ne=4, 0.14346767, 0.69593561, 0.014633
# ne=1024, 0.15097103, 0.71056837, 2.280e–7
#Exudyn: ksFact=1
# ne= 1 , ux= 0.07140273975041422 , uy= 0.5422582285449739
# ne= 2 , ux= 0.12379212054619537 , uy= 0.6568711099777776
# ne= 4 , ux= 0.14346766617229956 , uy= 0.695935613449867
# ne= 8 , ux= 0.14904162148449163 , uy= 0.7068152604035266
# ne= 16 , ux= 0.15048521526298897 , uy= 0.709623891842095
# ne= 32 , ux= 0.15084943688011565 , uy= 0.7103320154655514
# ne= 64 , ux= 0.15094070328691145 , uy= 0.7105094267817303
# ne= 128 , ux= 0.15096353326024237 , uy= 0.7105538037895819
# ne= 256 , ux= 0.15096924149743085 , uy= 0.7105648993600513
# ne= 512 , ux= 0.15097066651939461 , uy= 0.7105676689547459
# ne= 1024 , ux= 0.15097102364723924 , uy= 0.7105683631862169
# case = 2:
#2013 CND, Nachbagauer Gruber Gerstmayr (static, 3D); "Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples"
#Table 4:
# SMF
# 8, 1.0943e-4, 1.8638e-4, 1.8117e-2
# 32, 1.0943e-4, 1.8625e-4, 1.8117e-2
# ANSYS
# 40, 1.0939e-4, 1.8646e-4, 1.8117e-2
#Exudyn, ksFact=10:
# ne= 8 , u= [-0.00010900977088157404, -0.0001902100873246334, -0.01811732779800177]
# ne= 32 , u= [-0.00010941122286522997, -0.00018667435478355072, -0.01811739809277171]
# ne= 128 , u= [-0.00010943631319815239, -0.000186451835025629, -0.018117402461210096]
#==> in 2013 paper, element performed slightly better, especially in ux and uy terms
# case = 3:
#Princeton beam with ANSYS (Leonid Obrezkov / Aki Mikkola / Marko Matikainen et al.,
# Performance review of locking alleviation methods for continuum ANCF beam elements,
# Nonlinear Dynamics, Vol. 109, pp. 31–546, May 2022
# beta=[0 15 30 45 60 75 90];
if (case==3 or case == 4) and False:
# F2=8.896
# % ANSYS beam (10-199 el)
ANSYSF2y=np.array([1.071417630E-002, 1.061328706E-002, 1.011169630E-002, 8.837226265E-003, 6.604665004E-003, 3.538889001E-003, 0])
ANSYSF2z=np.array([0, 4.208232124E-002, 7.939482948E-002, 0.108987937, 0.129887616, 0.142194370, 0.146245978])
exu.Print('refsol ANSYS F2=8.896:\n',ANSYSF2y.round(6), '\n', ANSYSF2z.round(6))
# % ANSYS solid (el) (4x12x500) - finer mesh doesn't have much influence see in Size effect file
# ANSYS_solid_y=[1.069752828E-002 1.057180106E-002 9.938278402E-003 8.686786771E-003 6.500006282E-003 3.481999513E-003 0];
# ANSYS_solid_z=[0 4.101165651E-002 7.696749069E-002 0.105976311 0.127251299 0.139594740 0.143848652];
# F3=13.345
# % ANSYS beam (10-199 el)
ANSYSF3y=np.array([1.606423724E-002, 1.645825752E-002, 1.665873206E-002, 1.518618440E-002, 1.157837500E-002, 6.248967384E-003, 0])
ANSYSF3z=np.array([0, 6.435812858E-002, 0.117735994, 0.156467239, 0.181861627, 0.196097131, 0.200677707])
# % ANSYS solid (el) (4x12x500) - finer mesh see in Size effect file
#ANSYS_solid_y=[1.603700622E-002 1.637026068E-002 1.640440775E-002 1.485055210E-002 1.127173264E-002 6.062461977E-003 0])
#ANSYS_solid_z=[0 6.270699533E-002 0.113752002 0.153554457 0.179978534 0.192972233 0.197669499])
exu.Print('refsol ANSYS F3=13.345:\n',ANSYSF3y.round(6), '\n', ANSYSF3z.round(6))
#Exudyn results for Princeton beam:
#not exactly the same, but around the previous values with HOTINT
#using 16 elements, csFact=10 (no influence)
# F2=8.896
# case= 3, PrincetonBeam
# ne= 16 , beta= 0.0 , u= [-0.0001352 -0.0107023 0. ]
# ne= 16 , beta= 15.0 , u= [-0.0022414 -0.0106295 0.0421374]
# ne= 16 , beta= 30.0 , u= [-0.0076567 -0.0101861 0.0794434]
# ne= 16 , beta= 45.0 , u= [-0.0143664 -0.0089529 0.1089703]
# ne= 16 , beta= 60.0 , u= [-0.0204225 -0.0067182 0.1297877]
# ne= 16 , beta= 75.0 , u= [-0.0245093 -0.0036079 0.1420319]
# ne= 16 , beta= 90.0 , u= [-0.0259403 -0. 0.1460608]
# F3=13.345
# case= 4, PrincetonBeam
# ne= 16 , beta= 0.0 , u= [-0.0003039 -0.0160454 0. ]
# ne= 16 , beta= 15.0 , u= [-0.005319 -0.0165469 0.064622 ]
# ne= 16 , beta= 30.0 , u= [-0.0171901 -0.0169316 0.1179818]
# ne= 16 , beta= 45.0 , u= [-0.0303357 -0.0155488 0.1565214]
# ne= 16 , beta= 60.0 , u= [-0.0411035 -0.0118996 0.1817173]
# ne= 16 , beta= 75.0 , u= [-0.0479101 -0.0064334 0.1958343]
# ne= 16 , beta= 90.0 , u= [-0.0502184 -0. 0.2003738]