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geometricallyExactBeamTest.py

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]