lecture ix: finite-element methods - cs.uu.nl 9 - finite element... · lagrange fem basis •...
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LectureIX:Finite-ElementMethods
Motivation
• Discretizedeformations,vectors,andtensors• Derivativeandintegrals=>linearoperators(matrices)• SolvingPDEsó solvinglinearequations• Eitherdirectly,orbyiterations.
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http://fetk.org/codes/mc/images/bar3ndef_wbg.gifIrvingetal.“VolumeConservingFiniteElement SimulationsofDeformableModels”
https://www.youtube.com/watch?v=Rbq2CdUIvw4
LagrangeFEMBasis• Deformations:vector-valuedfunctionspervertex𝑢" =
𝑢$(𝑝)𝑢((𝑝)𝑢)(𝑝)
.
• Assumedtointerpolatelinearlyinsideelements:
𝑢 𝑝 =𝑢$(𝑝)𝑢((𝑝)𝑢)(𝑝)
=*𝜉" (𝑝)𝑢"
,
"-.
• 𝐵"(𝑝):Barycentric coordinatesofp inelement𝑒 ofdimension𝑑.• Triangles:𝑑 = 2,Tets:𝑑 = 3.
• Matrixrepresentationinsideelemente:usearowvector𝑢4:ℝ7,:
• 𝑢 𝑝 =𝜉8(𝑝)
𝜉8(𝑝)𝜉8(𝑝)
⋯𝜉,(𝑝)
𝜉,(𝑝)𝜉,(𝑝)
𝑢8,$𝑢8,(𝑢8,)⋮
𝑢,,$𝑢,,(𝑢,,)
= 𝐻4𝑢4
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LinearElasticity• Reminder:
Jacobian:𝐽𝑢 𝑝 :ℝ7×7 =𝛻𝑢$(𝑝)𝛻𝑢((𝑝)𝛻𝑢)(𝑝)
• FullLagraingian straintensor:
𝐸 =12 𝐽𝑢B𝐽𝑢 + 𝐽𝑢 + 𝐽𝑢B
• Notlinearinsidetets!• Linear elasticityapproximation:
𝜀 ≈12𝐽𝑢 + 𝐽𝑢B
• Goodforsmalldeformationswithoutmuchrotation.
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DiscreteStrainTensor• Writtenexplicitly:
𝜀 =12 𝐽𝑢 + 𝐽𝑢B =
12
2𝜕𝑢$𝜕𝑥
𝜕𝑢$𝜕𝑦 +
𝜕𝑢(𝜕𝑥
𝜕𝑢$𝜕𝑧 +
𝜕𝑢)𝜕𝑥
𝜕𝑢(𝜕𝑥 +
𝜕𝑢$𝜕𝑦 2
𝜕𝑢(𝜕𝑦
𝜕𝑢(𝜕𝑧 +
𝜕𝑢)𝜕𝑦
𝜕𝑢)𝜕𝑥 +
𝜕𝑢$𝜕𝑧
𝜕𝑢)𝜕𝑦 +
𝜕𝑢(𝜕𝑧 2
𝜕𝑢)𝜕𝑧
• Only6relevantelements (restaresymmetric):
𝜀88𝜀JJ𝜀77𝜀8J𝜀J7𝜀78
=
𝜕𝜕𝑥
𝜕𝜕𝑦
𝜕𝜕𝑧
0.5𝜕𝜕𝑦 0.5
𝜕𝜕𝑥
0.5𝜕𝜕𝑧 0.5
𝜕𝜕𝑦
0.5𝜕𝜕𝑧 0.5
𝜕𝜕𝑥
𝑢$(𝑝)𝑢((𝑝)𝑢)(𝑝)
= 𝐷𝑢(𝑝)
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DiscreteStrainTensor
• Straintensorperface:𝜀4 = 𝐷𝐻4𝑢4 = 𝐵4𝑢4
• Note:𝐻4 :ℝO$7, = ℝO$7×ℝ7$7,
• 𝐵4 containsderivativesof𝜉"(𝑝)• 𝜉" arelinear insidee.• Derivativesof𝜉" areconstant insidee.
• 𝐵4 isconstantinsidetheelement!
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𝜉8
DiscreteStressTensor
• Stress andstrain arerelatedbyHooke’slaw• Remember𝐹 = −𝑘𝑥?
• Inourcase,thediscretestiffnesstensor𝐶4:ℝO×Oholds:
𝜎4 = 𝐶4𝜀4 = 𝐶4𝐵4𝑢4
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StrainEnergy• Potentialenergygainedwhenapplyingstraintoobject:
𝑈4 =12W 𝜎4, 𝜀4 𝑑𝑉B
• Wehavethat𝜎4 = 𝐶4𝐵4𝑢4 and𝜀4 = 𝐵4𝑢4.• Then:
𝜎4, 𝜀4 = 𝜎4 B𝜀4 = 𝑢4B𝐵4B𝐶4𝐵4𝑢4• Bothareconstantinsidevolume,so:
𝑈4 =12𝑉𝑜𝑙 𝑒 ×𝑢4
B𝐵4B𝐶4𝐵4𝑢4 =12 𝑢4
B𝐾4𝑢4• 𝐾4:stiffnessmatrix.
• Onlydependsontheoriginalgeometryandmaterialproperties!
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http://www.engineeringarchives.com/img/les_mom_strainenergydensity_1.png
ElasticForces
• Derivatives ofthepotentialenergy:𝑓4 =
𝜕𝑈4𝜕𝑢4
= 𝐾4𝑢4
• Interpretation:a“forceLaplacian”.• Tryingtoreachan“averageequilibrium”.
9RestState Deformation
𝑓4
DynamicDeformationEquation
• Computingnewpositions𝑥(𝑡):𝑀𝑥__ 𝑡 + 𝐶𝑥_ 𝑡 + 𝐾 𝑥 − 𝑥. = 𝑓4$`
• M:Massmatrix• C:Dampingmatrix• K:ourstiffnessmatrix(aggregated)
• Solvedusingtimeintegrationmethods(implicitorexplicit).• Advantages oflinearelasticity:constantmatrices.• Disadvantages:manyartifacts.
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Corotational Elements
• Insight:rotatinganelementdoesnotchangethestrainenergy.• Conclusion:Givenanelementinposition𝑥,withstrainenergy𝑈4 ,andconsequentelasticforces 𝑓4 ,theforcesonarotatedelement𝑅4𝑥 are𝑅4𝑓4!
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𝑓4 𝑅4𝑓4
MüllerandGross,“InteractiveVirtualMaterials”
Corotational Elements
• Method:• Estimaterotation𝑅4,andfactoroutrotationfromthedeformedobject𝑥:𝑅4b8x• Computeelasticforcesofunrotatedobject:
𝐾4 𝑅4b8x− 𝑥.• Rotatebacktodeformedstatetogetactualforces:
𝑓4 = 𝑅4𝐾4 𝑅4b8x− 𝑥. = 𝐾4′𝑢4
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Corotational Elements
• Advantages:abletoworkwithlargerotations• Disadvantages:stiffnessmatrixnotconstantanymore.• Howtoestimaterotation𝑅4?
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Finding𝑅4• Originalpositionsx,deformedpositionsx’.• Create stackedcoordinatesofedgesoforiginalpoints:
𝑃 =𝑥8 − 𝑥J
𝑥8 − 𝑥,, 𝑄 =
𝑥8′ − 𝑥J′
𝑥8′ − 𝑥,′• Computematrix:𝑆 = 𝑃B𝑄 ∈ ℝ7×7
• Singularvaluedecomposition(SVD)extractsrotationfrom𝑆
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SingularValueDecomposition• Everylinearoperator(=matrix𝑀h×i)canbedecomposedto:• Rotation (Changeofbasis):𝑉i×i.• Stretch inthenewbasis:Σh×i
• Note(possible)changeindimension.• Rotation (anotherchangeofbasis):𝑈h×h
• Forvector𝑝 weget𝑀𝑝 = 𝑈Σ𝑉B𝑝.
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Examples
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https://www.youtube.com/watch?v=4Wl0ksysYKMhttps://www.youtube.com/watch?v=6f3UYHnR4zUhttps://www.youtube.com/watch?v=p5uhnSw8_Xw