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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1

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4

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