Character Animation CS 6967
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Date Created: 10/26/15
Finite Element Notes Adam W Bargteil School of Computing University of Utah 1 Green39s Strain We de ne a function Which describes the mapping from points in a rest con guration or material space to World or deformed space For each point ui is the correspond ing point after applying the deformation described by z u If We Want to understand how an in nitesimal region around a point is deformed by We look at the deformation grar dient 2 The deformation gradient describes how you move in World space as you move through material space We Will see later Why it is called the deformation gradient If our goal is to nd elastic forces that undo the deformation We must rst devise a Way of measuring deformation For this We de ne Green s strain tensor 1 8m 8m 7 7 7 6H 1 627 2 lt8ui Buj 27gt This metric is nonlinear We ll look at a linear approximar tion a bit later and consequently is invariant to rotations lnvariance to rotations is very good because this means our metric Will not respond to rigid body motion of the object 2 Stress From Hooke s laW iven strain We can de ne stress 039 as 7 7 7 039 Ce 2 Where C is a rank4 tensor ie a 4 dimensional matrix With 81 entries that relates the 9 entries in e to the 9 entries in 039 Of course since 6 and 039 are symmetric these entries are not independent In fact if We assume that the material is isotropic there are only tWo independent parameters and We can Write 727 Aekk ij 2M5 3 Where A and L are the Lame constants Of course this is a linear stressstrain relationship Other relationships are possible and required When dealing With complex materials 3 Elastic Potential Traction and Force We can noW de ne the elastic potential energy density 77 as 1 77 5027627quot 4 We can also de ne traction r or force per unit area as r an 5 Where n is the unit surface normal Force Will seek to reduce the energy of the system thus they Will be in the direction of the negative gradient of the energy that is the force at a point mi Will 077 12 6 lt6 Finite volume methods instead de ne force by integrating traction over some region f f ands 7 8R It has been shown that for the types of nite elements We Will be concerned With in this class these forces are equivalent and the second form is faster to compute 4 Damping Similar functions can be de ned for damping Which is de pendent on velocity rather than deformation We de ne these by taking a time derivative of the above to get v71 8 ai g 2 7 2 8m 6m 8m 6m 7 vkk5u WW 9 n gay 10 5 Linear Finite Elements As the name implies nite element methods take an object We Wish to simulate and break it up into a nite set of pieces While arbitrary elements are possible We ll stick to simpli cies triangles in 2D tetrahedra in 3D U3 11 U1 U2 x2 Finite elements work by essentially limiting the types of functions that can be represented do this we de ne basis over each element We can then work with func tions expressed in this basis The obvious basis to use with simplicies is the linear basis we should all be familiar with barycentric coordinates Recall that a point u in a trian le can be expressed as a convex combination of the vertices of the triangle ub1u1b2u2b3u3 11 However this is redundant since we have the additional con straint that 1 2 3 1 So we can alternately write u b1u1b2u217b17b2u3 u3b1u17u3b2u2iu37 12 E22313 lt 2 lt12 hus we construct a matrix which we call B that contains vectors along the edges of the triangle as its columns and this matrix describes the mapping from barycentric coordinates to material coordinates If we wish to go the other way we will need to invert this matrix or in matrix form uu3lt 7117713 m 7 u3y 2 B lm 7 us mu ns 14 Letting 8 Bil Similarly when mapping from barycen tric coordinates to world coordinates we have b zzaXltbgt7 15 where X is a matrix made up of vectors along the edges of the triangle in world space Now we can de ne the entire mapping as zuz3X uiu3 16 Taking the gradient derivative with respect to u of this function we a z F 7 X 17 an 8 lt gt This is the deformation gradient For linear nite elements it is a matrix which we call F We can similarly de ne the time derivative as a z i V 18 an 8 lt gt Where V contains velocity differences rather than posi tion differences in world coordinates Finally we can write Green s strain as e FTP 7 I 19 6 Cauchy39s Infinitesimal Strain Suppose we write our deformation gradient as 8m 7 I D 20 an That is the deformation gradient is the identity no defor mation plus some amount of deformation Taking Green s strain we have 1 e 5IDTID7I 21 IIDTDDTD71 22 DT D 23 Now if D is ver small then DTD is much smaller than y D and 12D DT is a good estimate of Green s strain Furthermore this strain measure is linear which leads to all sorts of nice consequences Unfortunately it is not invariant to rotations which leads to artifacts if it is used for large deformations Cauchy s strain can be written as 71 962 an 7 H71 T 7 6 2ltauauigt 627720011gt I 24 7 Other Stress Measures We can see from the de nition of traction r an 25 that stress maps normals to forces However its important to distinguish where these normals and forces are de ned If both normals and forces are in world space the stress is know as a Cauchy stress and often written as 039 1f the stress maps normals in material space to forces in material space it is known as a Second PiolarKirchhoH stress and sometimes written as S A rst PiolarKirchhoH stress maps normals in material space to forces in world space as such it is especially convenient and is written P Now watch out n only one stress is being considered it is often written with 039 The stress we de ned earlier was actually a second PiolarKirchhoH stress Now its easy to convert between these stress since they all measure the same thing just in different coordinate systems Letting J detF we have P JaF T 26 and P FS 27 By having different ways of specifying stress we can choose whichever one is most convenient for a given application The rst PiolarKirchhoH is particularly attractive since it works with normals in the material space where they are constant and maps directly to forces in world space where they will be applied 8 Computing Forces Using Einstein s summation convention we can write an an 7 mm 28 1 8m 8m 6 7 5 lt0ui I Bu 76 29 1 1 5 mk kian nj 5ij 30 727 Aepp ij 2114627 31 A 5 kakaanBnp 3 32 HL ka kian m39 52739 33 1 77 5627027 34 1 1 1 5 lt ka5kian5nj 552739 35 A 5 kamxmm 7 3 36 Mka5kian5nj 527 37 077 7 l l aXba 7 225bm5ak5men5m 38 1 ka5ki5bm5w5njt7ij 39 1 A Eij 6bm6akkaanBWF6H 40 A ka5kp5bm5m5np5ij 41 M5bm5ak5kian5nj 42 Mka5ki5mb6m5nj 43 1 Zwaian m39ka5ki5ajt7ij 44 1 A 5ij 6aprn6np6ij 45 A ka5kp5ap5ij 46 M5aian5nj Mka ki aj 47 1 5 nb aianaij 48 1 6ijan5np5ap5ij 49 2Man5ai5nj 50 1 5 nb aianaij 51 1 2an5ai5m 52 A 5em6ij 627 53 1 1 an ai nj t7ij t 502739 54 an ai njaij 55 56 Reintroducing summations and integrating oVer the Volume of the element in material space 2 We have the force on node a is 3 3 3 fa 7v ZXM ZZ m aiaz j 57 n1 i1 j1 An alternate formulation from nite Volumes yields the following forces 1 fa igFafalnl 127L2 dang 58 here aim are the arearWeighted normals of the three faces incident to node a 9 A Few Other Useful Formulas We need the gradient of force for the stiffness matrix A deriVation similar to the above yields Spi ej akakj 59 AXib bk ak emXpd dm 60 MXib bj ak erpd dk 61 MXib bj akch cj ek 62 The change in force With respect to Velocity is aFui a Xib5bk ak5mXpd5dm 63 Xib5bj ak erpd5dk 64 Xib5bj mch5cj gk 65 For linear strain We have X I and zero stress the for mulas simplify to aFui m A ik ak em pm 66 Wij ak e pd 67 M Sij ak sm k 68 The change in force With respect to Velocity is aFui f 6i a gm5m 69 6ng gt W W p Wu ak e pd 70 Wu aww ek 71