Constitutive models. Constitutive model: determines P in terms of deformation

Transcription

1 Constitutive models Constitutive model: determines P in terms of deformation Elastic material: P depends only on current F Hyperelastic material: work is independent of path strain energy density function Ψ(F) analogous to f = U P = dψ/df P ij = dψ/df ij

2 Constitutive models Different choices of Ψ different models of elastic materials No deformation: Ψ(I) = 0 Rotation independence: Ψ(R F) = Ψ(F) Decompose F = R S. Ψ depends only on S, min when S = I

3 Corotated linear elasticity Quadratic in S: Ψ(F) = μ S I 2 + λ/2 tr 2 (S I) P(F) = 2 μ R (S I) + λ R tr (S I) μ, λ: Lamé parameters μ: resistance to stretching, shearing λ: resistance to volume change (because tr (S I) det F 1) Related to Young s modulus, Poisson s ratio

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5 Other constitutive models St. Venant Kirchhoff (with E = ½ (F T F I)): Easier to compute: no need for polar decomposition Ψ, P polynomial in F Ψ(F) = μ E 2 + λ/2 tr 2 E StVK corotated linear Poor behaviour in compression 1

6 Other constitutive models Neo-Hookean (with C = F T F, I C = tr C, J = det F): Ψ(F) = μ/2 (I C 3) μ log J + λ/2 (log J) 2 Correctly models volume change using J = det F Good for nearly incompressible materials (e.g. rubber, flesh) StVK corotated linear Undefined under collapse, inversion (J 0) Stable neo-hookean [Smith et al. 2018] 1 neo-hookean

7 Anisotropy All these materials are isotropic: independent of direction of stretching Ψ(F) = Ψ(F Q) Ψ(F) = Ψ(U Σ V T ) = Ψ(Σ): Ψ only depends on principal strains Equivalently, only depends on invariants of C = F T F Examples of anisotropic materials: woven fabrics, composites [Kharevych et al. 2009]

8 From theory to simulation State φ(x), φ (X) Deformation gradient F(X) = dφ/dx Stress P(F) = dψ/df Force density ρ φ = div P + f ext (everything in terms of material space!) How to discretize φ(x)? How to compute F? How to compute div P?

9 Meshes Divide space into elements (usually triangles, tetrahedra), put samples at vertices (a.k.a nodes) Reconstruction: linear / polynomial within each element Differentiation: naively, gradient only defined in element interior Elements should be well-shaped : close to regular Software to generate mesh for given shape: Triangle, TetGen

10 Meshes for elasticity X i x i Create triangulated mesh in material space Each vertex stores fixed X i, varying x i = φ(x i ), v i = φ (X i ) Each element stores indices of vertices i 1, i 2, i 3 (, i 4 ) Reconstruct φ(x) by piecewise linear interpolation. (Other shapes e.g. quads, higher-order interpolation also possible)

11 Shape functions Interpolation can be expressed as linear combination of shape functions: f(x) = f 1 N 1 (x) + f 2 N 2 (x) + + f n N n (x) where N i (x i ) = 1, N i (x j ) = 0 for j i = f i f(x) N i (x)

12 Shape functions In particular, φ(x) = x 1 N 1 (X) + x 2 N 2 (X) + F(X) = x 1 dn 1 /dx + x 2 dn 2 /dx + Piecewise linear interpolation F, P constant on each element But what is div P? Zero in element interior, undefined on element boundaries

13 The finite element method FEM in graphics: Sifakis and Barbič, Part 1, Ch 4: Discretization FEM theory: Bathe, Finite Element Procedures (especially Ch 3.3, 4.2)