Eurographics/ ACM SIGGRAPH Symposium on Computer Animation (2014) Vladlen Koltun and Eftychios Sifakis (Editors) Stable Orthotropic Materials Yijing Li and Jernej Barbicˇ University of Southern California, United States Abstract Isotropic Finite Element Method (FEM) deformable object simulations are widely used in computer graphics. Several applications (wood, plants, muscles) require modeling the directional dependence of the material elas- tic properties in three orthogonal directions. We investigate orthotropic materials, a special class of anisotropic materials where the shear stresses are decoupled from normal stresses. Orthotropic materials generalize trans- versely isotropic materials, by exhibiting different stiffnesses in three orthogonal directions. Orthotropic materials are, however, parameterized by nine values that are difficult to tune in practice, as poorly adjusted settings easily lead to simulation instabilities. We present a user-friendly approach to setting these parameters that is guaran- teed to be stable. Our approach is intuitive as it extends the familiar intuition known from isotropic materials. We demonstrate our technique by augmenting linear corotational FEM implementations with orthotropic materials. Categories and Subject Descriptors (according to ACM CCS): I.3.6 [Computer Graphics]: Methodology and Techniques—Interaction Techniques, I.6.8 [Simulation and Modeling]: Types of Simulation—Animation 1. Introduction parameters, by parameterizing the six Poisson’s ratios using a stable one-dimensional parameter family, similar to the in- Simulation of three-dimensional solid deformable models tuition from isotropic simulation. This makes it possible to is important in many applications in computer graphics, easily augment existing simulation solvers with stable and robotics, special effects and virtual reality. Most applications intuitive anisotropic effects. We support large deformations in these fields have been limited to isotropic materials, i.e., using corotational Finite Element Method simulation. materials that are equally elastic in all directions. Many real materials are, however, stiffer in some directions than oth- 2. Related Work ers. The space of such anisotropic materials is vast and not easy to navigate, tune or control. In this paper, we study or- Anisotropic materials are discussed in many references, see, thotropic materials. Orthotropic materials exhibit different e.g. [Bow11]. Transversely isotropic hyperelastic materials stiffness in three orthogonal directions; formally, they pos- were presented by Bonet and Burton [BB98]. Picinbono sess three orthogonal planes of rotational symmetry. They et al. [PDA01] proposed a non-linear FEM model to sim- form an intuitive subset of all anisotropic materials, as they ulate soft tissues with large deformations and transversely generalize the familiar isotropic, and transversely isotropic, isotropic behavior. Thije et al. [TTAH07] addressed the in- materials to materials with three different stiffness values stabilities that occur under strong anisotropy, and provided a in some three orthogonal directions. Although simpler than simple updated Lagrangian FEM scheme to handle the prob- fully general anisotropic materials, orthotropic materials still lem. Picinbono et al. [PLDA00] described a surgery simu- require tuning nine independent parameter values. In prac- lator that can model linear transversely isotropic materials tice, this task is difficult due to the large number of param- at haptic rates and also presented a nonlinear transversely eters and because many of the settings lead to unstable sim- isotropic model for medical simulation [PDA03]. Sermesant ulations in a non-obvious way. In this paper, we study or- et al. [SCD∗01,SDA06] and Talbot et al. [TMD∗13] adopted thotropic materials from the point of view of practical sim- a transversely isotropic material in constructing an electro- ulation in computer graphics and related fields. We demon- mechanical model of the heart. Allard et al. [AMC∗09] used strate how to intuitively and stably tune orthotropic material a 2D anisotropic material to simulate thin soft tissue tearing, c The Eurographics Association 2014. Yijing Li & Jernej Barbiˇc/ Stable Orthotropic Materials Comas et al. [CTA∗08] implemented a transversely isotropic volume of tet e; and Be is a 6 × 12 matrix determined by the visco-hyperelastic model on the GPU, and Teran [TSB∗05] initial shape of tet e (see [Sha90] or [MG04]). and Sifakis [SNF05, Sif07] simulated human muscles with a transversely isotropic, quasi-incompressible model. Irv- Unlike isotropic materials that are parameterized by a sin- ing et al. [ITF04] proposed a robust, large-deformation in- gle Young’s modulus and Poisson’s ratio, orthotropic mate- vertible simulation method and demonstrated it with trans- rials have three different Young’s moduli E1;E2;E3, one for versely isotropic models. Previous anisotropic applications each orthogonal direction, and six Poisson’s ratios ni j; for in computer graphics focused on transversely isotropic ma- i 6= j; only three of which are independent. Young’s mod- terials where two directions have equal stiffness, leading ulus Ei gives the stiffness of the material when loaded in to five tunable parameters. We generalize materials to or- orthogonal direction i: Poisson’s ratio ni j gives the contrac- thotropic materials with three distinct stiffnesses in three tion in direction j when the extension is applied in direc- orthogonal directions, and present an intuitive approach to tion i: In a general anisotropic material, both the normal and tune the resulting nine parameters. To the best of our knowl- shear components of strain affect both the normal and shear edge, we are first work in computer graphics to analyze components of stress, i.e., matrix C is dense. In orthotropic orthotropic materials in substantial detail. Previous papers materials, however, the normal and shear components are on orthotropic materials in engineering assumed that the decoupled: normal stresses only cause normal strains, and nine orthotropic parameters are given or measured from shear stresses only cause shear strains. Furthermore, individ- real materials [VBCW81], whereas we provide an intuitive ual shear stresses in the 12;23;31 planes are decoupled from way for the users to tune them and ensure they are stable. each other: strain ei j (i 6= j) only depends on stress si j via a Our work uses corotational linear FEM materials introduced scalar parameter (shear modulus) mi j: Under these assump- in [MG04]. Construction of the stiffness matrix for linear tions, the elasticity tensor has 9 free parameters and takes a FEM materials can also be found, for example, in [Sha90]. block-diagonal form. It is easiest to first state its inverse 2 1 − n21 − n31 0 0 0 3 E1 E2 E3 3. Orthotropic Materials 6− n12 1 − n32 0 0 0 7 6 E1 E2 E3 7 6− n13 − n23 1 0 0 0 7 We now introduce orthotropic materials. Given the defor- 6 E1 E2 E3 7 Sortho = 6 7: (4) mation gradient F; the Green-Lagrange strain is defined as 6 0 0 0 1 0 0 7 6 m12 7 e3×3 = (FT F − I)=2; and the Cauchy stress s 3×3 gives 6 0 0 0 0 1 0 7 4 m23 5 the elastic forces per surface area in a unit direction n; 0 0 0 0 0 1 as s 3×3n [Sha90]. Note that we can operate with Cauchy m31 stresses here as they are equivalent to other forms of stresses The orthotropic elasticity tensor is then (Piola) due to the small-deformation analysis; we achieve large deformations via co-rotational linear FEM [MG04]. A 0 The 6 × 6 elasticity tensor S relates strain e to stress s via C = S −1 = ; for (5) ortho ortho 0 B e = S s; where we have unrolled the 3×3 symmetric matri- 3×3 3×3 2 3 ces e = [ei j]i j and s = [si j]i j into 6-vectors, using the E1(1 − n23n32) E2(n12 + n32n13) E3(n13 + n12n23) 12, 23, 31 ordering of the shear components as in [Sha90]: A = ¡4E1(n21 + n31n23) E2(1 − n13n31) E3(n23 + n21n13)5; E ( + ) E ( + ) E ( − ) T 1 n31 n21n32 2 n32 n12n31 3 1 n12n21 e = [e11 e22 e33 2e12 2e23 2e31] ; (1) (6) T s = [s11 s22 s33 s12 s23 s31] : (2) 2 3 m12 0 0 Components 11;22;33 are called normal components, B = 4 0 m23 0 5; and (7) whereas 12;23;31 are referred to as shear components. The 0 0 m31 inverse elasticity tensor C = S −1 relates s to e; via s = 1 ¡ = : (8) C e: The elasticity tensor must be symmetric and therefore it 1 − n12n21 − n23n32 − n31n13 − 2n21n32n13 has 21 independent entries for a general anisotropic material, Equations4 and5 give elasticity tensors with respect to the world coordinate axes. A general orthotropic material, 2C C C C C C 3 11 12 13 14 15 16 however, assumes the block-diagonal form given in Equa- 6 C22 C23 C24 C25 C267 6 7 tions4 and5 only in a special orthogonal basis, given by 6 C33 C34 C35 C367 C = 6 7: (3) the three principal axes where the stiffnesses are E1;E2;E3: 6 C44 C C 7 6 45 467 In other bases (including world-coordinate axes), its form 4 Sym: C55 C565 looks generic, as in Equation3. Therefore, to model or- C 66 thotropic materials whose principal axes are not aligned with Once C is known, the stiffness matrix for a linear tetrahe- the world axes, we need to convert elasticity tensors from dral element is computed as Ke = V eBeT C eBe; where V e is one basis to another. For a basis given by a rotation Q; the c The Eurographics Association 2014. Yijing Li & Jernej Barbiˇc/ Stable Orthotropic Materials elasticity tensor C transforms as follows: practice, it is very tedious to tune these parameters, as sub- (1) (2) optimal values easily cause the simulation to explode, or in- T K 2K troduce undue stiffness or other poor simulation behavior.