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Anisotropic hyperelastic distortion energies are used to solve many problems in fields like computer graphics and engineering with applications in shape analysis, deformation, design, mesh parameterization, biomechanics, and more. However, formulating a robust anisotropic energy that is low order and yet sufficiently non-linear remains a challenging problem for achieving the convergence promised by Newton-type methods in numerical optimization. In this article, we propose a novel analytic formulation of an anisotropic energy that is smooth everywhere, low order, rotationally invariant, and at least twice differentiable. At its core, our approach utilizes implicit rotation factorizations with invariants of the Cauchy-Green tensor that arises from the deformation gradient. The versatility and generality of our analysis is demonstrated through a variety of examples, where we also show that the constitutive law suggested by the anisotropic version of the well-known As-Rigid-As-Possible energy is the foundational parametric description of both passive and active elastic materials. The generality of our approach means that we can systematically derive the force and force-Jacobian expressions for use in implicit and quasistatic numerical optimization schemes, and we can also use our analysis to rewrite, simplify, and speed up several existing anisotropic and isotropic distortion energies with guaranteed inversion safety. [ABSTRACT FROM AUTHOR]

Isotropic As-Rigid-As-Possible (ARAP) energy has been popular for shape editing, mesh parametrisation and soft-body simulation for almost two decades. However, a formulation using Cauchy-Green (CG) invariants has always been unclear, due to a rotation-polluted trace term that cannot be directly expressed using these invariants. We show how this incongruent trace term can be understood via an implicit relationship to the CG invariants. Our analysis reveals this relationship to be a polynomial where the roots equate to the trace term, and where the derivatives also give rise to closed-form expressions of the Hessian to guarantee positive semi-definiteness for a fast and concise Newton-type implicit time integration. A consequence of this analysis is a novel analytical formulation to compute rotations and singular values of deformation-gradient tensors without explicit/numerical factorization which is significant, resulting in up-to 3.5× speedup and benefits energy function evaluation for reducing solver time. We validate our energy formulation by experiments and comparison, demonstrating that our resulting eigendecomposition using the CG invariants is equivalent to existing ARAP formulations. We thus reveal isotropic ARAP energy to be a member of the "Cauchy-Green club", meaning that it can indeed be defined using CG invariants and therefore that the closed-form expressions of the resulting Hessian are shared with other energies written in their terms. [ABSTRACT FROM AUTHOR]