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 paper, 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 speedup several existing anisotropic and isotropic distortion energies with guaranteed inversion-safety.

各向异性超弹性畸变能用于解决计算机图形学和工程等领域的许多问题，并应用于形状分析、变形、设计、网格参数化、生物力学等。然而，制定低阶且足够非线性的鲁棒各向异性能量对于实现数值优化中牛顿型方法所承诺的收敛仍然是一个具有挑战性的问题。在本文中，我们提出了一种新颖的各向异性能量解析公式，该公式处处平滑、低阶、旋转不变且至少两次可微。我们的方法的核心是利用隐式旋转分解以及由变形梯度产生的柯西-格林张量的不变量。我们的分析的多功能性和普遍性通过各种例子得到证明，其中我们还表明，众所周知的尽可能刚性能量的各向异性版本所提出的本构定律是被动和被动能量的基本参数描述。活性弹性材料。我们的方法的通用性意味着我们可以系统地导出用于隐式和准静态数值优化方案的力和力雅可比表达式，并且我们还可以使用我们的分析来重写、简化和加速几个现有的各向异性和各向同性畸变能量，并保证反转安全。