We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an r -dimensional deformation subspace with O ( r ) cubature points, our method is able to evaluate sub-space forces at O ( r 2 ) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multi-modal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.

中文翻译：

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优化容积以有效整合子空间变形

我们提出了一种有效的方案来评估与子空间变形相关的非线性子空间力（和雅可比行列式）。我们解决的核心问题是在 3D 空间域上有效整合子空间力密度。与有效整合特定多项式子空间中的函数的高斯正交方案类似，我们提出了优化的容积方案（多维正交），以有效整合与特定子空间变形、特定材料和特定几何域相关的力密度。我们支持通用子空间变形运动学和非线性超弹性材料。为r维变形子空间○(r) 容积点，我们的方法能够评估子空间力○(r 2） 成本。我们还描述了用于运行时误差估计的复合容积规则。为各种子空间变形模型、几种超弹性材料（St.Venant-Kirchhoff、Mooney-Rivlin、Arruda-Boyce）和多模态（图形、触觉、声音）应用提供了结果。我们展示了比传统的蒙特卡洛积分更好的效率。