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Higher-order finite elements for embedded simulation
ACM Transactions on Graphics  ( IF 6.2 ) Pub Date : 2020-11-27 , DOI: 10.1145/3414685.3417853
Andreas Longva 1 , Fabian Löschner 1 , Tassilo Kugelstadt 1 , José Antonio Fernández-Fernández 1 , Jan Bender 1
Affiliation  

As demands for high-fidelity physics-based animations increase, the need for accurate methods for simulating deformable solids grows. While higherorder finite elements are commonplace in engineering due to their superior approximation properties for many problems, they have gained little traction in the computer graphics community. This may partially be explained by the need for finite element meshes to approximate the highly complex geometry of models used in graphics applications. Due to the additional perelement computational expense of higher-order elements, larger elements are needed, and the error incurred due to the geometry mismatch eradicates the benefits of higher-order discretizations. One solution to this problem is the embedding of the geometry into a coarser finite element mesh. However, to date there is no adequate, practical computational framework that permits the accurate embedding into higher-order elements. We develop a novel, robust quadrature generation method that generates theoretically guaranteed high-quality sub-cell integration rules of arbitrary polynomial accuracy. The number of quadrature points generated is bounded only by the desired degree of the polynomial, independent of the embedded geometry. Additionally, we build on recent work in the Finite Cell Method (FCM) community so as to tackle the severe ill-conditioning caused by partially filled elements by adapting an Additive-Schwarz-based preconditioner so that it is suitable for use with state-of-the-art non-linear material models from the graphics literature. Together these two contributions constitute a general-purpose framework for embedded simulation with higher-order finite elements. We finally demonstrate the benefits of our framework in several scenarios, in which second-order hexahedra and tetrahedra clearly outperform their first-order counterparts.

中文翻译:

用于嵌入式仿真的高阶有限元

随着对基于物理的高保真动画的需求增加,对模拟可变形实体的精确方法的需求也在增长。虽然高阶有限元在工程中很常见,因为它们对许多问题具有出色的近似特性,但它们在计算机图形学界几乎没有受到关注。这可以部分解释为需要有限元网格来近似图形应用程序中使用的模型的高度复杂的几何形状。由于高阶元素的额外每元素计算费用,需要更大的元素,并且由于几何不匹配而产生的误差消除了高阶离散化的好处。该问题的一种解决方案是将几何嵌入到较粗的有限元网格中。然而,迄今为止还没有足够的,实用的计算框架,允许准确嵌入到高阶元素中。我们开发了一种新颖、稳健的正交生成方法,可以生成理论上保证的任意多项式精度的高质量子单元积分规则。生成的正交点的数量仅受多项式所需次数的限制,与嵌入的几何图形无关。此外,我们在有限细胞法 (FCM) 社区的最新工作的基础上,通过调整基于 Additive-Schwarz 的预处理器来解决由部分填充元素引起的严重病态,使其适用于 state-of - 来自图形文献的最先进的非线性材料模型。这两个贡献共同构成了用于具有高阶有限元的嵌入式仿真的通用框架。
更新日期:2020-11-27
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