Affiliation:
1. The University of Hong Kong, Hong Kong
Abstract
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.
Funder
Research Grant Council of Hong Kong
Publisher
Association for Computing Machinery (ACM)
Subject
Computer Graphics and Computer-Aided Design
Reference52 articles.
1. Technical Section;Bender Jan;Adaptive Cloth Simulation Using Corotational Finite Elements. Comput. Graph.,2013
2. Ebb
3. Maurice A. Biot. 1938. Theory of elasticity with large displacements and rotations. Maurice A. Biot. 1938. Theory of elasticity with large displacements and rotations.
4. Javier Bonet and Richard D . Wood . 2008 . Nonlinear Continuum Mechanics for Finite Element Analysis (2 ed.). Cambridge University Press . Javier Bonet and Richard D. Wood. 2008. Nonlinear Continuum Mechanics for Finite Element Analysis (2 ed.). Cambridge University Press.
5. A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability
Cited by
5 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Analytic rotation-invariant modelling of anisotropic finite elements;ACM Transactions on Graphics;2024-08-09
2. Stabler Neo-Hookean Simulation: Absolute Eigenvalue Filtering for Projected Newton;Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers '24;2024-07-13
3. GIPC: Fast and Stable Gauss-Newton Optimization of IPC Barrier Energy;ACM Transactions on Graphics;2024-03-23
4. A Unified Analysis of Penalty-Based Collision Energies;Proceedings of the ACM on Computer Graphics and Interactive Techniques;2023-08-16
5. Data-Free Learning of Reduced-Order Kinematics;Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Proceedings;2023-07-23