Micropolar Elasticity in Physically-Based Animation

Fabian Löschner, José Antonio Fernández-Fernández, Stefan Rhys Jeske, Andreas Longva, Jan Bender
Proceedings of the ACM on Computer Graphics and Interactive Techniques (SCA)

We explore micropolar materials for the simulation of volumetric deformable solids. In graphics, micropolar models have only been used in the form of one-dimensional Cosserat rods, where a rotating frame is attached to each material point on the one-dimensional centerline. By carrying this idea over to volumetric solids, every material point is associated with a microrotation, an independent degree of freedom that can be coupled to the displacement through a material's strain energy density. The additional degrees of freedom give us more control over bending and torsion modes of a material. We propose a new orthotropic micropolar curvature energy that allows us to make materials stiff to bending in specific directions.

For the simulation of dynamic micropolar deformables we propose a novel incremental potential formulation with a consistent FEM discretization that is well suited for the use in physically-based animation. This allows us to easily couple micropolar deformables with dynamic collisions through a contact model inspired from the Incremental Potential Contact (IPC) approach. For the spatial discretization with FEM we discuss the challenges related to the rotational degrees of freedom and propose a scheme based on the interpolation of angular velocities followed by quaternion time integration at the quadrature points.

In our evaluation we validate the consistency and accuracy of our discretization approach and demonstrate several compelling use cases for micropolar materials. This includes explicit control over bending and torsion stiffness, deformation through prescription of a volumetric curvature field and robust interaction of micropolar deformables with dynamic collisions.

» Show BibTeX

author = {L\"{o}schner, Fabian and Fern\'{a}ndez-Fern\'{a}ndez, Jos\'{e} Antonio and Jeske, Stefan Rhys and Longva, Andreas and Bender, Jan},
title = {Micropolar Elasticity in Physically-Based Animation},
year = {2023},
issue_date = {August 2023},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {6},
number = {3},
url = {https://doi.org/10.1145/3606922},
doi = {10.1145/3606922},
journal = {Proceedings of the ACM on Computer Graphics and Interactive Techniques},
month = {aug},
articleno = {46},
numpages = {24}

Higher-Order Time Integration for Deformable Solids

Fabian Löschner, Andreas Longva, Stefan Rhys Jeske, Tassilo Kugelstadt, Jan Bender
Computer Graphics Forum

Visually appealing and vivid simulations of deformable solids represent an important aspect of physically based computer animation. For the temporal discretization, it is customary in computer animation to use first-order accurate integration methods, such as Backward Euler, due to their simplicity and robustness. Although there is notable research on second-order methods, their use is not widespread. Many of these well-known methods have significant drawbacks such as severe numerical damping or scene-dependent time step restrictions to ensure stability. In this paper, we discuss the most relevant requirements on such methods in computer animation and motivate the interest beyond first-order accuracy. Keeping these requirements in mind, we investigate several promising methods from the families of diagonally implicit Runge-Kutta (DIRK) and Rosenbrock methods which currently do not appear to have considerable popularity in this field. We show that the usage of such methods improves the visual quality of physical animations. In addition, we demonstrate that they allow distinctly more control over damping at lower computational cost than classical methods. As part of our theoretical contribution, we review aspects of simulations that are often considered more intricate with higher-order methods, such as contact handling. To this end, we derive an implicit linearized contact model based on a predictor-corrector approach that leads to consistent behavior with higher-order integrators as predictors. Our contact model is well suited for the simulation of stiff, nonlinear materials with the integration methods presented in this paper and more common methods such as Backward Euler alike.

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author = {Fabian L{\"{o}}schner and Andreas Longva and Stefan Jeske and Tassilo Kugelstadt and Jan Bender},
title = {Higher-Order Time Integration for Deformable Solids},
year = {2020},
journal = {Computer Graphics Forum},
volume = {39},
number = {8}

Fast Corotated FEM using Operator Splitting

Tassilo Kugelstadt, Dan Koschier, Jan Bender
Computer Graphics Forum (SCA)

In this paper we present a novel operator splitting approach for corotated FEM simulations. The deformation energy of the corotated linear material model consists of two additive terms. The first term models stretching in the individual spatial directions and the second term describes resistance to volume changes. By formulating the backward Euler time integration scheme as an optimization problem, we show that the first term is invariant to rotations. This allows us to use an operator splitting approach and to solve both terms individually with different numerical methods. The stretching part is solved accurately with an optimization integrator, which can be done very efficiently because the system matrix is constant over time such that its Cholesky factorization can be precomputed. The volume term is solved approximately by using the compliant constraints method and Gauss-Seidel iterations. Further, we introduce the analytic polar decomposition which allows us to speed up the extraction of the rotational part of the deformation gradient and to recover inverted elements. Finally, this results in an extremely fast and robust simulation method with high visual quality that outperforms standard corotated FEMs by more than two orders of magnitude and even the fast but inaccurate PBD and shape matching methods by more than one order of magnitude without having their typical drawbacks. This enables a very efficient simulation of complex scenes containing more than a million elements.

» Show BibTeX

author = {Tassilo Kugelstadt and Dan Koschier and Jan Bender},
title = {Fast Corotated FEM using Operator Splitting},
year = {2018},
journal = {Computer Graphics Forum (SCA)},
volume = {37},
number = {8}

Direct Position-Based Solver for Stiff Rods

Crispin Deul, Tassilo Kugelstadt, Marcel Weiler, Jan Bender
Computer Graphics Forum

In this paper, we present a novel direct solver for the efficient simulation of stiff, inextensible elastic rods within the Position-Based Dynamics (PBD) framework. It is based on the XPBD algorithm, which extends PBD to simulate elastic objects with physically meaningful material parameters. XPBD approximates an implicit Euler integration and solves the system of non-linear equations using a non-linear Gauss-Seidel solver. However, this solver requires many iterations to converge for complex models and if convergence is not reached, the material becomes too soft. In contrast we use Newton iterations in combination with our direct solver to solve the non-linear equations which significantly improves convergence by solving all constraints of an acyclic structure (tree), simultaneously. Our solver only requires a few Newton iterations to achieve high stiffness and inextensibility. We model inextensible rods and trees using rigid segments connected by constraints. Bending and twisting constraints are derived from the well-established Cosserat model. The high performance of our solver is demonstrated in highly realistic simulations of rods consisting of multiple ten-thousand segments. In summary, our method allows the efficient simulation of stiff rods in the Position-Based Dynamics framework with a speedup of two orders of magnitude compared to the original XPBD approach.

» Show BibTeX

author = {Crispin Deul and Tassilo Kugelstadt and Marcel Weiler and Jan Bender},
title = {Direct Position-Based Solver for Stiff Rods},
year = {2018},
journal = {Computer Graphics Forum},
volume = {37},
number = {6},
pages = {313-324},
keywords = {physically based animation, animation, Computing methodologies → Physical simulation},
doi = {10.1111/cgf.13326},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/cgf.13326},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.13326},

A Robust Method to Extract the Rotational Part of Deformations

Matthias Müller, Jan Bender, Nuttapong Chentanez, Miles Macklin
ACM SIGGRAPH Motion in Games

We present a novel algorithm to extract the rotational part of an arbitrary 3x3 matrix. This problem lies at the core of two popular simulation methods in computer graphics, the co-rotational Finite Element Method and Shape Matching techniques. In contrast to the traditional method based on polar decomposition, degenerate configurations and inversions are handled robustly and do not have to be treated in a special way. In addition, our method can be implemented with only a few lines of code without branches which makes it particularly well suited for GPU-based applications. We demonstrate the robustness, coherence and efficiency of our method by comparing it to stabilized polar decomposition in several simulation scenarios.

» Show BibTeX

author = {Matthias M\"{u}ller and Jan Bender and Nuttapong Chentanez and Miles Macklin},
title = {A Robust Method to Extract the Rotational Part of Deformations},
booktitle = {Proceedings of ACM SIGGRAPH Conference on Motion in Games},
series = {MIG '16},
year = {2016},
publisher = {ACM}

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