Kirchhoff-Love Shells with Arbitrary Hyperelastic Materials

Jiahao Wen, Jernej Barbič

Kirchhoff-Love shells are commonly used in many branches of engineering, including in computer graphics, but have so far been simulated only under limited nonlinear material options. We derive the Kirchhoff-Love thin-shell mechanical energy for an arbitrary 3D volumetric hyperelastic material, including isotropic materials, anisotropic materials, and materials whereby the energy includes both even and odd powers of the principal stretches. We do this by starting with any 3D hyperelastic material, and then analytically computing the corresponding thin-shell energy limit. This explicitly identifies and separates in-plane stretching and bending terms, and avoids numerical quadrature. Thus, in-plane stretching and bending are shown to originate from one and the same process (volumetric elasticity of thin objects), as opposed to from two separate processes as done traditionally in cloth simulation. Because we can simulate materials that include both even and odd powers of stretches, we can accommodate standard mesh distortion energies previously employed for 3D solid simulations, such as Symmetric ARAP and Co-rotational materials. We relate the terms of our energy to those of prior work on Kirchhoff-Love thin-shells in computer graphics that assumed small in-plane stretches, and demonstrate the visual difference due to the presence of our exact stretching and bending terms. Furthermore, our formulation allows us to categorize all distinct hyperelastic Kirchhoff-Love thin-shell energies. Specifically, we prove that for Kirchhoff-Love thin-shells, the space of all hyperelastic materials collapses to two-dimensional hyperelastic materials. This observation enables us to create an interface for the design of thin-shell Kirchhoff-Love mechanical energies, which in turn enables us to create thin-shell materials that exhibit arbitrary stiffness profiles under large deformations.

Kirchhoff-Love Shells with Arbitrary Hyperelastic Materials

Neural Stress Fields for Reduced-order Elastoplasticity and Fracture

Zeshun Zong, Xuan Li, Minchen Li, Maurizio M. Chiaramonte, Wojciech Matusik, Eitan Grinspun, Kevin Carlberg, Chenfanfu Jiang, Peter Yichen Chen

We propose a hybrid neural network and physics framework for reduced-order modeling of elastoplasticity and fracture. State-of-the-art scientific computing models like the Material Point Method (MPM) faithfully simulate large-deformation elastoplasticity and fracture mechanics. However, their long runtime and large memory consumption render them unsuitable for applications constrained by computation time and memory usage, e.g., virtual reality. To overcome these barriers, we propose a reduced-order framework. Our key innovation is training a low-dimensional manifold for the Kirchhoff stress field via an implicit neural representation. This low-dimensional neural stress field (NSF) enables efficient evaluations of stress values and, correspondingly, internal forces at arbitrary spatial locations. In addition, we also train neural deformation and affine fields to build low-dimensional manifolds for the deformation and affine momentum fields. These neural stress, deformation, and affine fields share the same low-dimensional latent space, which uniquely embeds the high-dimensional simulation state. After training, we run new simulations by evolving in this single latent space, which drastically reduces the computation time and memory consumption. Our general continuum-mechanics-based reduced-order framework is applicable to any phenomena governed by the elasto-dynamics equation. To showcase the versatility of our framework, we simulate a wide range of material behaviors, including elastica, sand, metal, non-Newtonian fluids, fracture, contact, and collision. We demonstrate dimension reduction by up to 100,000× and time savings by up to 10×.

Neural Stress Fields for Reduced-order Elastoplasticity and Fracture

Power Plastics: A Hybrid Lagrangian/Eulerian Solver for Mesoscale Inelastic Flows

Ziyin Qu, Minchen Li, Yin Yang, Chenfanfu Jiang, Fernando de Goes

We present a novel hybrid Lagrangian/Eulerian method for simulating inelastic flows that generates high-quality particle distributions with adaptive volumes. At its core, our approach integrates an updated Lagrangian time discretization of continuum mechanics with the Power Particle-In-Cell geometric representation of deformable materials. As a result, we obtain material points described by optimized density kernels that precisely track the varying particle volumes both spatially and temporally. For efficient CFL-rate simulations, we also propose an implicit time integration for our system using a non-linear Gauss-Seidel solver inspired by X-PBD, viewing Eulerian nodal velocities as primal variables. We demonstrate the versatility of our method with simulations of mesoscale bubbles, sands, liquid, and foams.

Power Plastics: A Hybrid Lagrangian/Eulerian Solver for Mesoscale Inelastic Flows

A Physically-inspired Approach to the Simulation of Plant Wilting

F. Maggioli, J. Klein, T. Hädrich, E. Rodolà, W. Pałubicki, S. Pirk, D. L. Michels.

Plants are among the most complex objects to be modeled in computer graphics. While a large body of work is concerned with structural modeling and the dynamic reaction to external forces, our work focuses on the dynamic deformation caused by plant internal wilting processes. To this end, we motivate the simulation of water transport inside the plant which is a key driver of the wilting process. We then map the change of water content in individual plant parts to branch stiffness values and obtain the wilted plant shape through a position based dynamics simulation. We show, that our approach can recreated measured wilting processes and does so with a higher fidelity than approaches ignoring the internal water flow. Realistic plant wilting is not only important in a computer graphics context but can also aid the development of machine learning algorithms in agricultural applications through the generation of synthetic training data.

A Physically-inspired Approach to the Simulation of Plant Wilting

Real-time Height-field Simulation of Sand and Water Mixtures

Haozhe Su, Siyu Zhang, Zherong Pan, Mridul Aanjaneya, Xifeng Gao, Kui Wu

We propose a height-field-based real-time simulation method for sand and water mixtures. Inspired by the shallow-water assumption, our approach extends the governing equations to handle two-phase flows of sand and water using height fields. Our depth-integrated governing equations can model the elastoplastic behavior of sand, as well as sand-water-mixing phenomena such as friction, diffusion, saturation, and momentum exchange. We further propose an operator-splitting time integrator that is both GPU-friendly and stable under moderate time step sizes. We have evaluated our method on a set of benchmark scenarios involving large bodies of heterogeneous materials, where our GPU-based algorithm runs at real-time frame rates. Our method achieves a desirable trade-off between fidelity and performance, bringing an unprecedentedly immersive experience for real-time applications.

Real-time Height-field Simulation of Sand and Water Mixtures

High-Order Moment-Encoded Kinetic Simulation of Turbulent Flows

Wei Li, Tongtong Wang, Zherong Pan, Xifeng Gao, Kui Wu, Mathieu Desbrun

Kinetic solvers for incompressible fluid simulation were designed to run efficiently on massively parallel architectures such as GPUs. While these lattice Boltzmann solvers have recently proven much faster and more accurate than the macroscopic Navier-Stokes-based solvers traditionally used in graphics, it systematically comes at the price of a very large memory requirement: a mesoscopic discretization of statistical mechanics requires over an order of magnitude more variables per grid node than most fluid solvers in graphics. In order to open up kinetic simulation to gaming and simulation software packages on commodity hardware, we propose a High-Order Moment-Encoded Lattice-Boltzmann-Method solver which we coined HOME-LBM, requiring only the storage of a few moments per grid node, with little to no loss of accuracy in the typical simulation scenarios encountered in graphics. We show that our lightweight and lightspeed fluid solver requires three times less memory and runs ten times faster than state-of-the-art kinetic solvers, for a nearly-identical visual output.

High-Order Moment-Encoded Kinetic Simulation of Turbulent Flows

Subspace-Preconditioned GPU Projective Dynamics with Contact for Cloth Simulation

Xuan Li, Yu Fang, Lei Lan, Huamin Wang, Yin Yang, Minchen Li, Chenfanfu Jiang

We propose an efficient cloth simulation method that combines the merits of two drastically different numerical procedures, namely the subspace integration and parallelizable iterative relaxation. We show those two methods can be organically coupled within the framework of projective dynamics (PD), where both low- and high-frequency cloth motions are effectively and efficiently computed. Our method works seamlessly with the state-of-the-art contact handling algorithm, the incremental potential contact (IPC), to offer the non-penetration guarantee of the resulting animation. Our core ingredient centers around the utilization of subspace for the expedited convergence of Jacobi-PD. This involves solving the reduced global system and smartly employing its precomputed factorization. In addition, we incorporate a time-splitting strategy to handle the frictional self-contacts. Specifically, during the PD solve, we employ a quadratic proxy to approximate the contact barrier. The prefactorized subspace system matrix is exploited in a reduced-space LBFGS. The LBFGS method starts with the reduced system matrix of the rest shape as the initial
Hessian approximation, incorporating contact information into the reduced system progressively, while the full-space Jacobi iteration captures high-frequency details. Furthermore, we address penetration issues through a penetration correction step. It minimizes an incremental potential without elasticity using Newton-PCG. Our method can be efficiently executed on modern GPUs. Experiments show significant performance improvements over existing GPU solvers for high-resolution cloth simulation.

Subspace-Preconditioned GPU Projective Dynamics with Contact for Cloth Simulation

The Design Space of Kirchhoff Rods

Christian Hafner, Bernd Bickel

The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions, and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods, and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.

The Design Space of Kirchhoff Rods

Constrained Delaunay Tetrahedrization: A Robust and Practical Approach

Lorenzo Diazzi, Daniele Panozzo, Amir Vaxman, Marco Attene

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known TetGen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in TetGen’s theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate.

Constrained Delaunay Tetrahedrization: A Robust and Practical Approach

SIGGRAPH Asia 2023