The Extended Position Based Dynamics (XPBD) approach of Macklin et al. (2016) addresses issues with iteration-dependent behavior in the original Position Based Dynamics (Müller et al., 2007) (PBD). PBD itself is a powerful method for the real-time simulation of elastic objects, however, it is limited in its application to hyperelastic solids. It can only treat models with a strain energy density that is quadratic in some notion of constraint. Furthermore, we show that even when applicable the formulation does not always lead to convergent behaviors with hyperelasticity. We isolate the root cause to be the approximate linearization of the nonlinear backward Euler systems utilized by XPBD. We provide two fixes to these terms that allow for convergent behavior. The first (B-PXPBD) is a small modification to an existing XPBD code, but can only be used with models addressable by the original XPBD. The second (FP-PXPBD) is a more general formulation that extends XPBD (and our residual correction) to arbitrary hyperelasticity. We show that our modifications allow for convergent behavior that rivals accurate techniques like Newton’s method when the computational budget is large without sacrificing the stable and robust behavior exhibited by the original PBD and XPBD when the computational budget is limited.

Position based dynamics [Müller et al. 2007] is a powerful technique for simulating a variety of materials. Its primary strength is its robustness when run with limited computational budget. Even though PBD is based on the projection of static constraints, it does not work well for quasistatic problems. This is particularly relevant since the efficient creation of large data sets of plausible, but not necessarily accurate elastic equilibria is of increasing importance with the emergence of quasistatic neural networks [Bailey et al. 2018; Chentanez et al. 2020; Jin et al. 2022; Luo et al. 2020]. Recent work [Macklin et al. 2016] has shown that PBD can be related to the Gauss-Seidel approximation of a Lagrange multiplier formulation of backward Euler time stepping, where each constraint is solved/projected independently of the others in an iterative fashion. We show that a position-based, rather than constraint-based nonlinear Gauss-Seidel approach resolves a number of issues with PBD, particularly in the quasistatic setting. Our approach retains the essential PBD feature of stable behavior with constrained computational budgets, but also allows for convergent behavior with expanded budgets. We demonstrate the efficacy of our method on a variety of representative hyperelastic problems and show that both successive over relaxation (SOR), Chebyshev and multiresolution-based acceleration can be easily applied.

We present a comprehensive neural network to model the deformation of human soft tissues including muscle, tendon, fat and skin. Our approach provides kinematic and active correctives to linear blend skinning [Magnenat-Thalmann et al. 1989] that enhance the realism of soft tissue deformation at modest computational cost. Our network accounts for deformations induced by changes in the underlying skeletal joint state as well as the active contractile state of relevant muscles. Training is done to approximate quasistatic equilibria produced from physics-based simulation of hyperelastic soft tissues in close contact. We use a layered approach to equilibrium data generation where deformation of muscle is computed first, followed by an inner skin/fascia layer, and lastly a fat layer between the fascia and outer skin. We show that a simple network model which decouples the dependence on skeletal kinematics and muscle activation state can produce compelling behaviors with modest training data burden. Active contraction of muscles is estimated using inverse dynamics where muscle moment arms are accurately predicted using the neural network to model kinematic musculotendon geometry. Results demonstrate the ability to accurately replicate compelling musculoskeletal and skin deformation behaviors over a representative range of motions, including the effects of added weights in body building motions.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search direction at each iteration. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We demonstrate the efficacy of our approach on spatially discretized Poisson equations, which arise in computational fluid dynamics applications, with millions of degrees of freedom. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

The Extended Position Based Dynamics (XPBD) approach of Macklin et al. [2016] addresses the issues with iteration-dependent behavior in the original Position Based Dynamics [2007] (PBD) which itself is a powerful method for the real-time simulation of elastic objects. However, it is limited in its application to hyperelastic solids. It can only treat models with a strain energy density that is quadratic in some notion of constraint. Furthermore, we show that even when applicable the formulation does not always lead to convergent behaviors with hyperelasticity. We isolate the root cause in the approximate linearization of the nonlinear backward Euler systems utilized by XPBD. We provide two fixes to these terms that allow for convergent behavior. The first (B-PXPBD) is a small modification to an existing XPBD code, but can only be used with models addressable by the original XPBD. The second (FP-PXPBD) is a more general formulation that extends XPBD (and our residual correction) to arbitrary hyperelasticity. We show that our modifications allow for convergent behavior that rivals accurate techniques like Newton’s method when the computational budget is large without sacrificing the stable and robust behavior exhibited by the original PBD and XPBD when the computational budget is limited.

We present a novel Material Point Method (MPM) discretization of surface tension forces that arise from spatially varying surface energies. These variations typically arise from surface energy dependence on temperature and/or concentration. Furthermore, since the surface energy is an interfacial property depending on the types of materials on either side of an interface, spatial variation is required for modeling the contact angle at the triple junction between a liquid, solid and surrounding air. Our discretization is based on the surface energy itself, rather than on the associated traction condition most commonly used for discretization with particle methods. Our energy based approach automatically captures surface gradients without the explicit need to resolve them as in traction condition based approaches. We include an implicit discretization of thermomechanical material coupling with a novel particle-based enforcement of Robin boundary conditions associated with convective heating. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with Affine-Particle-In-Cell (APIC) [Jiang et al. 2015]. We show that our approach enables implicit time stepping for complex behaviors like the Marangoni effect and hydrophobicity/hydrophilicity. We demonstrate the robustness and utility of our method by simulating materials that exhibit highly diverse degrees of surface tension and thermomechanical effects, such as water, wine and wax.