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 10 000 segments. In summary, our method allows the efficient simulation of stiff rods in the PBD framework with a speedup of two orders of magnitude compared to the original XPBD approach.

We develop a new operator splitting formulation for the simulation of corotated linearly elastic solids with Smoothed Particle Hydrodynamics (SPH). Based on the technique of Kugelstadt et al. [2018] originally developed for the Finite Element Method (FEM), we split the elastic energy into two separate terms corresponding to stretching and volume conservation, and based on this principle, we design a splitting scheme compatible with SPH. The operator splitting scheme enables us to treat the two terms separately, and because the stretching forces lead to a stiffness matrix that is constant in time, we are able to prefactor the system matrix for the implicit integration step. Solid-solid contact and fluid-solid interaction is achieved through a unified pressure solve. We demonstrate more than an order of magnitude improvement in computation time compared to a state-of-the-art SPH simulator for elastic solids. We further improve the stability and reliability of the simulation through several additional contributions. We introduce a new implicit penalty mechanism that suppresses zero-energy modes inherent in the SPH formulation for elastic solids, and present a new, physics-inspired sampling algorithm for generating high-quality particle distributions for the rest shape of an elastic solid. We finally also devise an efficient method for interpolating vertex positions of a high-resolution surface mesh based on the SPH particle positions for use in high-fidelity visualization.

In this paper we introduce a novel micropolar material model for the simulation of turbulent inviscid fluids. The governing equations are solved by using the concept of Smoothed Particle Hydrodynamics (SPH). SPH fluid simulations suffer from numerical diffusion which leads to a lower vorticity, a loss in turbulent details and finally in less realistic results. To solve this problem we propose a micropolar fluid model. The micropolar fluid model is a generalization of the classical Navier-Stokes equations, which are typically used in computer graphics to simulate fluids. In contrast to the classical Navier-Stokes model, micropolar fluids have a microstructure and therefore consider the rotational motion of fluid particles. In addition to the linear velocity field these fluids have a field of microrotation which represents existing vortices and provides a source for new ones. Our novel micropolar model can generate realistic turbulences, is linear and angular momentum conserving, can be easily integrated in existing SPH simulation methods and its computational overhead is negligible. Another important visual feature of turbulent liquids is foam. Therefore, we present a post-processing method which considers microrotation in the foam generation. It works completely automatic and requires only one user-defined parameter to control the amount of foam.

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.

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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.

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Figure 1: Our novel micropolar uid model allows the simulation of complex turbulent uids with static and dynamic boundaries. Left: A turbulent river with 12 million uid particles ows through a complex static boundary. Right: One million uid particles interact with a fast rotating propeller.