Efficient Deformations Using Custom Coordinate Systems

Physics-based deformable object simulations have been playing an increasingly important role in 3D computer graphics. They have been adopted for humanoid character animations as well as special effects such as fire and explosion. However, simulations of large, complex systems can consume large amounts of computation and mostly remain offline, which prohibits their use for interactive applications.We present several highly efficient schemes for deformable object simulation using custom spatial coordinate systems. Our choices span the spectrum of subspace to full space and both Lagrangian and Eulerian viewpoints.Subspace methods achieve massive speedups over their “full space” counterparts by drastically reducing the degrees of freedom involved in the simulation. A long standing difficulty in subspace simulation is incorporating various non-linearities. They introduce expensive computational bottlenecks and quite often cause novel deformations that are outside the span of the subspace.We address these issues in articulated deformable body simulations from a Lagrangian viewpoint. We remove the computational bottleneck of articulated self-contact handling by deploying a pose-space cubature scheme, a generalization of the standard “cubature” approximation. To handle novel deformations caused by arbitrary external collisions, we introduce a generic approach called subspace condensation, which activates full space simulation on the fly when an out-of-basis event is encountered. Our proposed frameworkefficiently incorporates various non-linearities and allows subspace methods to be used in cases where they previously would not have been considered.Deformable solids can interact not only with each other, but also with fluids. Wedesign a new full space method that achieves a two-way coupling between deformable solids and an incompressible fluid where the underlying geometric representation is entirely Eulerian. No-slip boundary conditions are automatically satisfied by imposing a global divergence-free condition. We are able to simulate multiple solids undergoing complex, frictional contact while simultaneously interacting with a fluid. The complexity of the scenarios we are able to simulate surpasses those that we have seen from any previous method

Multiscale Modeling and Simulation of Deformation Accumulation in Fault Networks

Strain accumulation and stress release along multiscale geological fault networks are fundamental mechanisms for earthquake and rupture processes in the lithosphere. Due to long periods of seismic quiescence, the scarcity of large earthquakes and incompleteness of paleoseismic, historical and instrumental record, there is a fundamental lack of insight into the multiscale, spatio-temporal nature of earthquake dynamics in fault networks. This thesis constitutes another step towards reliable earthquake prediction and quantitative hazard analysis. Its focus lies on developing a mathematical model for prototypical, layered fault networks on short time scales as well as their efficient numerical simulation. This exposition begins by establishing a fault system consisting of layered bodies with viscoelastic Kelvin-Voigt rheology and non-intersecting faults featuring rate-and-state friction as proposed by Dieterich and Ruina. The individual bodies are assumed to experience small viscoelastic deformations, but possibly large relative tangential displacements. Thereafter, semi-discretization in time with the classical Newmark scheme of the variational formulation yields a sequence of continuous, nonsmooth, coupled, spatial minimization problems for the velocities and states in each time step, that are decoupled by means of a fixed point iteration. Subsequently, spatial discretization is based on linear and piecewise constant finite elements for the rate and state problems, respectively. A dual mortar discretization of the non-penetration constraints entails a hierarchical decomposition of the discrete solution space, that enables the localization of the non-penetration condition. Exploiting the resulting structure, an algebraic representation of the parametrized rate problem can be solved efficiently using a variant of the Truncated Nonsmooth Newton Multigrid (TNNMG) method. It is globally convergent due to nonlinear, block Gauß–Seidel type smoothing and employs nonsmooth Newton and multigrid ideas to enhance robustness and efficiency of the overall method. A key step in the TNNMG algorithm is the efficient computation of a correction obtained from a linearized, inexact Newton step. The second part addresses the numerical homogenization of elliptic variational problems featuring fractal interface networks, that are structurally similar to the ones arising in the linearized correction step of the TNNMG method. Contrary to the previous setting, this model incorporates the full spatial complexity of geological fault networks in terms of truly multiscale fractal interface geometries. Here, the construction of projections from a fractal function space to finite element spaces with suitable approximation and stability properties constitutes the main contribution of this thesis. The existence of these projections enables the application of well-known approaches to numerical homogenization, such as localized orthogonal decomposition (LOD) for the construction of multiscale discretizations with optimal a priori error estimates or subspace correction methods, that lead to algebraic solvers with mesh- and scale-independent convergence rates. Finally, numerical experiments with a single fault and the layered multiscale fault system illustrate the properties of the mathematical model as well as the efficiency, reliability and scale-independence of the suggested algebraic solver

Animation, Simulation, and Control of Soft Characters using Layered Representations and Simplified Physics-based Methods

Realistic behavior of computer generated characters is key to bringing virtual environments, computer games, and other interactive applications to life. The plausibility of a virtual scene is strongly influenced by the way objects move around and interact with each other. Traditionally, actions are limited to motion capture driven or pre-scripted motion of the characters. Physics enhance the sense of realism: physical simulation is required to make objects act as expected in real life. To make gaming and virtual environments truly immersive,it is crucial to simulate the response of characters to collisions and to produce secondary effects such as skin wrinkling and muscle bulging. Unfortunately, existing techniques cannot generally achieve these effects in real time, do not address the coupled response of a character's skeleton and skin to collisions nor do they support artistic control. In this dissertation, I present interactive algorithms that enable physical simulation of deformable characters with high surface detail and support for intuitive deformation control. I propose a novel unified framework for real-time modeling of soft objects with skeletal deformations and surface deformation due to contact, and their interplay for object surfaces with up to tens of thousands of degrees of freedom.I make use of layered models to reduce computational complexity. I introduce dynamic deformation textures, which map three dimensional deformations in the deformable skin layer to a two dimensional domain for extremely efficient parallel computation of the dynamic elasticity equations and optimized hierarchical collision detection. I also enhance layered models with responsive contact handling, to support the interplay between skeletal motion and surface contact and the resulting two-way coupling effects. Finally, I present dynamic morph targets, which enable intuitive control of dynamic skin deformations at run-time by simply sculpting pose-specific surface shapes. The resulting framework enables real-time and directable simulation of soft articulated characters with frictional contact response, capturing the interplay between skeletal dynamics and complex,non-linear skin deformations

A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6

Doctor of Philosophy

dissertationPhysical simulation has become an essential tool in computer animation. As the use of visual effects increases, the need for simulating real-world materials increases. In this dissertation, we consider three problems in physics-based animation: large-scale splashing liquids, elastoplastic material simulation, and dimensionality reduction techniques for fluid simulation. Fluid simulation has been one of the greatest successes of physics-based animation, generating hundreds of research papers and a great many special effects over the last fifteen years. However, the animation of large-scale, splashing liquids remains challenging. We show that a novel combination of unilateral incompressibility, mass-full FLIP, and blurred boundaries is extremely well-suited to the animation of large-scale, violent, splashing liquids. Materials that incorporate both plastic and elastic deformations, also referred to as elastioplastic materials, are frequently encountered in everyday life. Methods for animating such common real-world materials are useful for effects practitioners and have been successfully employed in films. We describe a point-based method for animating elastoplastic materials. Our primary contribution is a simple method for computing the deformation gradient for each particle in the simulation. Given the deformation gradient, we can apply arbitrary constitutive models and compute the resulting elastic forces. Our method has two primary advantages: we do not store or compare to an initial rest configuration and we work directly with the deformation gradient. The first advantage avoids poor numerical conditioning and the second naturally leads to a multiplicative model of deformation appropriate for finite deformations. One of the most significant drawbacks of physics-based animation is that ever-higher fidelity leads to an explosion in the number of degrees of freedom