In this paper, we present an approach for fast subspace integration of reduced-coordinate nonlinear deformable models that is suitable for interactive applications in computer graphics and haptics. Our approach exploits dimensional model reduction to build reduced-coordinate deformable models for objects with complex geome-try. We exploit the fact that model reduction on large deforma-tion models with linear materials (as commonly used in graphics) result in internal force models that are simply cubic polynomials in reduced coordinates. Coefficients of these polynomials can be precomputed, for efficient runtime evaluation. This allows simula-tion of nonlinear dynamics using fast implicit Newmark subspace integrators, with subspace integration costs independent of geomet-ric complexity. We present two useful approaches for generating low-dimensional subspace bases: modal derivatives and an interac-tive sketching technique. Mass-scaled principal component analy-sis (mass-PCA) is suggested for dimensionality reduction. Finally, several examples are given from computer animation to illustrate high performance, including force-feedback haptic rendering of a complicated object undergoing large deformations.

of nonlinear control systems

dynamics by driving the scene with parameterized interactions representative of runtime usage. (b) Model reduction on observed dynamic deformations yields a low-rank approximation to the system’s parameterized impulse response functions. (c) Deformed state geometries are then sampled and used to precompute and coparameterize a radiance transfer model for deformable objects. (d) The final simulation responds plausibly to interactions similar to those precomputed, includes complex collision and global illumination effects, and runs in real time. We present an approach for precomputing data-driven models of interactive physically based deformable scenes. The method permits real-time hardware synthesis of nonlinear deformation dynamics, including self-contact and global illumination effects, and supports real-time user interaction. We use data-driven tabulation of the system’s deterministic state space dynamics, and model reduction to build efficient low-rank parameterizations of the deformed shapes. To support runtime interaction, we also tabulate impulse response functions for a palette of external excitations. Although our approach simulates particular systems under very particular interaction conditions, it has several advantages. First, parameterizing all possible scene deformations enables us to precompute novel reduced coparameterizations of global scene illumination for lowfrequency lighting conditions. Second, because the deformation dynamics are precomputed and parameterized as a whole, collisions are resolved within the scene during precomputation so that runtime self-collision handling is implicit. Optionally, the data-driven models can be synthesized on programmable graphics hardware, leaving only the low-dimensional state space dynamics and appearance data models to be computed by the main CPU.

This thesis presents a framework for low-latency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differention equations. This approach makes it feasible to interactively simulate large-scale physical models. Linearity is exploited by formulating the boundary value problem (BVP) solution in terms of Green’s functions (GFs) which may be precomputed to provide speed and cheap lookup operations. Runtime BVPs are solved using a collection of Capacitance Matrix Algorithms (CMAs) based on the Sherman-Morrison-Woodbury formula. Temporal coherence is exploited by caching and reusing, as well as sequentially updating, previous capacitance matrix inverses. Multiresolution enhancements make it practical to simulate and store very large models. Efficient compressed representations of precomputed GFs are obtained using secondgeneration wavelets defined on surfaces. Fast inverse wavelet transforms allow fast summation methods to be used to accelerate runtime BVP solution. Wavelet GF compression factors are directly related to interactive simulation speedup, and examples are provided with

Finite element simulations of nonlinear deformable models are computationally costly, routinely taking hours or days to compute the motion of detailed meshes. Dimensional model reduction can make simulations orders of magnitude faster, but is unsuitable for general deformable body simulations because it requires expensive precomputations, and it can suppress motion that lies outside the span of a pre-specified low-rank basis. We present an online model reduction method that does not have these limitations. In lieu of precomputation, we analyze the motion of the full model as the simulation progresses, incrementally building a reduced-order nonlinear model, and detecting when our reduced model is capable of performing the next timestep. For these subspace steps, full-model computation is “skipped ” and replaced with a very fast (on the order of milliseconds) reduced order step. We present algorithms for both dynamic and quasistatic simulations, and a “throttle ” parameter that allows a user to trade off between faster, approximate previews and slower, more conservative results. For detailed meshes undergoing low-rank motion, we have observed speedups of over an order of magnitude with our method.

We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an r-dimensional deformation subspace with O(r) cubature points, our method is able to evaluate subspace forces at O(r2) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.

Figure 1: A result of our method: given a character rig and a set of keyframes for some of its parameters, our method automatically produces animation curves for the remaining parameters by solving the equations of motion in the space of deformations defined by the rig. The resulting motion is physically plausible, maintains the original artistic intent, and is easily editable. We present a method that brings the benefits of physics-based sim-ulations to traditional animation pipelines. We formulate the equa-tions of motions in the subspace of deformations defined by an animator’s rig. Our framework fits seamlessly into the workflow typically employed by artists, as our output consists of animation curves that are identical in nature to the result of manual keyfram-ing. Artists can therefore explore the full spectrum between hand-crafted animation and unrestricted physical simulation. To enhance the artist’s control, we provide a method that transforms stiffness values defined on rig parameters to a non-homogeneous distribu-tion of material parameters for the underlying FEM model. In ad-dition, we use automatically extracted high-level rig parameters to intuitively edit the results of our simulations, and also to speed up computation. To demonstrate the effectiveness of our method, we create compelling results by adding rich physical motions to coarse input animations. In the absence of artist input, we create realistic passive motion directly in rig space.

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details.

This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

We propose a procedural method for synthesizing realistic sounds due to nonlinear thin-shell vibrations. We use linear modal analysis to generate a small-deformation displacement basis, then couple the modes together using nonlinear thin-shell forces. To enable audiorate time-stepping of mode amplitudes with mesh-independent cost, we propose a reduced-order dynamics model based on a thin-shell cubature scheme. Limitations such as mode locking and pitch glide are addressed. To support fast evaluation of mid-frequency modebased sound radiation for detailed meshes, we propose far-field acoustic transfer maps (FFAT maps) which can be precomputed using state-of-the-art fast Helmholtz multipole methods. Familiar examples are presented including rumbling trash cans and plastic bottles, crashing cymbals, and noisy sheet metal objects, each with increased richness over linear modal sound models.