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57
Realtime subspace integration for St. VenantKirchhoff deformable models
 ACM Transactions on Graphics
, 2005
"... In this paper, we present an approach for fast subspace integration of reducedcoordinate nonlinear deformable models that is suitable for interactive applications in computer graphics and haptics. Our approach exploits dimensional model reduction to build reducedcoordinate deformable models for ob ..."
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Cited by 121 (13 self)
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In this paper, we present an approach for fast subspace integration of reducedcoordinate nonlinear deformable models that is suitable for interactive applications in computer graphics and haptics. Our approach exploits dimensional model reduction to build reducedcoordinate deformable models for objects with complex geometry. We exploit the fact that model reduction on large deformation 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 simulation of nonlinear dynamics using fast implicit Newmark subspace integrators, with subspace integration costs independent of geometric complexity. We present two useful approaches for generating lowdimensional subspace bases: modal derivatives and an interactive sketching technique. Massscaled principal component analysis (massPCA) is suggested for dimensionality reduction. Finally, several examples are given from computer animation to illustrate high performance, including forcefeedback haptic rendering of a complicated object undergoing large deformations.
A subspace approach to balanced truncation for model reduction of nonlinear control systems
 International Journal on Robust and Nonlinear Control
, 2002
"... of nonlinear control systems ..."
Precomputing interactive dynamic deformable scenes
 ACM Trans. Graph
, 2003
"... dynamics by driving the scene with parameterized interactions representative of runtime usage. (b) Model reduction on observed dynamic deformations yields a lowrank approximation to the system’s parameterized impulse response functions. (c) Deformed state geometries are then sampled and used to pre ..."
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Cited by 90 (8 self)
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dynamics by driving the scene with parameterized interactions representative of runtime usage. (b) Model reduction on observed dynamic deformations yields a lowrank 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 datadriven models of interactive physically based deformable scenes. The method permits realtime hardware synthesis of nonlinear deformation dynamics, including selfcontact and global illumination effects, and supports realtime user interaction. We use datadriven tabulation of the system’s deterministic state space dynamics, and model reduction to build efficient lowrank 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 selfcollision handling is implicit. Optionally, the datadriven models can be synthesized on programmable graphics hardware, leaving only the lowdimensional state space dynamics and appearance data models to be computed by the main CPU.
Multiresolution green’s function methods for interactive simulation of largescale elastostatic objects
 ACM Trans. Graph
, 2003
"... This thesis presents a framework for lowlatency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differention equations. This approach makes it feasible to interactively simulate largescale physical models. Linearity is exploited by for ..."
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Cited by 53 (12 self)
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This thesis presents a framework for lowlatency interactive simulation of linear elastostatic models and other systems associated with linear elliptic partial differention equations. This approach makes it feasible to interactively simulate largescale 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 ShermanMorrisonWoodbury 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
Skipping Steps in Deformable Simulation with Online Model Reduction
"... 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 beca ..."
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Cited by 33 (4 self)
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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 prespecified lowrank 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 reducedorder nonlinear model, and detecting when our reduced model is capable of performing the next timestep. For these subspace steps, fullmodel 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 lowrank motion, we have observed speedups of over an order of magnitude with our method.
Optimizing cubature for efficient integration of subspace deformations
 ACM Transactions on Graphics (SIGGRAPH Asia
, 2008
"... 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 integr ..."
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Cited by 32 (5 self)
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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 (multidimensional 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 rdimensional 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.VenantKirchhoff, MooneyRivlin, ArrudaBoyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration.
Particle Physics
, 1997
"... 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 physi ..."
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Cited by 31 (4 self)
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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 physicsbased simulations to traditional animation pipelines. We formulate the equations 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 keyframing. Artists can therefore explore the full spectrum between handcrafted 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 nonhomogeneous distribution of material parameters for the underlying FEM model. In addition, we use automatically extracted highlevel 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.
Geometric, Variational Integrators for Computer Animation
 EUROGRAPHICS/ACM SIGGRAPH SYMPOSIUM ON COMPUTER ANIMATION, M.P. CANI, J. O’BRIEN (EDITORS)
, 2006
"... We present a generalpurpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physicsbased animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically pre ..."
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Cited by 29 (3 self)
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We present a generalpurpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physicsbased 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 highorder 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 PontryaginHamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of nonlinear elasticity with implementation details.
Structurepreserving model reduction for mechanical systems
, 2003
"... This paper focuses on methods of constructing of reducedorder 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 wavel ..."
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Cited by 26 (4 self)
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This paper focuses on methods of constructing of reducedorder 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 threedimensional finiteelement 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.
Harmonic Shells: A Practical Nonlinear Sound Model for NearRigid Thin Shells
"... We propose a procedural method for synthesizing realistic sounds due to nonlinear thinshell vibrations. We use linear modal analysis to generate a smalldeformation displacement basis, then couple the modes together using nonlinear thinshell forces. To enable audiorate timestepping of mode ampli ..."
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Cited by 23 (6 self)
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We propose a procedural method for synthesizing realistic sounds due to nonlinear thinshell vibrations. We use linear modal analysis to generate a smalldeformation displacement basis, then couple the modes together using nonlinear thinshell forces. To enable audiorate timestepping of mode amplitudes with meshindependent cost, we propose a reducedorder dynamics model based on a thinshell cubature scheme. Limitations such as mode locking and pitch glide are addressed. To support fast evaluation of midfrequency modebased sound radiation for detailed meshes, we propose farfield acoustic transfer maps (FFAT maps) which can be precomputed using stateoftheart 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.