Space-time editing of elastic motion through material optimization and reduction

We present a novel method for elastic animation editing with space-time constraints. In a sharp departure from previous approaches, we not only optimize control forces added to a linearized dynamic model, but also optimize material properties to better match user constraints and provide plausible and consistent motion. Our approach achieves efficiency and scalability by performing all computations in a reduced rotation-strain (RS) space constructed with both cubature and geometric reduction, leading to two orders of magnitude improvement over the original RS method. We demonstrate the utility and versatility of our method in various applications, including motion editing, pose interpolation, and estimation of material parameters from existing animation sequences

Data-driven finite elements for geometry and material design

Crafting the behavior of a deformable object is difficult---whether it is a biomechanically accurate character model or a new multimaterial 3D printable design. Getting it right requires constant iteration, performed either manually or driven by an automated system. Unfortunately, Previous algorithms for accelerating three-dimensional finite element analysis of elastic objects suffer from expensive precomputation stages that rely on a priori knowledge of the object's geometry and material composition. In this paper we introduce Data-Driven Finite Elements as a solution to this problem. Given a material palette, our method constructs a metamaterial library which is reusable for subsequent simulations, regardless of object geometry and/or material composition. At runtime, we perform fast coarsening of a simulation mesh using a simple table lookup to select the appropriate metamaterial model for the coarsened elements. When the object's material distribution or geometry changes, we do not need to update the metamaterial library---we simply need to update the metamaterial assignments to the coarsened elements. An important advantage of our approach is that it is applicable to non-linear material models. This is important for designing objects that undergo finite deformation (such as those produced by multimaterial 3D printing). Our method yields speed gains of up to two orders of magnitude while maintaining good accuracy. We demonstrate the effectiveness of the method on both virtual and 3D printed examples in order to show its utility as a tool for deformable object design.National Science Foundation (U.S.) (Grant CCF-1138967)United States. Defense Advanced Research Projects Agency (N66001-12-1-4242

Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy.Peer ReviewedPostprint (published version

Model-reduced variational fluid simulation

We present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness of coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. At the core of our contributions is a functional map approach to fluid simulation for which scalar- and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time to preserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computational time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer. We also demonstrate the versatility of our approach by showing how it easily extends to magnetohydrodynamics and turbulence modeling in 2D, 3D and curved domains

Computational design of skinned Quad-Robots

We present a computational design system that assists users to model, optimize, and fabricate quad-robots with soft skins. Our system addresses the challenging task of predicting their physical behavior by fully integrating the multibody dynamics of the mechanical skeleton and the elastic behavior of the soft skin. The developed motion control strategy uses an alternating optimization scheme to avoid expensive full space time-optimization, interleaving space-time optimization for the skeleton, and frame-by-frame optimization for the full dynamics. The output are motor torques to drive the robot to achieve a user prescribed motion trajectory. We also provide a collection of convenient engineering tools and empirical manufacturing guidance to support the fabrication of the designed quad-robot. We validate the feasibility of designs generated with our system through physics simulations and with a physically-fabricated prototype

Space-time sculpting of liquid animation

International audienceWe propose an interactive sculpting system for seamlessly editing pre-computed animations of liquid, without the need for any re-simulation. The input is a sequence of meshes without correspondences representing the liquid surface over time. Our method enables the efficient selection of consistent space-time parts of this animation, such as moving waves or droplets, which we call space-time features. Once selected, a feature can be copied, edited, or duplicated and then pasted back anywhere in space and time in the same or in another liquid animation sequence. Our method circumvents tedious user interactions by automatically computing the spatial and temporal ranges of the selected feature. We also provide space-time shape editing tools for non-uniform scaling, rotation, trajectory changes, and temporal editing to locally speed up or slow down motion. Using our tools, the user can edit and progressively refine any input simulation result, possibly using a library of pre-computed space-time features extracted from other animations. In contrast to the trial-and-error loop usually required to edit animation results through the tuning of indirect simulation parameters, our method gives the user full control over the edited space-time behaviors

Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accurac