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 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 low-dimensional subspace bases: modal derivatives and an interactive sketching technique. Mass-scaled principal component analysis (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.

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 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 low-dimensional subspace bases: modal derivatives and an interactive sketching technique. Mass-scaled principal component analysis (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.

Abstract-Real-time evaluation of distributed contact forces between rigid or deformable 3D objects is a key ingredient of 6-DoF force-feedback rendering. Unfortunately, at very high temporal rates, there is often insufficient time to resolve contact between geometrically complex objects. We propose a spatially and temporally adaptive approach to approximate distributed contact forces under hard real-time constraints. Our method is CPU-based and supports contact between rigid or reduced deformable models with complex geometry. We propose a contact model that uses a point-based representation for one object and a signed-distance field for the other. This model is related to the Voxmap-PointShell (VPS) method, but gives continuous contact forces and torques, enabling stable rendering of stiff penalty-based distributed contacts. We demonstrate that stable haptic interactions can be achieved by point-sampling offset surfaces to input "polygon soup" geometry using particle repulsion. We introduce a multiresolution nested pointshell construction that permits level-of-detail contact forces and enables graceful degradation of contact in close-proximity scenarios. Parametrically deformed distance fields are proposed for contact between reduced deformable objects. We present several examples of 6-DoF haptic rendering of geometrically complex rigid and deformable objects in distributed contact at real-time kilohertz rates.

Figure 1: Our linear subspaces are very fast to compute. This enables the users to add (or remove) control handles very quickly, allowing them to realize their creative intent in a single interactive session. AbstractWe propose a method to design linear deformation subspaces, unifying linear blend skinning and generalized barycentric coordinates. Deformation subspaces cut down the time complexity of variational shape deformation methods and physics-based animation (reduced-order physics). Our subspaces feature many desirable properties: interpolation, smoothness, shape-awareness, locality, and both constant and linear precision. We achieve these by minimizing a quadratic deformation energy, built via a discrete Laplacian inducing linear precision on the domain boundary. Our main advantage is speed: subspace bases are solutions to a sparse linear system, computed interactively even for generously tessellated domains. Users may seamlessly switch between applying transformations at handles and editing the subspace by adding, removing or relocating control handles. The combination of fast computation and good properties means that designing the right subspace is now just as creative as manipulating handles. This paradigm shift in handle-based deformation opens new opportunities to explore the space of shape deformations.

Keyframe animation is a common technique to generate animations of deformable characters and other soft bodies. With spline interpolation, however, it can be difficult to achieve secondary motion effects such as plausible dynamics when there are thousands of degrees of freedom to animate. Physical methods can provide more realism with less user effort, but it is challenging to apply them to quickly create specific animations that closely follow prescribed animator goals. We present a fast space-time optimization method to author physically based deformable object simulations that conform to animator-specified keyframes. We demonstrate our method with FEM deformable objects and mass-spring systems.Our method minimizes an objective function that penalizes the sum of keyframe deviations plus the deviation of the trajectory from physics. With existing methods, such minimizations operate in high dimensions, are slow, memory consuming, and prone to local minima. We demonstrate that significant computational speedups and robustness improvements can be achieved if the optimization problem is properly solved in a low-dimensional space. Selecting a low-dimensional space so that the intent of the animator is accommodated, and that at the same time space-time optimization is convergent and fast, is difficult. We present a method that generates a quality low-dimensional space using the given keyframes. It is then possible to find quality solutions to difficult space-time optimization problems robustly and in a manner of minutes.

Figure 1: Design of isotropic nonlinear materials: The soft-body motion of the wrestler was computed using FEM, constrained to a motion capture skeletal dancing animation. Using our method, we designed a nonlinear isotropic material that performs well both during impulsive and gentle animation phases. Top row: the wrestler is performing high jumps. The soft Neo-Hookean material exhibits artifacts (belly, thighs) when the character moves abruptly. Our material and the stiff Neo-Hookean material produce good deformations. Bottom row: deformations during a gentle phase (walking while dancing) of the same motion sequence. The soft Neo-Hookean material and our method produce rich small-deformation dynamics, whereas the stiff Neo-Hookean material inhibits it. The Young's modulus of material (a) was chosen to produce good dynamics during gentle motion. We then edited it to address impulsive motion, producing (c). The stiff material in (b) is the best matching material to (c) among Neo-Hookean materials, minimizing the L 2 material curve difference to (c). AbstractThe Finite Element Method is widely used for solid deformable object simulation in film, computer games, virtual reality and medicine. Previous applications of nonlinear solid elasticity employed materials from a few standard families such as linear corotational, nonlinear St.Venant-Kirchhoff, Neo-Hookean, Ogden or Mooney-Rivlin materials. However, the spaces of all nonlinear isotropic and anisotropic materials are infinite-dimensional and much broader than these standard materials. In this paper, we demonstrate how to intuitively explore the space of isotropic and anisotropic nonlinear materials, for design of animations in computer graphics and related fields. In order to do so, we first formulate the internal elastic forces and tangent stiffness matrices in the space of the principal stretches of the material. We then demonstrate how to design new isotropic materials by editing a single stress-strain curve, using a spline interface. Similarly, anisotropic (orthotropic) materials can be designed by editing three curves, one for each material direction. We demonstrate that modifying these curves using our proposed interface has an intuitive, visual, effect on the simulation. Our materials accelerate simulation design and enable visual effects that are difficult or impossible to achieve with standard nonlinear materials.

We demonstrate an interactive method to create heterogeneous continuous deformable materials on complex three-dimensional meshes. The user specifies displacements and internal elastic forces at a chosen set of mesh vertices. Our system then rapidly solves an optimization problem to compute a corresponding heterogeneous spatial distribution of material properties using the Finite Element Method (FEM) analysis. We apply our method to linear and nonlinear isotropic deformable materials. We demonstrate that solving the problem interactively in the full-dimensional space of individual tetrahedron material values is not practical. Instead, we propose a new model reduction method that projects the material space to a low-dimensional space of material modes. Our model reduction accelerates optimization by two orders of magnitude and makes the convergence much more robust, making it possible to interactively design material distributions on complex meshes. We apply our method to precise control of contact forces and control of pressure over large contact areas between rigid and deformable objects for ergonomics. Our tetrahedron-based dithering method can efficiently convert continuous material distributions into discrete ones and we demonstrate its precision via FEM simulation. We physically display our distributions using haptics, as well as demonstrate how haptics can aid in the material design. The produced heterogeneous material distributions can also be used in computer animation applications.

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