Multi-resolution isotropic strain limiting

In this paper we describe a fast strain-limiting method that allows stiff, incompliant materials to be simulated efficiently. Unlike prior approaches, which act on springs or individual strain components, this method acts on the strain tensors in a coordinate-invariant fashion allowing isotropic behavior. Our method applies to both two-and three-dimensional strains, and only requires computing the singular value decomposition of the deformation gradient, either a small 2x2 or 3x3 matrix, for each element. We demonstrate its use with triangular and tetrahedral linear-basis elements. For triangulated surfaces in three-dimensional space, we also describe a complementary edge-angle-limiting method to limit out-of-plane bending. All of the limits are enforced through an iterative, non-linear, Gauss-Seidel-like constraint procedure. To accelerate convergence, we propose a novel multi-resolution algorithm that enforces fitted limits at each level of a non-conforming hierarchy. Compared with other constraint-based techniques, our isotropic multi-resolution strain-limiting method is straightforward to implement, efficient to use, and applicable to a wide range of shell and solid materials. © 2010 ACM

Strain limiting for clustered shape matching

journal articleIn this paper, we advocate explicit symplectic Euler integration and strain limiting in a shape matching simulation framework. The resulting approach resembles not only previous work on shape matching and strain limiting, but also the recently popular position-based dynamics. However, unlike this previous work, our approach reduces to explicit integration under small strains, but remains stable in the presence of non-linearities

Interactive procedural simulation of paper tearing with sound

International audienceWe present a phenomenological model for the real-time simulation of paper tearing and sound. The model uses as input rotations of the hand along with the index and thumb of left and right hands to drive the position and orientation of two regions of a sheet of paper. The motion of the hands produces a cone shaped deformation of the paper and guides the formation and growth of the tear. We create a model for the direction of the tear based on empirical observation, and add detail to the tear with a directed noise model. Furthermore, we present a procedural sound synthesis method to produce tearing sounds during interaction. We show a variety of paper tearing examples and discuss applications and limitations

Interactive Thin Elastic Materials

Despite great strides in past years are being made to generate motions of elastic 1 materials such as cloth and biological skin in virtual world, unfortunately, the computational cost of realistic high-resolution simulations currently precludes their use in interactive applications. Thin elastic materials such as cloth and biological skin often exhibit complex nonlinear elastic behaviors. However, modeling elastic nonlinearity can be computationally expensive and numerically unstable, imposing significant challenges for their use in interactive applications. This paper presents a novel simulation framework for simulating realistic material behaviours with interactive frame rate. Central to the framework is the use of a constraint-based multi-resolution solver for efficient and robust modelling of the material nonlinearity. We extend a strain limiting method to work on deformation gradients of triangulated surface models in three dimensional space with a novel data structure. The simulation framework utilises an iterative nonlinear Gauss-Seidel procedure and a multilevel hierarchy structure to achieve computational speed ups. As material non-linearity are generated by enforcing strain limiting constraints at a multilevel hierarchy, our simulation system can rapidly accelerate the convergence of the large constraint system with simultaneous enforcement of boundary conditions. The simplicity and efficiency of the framework makes simulations of highly realistic thin elastic materials substantially fast and is applicable of simulations for interactive applications

Fast Simulation of Mass-Spring Systems

We describe a scheme for time integration of mass-spring systems that makes use of a solver based on block coordinate descent. This scheme provides a fast solution for classical linear (Hookean) springs. We express the widely used implicit Euler method as an energy minimization problem and introduce spring directions as auxiliary unknown variables. The system is globally linear in the node positions, and the non-linear terms involving the directions are strictly local. Because the global linear system does not depend on run-time state, the matrix can be pre-factored, allowing for very fast iterations. Our method converges to the same final result as would be obtained by solving the standard form of implicit Euler using Newton’s method. Although the asymptotic convergence of Newton’s method is faster than ours, the initial ratio of work to error reduction with our method is much faster than Newton’s. For real-time visual applications, where speed and stability are more important than precision, we obtain visually acceptable results at a total cost per timestep that is only a fraction of that required for a single Newton iteration. When higher accuracy is required, our algorithm can be used to compute a good starting point for subsequent Newton’s iteration

Anisotropic Strain Limiting

Many materials exhibit a highly nonlinear elastic behavior, such as textiles or finger flesh. An efficient way of enforcing the nonlinearity of these materials is through strain-limiting constraints, which is often the model of choice in computer graphics. Strain-limiting allows to model highly non-linear stiff materials by eliminating degrees of freedom from the computations and by enforcing a set of constraints. However, many nonlinear elastic materials, such as composites, wood or flesh, exhibit anisotropic behaviors, with different material responses depending on the deformation direction. This anisotropic behavior has not been addressed in the past in the context of strain limiting, and naïve approaches, such as applying a different constraint on each component of the principal axes of deformation, produce unrealistic results. In this paper, we enable anisotropic behaviors when using strain-limiting constraints to model nonlinear elastic materials. We compute the limits for each principal axis of deformation through the rotation and hyperbolic projection of the deformation limits defined in the global reference frame. The limits are used to formulate the strain-limiting constraints, which are then seamlessly combined with frictional contact constraints in a standard constrained dynamics solver. Categories and Subject Descriptors (according to ACM CCS): modeling

Stable Constrained Dynamics

International audienceWe present a unification of the two main approaches to simulate deformable solids, namely elasticity and constraints. Elasticity accurately handles soft to moderately stiff objects, but becomes numerically hard as stiffness increases. Constraints efficiently handle high stiffness, but when integrated in time they can suffer from instabilities in the nullspace directions, generating spurious transverse vibrations when pulling hard on thin inextensible objects or articulated rigid bodies. We show that geometric stiffness, the tensor encoding the change of force directions (as opposed to intensities) in response to a change of positions, is the missing piece between the two approaches. This previously neglected stiffness term is easy to implement and dramatically improves the stability of inextensible objects and articulated chains, without adding artificial bending forces. This allows time step increases up to several orders of magnitude using standard linear solvers

Modeling and estimation of internal friction in cloth

Force-deformation measurements of cloth exhibit significant hysteresis, and many researchers have identified internal friction as the source of this effect. However, it has not been incorporated into computer animation models of cloth. In this paper, we propose a model of internal friction based on an augmented reparameterization of Dahl's model, and we show that this model provides a good match to several important features of cloth hysteresis even with a minimal set of parameters. We also propose novel parameter estimation procedures that are based on simple and inexpensive setups and need only sparse data, as opposed to the complex hardware and dense data acquisition of previous methods. Finally, we provide an algorithm for the efficient simulation of internal friction, and we demonstrate it on simulation examples that show disparate behavior with and without internal friction

Projective Dynamics: Fusing Constraint Projections for Fast Simulation

We present a new method for implicit time integration of physical systems. Our approach builds a bridge between nodal Finite Element methods and Position Based Dynamics, leading to a simple, efficient, robust, yet accurate solver that supports many different types of constraints. We propose specially designed energy potentials that can be solved efficiently using an alternating optimization approach. Inspired by continuum mechanics, we derive a set of continuumbased potentials that can be efficiently incorporated within our solver. We demonstrate the generality and robustness of our approach in many different applications ranging from the simulation of solids, cloths, and shells, to example-based simulation. Comparisons to Newton-based and Position Based Dynamics solvers highlight the benefits of our formulation