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

Large steps in cloth simulation

The bottle-neck in most cloth simulation systems is that time steps must be small to avoid numerical instability. This paper describes a cloth simulation system that can stably take large time steps. The simulation system couples a new technique for enforcing constraints on individual cloth particles with an implicit integration method. The simulator models cloth as a triangular mesh, with internal cloth forces derived using a simple continuum formulation that supports modeling operations such as local anisotropic stretch or compression; a unified treatment of damping forces is included as well. The implicit integration method generates a large, unbanded sparse linear system at each time step which is solved using a modified conjugate gradient method that simultaneously enforces particles ’ constraints. The constraints are always maintained exactly, independent of the number of conjugate gradient iterations, which is typically small. The resulting simulation system is significantly faster than previous accounts of cloth simulation systems in the literature. Keywords—Cloth, simulation, constraints, implicit integration, physically-based modeling.

Multi-Scale Models to Simulate Interactions between Liquid and Thin Structures

In this dissertation, we introduce a framework for simulating the dynamics between liquid and thin structures, including the effects of buoyancy, drag, capillary cohesion, dripping, and diffusion. After introducing related works, Part I begins with a discussion on the interactions between Newtonian fluid and fabrics. In this discussion, we treat both the fluid and the fabrics as continuum media; thus, the physical model is built from mixture theory. In Part II, we discuss the interactions between Newtonian fluid and hairs. To have more detailed dynamics, we no longer treat the hairs as continuum media. Instead, we treat them as discrete Kirchhoff rods. To deal with the thin layer of liquid that clings to the hairs, we augment each hair strand with a height field representation, through which we introduce a new reduced-dimensional flow model to solve the motion of liquid along the longitudinal direction of each hair. In addition, we develop a faithful model for the hairs' cohesion induced by surface tension, where a penalty force is applied to simulate the collision and cohesion between hairs. To enable the discrete strands interact with continuum-based, shear-dependent liquid, in Part III, we develop models that account for the volume change of the liquid as it passes through strands and the momentum exchange between the strands and the liquid. Accordingly, we extend the reduced-dimensional flow model to simulate liquid with elastoviscoplastic behavior. Furthermore, we use a constraint-based model to replace the penalty-force model to handle contact, which enables an accurate simulation of the frictional and adhesive effects between wet strands. We also present a principled method to preserve the total momentum of a strand and its surface flow, as well as an analytic plastic flow approach for Herschel-Bulkley fluid that enables stable semi-implicit integration at larger time steps. We demonstrate a wide range of effects, including the challenging animation scenarios involving splashing, wringing, and colliding of wet clothes, as well as flipping of hair, animals shaking, spinning roller brushes from car washes being dunked in water, and intricate hair coalescence effects. For complex liquids, we explore a series of challenging scenarios, including strands interacting with oil paint, mud, cream, melted chocolate, and pasta sauce

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

Optimization Of Strategic Planning Processes For Configurable Products: Considerations For Global Supply, Demand, And Sustainability Issues

The assortment planning problem is to decide on the set of products that a retailer or manufacturer will offer to its customers to maximize profitability. While assortment planning research has been expanding in recent years, the current models are inadequate for the needs of a configurable product manufacturer. In particular, we address assortment planning for an automobile manufacturer. We develop models to integrate assortment planning and supply chain management, designed for use by a large automaker in its strategic planning phase. Our model utilizes a multinomial logit model transformed into a mixed integer linear program through the Charnes-Cooper transformation. It is able to scale to problems that contain thousands of configurations to possibly be offered, a necessity given the number of possible configurations an automaker can build. In addition, most research in assortment planning contains simplified costs associated with product complexity. We model a full supply chain and give a rich treatment of the complexity associated with product complexity. We believe that our model can significantly aid automotive manufacturers to balance their product complexity with supply chain complexity, thus increasing profitability. In addition, we study the effect of packaging on the assortment and supply chain of an automaker. We develop a new model for mathematically expressing the effect that packaging has on the way in which customers choose products. Packaging significantly complicates the search space of the assortment planning problem. We introduce a heuristic method based on our packaging model that speeds up the solve times of the models while finding reasonably good solutions. Finally, we extend our initial model to study the effects of sustainability requirements on an automaker\u27s assortment and supply chain. We introduce constraints on the vehicle program average fuel economy, greenhouse gas emissions in the supply chain, and greenhouse gas emissions in the product use phase. We dive deep into each case to glean insights about how automakers can change their decision-making process to balance making their companies more sustainable with profit maximization. While all the examples discussed are from the automotive industry, the models developed can be adapted to address assortment planning for other types of configurable products (e.g., computers, printers, phones)

Towards Real-Time Simulation Of Hyperelastic Materials

We propose a new method for physics-based simulation supporting many different types of hyperelastic materials from mass-spring systems to three-dimensional finite element models, pushing the performance of the simulation towards real-time. Fast simulation methods such as Position Based Dynamics exist, but support only limited selection of materials; even classical materials such as corotated linear elasticity and Neo-Hookean elasticity are not supported. Simulation of these types of materials currently relies on Newton\u27s method, which is slow, even with only one iteration per timestep. In this work, we start from simple material models such as mass-spring systems or as-rigid-as-possible materials. We express the widely used implicit Euler time integration as an energy minimization problem and introduce auxiliary projection variables as extra unknowns. After our reformulation, the minimization problem becomes linear in the node positions, while all the non-linear terms are isolated in individual elements. We then extend this idea to efficiently simulate a more general spatial discretization using finite element method. We show that our reformulation can be interpreted as a quasi-Newton method. This insight enables very efficient simulation of a large class of hyperelastic materials. The quasi-Newton interpretation also allows us to leverage ideas from numerical optimization. In particular, we show that our solver can be further accelerated using L-BFGS updates (Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm). Our final method is typically more than ten times faster than one iteration of Newton\u27s method without compromising quality. In fact, our result is often more accurate than the result obtained with one iteration of Newton\u27s method. Our method is also easier to implement, implying reduced software development costs

GIPC: Fast and stable Gauss-Newton optimization of IPC barrier energy

Barrier functions are crucial for maintaining an intersection and inversion free simulation trajectory but existing methods which directly use distance can restrict implementation design and performance. We present an approach to rewriting the barrier function for arriving at an efficient and robust approximation of its Hessian. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, from which analytic eigensystems are derived and enhanced with filtering and stiffening terms that ensure robustness with respect to the convergence of a Project-Newton solver. A further advantage of our rewriting of the barrier function is that it naturally caters to the notorious case of nearly-parallel edge-edge contacts for which we also present a novel analytic eigensystem. Our approach is thus well suited for standard second order unconstrained optimization strategies for resolving contacts, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The efficiency of our eigensystems alone yields a 3x speedup over the standard IPC barrier formulation. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration