Simulating deformable objects for computer animation: A numerical perspective

Abstract

<p style='text-indent:20px;'>We examine a variety of numerical methods that arise when considering dynamical systems in the context of physics-based simulations of deformable objects. Such problems arise in various applications, including animation, robotics, control and fabrication. The goals and merits of suitable numerical algorithms for these applications are different from those of typical numerical analysis research in dynamical systems. Here the mathematical model is not fixed <i>a priori</i> but must be adjusted as necessary to capture the desired behaviour, with an emphasis on effectively producing lively animations of objects with complex geometries. Results are often judged by how realistic they appear to observers (by the "eye-norm") as well as by the efficacy of the numerical procedures employed. And yet, we show that with an adjusted view numerical analysis and applied mathematics can contribute significantly to the development of appropriate methods and their analysis in a variety of areas including finite element methods, stiff and highly oscillatory ODEs, model reduction, and constrained optimization.</p>

Second-order Rosenbrock-Exponential (ROSEXP) Methods for Partitioned Differential Equations

“…Other ideas of partitioned nonlinear-nonlinear schemes were also explored in [12,13], but both of the methods derived in these publications are limited to first-order accuracy. We will include the SIERE and SBDF2ERE schemes derived in these papers in our comparisons:…”

“…For such problems, however, more attention has been dedicated to systems where one of the forcing terms is linear. Fewer options have been introduced for problems with nonlinear-nonlinear additive stiff forcing structure [12,13].…”

“…IMEX techniques treat one component of the forcing term implicitly and the other explicitly, thus these methods are appropriate for problems where one of the forcing terms is responsible for stiffness in the system. For problems with stiffness present in both forcing terms, partitioning approach has been extended to implicit-implicit methods [10] and implicit-exponential (IMEXP) integrators [11][12][13]. For such problems, however, more attention has been dedicated to systems where one of the forcing terms is linear.…”