\`x^2+y_1+z_12^34\`
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Simulating deformable objects for computer animation: A numerical perspective

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    * Corresponding author 

The first and last authors are supported by NSERC Discovery grants 84306 and RGPIN/2017-04604 respectively. Pai's research was also supported by a Canada Research Chair and an NSERC Idea-to-Innovation grant co-sponsored by Vital Mechanics

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  • 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 a priori 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.

    Mathematics Subject Classification: Primary: 65D18, 68U05; Secondary: 65P99.

    Citation:

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  • Figure 1.  Deformable articulated objects: a swaying tree and a constrained jelly brick; cf. [13]

    Figure 2.  Moving tetrahedral FEM mesh for position coordinates $ {\bf q}(t) $

    Figure 3.  Damping curves for the SDIRK method (solid line), TR-BDF2 (dashed) and BDF2 (dash-dot). The two DIRK methods behave similarly, and they differ significantly from BDF2

    Figure 4.  Computational costs for a swinging armadillo simulation [13]. The cost of exponential integrators including ERE becomes prohibitive as the stiffness parameter increases. By contrast, the cost of SIERE does not grow significantly with stiffness

    Figure 5.  Plot of of the first 1000 eigenvalues of a soft body problem

    Figure 6.  Potential energy plots for different integrators applied to a soft object: BE (thick solid line), SIERE with $ s = 10 $ (thin solid line), STR-SBDF2ERE with $ s = 10 $ (dash-dot), TR-BDF2 (dotted), and SDIRK (dashed). A soft beam is fixed at its ends and is subjected to gravity. Notice that the TR-BDF2 and SDIRK energies do not decay by much, whereas BE dissipates energy quickly. SIERE is less damping than BE but still much more damping than STR-SBDF2ERE, which in turn is still more damping than the two DIRK methods

    Figure 7.  With large enough time steps, velocity based contact constraints may reject plausible steps. If a vertex in blue is constrained to have a strictly positive velocity with respect to the convex gray contact surface, then plausibly valid next-step configurations (right) may be erroneously rejected

    Figure 8.  Barrier function $ b = b(x; \delta) $ for different values of $ \delta $. It is used in (30) to approximate the contact force

    Figure 9.  Plot of the smoothing function $ s = s(x;\epsilon) $ for different values of $ \epsilon $. It is used in (34) through (32) to approximate the Coulomb friction force

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