Energy Conservation for the Simulation of Deformable Bodies

Su, Jonathan; Sheth, Rahul; Fedkiw, Ronald

doi:10.1109/tvcg.2012.132

Cited by 19 publications

(16 citation statements)

References 36 publications

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“…This motivated the development of symplectic integrators [Hairer et al 2002;Kharevych et al 2006] and mixed implicit-explicit methods (IMEX) [Bridson et al 2003;Stern and Grinspun 2009], featuring better energy conservation properties. Another approach is energy budgeting [Su et al 2013] which enforces energy conservation explicitly. However, implicit Euler integration continues to be one of the popular choices in applications of physics-based animation where robustness is an important criterion and numerical damping is not a major concern.…”

Section: Related Workmentioning

confidence: 99%

Projective dynamics

Martín²,

et al. 2014

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EPFL Figure 1: We propose a new "projection-based" implicit Euler integrator that supports a large variety of geometric constraints in a single physical simulation framework. In this example, all the elements including building, grass, tree, and clothes (49k DoFs, 43k constraints), are simulated at 3.1ms/iteration using 10 iterations per frame (see also accompanying video). AbstractWe 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.

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Section: Related Workmentioning

confidence: 99%

Projective dynamics

Martín²,

et al. 2014

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“…However, numerical damping may be tolerable in applications where energy conservation is not critical and many recent methods continue to rely on the implicit Euler method [Martin et al 2011;Hahn et al 2012]. A recently proposed technique that allows for a more direct control of damping is energy budgeting [Su et al 2013]. Energy budgeting can be applied on top of any numerical time integration method, including ours.…”

Section: Related Workmentioning

confidence: 99%

“…(3) with a damped velocityṽn. We use only a very simple damping modelether drag [Su et al 2013], which setsṽn := αvn, where α ∈ [0, 1] is a parameter, typically very close to 1. However, any damping model can be used with our method, such as the rigid-body modes preserving drag [Müller et al 2007] or truly material-only stiffnessproportional damping [Nealen et al 2005].…”

Section: Damping and Collisionsmentioning

confidence: 99%

Fast simulation of mass-spring systems

et al. 2013

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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. AbstractWe 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.

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“…This paper extends the Gauss‐Seidel iteration of the PBD/XPBD algorithm while maintaining its core. The proposed extension is inspired by the idea of conserved energy differences from [SSF13], and utilizes it within the Gauss‐Seidel iterations by applying the potential energies of the materials as projected constraints. Furthermore, we keep the velocities in their implicit state similar to [Jak01]; therefore, the iterations do not suffer from the extreme values of velocities due to the step‐size division.…”

Section: Introductionmentioning

confidence: 99%

Position‐Based Simulation of Elastic Models on the GPU with Energy Aware Gauss‐Seidel Algorithm

Cetinaslan

2019

Computer Graphics Forum

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In this paper, we provide a smooth extension of the energy aware Gauss‐Seidel iteration to the Position‐Based Dynamics (PBD) method. This extension is inspired by the kinetic and potential energy changes equalization and uses the foundations of the recent extended version of PBD algorithm (XPBD). The proposed method is not meant to conserve the total energy of the system and modifies each position constraint based on the equality of the kinetic and potential energy changes within the Gauss‐Seidel process of the XPBD algorithm. Our extension provides an implicit solution for relatively better stiffness during the simulation of elastic objects. We apply our solution directly within each Gauss‐Seidel iteration and it is independent of both simulation step‐size and integration methods. To demonstrate the benefits of our proposed extension with higher frame rates, we develop an efficient and practical mesh coloring algorithm for the XPBD method which provides parallel processing on a GPU. During the initialization phase, all mesh primitives are grouped according to their connectivity. Afterwards, all these groups are computed simultaneously on a GPU during the simulation phase. We demonstrate the benefits of our method with many spring potential and strain‐based continuous material constraints. Our proposed algorithm is easy to implement and seamlessly fits into the existing position‐based frameworks.

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Energy Conservation for the Simulation of Deformable Bodies

Cited by 19 publications

References 36 publications

Projective dynamics

Projective dynamics

Fast simulation of mass-spring systems

Position‐Based Simulation of Elastic Models on the GPU with Energy Aware Gauss‐Seidel Algorithm

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