Structure-preserving discretization of incompressible fluids

Pavlov, Dmitry; Mullen, Patrick; Tong, Yiying; Kanso, Eva; Marsden, Jerrold E.; Desbrun, Mathieu

doi:10.1016/j.physd.2010.10.012

Cited by 110 publications

(214 citation statements)

References 35 publications

Supporting

Mentioning

212

Contrasting

Order By: Relevance

“…Other ways to think of vector fields on manifolds in the continuous setting can also be leveraged to derive discrete representations. For instance, vector fields can be characterized as derivations of smooth functions on a manifold using directional derivative; this has led to an operator representation of vector fields, used first in fluid animation [Mullen et al 2009;Pavlov et al 2011;Gawlik et al 2011], and more recently in geometry processing [Azencot et al 2013;Azencot et al 2015].…”

Section: Representationsmentioning

confidence: 99%

“…Moreover, since a vector field represents an infinitesimal flow, this Lie derivative operator can be seen as an infinitesimal action of a Lie algebra element of the diffeomorphism group on the space of scalar functions (in the study of dynamical systems, this idea was brought forth nearly a century ago by Koopman [1931]). In the discrete setting, Pavlov et al [2011] showed that if one defines a discrete notion of diffeormorphism that transfers scalar values between dual cells, then the resulting Lie derivative turns out to be (the Hodge star of) the discrete 1-form we used above, since it represents fluxes across dual cells. Since then, this functional point of view was shown to be very relevant to, e.g., the problem of finding correspondences between meshes [Azencot et al 2013;Azencot et al 2015].…”

Section: Link To Discrete Diffeomorphismsmentioning

confidence: 99%

See 1 more Smart Citation

Vector field processing on triangle meshes

Goes¹,

Desbrun

Tong

2016

ACM SIGGRAPH 2016 Courses

View full text Add to dashboard Cite

While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application.This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finitedimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians. PrefaceT hese course notes provide a review of the various representations of tangent vector fields on triangulated surfaces. Over the past decades, a number of approaches to express, design, and analyze vector fields on arbitrary meshes have been proposed in geometry processing, targeting a series of applications varying from rendering to animation. Some describe vector fields through values on vertices, some on edges, and some on faces; some treat vector fields as piecewise constant, some as piecewise linear, and some even involve non-linear finite elements. This lack of a unified approach to vector field representation and manipulation can be confusing for practitioners, and rare are the references that discuss the pros and cons of these various representations. The chapters in this document were written with this fact in mind, as a way to offer an introduction to the motivations, benefits, and shortcomings of the most common discretizations of vector fields in graphics-along with their associated discrete differential operators (curl, divergence, Laplacian, etc). Emphasis is placed on the tradeoffs between simplicity, efficiency, and accuracy of these geometry processing tools for applications ranging from texture synthesis to meshing.Course rationale. In many ways, this course is a continuation of ACM SIGGRAPH courses on Discrete Differential Geometry (SIGGRAPH '05 and '06, and SIGGRAPH Asia '09) and Discrete Exte...

show abstract

Section: Representationsmentioning

confidence: 99%

Section: Link To Discrete Diffeomorphismsmentioning

confidence: 99%

Vector field processing on triangle meshes

Goes¹,

Desbrun

Tong

2016

ACM SIGGRAPH 2016 Courses

View full text Add to dashboard Cite

show abstract

“…In order to provide fluid simulations with stable long-term behavior across different space or time resolutions, Pavlov et al [2011] introduced a variational integrator for fluids in Eulerian representation by discretizing the fluid motion as a Lie group acting on the space of functions, and formulating the kinetic energy on its Lie algebra. The motion of an incompressible, inviscid fluid is described in the continuous setting by a volume-preserving flow φt, i.e., a particle which is at a point p at time t = 0 will be found at φt(p) after being advected by the flow.…”

Section: Recap Of Variational Eulerian Integrationmentioning

confidence: 99%

Model-reduced variational fluid simulation

Liu¹,

Mason

Hodgson

et al. 2015

ACM Trans. Graph.

View full text Add to dashboard Cite

We present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness of coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. At the core of our contributions is a functional map approach to fluid simulation for which scalar-and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time to preserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computational time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer. We also demonstrate the versatility of our approach by showing how it easily extends to magnetohydrodynamics and turbulence modeling in 2D, 3D and curved domains.

show abstract

“…This requires a full-dimensional fluid solver that works for tetrahedral meshes. We considered several classes of solvers including finite element methods (e. g. [Feldman et al 2005a;Feldman et al 2005b]), methods based on discrete exterior calculus [Mullen et al 2009;Pavlov et al 2011], and ALE methods (such as [Klingner et al 2006]). We chose the residual distribution scheme [Sewall et al 2007;Dobes and Deconinck 2006;Deconinck and Ricchiuto 2007], which is akin to a finite-difference fluid approximation [Foster and Metaxas 1996], but generalized to a tetrahedral mesh, due to its amenability to our non-polynomial Galerkin projection and the fact that it can be stably integrated in the reduced space.…”

Section: Related Workmentioning

confidence: 99%

Non-polynomial Galerkin projection on deforming meshes

et al. 2013

View full text Add to dashboard Cite

Figure 1: Our method enables reduced simulation of fluid flow around this flying bird over 2000 times faster than the corresponding full simulation and reduced radiosity computation in this architectural scene over 113 times faster than the corresponding full radiosity. AbstractThis paper extends Galerkin projection to a large class of nonpolynomial functions typically encountered in graphics. We demonstrate the broad applicability of our approach by applying it to two strikingly different problems: fluid simulation and radiosity rendering, both using deforming meshes. Standard Galerkin projection cannot efficiently approximate these phenomena. Our approach, by contrast, enables the compact representation and approximation of these complex non-polynomial systems, including quotients and roots of polynomials. We rely on representing each function to be model-reduced as a composition of tensor products, matrix inversions, and matrix roots. Once a function has been represented in this form, it can be easily model-reduced, and its reduced form can be evaluated with time and memory costs dependent only on the dimension of the reduced space.

show abstract

Structure-preserving discretization of incompressible fluids

Cited by 110 publications

References 35 publications

Vector field processing on triangle meshes

Vector field processing on triangle meshes

Model-reduced variational fluid simulation

Non-polynomial Galerkin projection on deforming meshes

Contact Info

Product

Resources

About