Results 1  10
of
11
FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY
, 2009
"... Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to diff ..."
Abstract

Cited by 84 (6 self)
 Add to MetaCart
(Show Context)
Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the wellposedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures that ensure wellposedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation selfcontained for a variety of readers.
Rotational Symmetry Field Design on Surfaces
"... tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). ..."
Abstract

Cited by 58 (7 self)
 Add to MetaCart
tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). In addition, both topological editing (c) and global smoothing (d) can be used to remove more singularities from (b). However, topological editing (c) provides local control while excessive global smoothing (d) can cause hatch directions to deviate from their natural orientations (neck and chest). Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and penandink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces. As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.
NSymmetry Direction Field Design
, 2008
"... Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and ..."
Abstract

Cited by 57 (1 self)
 Add to MetaCart
Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to use nonoptimum local relaxation procedures. In this paper, we formalize Nsymmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the PoincaréHopf theorem in the case of Nsymmetry direction fields on 2manifolds. Based on this theorem, we explain how to control the topology of Nsymmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth eld interpolating user de ned singularities and directions.
Discrete Quadratic Curvature Energies
"... We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assemble ..."
Abstract

Cited by 25 (3 self)
 Add to MetaCart
We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast timeintegration of cloth dynamics with a two to threefold net speedup over existing nonlinear methods, and nearinteractive rates for Willmore smoothing of large meshes.
GEOMETRIC COMPUTATIONAL ELECTRODYNAMICS WITH VARIATIONAL INTEGRATORS AND DISCRETE DIFFERENTIAL FORMS
, 2009
"... In this paper, we develop a structurepreserving discretization of the Lagrangian framework for electromagnetism, combining techniques from variational integrators and discrete differential forms. This leads to a general family of variational, multisymplectic numerical methods for solving Maxwell’s ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
In this paper, we develop a structurepreserving discretization of the Lagrangian framework for electromagnetism, combining techniques from variational integrators and discrete differential forms. This leads to a general family of variational, multisymplectic numerical methods for solving Maxwell’s equations that automatically preserve key symmetries and invariants. In doing so, we demonstrate several new results, which apply both to some wellestablished numerical methods and to new methods introduced here. First, we show that Yee’s finitedifference timedomain (FDTD) scheme, along with a number of related methods, are multisymplectic and derive from a discrete Lagrangian variational principle. Second, we generalize the Yee scheme to unstructured meshes, not just in space but in 4dimensional spacetime. This relaxes the need to take uniform time steps, or even to have a preferred time coordinate at all. Finally, as an example of the type of methods that can be developed within this general framework, we introduce a new asynchronous variational integrator (AVI) for solving Maxwell’s equations. These results are illustrated with some prototype simulations that show excellent energy and conservation behavior and lack of spurious modes, even for an irregular mesh with asynchronous time stepping.
Robust Polylines Tracing for NSymmetry Direction Field on Triangulated Surfaces
"... We are proposing an algorithm for tracing polylines that are oriented by a direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. T ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We are proposing an algorithm for tracing polylines that are oriented by a direction field defined on a triangle mesh. The challenge is to ensure that two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is impossible to enforce with existing algorithms. The core of our contribution is to determine how polylines cross each triangle. Our solution is inspired by EdgeMaps where each triangle boundary is decomposed into inflow and outflow intervals such that each inflow interval is mapped onto an outflow interval. To cross a triangle, we find the inflow interval that contains the entry point, and link it to the corresponding outflow interval, with the same barycentric coordinate. To ensure that polylines cannot merge or cross, we introduce a new direction field representation, we resolve the inflow/outflow interval pairing with a guaranteed combinatorial algorithm, and propagate the barycentric positions with arbitrary precision number representation. Using these techniques, two streamlines crossing the same triangle cannot merge or cross, but only locally overlap when all streamline extremities are located on the same edge. Crossfree and mergefree polylines can be traced on the mesh by iteratively crossing triangles. Vector field singularities and polyline/vertex crossing are characterized and consistently handled.
Design of Flows on Smooth Surfaces
"... Vector fields tangent to a 2D manifold surface have many applications in Computer Graphics, Scientific Visualization and Geometric Design. These Tangent Vector Fields (TVF) may be used for generating textures on a surface, computing expressive renderings of a surface, modeling and visualizing a shea ..."
Abstract
 Add to MetaCart
(Show Context)
Vector fields tangent to a 2D manifold surface have many applications in Computer Graphics, Scientific Visualization and Geometric Design. These Tangent Vector Fields (TVF) may be used for generating textures on a surface, computing expressive renderings of a surface, modeling and visualizing a shear stress, simulating the flow of fluids or fire on a surface. These methods usually require the design of the TVF over the surface. In the past years there has been several papers on the design of TVF on a given 2D manifold either defined as a triangulation ([1], [2], [3]) or as a subdivision surface ([4]). At the kernel of each of these methods is a function that interpolates continuously discrete values of a tangent vector field over a surface. Objectives: We have two objectives. First we would like to establish a proper comparison between the previously proposed interpolation methods. Second we want to establish a new TVF design method targeted to smooth piecewise polynomial surfaces. Such surfaces are the standard model for manufactured objects, and can also be used to model arbitrary topology surfaces ([5]). We expect that a dedicated method for smooth polynomial surfaces will speed up in an order of magnitude the design of TVF on such surfaces, thereby allowing realtime design of TVF.