Discrete multiscale vector field decomposition
ACM SIGGRAPH 2003 Papers
Discrete one-forms on meshes and applications to 3D mesh parameterization
Computer Aided Geometric Design
Discrete conformal mappings via circle patterns
ACM Transactions on Graphics (TOG)
Discrete differential forms for computational modeling
ACM SIGGRAPH 2006 Courses
Stable, circulation-preserving, simplicial fluids
ACM SIGGRAPH 2006 Courses
Stable, circulation-preserving, simplicial fluids
ACM Transactions on Graphics (TOG)
Discrete conformal mappings via circle patterns
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Discrete differential forms for computational modeling
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Discrete, vorticity-preserving, and stable simplicial fluids
SIGGRAPH '05 ACM SIGGRAPH 2005 Courses
Meshing genus-1 point clouds using discrete one-forms
Computers and Graphics
Design of tangent vector fields
ACM SIGGRAPH 2007 papers
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Mesh parameterization: theory and practice
ACM SIGGRAPH ASIA 2008 courses
Oriented Morphometry of Folds on Surfaces
IPMI '09 Proceedings of the 21st International Conference on Information Processing in Medical Imaging
Recent Advances in Computational Conformal Geometry
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
Surface Quasi-Conformal Mapping by Solving Beltrami Equations
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
Non-negative mixed finite element formulations for a tensorial diffusion equation
Journal of Computational Physics
Generalized Koebe's method for conformal mapping multiply connected domains
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Discrete one-forms on meshes and applications to 3D mesh parameterization
Computer Aided Geometric Design
Discrete surface Ricci flow: theory and applications
Proceedings of the 12th IMA international conference on Mathematics of surfaces XII
A generalization for stable mixed finite elements
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
SIAM Journal on Scientific Computing
HOT: Hodge-optimized triangulations
ACM SIGGRAPH 2011 papers
Combinatorial Continuous Maximum Flow
SIAM Journal on Imaging Sciences
IPMI'05 Proceedings of the 19th international conference on Information Processing in Medical Imaging
Journal of Scientific Computing
Dual formulations of mixed finite element methods with applications
Computer-Aided Design
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
ACM Transactions on Mathematical Software (TOMS)
Technical note: Delaunay Hodge star
Computer-Aided Design
Computational Geometry: Theory and Applications
Explicit simplicial discretization of distributed-parameter port-Hamiltonian systems
Automatica (Journal of IFAC)
Tool-adaptive offset paths on triangular mesh workpiece surfaces
Computer-Aided Design
An operator approach to tangent vector field processing
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
Dirichlet energy for analysis and synthesis of soft maps
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex and its geometric dual. The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. Definitions are given for discrete versions of all the usual operators of exterior calculus. The presence of forms and vector fields allows us to address their various interactions, which are important in applications. In many examples we find that the formulas derived from DEC are identical to the existing formulas in the literature. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. One potential application of DEC is to variational problems which come equipped with a rich exterior calculus structure. On the discrete level, such structures will be enhanced by the availability of DEC. One of the objectives of this thesis is to fill this gap. There are many constraints in numerical algorithms that naturally involve differential forms. Preserving such features directly on the discrete level is another goal, overlapping with our goals for variational problems. In this thesis we have tried to push a purely discrete point of view as far as possible. We argue that this can only be pushed so far, and that interpolation is a useful device. For example, we found that interpolation of functions and vector fields is a very convenient. In future work we intend to continue this interpolation point of view, extending it to higher degree forms, especially in the context of the sharp, Lie derivative and interior product operators. Some preliminary ideas on this point of view are presented in the thesis. We also present some preliminary calculations of formulas on regular nonsimplicial complexes.