Introduction to Solid Modeling
Introduction to Solid Modeling
Geometric and solid modeling: an introduction
Geometric and solid modeling: an introduction
Implementing discrete mathematics: combinatorics and graph theory with Mathematica
Implementing discrete mathematics: combinatorics and graph theory with Mathematica
Geometric programming: a programming approach to geometric design
ACM Transactions on Graphics (TOG)
Splitting a complex of convex polytopes in any dimension
Proceedings of the twelfth annual symposium on Computational geometry
Journal of Computational Physics
Primitives for the manipulation of general subdivisions and the computation of Voronoi
ACM Transactions on Graphics (TOG)
Representations for Rigid Solids: Theory, Methods, and Systems
ACM Computing Surveys (CSUR)
A multivector data structure for differential forms and equations
Mathematics and Computers in Simulation
Nonmanifold Topology Based on Coupling Entities
IEEE Computer Graphics and Applications
Winged edge polyhedron representation.
Winged edge polyhedron representation.
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Direct Methods for Sparse Linear Systems (Fundamentals of Algorithms 2)
Progressive dimension-independent Boolean operations
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Discrete physics using metrized chains
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Interaction based simulation of dynamical system with a dynamical structure (DS)2 in MGS
Proceedings of the 2011 Summer Computer Simulation Conference
Q-Complex: Efficient non-manifold boundary representation with inclusion topology
Computer-Aided Design
Technical note: Linear algebraic representation for topological structures
Computer-Aided Design
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In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multi-linear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling---which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures---under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are employed. Last but not least, our tensorbased approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior.