Discrete physics using metrized chains

DiCarlo, Antonio; Milicchio, Franco; Paoluzzi, Alberto; Shapiro, Vadim

doi:10.1145/1629255.1629273

Cited by 15 publications

(17 citation statements)

References 41 publications

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“…Moreover, in algebraic topology the topological equations of a fundamental problem are described by using the coboundary operators: a coboundary operator is any map from a subset of n p−cells to a subset of m (p + 1)-cells [56][57][58][59][60]: (4) where e i p is the i-th p-cell and e j p+1 is the j-th (p + 1)-cell. When m = 1 and n equals the number of cofaces of the (p + 1) −cell, the coboundary operator is indicated with the symbol δ: (5) and δ p defines the coboundary of e p+1 .…”

Section: Introductionmentioning

confidence: 99%

On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

Ferretti

2019

Curved and Layered Structures

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The Cell Method (CM) is an algebraic numerical method based on the use of global variables: the configuration, source and energetic global variables. The configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, define two vector spaces that are a bialgebra and its dual algebra. The operators of these topological equations are generated by the outer product of the geometric algebra, for the primal vector space, and by the dual product of the dual algebra, for the dual vector space. The topological equations in the primal cell complex are coboundary processes on even exterior discrete p−forms, whereas the topological equations in the dual cell complex are coboundary processes on odd exterior discrete p−forms. Being expressed by coboundary processes in two different vector spaces, compatibility and equilibrium can be enforced at the same time, with compatibility enforced on the primal cell complex and equilibrium enforced on the dual cell complex. By way of example, in the present paper compatibility and equilibrium are enforced on a cantilever elastic beam with elastic inclusion. In effect, the CM shows its maximum potentialities right in domains made of several materials, as, being an algebraic approach, can treat any kind of discontinuities of the domain easily.

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Section: Introductionmentioning

confidence: 99%

On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

Ferretti

2019

Curved and Layered Structures

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“…e groups of p-chains and those of p-cochains may be identi ed with each other in in nitely di erent ways. Di erent legitimate identi cations, while a ecting the metric properties of the chain-cochain complex [16], do not change the topology of nite complexes. Since we shall only use the topological properties of nite chain-cochain complexes, we feel free to chose the simplest possible identi cation, obtained by identifying the elements of the standard chain bases with the corresponding elements of the standard cochain bases.…”

Section: Chain Addition Is De Ned By Addition Of Chain Valuesmentioning

confidence: 99%

Regularized arrangements of cellular complexes

Paoluzzi¹,

Shapiro²,

DiCarlo³

2017

Preprint

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In this paper we propose a novel algorithm to combine two or more cellular complexes, providing a minimal fragmentation of the cells of the resulting complex. We introduce here the idea of arrangement generated by a collection of cellular complexes, producing a cellular decomposition of the embedding space. e algorithm that executes this computation is called Merge of complexes. e arrangements of line segments in 2D and polygons in 3D are special cases, as well as the combination of closed triangulated surfaces or meshed models.is algorithm has several important applications, including Boolean and other set operations over large geometric models, the extraction of solid models of biomedical structures at the cellular scale, the detailed geometric modeling of buildings, the combination of 3D meshes, and the repair of graphical models. e algorithm is e ciently implemented using the Linear Algebraic Representation (LAR) of argument complexes, i.e., on sparse representation of binary characteristic matrices of d-cell bases, well-suited for implementation in last generation accelerators and GPGPU applications.

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“…Dodziuk [11] originally proposed the definition of M W hit k but it has been called the Galerkin Hodge [7] for its relation to finite element methods. Bell [5] has implemented linear solvers in a DEC context using M W hit k for various k. Many other discrete Hodge stars appear in the literature, including the combinatorial discrete Hodge star of Wardetzsky and Wilson [25,28] and the metrized chain Hodge star of DiCarlo et al [10]. To our knowledge, no authors have defined a discrete Hodge star using dual interpolatory functions as we propose in this work.…”

Section: Prior Work and Notationmentioning

confidence: 99%

Dual formulations of mixed finite element methods with applications

Gillette

Bajaj

2011

Computer-Aided Design

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Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.

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Discrete physics using metrized chains

Cited by 15 publications

References 41 publications

On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

Regularized arrangements of cellular complexes

Dual formulations of mixed finite element methods with applications

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