Chain-Based Representations for Solid and Physical Modeling

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

doi:10.1109/tase.2009.2021342

Cited by 24 publications

(30 citation statements)

References 29 publications

(39 reference statements)

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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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“…The Linear Algebraic Representation (lar), introduced in [9], aims at representing the chain complex [8,17] generated by a piecewise-linear geometric complex embedded either in 2D or in 3D. This representation provides a minimal characterization of geometry and topology of a cellular complex, through (a) the embedding mapping µ : C 0 → E d of 0-cells (vertices), and (b) a description of d-cells and/or (d − 1)-cells as subsets of vertices.…”

Section: Linear Algebraic Representationmentioning

confidence: 99%

Algebraic Filtering of Surfaces from 3D Medical Images with Julia

Jiřík¹,

Paoluzzi²

2020

Cad'20

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In this paper we introduce a novel algebraic filter, based on algebraic topology methods, to extract and smooth the boundary surface of any subset of voxels arising from the segmentation of a 3D medical image. The input of the Linear Algebraic Representation (lar) Surface extraction filter (lar-surf) is defined as a chain, i.e., an element of a linear space of chains here subsets of voxels represented in coordinates as a sparse binary vector. The output is produced by a linear mapping between spaces of 3-and 2-chains, given by the boundary operator ∂ 3 : C 3 → C 2 . The only data structures used in this approach are sparse arrays with one or two indices, i.e., sparse vectors and sparse matrices. This work is based on lar algebraic methods and is implemented in Julia language, natively supporting parallel computing on hybrid hardware architectures.

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“…Increasingly, these algebraic topological models are finding applications in computational modeling of engineering systems. For example, several researchers proposed that all geometric and physical computations should be based on a common (co)chain complex model [27,28,29], and that a vast majority of physical laws may be enforced as topological invariants of common transformations used in automated design systems [30]. Algebraic topological structure has been used explicitly in variety of modeling systems spanning diverse areas such interactive physics [31], computer graphics [32], and manufacturing systems [33].…”

Section: Brief History Of Lumped-parameter Modelingmentioning

confidence: 99%

Topological semantics for lumped parameter systems modeling

Wang

Shapiro

2019

Advanced Engineering Informatics

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Behaviors of many engineering systems are described by lumped parameter models that encapsulate the spatially distributed nature of the system into networks of lumped elements; the dynamics of such a network is governed by a system of ordinary differential and algebraic equations. Languages and simulation tools for modeling such systems differ in syntax, informal semantics, and in the methods by which such systems of equations are generated and simulated, leading to numerous interoperability challenges. Logical extensions of SysML aim specifically at unifying a subset of the underlying concepts in such languages.We propose to unify semantics of all such systems using standard notions from algebraic topology. In particular, Tonti diagrams classify all physical theories in terms of physical laws (topological and constitutive) defined over a pair of dual cochain complexes and may be used to describe different types of lumped parameter systems. We show that all possible methods for generating the corresponding state equations within each physical domain correspond to paths over Tonti diagrams. We further propose a generalization of Tonti diagram that captures the behavior and supports canonical generation of state equations for multi-domain lumped parameter systems.The unified semantics provides a basis for greater interoperability in systems modeling, supporting automated translation, integration, reuse, and numerical simulation of models created in different authoring systems and applications. Notably, the proposed algebraic topological semantics is also compatible with spatially and temporally distributed models that are at the core of modern CAD and CAE systems.

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Chain-Based Representations for Solid and Physical Modeling

Cited by 24 publications

References 29 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

Algebraic Filtering of Surfaces from 3D Medical Images with Julia

Topological semantics for lumped parameter systems modeling

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