Adaptive multiresolution refinement with distance fields

Tsukanov, Igor; Shapiro, Vadim

doi:10.1002/nme.2087

Cited by 6 publications

(3 citation statements)

References 33 publications

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“…Both p and h refinements may be supported, but the global problem must be solved for each refinement, and the shape of refinement regions is determined and limited by the type of basis functions used in Ψ. Another approach to refinement in Tsukanov and Shapiro (2007) advocates representing the refined solution as a series of localized structures, each requiring the solving of a local boundary value problem. Each localization is specified by a refinement window of arbitrary shape, represented implicitly by a normalized window function (ω 1 ≥ 0).…”

Section: Meshfree Modelling and Analysis With Rfmmentioning

confidence: 99%

“…For example, if the refinement region is contained in the interior of a domain Ω, the Dirichlet solution structure (2.6) is modified as u = ϕ + ωΨ + ω 2 1 H(ω 1 )Ψ 1 , where ω 2 1 ensures C 1 -continuity of the solution field, H(ω 1 ) guarantees that the refined solution does not modify the solution outside the refinement window, and Ψ 1 is a refinement polynomial constructed from a set of additional basis functions. See Tsukanov and Shapiro (2007) for further details and application to more general boundary value problems and refinement windows.…”

Section: Meshfree Modelling and Analysis With Rfmmentioning

confidence: 99%

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Semi-analytic geometry with R-functions

2007

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V. L. Rvachev called R-functions 'logically charged functions' because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework -in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of Rfunctions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use. CONTENTS1 From Descartes to Rvachev 240 2 Functions for shapes with corners 241 3 R-functions 246 4 From inequalities to normalized functions 258 5 The inverse problem of analytic geometry 263 6 Geometric modelling 275 7 Boundary value problems 284 8 Conclusions 294 References 296 240 V. ShapiroHowever, he could not explain his own constructions using the classical Semi-analytic geometry with R-functions 241 methods of analytic and algebraic geometry that focus on direct problems of investigating given equations and inequalities. In contrast, Rvachev wanted to devise a methodology for solving what he termed the inverse problem of analytic geometry: constructing equations and inequalities for given geometric objects. This quest resulted in his seminal publication, Rvachev (1963), followed by the comprehensive theory of R-functions that has been developed over the last forty-plus years. 1 In a nutshell, R-functions operate on real-valued inequalities as differentiable logic operations; the resulting theory solves the inverse problem of analytic geometry and has a wide range of applications, with particular emphasis on solutions of boundary value problems. As of 2001, the bibliography on the theory of R-functions included more than fifteen monographs and over five hundred technical articles coauthored by Rvachev and his followers (Matsevity 2001). R-functions were introduced into the Western literature by the author (Shapiro 1988), and are now widely used in geometric modelling, computer graphics, robotics, engineering analysis, and other computational applications. The goal of this paper is to expose the key concepts in the theory of R-functions, without trying to be comprehensive. This subject will take us up to Section 4. Additional references in English are now accessible, notably the review by Rvachev and Sheiko (1995). Among the references in Russian, the monograph by Rvachev (1982) continues to serve as an encyclopedic source of many key ideas and results. The utility of R-functions cannot be fully appreciated without some discussion of...

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Section: Meshfree Modelling and Analysis With Rfmmentioning

confidence: 99%

Section: Meshfree Modelling and Analysis With Rfmmentioning

confidence: 99%

Semi-analytic geometry with R-functions

2007

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“…how information is transferred among scales? Among the recent works, we can mention the method of Guidault et al [28] based on the LATIN method and domain decomposition concepts, the multi-grid method proposed in [29], the method of Cloirec et al [30] based on Lagrange multipliers, the multi-scale projection method of Belytschko and coworkers [31,32], the concurrent multi-scale approach of Liu and coworkers [33][34][35], the hp FEM method of Krause et al [36,37], the concurrent multi-level method of Gosh et al [38,39] based on the Voronoi Cell FEM; the multi-resolution approach proposed by Tsukanov and Shapiro [40] based on distance fields. The proposed GFEM gl is also related to the refined global-local FEM proposed by Mao and Sun [41] and based on linear combinations of global and local approximations.…”

Section: Introductionmentioning

confidence: 99%

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

Kim

Pereira

Duarte

2009

Numerical Meth Engineering

109

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SUMMARYThis paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)-scale problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of three-dimensional cracks, while the global problem addresses the macro-scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp-GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three-dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three-dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented.

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Mesh independent analysis of shell‐like structures

Kumar

Periyasamy

2012

Numerical Meth Engineering

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SUMMARY Mesh independent analysis is motivated by the desire to use accurate geometric models represented as equations rather than approximated by a mesh. The trial and test functions are approximated or interpolated on a background mesh that is independent of the geometry. This background mesh is easy to generate because it does not have to conform to the geometry. Essential boundary conditions can be applied using the implicit boundary method where the trial and test functions are constructed utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. This approach has been demonstrated for two‐dimensional (2D) and three‐dimensional (3D) structural analysis and is extended in this paper to model shell‐like structures. The background mesh consists of 3D elements that use uniform B‐spline approximations, and the shell geometry is assumed to be defined as parametric surfaces to allow arbitrarily complex shell‐like structures to be modeled. Several benchmark problems are used to study the validity of these 3D B‐spline shell elements. Copyright © 2012 John Wiley & Sons, Ltd.

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Adaptive multiresolution refinement with distance fields

Cited by 6 publications

References 33 publications

Semi-analytic geometry with R-functions

Semi-analytic geometry with R-functions

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

Mesh independent analysis of shell‐like structures

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