A unified algorithm for the selection of collocation stencils for convex, concave, and singular problems

Jacquemin, Thibault; Bordas, Stéphane P.A.

doi:10.1002/nme.6703

Cited by 15 publications

(8 citation statements)

References 43 publications

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“…We used this approach. The algorithms presented in reference [33] or in reference [2] are alternative algorithms that use boundary nodes and elements to decide upon the inclusion of nodes in the domain. Other algorithms such as the "crossing number" or the "winding number" methods, described in references [34,35,36], can be used in 2D.…”

Section: Domain Discretizationmentioning

confidence: 99%

“…• the connections to other boundary nodes if boundary elements are used to enforce the generalized visibility criterion [2] or to speed-up the refinement of the surface.…”

Section: From Discretization To Smart Cloudmentioning

confidence: 99%

“…In this work, we selected the stencil nodes based on the distance criterion [4]. Close to concave boundaries of the domain, we used the visibility criterion with a threshold angle of 5.0 • as presented in reference [2]. Based on these results, we selected in this work a threshold value of 0.3 as it leads to the lowest error for the body with a cylindrical hole problem.…”

Section: Threshold Sensitivity Analysismentioning

confidence: 99%

“…The selection of collocation stencil is a key aspect of this family of methods. To ease their application to all types of problems and domains, a unified approach was introduced in reference [2]. The finite difference method was the first such collocation method.…”

Section: Introductionmentioning

confidence: 99%

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Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Jacquemin

Suchde

Bordas

2023

Computer-Aided Design

Self Cite

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Section: Domain Discretizationmentioning

confidence: 99%

“…• the connections to other boundary nodes if boundary elements are used to enforce the generalized visibility criterion [2] or to speed-up the refinement of the surface.…”

Section: From Discretization To Smart Cloudmentioning

confidence: 99%

Section: Threshold Sensitivity Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Jacquemin

Suchde

Bordas

2023

Computer-Aided Design

Self Cite

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“…In context of RBF-FD the topic has been thoroughly investigated in adaptive solution of Poisson's equation [26], including the problems with point singularities [23]. In [28], the authors present a method for stencil selection, based on the visibility criterion, for convex, concave, and singular problems in 2D and 3D. In [26,23] authors demonstrated effective stencils of 6 or 5 nearby points that are sufficiently uniformly distributed around central node to support RBF-FD, and successfully solved several test problems.…”

Section: Introductionmentioning

confidence: 99%

Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs

Depolli¹,

Slak²,

Kosec³

2022

Preprint

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In this paper, we present a novel parallel dimension-independent node positioning algorithm that is capable of generating nodes with variable density, suitable for meshless numerical analysis. A very efficient sequential algorithm based on Poisson disc sampling is parallelized for use on shared-memory computers, such as the modern workstations with multi-core processors. The parallel algorithm uses a global spatial indexing method with its data divided into two levels, which allows for an efficient multi-threaded implementation. The addition of bootstrapping enables the algorithm to use any number of parallel threads while remaining as general as its sequential variant. We demonstrate the algorithm performance on six complex 2-and 3-dimensional domains, which are either of non rectangular shape or have varying nodal spacing or both. We perform a run-time analysis of the algorithm, to demonstrate its ability to reach high speedups regardless of the domain and to show how well it scales on the experimental hardware with 16 processor cores. We also analyse the algorithm in terms of the effects of domain shape, quality of point placement, and various parallelization overheads.

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Strong form mesh-free hp-adaptive solution of linear elasticity problem

Jančič

Kosec

2023

Engineering with Computers

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We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the need for user intervention.

show abstract

A unified algorithm for the selection of collocation stencils for convex, concave, and singular problems

Cited by 15 publications

References 43 publications

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Smart Cloud Collocation: Geometry-Aware Adaptivity Directly From CAD

Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs

Strong form mesh-free hp-adaptive solution of linear elasticity problem

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