Reproducing Kernel Particle Method for Solving Partial Differential Equations

Chen, Jiun‐Shyan; Liu, Wing Kam; Hillman, Michael; Chi, Sheng‐Wei; Lian, Yuan; Bessa, Miguel A.

doi:10.1002/9781119176817.ecm2104

Cited by 13 publications

(3 citation statements)

References 103 publications

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“…Concerning mesh-free strategies for the approximation of the gradient and Hessian operators, we refer to [23] for applications in structural dynamics as well as [24] for applications in fluid dynamics. The reader can also refer to [1,26] for applications to heat conduction problems, while [7] provides a state-of-the art review of the subject. We provide a here a systematic standpoint which encompasses a wide variety of discrete operators and aims at applications beyond the field of partial differential equations.…”

Section: Definition Of Discrete Projection and Differential Operatorsmentioning

confidence: 99%

A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

LeFloch

Jean-Marc²

2021

SSRN Journal

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Section: Definition Of Discrete Projection and Differential Operatorsmentioning

confidence: 99%

A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

LeFloch

Jean-Marc²

2021

SSRN Journal

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“…In this paper, RKPM is employed due to its flexibility in adopting arbitrary smoothness and domain discretization in solving the variational multiscale equations. Interested readers can refer to [39][40][41] for a comprehensive review of RKPM and its applications.…”

Section: Introductionmentioning

confidence: 99%

A variational multiscale immersed meshfree method for heterogeneous materials

et al. 2021

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We introduce an immersed meshfree formulation for modeling heterogeneous materials with flexible non-body-fitted discretizations, approximations, and quadrature rules. The interfacial compatibility condition is imposed by a volumetric constraint, which avoids a tedious contour integral for complex material geometry. The proposed immersed approach is formulated under a variational multiscale based formulation, termed the variational multiscale immersed method (VMIM). Under this framework, the solution approximation on either the foreground or the background can be decoupled into coarse-scale and fine-scale in the variational equations, where the fine-scale approximation represents a correction to the residual of the coarse-scale equations. The resulting fine-scale solution leads to a residual-based stabilization in the VMIM discrete equations. The employment of reproducing kernel (RK) approximation for the coarse- and fine-scale variables allows arbitrary order of continuity in the approximation, which is particularly advantageous for modeling heterogeneous materials. The effectiveness of VMIM is demonstrated with several numerical examples, showing accuracy, stability, and discretization efficiency of the proposed method.

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“…While originally developed purely for visual effects, and associated with evolution laws focused on the final appearance rather than the physical correctness of the behavior, particle systems also form the digital infrastructure for the implementation of a class of numerical methods (known as meshless, meshfree, or particle methods) that have started emerging since the late 1970s as alternatives to the traditional grid-based numerical methods (finite differences, finite volumes, finite elements). These methods-smoothed particle hydrodynamics (SPH) [4], reproducing kernel particle method (RKPM) [5], finite pointset method (FPM) [6], discrete element method (DEM) [7], etc.-provide rigorous methods to discretize the physical laws governing the continuum and thus provide physics-based evolution law for the properties of the particles that act both as interpolation nodes in the mathematical sense and as virtual volumes of infinitesimal size carrying the properties of the macroscopic mass they represent.…”

Section: Introductionmentioning

confidence: 99%

Design and Implementation of Particle Systems for Meshfree Methods with High Performance

Bilotta¹,

Zago²,

Hérault³

2019

High Performance Parallel Computing

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Particle systems, commonly associated with computer graphics, animation, and video games, are an essential component in the implementation of numerical methods ranging from the meshfree methods for computational fluid dynamics and related applications (e.g., smoothed particle hydrodynamics, SPH) to minimization methods for arbitrary problems (e.g., particle swarm optimization, PSO). These methods are frequently embarrassingly parallel in nature, making them a natural fit for implementation on massively parallel computational hardware such as modern graphics processing units (GPUs). However, naive implementations fail to fully exploit the capabilities of this hardware. We present practical solutions to the challenges faced in the efficient parallel implementation of these particle systems, with a focus on performance, robustness, and flexibility. The techniques are illustrated through GPUSPH, the first implementation of SPH to run completely on GPU, and currently supporting multi-GPU clusters, uniform precision independent of domain size, and multiple SPH formulations.

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Reproducing Kernel Particle Method for Solving Partial Differential Equations

Cited by 13 publications

References 103 publications

A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

A variational multiscale immersed meshfree method for heterogeneous materials

Design and Implementation of Particle Systems for Meshfree Methods with High Performance

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