Implementation of boundary conditions for meshless methods

Günther, Frank; Liu, Wing Kam

doi:10.1016/s0045-7825(98)00014-0

Cited by 144 publications

(69 citation statements)

References 25 publications

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“…There are many different approaches toward the treatment of Dirichlet boundary conditions with meshfree methods [4,10,16,27], e.g.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…Many different approaches toward the treatment of Dirichlet problems have been proposed over the years [4,10,12,13,16,17,19,22,27]. Most of them, however, suffer from one or several drawbacks (restrictions on the distribution of the discretization points, need for an additional boundary function space, saddle-point structure, non-optimal convergence rates, etc.).…”

Section: Introductionmentioning

confidence: 99%

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A Particle-Partition of Unity Method Part V: Boundary Conditions

Griebel

Schweitzer

2003

Geometric Analysis and Nonlinear Partial Differential Equations

100

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In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970's allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

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“…There are many different approaches toward the treatment of Dirichlet boundary conditions with meshfree methods [4,10,16,27], e.g.…”

Section: Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Particle-Partition of Unity Method Part V: Boundary Conditions

Griebel

Schweitzer

2003

Geometric Analysis and Nonlinear Partial Differential Equations

100

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“…The treatment of essential boundary conditions in meshfree methods is not straightforward and a number of different approaches have been suggested [1,12,14,15,17,[19][20][21][22][23]26]. In [17] we have presented how Nitsche's method [25] can be applied successfully in the meshfree context.…”

Section: Essential Boundary Conditionsmentioning

confidence: 99%

“…Many different techniques have been suggested to overcome this issue, see e.g. [1,12,14,15,17,[19][20][21][22][23]26] and the references within.…”

Section: Introductionmentioning

confidence: 99%

An Algebraic Treatment of Essential Boundary Conditions in the Particle–Partition of Unity Method

Schweitzer¹

2009

SIAM J. Sci. Comput.

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Summary. This paper is concerned with the treatment of essential boundary conditions in meshfree methods. In particular we focus on the particle-partition of unity method (PPUM). However, the proposed technique is applicable to any partition of unnity based approach.We present an efficient scheme for the automatic construction of a direct splitting of a PPUM function space into the degrees of freedom suitable for the approximation of the Dirichlet data and the degrees of freedom that remain for the approximation of the PDE by simple linear algebra. Notably, our approach requires no restrictions on the distribution of the discretization points nor on the employed (local) approximation spaces.We attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces. Hence, the global splitting can be computed with (sub-)linear complexity. Due to this direct splitting of the meshfree PPUM function space we can implement a conforming local treatment of essential boundary data so that the realization of Dirichlet boundary values in the meshfree PPUM is straightforward. The presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results.

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“…There is a considerable amount of research on this topic, and many methods have been proposed. Research on the application of boundary conditions in meshless methods can be found for instance in Gosz and Liu [57], Günther and Liu [60], Chen and Wang [32], Pannachet and Askes [100], Fernández-Méndez and Huerta [50] and Guedes and César de Sá [59].…”

Section: Applying Boundary Conditionsmentioning

confidence: 99%

On meshless and nodel-based numerical simulations of forming processes

Quak¹

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Implementation of boundary conditions for meshless methods

Cited by 144 publications

References 25 publications

A Particle-Partition of Unity Method Part V: Boundary Conditions

A Particle-Partition of Unity Method Part V: Boundary Conditions

An Algebraic Treatment of Essential Boundary Conditions in the Particle–Partition of Unity Method

On meshless and nodel-based numerical simulations of forming processes

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