A new implementation of the element free Galerkin method

Lu, Ye; Belytschko, Ted; Gu, Linxia

doi:10.1016/0045-7825(94)90056-6

Cited by 635 publications

(347 citation statements)

References 7 publications

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“…coupling to mesh-based methods close to the boundary [19], 2. penalty or perturbation methods [17,22], 3. the Lagrange multiplier method [12,27], 4. and Nitsche's method [4,28].…”

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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“…coupling to mesh-based methods close to the boundary [19], 2. penalty or perturbation methods [17,22], 3. the Lagrange multiplier method [12,27], 4. and Nitsche's method [4,28].…”

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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“…If p x is a complete polynomial of length and a x contains non-constant coefficients that depend on x: a x = 8 x 9 x : x … < x = (2) then the approximation ? is expressed as a polynomial of length with non-constant coefficients.…”

Section: Basic Ingredients Of the Meshless Efg Methodsmentioning

confidence: 99%

Efficient Shape Function Costruction of Efg Meshless Method

Metsis¹,

Papadrakakis²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…[6], text that coined the term Element-free Galerkin" and makes use of Lagrange Multipliers. A text by the same group [42] identi es the Lagrange Multipliers as the tractions on the boundary and adopts the same approximation used for the primary eld to interpolate these tractions. This modi ed variational principle is very similar to a weak form derived by Nitsche [47] but for a stabilization term.…”

Section: Meshless Approximationmentioning

confidence: 99%

“…Solutions ranged from coupling with FEs along the boundaries [7,29] to the use of the domain shape functions to approximate the Lagrange multipliers [42,43,9].…”

Section: Introductionmentioning

confidence: 99%

Essential boundary and interface conditions in the meshless analysis of shells.

Costa¹

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AcknowledgmentsFirstly, I would like to thank Prof. Paulo Pimenta for taking me as a student and guiding me through my research and more importantly through a real academic formation.Thank you for the many fruitfull and happy get-togethers and conversations, about my research and a whole variaty of other subjects. It has been an honor to be your student.Other masters were also kind enough to advise and help me, especially Prof. Carlos Tiago who welcomed me for a stay in Lisbon during my masters and became my referrence for all matters of meshless methods, and Prof. Eduardo Campello for great insights and helpful discussions.My sincere gratitute to Prof. Peter Wriggers for receiving me at the Institut für Kontinuumsmechanik in Hannover for a proli c and very pleasant year. Thank you for helping my research and formation.To my co-workers and o ce mates, thank you for providing a nice working environment, for attentive ears to obstacles and smily faces for the happy-hours. We use the EFG discretization technique for thick Reissner-Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local re nement on multi-region problems.

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A new implementation of the element free Galerkin method

Cited by 635 publications

References 7 publications

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

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

Efficient Shape Function Costruction of Efg Meshless Method

Essential boundary and interface conditions in the meshless analysis of shells.

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