A stabilized conforming nodal integration for Galerkin mesh-free methods

Chen, Jiun‐Shyan; Wu, C. T.; Yoon, Sangpil; You, Yong‐Ouk

doi:10.1002/1097-0207(20010120)50:2<435::aid-nme32>3.0.co;2-a

Cited by 1,186 publications

(777 citation statements)

References 23 publications

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“…However, direct nodal integration is instable because of under integration and vanishing derivatives of shape functions at the nodes [29]. A stabilized conforming nodal integration (SCNI) is proposed in [30] to eliminate spatial instability problems and to improve accuracy and convergence properties. The main idea of the method is that nodal strains are determined by spatially averaging strains using the divergence theorem.…”

Section: Stabilized Conforming Nodal Integrationmentioning

confidence: 99%

“…This is because additional variables need to be added, a total of 5n when Gauss integration is used compared with n when SCNI is used. In summary, SCNI appears to offer a good combination of accuracy and computational efficiency, not only for elastic analysis problems [30] but now also for plastic analysis problems. Next, the efficacy of various optimization algorithms was considered (using SCNI).…”

Section: Numerical Examplesmentioning

confidence: 99%

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Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

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Section: Stabilized Conforming Nodal Integrationmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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“…The need for this background grid can be removed with the move to node-based integration schemes. A number of these have already been developed [8,19,18], and this area remains an active topic of research. Methods making use of nodal integration schemes are often referred to as "truly meshfree".…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we have not studied effect of particle distribution on numerical errors; however, Dolbow and Belytschko [23] have shown that integration errors grow with grid-based integration schemes as the particle spacing becomes highly irregular. The proposed nodal integration schemes of Chen et al [19,18] display accuracy that appears less sensitive to particle distribution than grid-based integration. For complicated geometries or situations for which the particle distribution needs to be carefully controlled, software for generating high quality finite element meshes represent the best tools for producing regular or smoothly varying particle distributions.…”

Section: Introductionmentioning

confidence: 99%

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Klein

Foulk

Chen

et al. 2001

Theoretical and Applied Fracture Mechanics

151

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Simulation of generalized fracture and fragmentation remains an ongoing challenge in computational fracture mechanics. There are difficulties associated not only with the formulation of physically-based models of material failure, but also with the numerical methods required to treat geometries that change in time. The issue of fracture criteria is addressed in this work through a cohesive view of material, meaning that a finite material strength and work to fracture are included in the material description. In this study, we present both surface and bulk cohesive formulations for modeling brittle fracture, detailing the derivation of the formulations, fitting relations, and providing a critical assessment of their capabilities in numerical simulations of fracture. Due to their inherent adaptivity and robustness under severe deformation, meshfree methods are especially well-suited to modeling fracture behavior. We describe the application of meshfree methods to both bulk and surface approaches to cohesive modeling. We present numerical examples highlighting the capabilities and shortcomings of the methods in order to identify which approaches are best-suited to modeling different types of fracture phenomena.

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“…One meshless approach based on natural neighbor coordinates has been proposed in [8]. Another meshless technique is nodal integration, as introduced in [9]. Here, an alternative tessellation of the domain is considered, where the vertices v j are enclosed by polygons T j , with Ω = j T j and the above integral becomes j Tj ∇ψ · f, assigning a part of the integral to each node v j rather than to an element Ω i .…”

Section: Linear Elasticity and Nodal Integration In The Finite Elemenmentioning

confidence: 99%

Geometric Properties of the Adaptive Delaunay Tessellation

Bobach

Constantiniu

Steinmann

et al. 2010

Mathematical Methods for Curves and Surfaces

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Abstract. Recently, the Adaptive Delaunay Tessellation (Adt) was introduced in the context of computational mechanics as a tool to support Voronoi-based nodal integration schemes in the finite element method. While focusing on applications in mechanical engineering, the former presentation lacked rigorous proofs for the claimed geometric properties of the Adt necessary for the computation of the nodal integration scheme. This paper gives pending proofs for the three main claims which are uniqueness of the Adt, connectedness of the Adt, and coverage of the Voronoi tiles by adjacent Adt tiles. Furthermore, this paper provides a critical assessment of the Adt for arbitrary point sets.

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A stabilized conforming nodal integration for Galerkin mesh-free methods

Cited by 1,186 publications

References 23 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Geometric Properties of the Adaptive Delaunay Tessellation

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