A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

Zhu, Tengteng; Atluri, Satya N.

doi:10.1007/s004660050296

Cited by 385 publications

(200 citation statements)

References 16 publications

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“…By applying the variational principle and adding penalty term enforcing the displacement boundary conditions [43], the variation of stationary total potential energy for linear elastic materials can be obtained…”

Section: Governing Equationsmentioning

confidence: 99%

Meshfree formulation for modelling of orthogonal cutting of composites

Kahwash

Shyha

Maheri

2017

Composite Structures

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Please cite this article as: Kahwash, F., Shyha, I., Maheri, A., Meshfree formulation for modelling of orthogonal cutting of composites, Composite Structures (2017), doi: http://dx.doi.org/10. 1016/j.compstruct.2017.01.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Simulations show that the meshfree model is capable of predicting cutting forces as a function of the fibre orientation. Sensitivity analysis is conducted to investigate the effect of important meshfree parameters such as the domain of influence and weight function on forces. One of the strongest advantages of the proposed model is the simple and automatic set up process, as meshing for domain discretisation is not required.

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Section: Governing Equationsmentioning

confidence: 99%

Meshfree formulation for modelling of orthogonal cutting of composites

Kahwash

Shyha

Maheri

2017

Composite Structures

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“…To overcome this difficulty, collocation at nodes [33] is used to enforce essential boundary conditions:u h (x b ) =u I (x b ), x b are nodes on essential boundaries. Since essential boundaries are fixed, the boundary conditions are given aṡ…”

Section: Figure 1 Geometry Definition Of a Representative Nodal Domainmentioning

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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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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“…This unfortunately complicates matters when seeking to enforce boundary conditions. One way of addressing this is to use the collocation method proposed in [27]. Let M n , M nt and Q n denote the normal bending moment, twisting moment and transverse shear force at node x b on a free unloaded edge, where its normal vector n forms an angle α with the x-axis.…”

Section: Enforcement Of Boundary Conditionsmentioning

confidence: 99%

Limit analysis of plates and slabs using a meshless equilibrium formulation

Gilbert

Askes

2010

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2010) SUMMARYA meshless Element-Free Galerkin (EFG) equilibrium formulation is proposed to compute the limit loads which can be sustained by plates and slabs. In the formulation pure moment fields are approximated using a moving least squares technique, which means that the resulting fields are smooth over the entire problem domain. There is therefore no need to enforce continuity conditions at interfaces within the problem domain, which would be a key part of a comparable finite element formulation. The collocation method is used to enforce the strong form of the equilibrium equations and a stabilized conforming nodal integration scheme is introduced to eliminate numerical instability problems. The combination of the collocation method and the smoothing technique means that equilibrium only needs to be enforced at the nodes, and stable and accurate solutions can be obtained with minimal computational effort. The von Mises and Nielsen yield criteria which are used in the analysis of plates and slabs respectively are enforced by introducing second-order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plate and slab problems.

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A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

Cited by 385 publications

References 16 publications

Meshfree formulation for modelling of orthogonal cutting of composites

Meshfree formulation for modelling of orthogonal cutting of composites

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

Limit analysis of plates and slabs using a meshless equilibrium formulation

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