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American Society for Testing and Materials

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

The extended/generalized finite element method: An overview of the method and its applications

https://orbilu.uni.lu/bitstream/10993/13726/1/phu-meshless.pdf

Review: Meshless methods: A review and computer implementation aspects

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

http://arturo.imati.cnr.it/brezzi/papers/hitchhikers-preprint.pdf

The Hitchhiker's Guide to the Virtual Element Method

Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review

In this paper, we consider the discretization of a parabolic nonlocal problem within the framework of the virtual element method. Using the fixed point argument, we prove that the fully discrete scheme has a unique solution. The presence of the nonlocal term makes the problem nonlinear, and the resulting nonlinear equations are solved using the Newton method. The computational cost of the Jacobian of the nonlinear scheme increases in the presence of nonlocal coefficient. To reduce the computational burden in computing the Jacobian, which otherwise is inevitable in the usual approach, in this paper, we propose an equivalent formulation. A priori error estimates in the L2 and the H1 norms are derived. Furthermore, we employ a linearized scheme without compromising the rate of convergence in the respective norms. Finally, the theoretical convergence results are verified through numerical experiments over polygonal meshes.

Virtual element methods for nonlocal parabolic problems on general type of meshes

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

In this work, we propose a framework for an adaptive contact analysis in deformable solids using the effective error indicator from the scaled boundary finite element method (SBFEM) with a quadtree decomposition. Further, the SBFEM is implemented with the commercial finite element software, Abaqus, to perform the contact analysis by employing the user element subroutine (UEL) feature. The SBFEM error indicator coupled with the quadtree decomposition is implemented in Matlab and allowed to interact with the Abaqus using .inp file for an adaptive refinement. The detailed implementation of the framework, input data format, and the UEL subroutine which is one of the key features of the proposed work are clearly explained. The effectiveness of the proposed framework is demonstrated by solving several contact problems of engineering significance. The developed SBFEM code can be downloaded from https://github.com/nsundar/sbfem .

An adaptive scaled boundary finite element method for contact analysis

In this paper, we propose a smoothed stable extended finite element method (S2XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions and/or voids in hyperelastic matrix materials. The interface geometries are implicitly represented through level sets and a geometry based error indicator is used to resolve the geometry. For the unknown fields, the mesh is refined based on a recovery based error indicator combined with a quadtree decomposition guarantee the method's accuracy with respect to the computational costs. Elements with hanging nodes (due to the quadtree meshes) are treated as polygonal elements with mean value coordinates as the basis functions. The accuracy and the convergence properties are compared to similar approaches for several numerical examples. The examples indicate that S2XFEM is computationally the most efficient without compromising the accuracy.

/pdf/adaptive-smoothed-stable-extended-finite-element-method-for-4t9dihz18b.pdf

Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine-Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank-Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton's method. We derive a priori error estimations in $L^2$ and $H^1$ norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine-Gordon equation are demonstrated with academic numerical experiments.

Sundararajan Natarajan

Papers

Virtual element methods for nonlocal parabolic problems on general type of meshes

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

An adaptive scaled boundary finite element method for contact analysis

Adaptive smoothed stable extended finite element method for weak discontinuities for finite elasticity

Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique