In this paper, the virtual element method is employed to approximate semilinear elliptic problems over arbitrary polygonal meshes. The nonlinear load term is approximated by employing the orthogonal L2 projection. The finite dimensional formulation and its implementation are discussed in detail and optimal a priori error estimate in H1 norm is derived. Further, two numerical experiments are conducted in order to illustrate the performance of the proposed scheme and to numerically justify the theoretical convergence rate. It is observed that the proposed method yields optimal convergence rates in both L2 and H1 norm. © 2019 IMACS

Sundararajan Natarajan

Department of Mechanical Engineering

D. Adak

E. Natarajan

Computational mechanics

Differential equations

Error estimates

Numerical experiments

OPTIMAL A PRIORI ERROR ESTIMATES

OPTIMAL CONVERGENCE

POLYGONAL MESHES

SEMILINEAR ELLIPTIC EQUATION

SEMILINEAR ELLIPTIC PROBLEM

VIRTUAL ELEMENT METHOD

Nonlinear equations

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Applied Numerical Mathematics

In this paper, a displacement based finite element framework for general three-dimensional convex polyhedra is presented. The method is based on a semi-analytical framework, the scaled boundary finite element method. The method relies on the definition of a scaling center from which the entire boundary is visible. The salient feature of the method is that the discretizations are restricted to the surfaces of the polyhedron, thus reducing the dimensionality of the problem by one. Hence, an explicit form of the shape functions inside the polyhedron is not required. Conforming shape functions defined over arbitrary polygon, such as the Wachpress interpolants are used over each surface of the polyhedron. Analytical integration is employed within the polyhedron. The proposed method passes patch test to machine precision. The convergence and the accuracy properties of the method is discussed by solving few benchmark problems in linear elasticity. © 2017 Elsevier Ltd

Engineering Analysis with Boundary Elements

A scaled boundary finite element formulation over arbitrary faceted star convex polyhedra

This manuscript presents the development of novel high-order complete shape functions over star-convex polygons based on the scaled boundary finite element method. The boundary of a polygon is discretised using one-dimensional high order shape functions. Within the domain, the shape functions are analytically formulated from the equilibrium conditions of a polygon. These standard scaled boundary shape functions are augmented by introducing additional bubble functions, which renders them high-order complete up to the order of the line elements on the polygon boundary. The bubble functions are also semi-analytical and preserve the displacement compatibility between adjacent polygons. They are derived from the scaled boundary formulation by incorporating body force modes. Higher-order interpolations can be conveniently formulated by simultaneously increasing the order of the shape functions on the polygon boundary and the order of the body force mode. The resulting stiffness-matrices and mass-matrices are integrated numerically along the boundary using standard integration rules and analytically along the radial coordinate within the domain. The bubble functions improve the convergence rate of the scaled boundary finite element method in modal analyses and for problems with non-zero body forces. Numerical examples demonstrate the accuracy and convergence of the developed approach. Copyright © 2016 John Wiley &amp; Sons, Ltd. Copyright © 2016 John Wiley &amp; Sons, Ltd.

International Journal for Numerical Methods in Engineering

Construction of high-order complete scaled boundary shape functions over arbitrary polygons with bubble functions

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub-triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the O(dofs)1.1 in case of the conventional polygonal FEM, while it scales as O(dofs)0.7 in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. © 2015 John Wiley &amp; Sons, Ltd.

Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

Journal	Data powered by TypesetApplied Numerical Mathematics
Publisher	Data powered by TypesetElsevier B.V.
ISSN	01689274
Open Access	No