On finite element approximation of general boundary value problems in nonlinear elasticity

A set of sufficient conditions for the well posedness and the convergence of the finite element approximation of three-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting rotating objects with stationary boundaries or bianisotropic media is provided for the first time to the best of authors’ knowledge. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve rotating or bianisotropic materials. All details of the applications of the theory are provided for electromagnetic problems involving rotating axisymmetric objects.

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AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005556 ◽

2011 ◽

Vol 21 (08) ◽

pp. 1733-1760 ◽

Cited By ~ 30

Author(s):

XIANMIN XU ◽

DUVAN HENAO

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Numerical Method ◽

Nonlinear Elasticity ◽

Elastic Energy ◽

Finite Element Approximation ◽

Void Coalescence ◽

Element Approximation ◽

Nonconforming Finite Element ◽

First Time

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

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Iterative solution methods for elliptic boundary value problems

Russian Journal of Numerical Analysis and Mathematical Modelling ◽

10.1515/rnam-2020-0018 ◽

2020 ◽

Vol 35 (4) ◽

pp. 215-222

Author(s):

Georgy M. Kobelkov

Keyword(s):

Boundary Value Problems ◽

Elliptic Boundary ◽

Boundary Value ◽

Finite Element Approximation ◽

Iterative Solution ◽

Elliptic Boundary Value Problems ◽

Element Approximation ◽

Elliptic Boundary Value ◽

Varying Coefficients ◽

Solution Methods

AbstractFor elliptic boundary value problems (the diffusion equation and elasticity theory ones) with highly varying coefficients, there are proposed iterative methods with the number of iterations independent of the coefficient jumps. In the differential case these methods take solving the boundary value problem for the Poisson equation at each step of iterations while in the finite difference (finite element) approximation it is possible to use another operator as a preconditioner.

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