Element Method For Elasticity Research Articles

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.

Virtual boundary element-equivalent collocation method for the plane magnetoelectroelastic solids

This paper presents a virtual boundary element-equivalent collocation method (VBEM) for the plane magnetoelectroelastic solids, which is based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. Besides all the advantages of the conventional boundary element method (BEM) over domain discretization methods, this method avoids the computation of singular integral on the boundary by introducing the virtual boundary. In the end, several numerical examples are performed to demonstrate the performance of this method, and the results show that they agree well with the exact solutions. So the method is one of the efficient numerical methods used to analyze megnatoelectroelastic solids.

Matlab implementation of the finite element method in elasticity

A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.

The least‐squares finite element method in elasticity. Part II: Bending of thin plates

AbstractA least‐squares finite element method (LSFEM) for bending problems of thin plates is developed. This LSFEM is based on the first‐order deflection‐slope‐moment‐shear force formulation. Four compatibility conditions are added into the first‐order system; thus, the method can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples show that the method achieves an optimal rate of convergence for all eight variables. Copyright © 2002 John Wiley & Sons, Ltd.

The least‐squares finite element method in elasticity—Part I: Plane stress or strain with drilling degrees of freedom

AbstractA new least‐squares finite element method (LSFEM) for plane elasticity problems is developed based on the first‐order displacement–stress–rotation formulation which includes two new first‐order compatibility constraints among the stresses and the drilling rotation. This LSFEM can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples including incompressible materials show that the method achieves an optimal rate of convergence for all variables. Copyright © 2001 John Wiley & Sons, Ltd.

A unified formulation and error estimation measure for the direct and the indirect boundary element methods in elasticity

In this paper, given a boundary value problem for a finite elastic body in two-dimensions, a problem of representing the solution by layers of point forces ( F j ) and Somigliana dislocations ( B j ) is considered in the infinite homogeneous body that contains the original finite body. In the boundary element method (BEM) solution, either the boundary displacement or traction component at each node is specified, but not both. This provides us a degree of freedom to arbitrarily specify the proportion of the densities F j and B j to be used in the direct and the indirect BEM formulations. The nature of the BEM solution errors is identified and a unified error estimation measure with a mesh refinement scheme for both formulations is proposed.

Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second-order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in elasticity. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides and independent of the structure of a shape-regular mesh.

A hybrid coupled finite-boundary element method in elasticity

We present a hybrid coupled finite-boundary element method for a mixed boundary-value problem in linear elasticity. In this hybrid method, we consider, in addition to traditional finite elements, the Trefftz elements for which the governing equations of equilibrium are required to be satisfied. The Trefftz elements are then modelled with boundary potentials supported by the individual element boundaries, the so-called macro-elements. The coupling between the finite and macro-elements is accomplished by using a generalized compatibility condition in weak sense with mortar elements on the skeleton. The latter allows us to relax the continuity requirements for the global displacement field. In particular, the mesh points of the macro-elements can be chosen independently of the nodes of the FEM structure. This approach permits the combination of independent meshes and also the exploitation of modern parallel computing facilities. By following Hsiao et al. [G.C. Hsiao, E. Schnack and W.L. Wendland, Hybrid coupled finite-boundary element methods for elliptic systems of second order, in preparation], we give the precise formulation of the method, its functional analytic setting as well as corresponding discretizations and asymptotic error estimates. For illustrations, some computational results are also included.

Complex fundamental solutions and complex variables boundary element method in elasticity

The CVBEM for elasticity problems in its basic formulations (direct, indirect and displacement discontinuity (DD) ones) is presented. It is based on the use of complex fundamental solutions. The complex boundary integral equations (CBIEs) arising from the basic CVBEM formulations are considered. It was shown that the developed theory includes the main CBIEs obtained by using the different approaches. Besides, new real and complex integral equations were obtained. The revealed links between real variables BEM (RVBEM) and CVBEM serve to mutual enrichment of these two approaches.

Traction-based Completed Adjoint Double Layer Boundary Element Method in elasticity

This paper is concerned with a traction-based Completed Adjoint Double Layer Boundary Element Method to solve for the surface traction of a system of rigid particles embedded in an elastic matrix. The main feature of the method is a single layer representation of the displacement field, which leads to a system of second-kind integral equations for the traction field, the extreme eigenvalue of which could be deflated, allowing iterative solution strategies to be effectively applied. The method is therefore most suitable for large-scale simulations of particulate solids. The method is benchmarked against some known analytic solutions, including the difficult stress singularity problems at sharp edges. The effectiveness of the method in dealing with a large number of inclusions is also demonstrated with an elongational deformation problem involving up to 25 inclusions.

Stochastic boundary element method in elasticity

The stochastic boundary element method is developed to analyze elasticity problems with random material and/or geometrical parameters and randomly perturbed boundaries. Based on the first-order Taylor series expansion, the boundary integration equations concerning the mean and deviation of the displacements are derived, respectively. It is found that the randomness of material parameters is equivalent to a random body force, so the mean and covariance matrices of unknown boundary displacements and tractions can be obtained. Furthermore, the mean and covariance of displacements and stresses at inner points can also be obtained. Numerical examples show that the proposed stochastic boundary element method gives satisfactory solutions, as compared with those obtained by theoretical analysis or other numerical methods.

Completed double layer boundary element method in elasticity

This paper reports an indirect boundary element method in elasticity that is most suitable to deal with particulate solids. The method involves a distribution of a double layer potential and, after a suitable completion and deflation, is amenable to iterative solution techniques. It can therefore accommodate a large number of particles with complex geometries. Convergence of the method is significantly improved by the introduction of a simple domain decomposition to solve the system of equations. The method is illustrated by the translating sphere problem, the load transfer problem between two spheres at near contact, and the shear deformation of a cluster of 125 spheres initially in a simple cubic array.

The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary

AbstractThis paper presents a new boundary element approach to transform domain integrals into equivalent boundary integrals. The technique, called the Multiple Reciprocity Method, is applied to 2‐D elasticity problems and operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions. Numerical examples are presented to demonstrate the accuracy and efficiency of the method.

The boundary element method in elasticity

This paper describes the application of the boundary element method in elasticity starting by discussing the basic weighted residual expressions. Boundary elements of different orders are presented and discussed, together with the numerical integration schemes. The relationships are presented for three dimensional elasticity and specialized to the two dimensional case. Results are presented and discussed, specially in relation to finite elements and other “domain” type techniques.