Shape optimization in three-dimensional linear elasticity by the boundary contour method

A variant of the usual boundary element method (BEM), called the boundary contour method (BCM), has been presented in the literature in recent years. In the BCM in three dimensions, surface integrals on boundary elements of the usual BEM are transformed, through an application of Stokes’ theorem, into line integrals on the bounding contours of these elements. The BCM employs global shape functions with the weights, in the linear combinations of these shape functions, being defined piecewise on boundary elements. A very useful consequence of this approach is that stresses, at suitable points on the boundary of a body, can be easily obtained from a post-processing step of the standard BCM. The subject of this paper is shape optimization in three-dimensional (3D) linear elasticity by the BCM. This is achieved by coupling a 3D BCM code with a mathematical programming code based on the successive quadratic programming (SQP) algorithm. Numerical results are presented for several interesting illustrative examples.     

The meshless hypersingular boundary node method for three-dimensional potential theory and linear elasticity problems

The Boundary Node Method (BNM) represents a coupling between Boundary Integral Equations (BIEs) and Moving Least Squares (MLS) approximants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the latter. The result is a ‘meshfree’ method that decouples the mesh and the interpolation procedures. The BNM has been applied to solve 2-D and 3-D problems in potential theory and linear elasticity. The Hypersingular Boundary Element Method (HBEM) has diverse important applications in areas such as fracture mechanics, wave scattering, error analysis and adaptivity, and to obtain a symmetric Galerkin boundary element formulation. The present work presents a coupling of Hypersingular Boundary Integral Equations (HBIEs) with MLS approximants, to produce a new meshfree method — the Hypersingular Boundary Node Method (HBNM). Numerical results from this new method, for selected 3-D problems in potential theory and in linear elasticity, are presented and discussed in this paper.     

The traction boundary contour method for linear elasticity

This paper presents a further development of the boundary contour method. The boundary contour method is extended to cover the traction boundary integral equation. A traction boundary contour method is proposed for linear elastostatics. The formulation of traction boundary contour method is regular for points except the ends of the boundary element and corners. The present approach only requires line integrals for three‐dimensional problems and function evaluations at the ends of boundary elements for two‐dimensional cases. The implementation of the traction boundary contour method with quadratic boundary elements is presented for two‐dimensional problems. Numerical results are given for some two‐dimensional examples, and these are compared with analytical solutions. This method is shown to give excellent results for illustrative examples. Copyright © 1999 John Wiley & Sons, Ltd.     

Successive linear approximation for boundary value problems of nonlinear elasticity in relative-descriptional formulation

A method of successive linear approximation in current configuration for solving boundary value problems of large deformation in finite elasticity is proposed. Instead of using Lagrangian or Eulerian formulation, we can also formulated the problem relative to the current configuration, and linearize the constitutive function at the present state so that it leads to a linear boundary value problem for an incremental time step. Therefore, as linearization at present state proceed in time, problem for large deformation can be formulated. The idea is similar to the Euler’s method for differential equations. As an example for the proposed method, numerical simulation of bending a rectangular block into a circular section for Mooney-Rivlin material is given for comparison with the exact solution, which is one of the well-known universal solutions in finite elasticity.     

The hypersingular boundary contour method for two-dimensional linear elasticity

Summary This paper presents a novel method called the Hypersingular Boundary Contour Method (HBCM) for two-dimensional (2-D) linear elastostatics. This new method can be considered to be a variant of the standard Boundary Element Method (BEM) and the Boundary Contour Method (BCM) because: (a) a regularized form of the hypersingular boundary integral equation (HBIE) is employed as the starting point, and (b) the above regularized form is then converted to a boundary contour version based on the divergence free property of its integrand. Therefore, as in the 2-D BCM, numerical integrations are totally eliminated in the 2-D HBCM. Furthermore, the regularized HBIE can be collocated at any boundary point on a body where stresses are physically continuous. A full theoretical development for this new method is addressed in the present work. Selected examples are also included and the numerical results obtained are uniformly accurate.     

A Galerkin boundary contour method for two-dimensional linear elasticity

The Boundary Contour Method (BCM) is a recent variant of the Boundary Element Method (BEM) resting on the use of boundary approximations which a-priori satisfy the field equations. For two-dimensional problems, the evaluation of all the line-integrals involved in the collocation BCM reduces to function evaluations at the end-points of each element, thus completely avoiding numerical integrations. With reference to 2-D linear elasticity, this paper develops a variational version of BCM by transferring to the BCM context the ingredients which characterize the Galerkin-Symmetric BEM (GSBEM). The method proposed herein requires no numerical integrations: all the needed double line-integrals over boundary elements pairs can be evaluated by generating appropriate “potential functions” (in closed form) and computing their values at the element end-points. This holds for straight as well as curved elements; however the coefficient matrix of the equation system in the boundary unknowns turns out to be fully symmetric only when all the elements are straight. The numerical results obtained for some benchmark problems, for which analytical solutions are available, validate the proposed formulation and the corresponding solution procedure.     

The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements

This paper presents a further development of the Boundary Contour Method (BCM) for two-dimensional linear elasticity. The new developments are: (a) explicit use of the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor, (b) quadratic boundary elements compared to linear elements in previous work and (c) evaluation of stresses both inside and on the boundary of a body. This method allows boundary stress computations at regular points (i.e. at points where the boundary is locally smooth) inside boundary elements without the need of any special algorithms for the numerical evaluation of hypersingular integrals. Numerical solutions for illustrative examples are compared with analytical ones. The numerical results are uniformly accurate.     

A boundary element-free method (BEFM) for three-dimensional elasticity problems

This study combines the boundary integral equation (BIE) method and improved moving least-squares (IMLS) approximation to present a direct meshless boundary integral equation method, the boundary element-free method (BEFM) for three-dimensional elasticity. Based on the improved moving least-squares approximation and the boundary integral equation for three-dimensional elasticity, the formulae of the boundary element-free method are given, and the numerical procedure is also shown. Unlike other meshless boundary integral equation methods, the BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus giving it a greater computational precision. Three selected numerical examples are presented to demonstrate the method.Aknowledgement The work in this project was fully supported by a grant from the Research Grants Council (RGC) of the Hong Kong Special Administrative Region, China (Project No. CityU 1011/02E).The work that is described in this paper was supported by Project No. CityU 1011/02E, which was awarded by the Research Grants Council of the Hong Kong Special Administrative Region, China. The authors are grateful for the financial support.     

A fast solution method for three-dimensional many-particle problems of linear elasticity

A boundary element method for solving three-dimensional linear elasticity problems that involve a large number of particles embedded in a binder is introduced. The proposed method relies on an iterative solution strategy in which matrix–vector multiplication is performed with the fast multipole method. As a result the method is capable of solving problems with N unknowns using only 𝒪(N) memory and 𝒪(N) operations. Results are given for problems with hundreds of particles in which N=𝒪(10⁵). © 1998 John Wiley & Sons, Ltd.     

Singular boundary elements for three-dimensional elasticity problems

The focus of this paper is a set of semi-discontinuous, traction-singular surface elements introduced to help the rigorous boundary integral analysis of problems in three-dimensional solid mechanics. In contrast to the singular boundary elements developed for linear fracture mechanics where the square-root singularity is of primary interest, traction shape functions featuring the proposed four- and eight-node boundary elements can be used to represent power-type singularities of arbitrary order, such as those arising at non-smooth material boundaries and interfaces. Apart from being capable of rigorously handling traction singularities and discontinuities across the domain boundaries and interfaces, these elements also permit a smooth transition to adjacent regular elements. Complemented with a family of suitable displacement and geometry shape functions, the singular surface elements are incorporated into a regularized boundary integral equation method and shown, through a set of benchmark results, to perform well for both static and dynamic problems.     

The boundary node method for three‐dimensional linear elasticity

The Boundary Node Method (BNM) is developed in this paper for solving three‐dimensional problems in linear elasticity. The BNM represents a coupling between Boundary Integral Equations (BIE) and Moving Least‐Squares (MLS) interpolants. The main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure.For problems in linear elasticity, free rigid‐body modes in traction prescribed problems are typically eliminated by suitably restraining the body. However, an alternative approach developed recently for the Boundary Element Method (BEM) is extended in this work to the BNM. This approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix. Also, the BNM has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body. Copyright © 1999 John Wiley & Sons, Ltd.     

A dual reciprocity boundary face method for 3D non-homogeneous elasticity problems

The boundary face method is coupled with the dual reciprocity method (DRM) to solve non-homogeneous elasticity problems. We will analyze thin structures based on 3D solid elastic theory rather than the shell theory as in the finite element method (FEM). To circumvent the ill-conditioning problem that occurs in the radial basis function (RBF) approximation in thin structures, a special variation scheme for determining the RBF parameters is proposed. In addition, a new exponential RBF is used which has significantly improved the stability of the RBF, and its particular solution to the elasticity problem is derived for the first time. Comparisons of our method with the traditional DRM, the boundary element method (BEM) and the FEM have been made. Numerical examples have demonstrated that our method outperforms the BEM and FEM with respect to stability, accuracy and efficiency, especially when the structure in question has features of small size, such as thin shells.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

The meshless regular hybrid boundary node method for 2D linear elasticity

The meshless Regular Hybrid Boundary Node Method (RHBNM) is a promising method for solving boundary value problems, and is further developed and numerically implemented for 2D linear elasticity in this paper. The present method is based on a modified functional and the Moving Least Squares (MLS) approximation, and exploits the meshless attributes of the MLS and the reduced dimensionality advantages of the BEM. As a result, the RHBNM is truly meshless, i.e. it only requires nodes constructed on the surface, and absolutely no cells are needed either for interpolation of the solution variables or for the boundary integration. All integrals can be easily evaluated over regular shaped domains and their boundaries.Numerical examples show that the high convergence rates with mesh refinement and the high accuracy with a small node number is achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM, can be avoided.     

Preconditioned conjugate gradient methods for three-dimensional linear elasticity

Finite element modelling of three-dimensional elasticity problems give rise to large sparse matrices. Various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques. Incomplete factorizations based on levels of fill, drop tolerance, and a two-level hierarchical basis are developed. Various techniques for ensuring that the incomplete factors have positive pivots are presented. Computational tests are carried out for problems generated using unstructured tetrahedral meshes. Quadratic basis functions are used. The performance of the iterative methods is compared to a standard direct sparse matrix solver. Problems with up to 70 000 degrees of freedom and small (?1) element aspect ratio are considered.     

On design sensitivity analysis in linear elasticity by the boundary contour method

A new derivation for design sensitivity analysis using the concept of the material derivative (or total derivative), and boundary contour conversion, is presented in this paper. This derivation is carried out by first taking the material derivative of the regularized Boundary Integral Equation (BIE) with respect to a shape design variable, and then converting the resulting equations into their boundary contour version. As expected, the final design sensitivity equations are identical to those presented in Ref. [1] in which the opposite process, namely, conversion of the BIE into a boundary contour version, followed by material differentiation, had been carried out.     

On the evaluation of hyper-singular integrals arising in the boundary element method for linear elasticity

The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.     

On the corner tensor in three-dimensional linear elasticity

The subject of this paper is the corner tensor C that appears in the free term in a boundary integral equation formulation for three-dimensional linear elasticity. A general corner, locally composed of piecewise flat and curved surfaces, is considered in explicit fashion. The solid angle at the corner appears in the expression for C. A new formula for the solid angle at a general corner, in terms of line integrals, is derived in this paper. Finally, examples for cones are presented and discussed.     

A three-dimensional parabolic punch problem in linear elasticity

In this paper an analytic solution for a three-dimensional contact problem, in linear elasticity, is constructed through the separation of Laplace's equation in paraboloidal coordinates. A rigid punch under normal loading is applied to an isotropic elastic medium occupying an infinite half-space where the contact region is parabolic and the punch profile is prescribed. This treatment allows for a general punch profile provided it is physically reasonable so as to ensure the convergence of the solution.