Regularized Boundary Integral Equation Research Articles

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.

A meshless integral method based on regularized boundary integral equation

A meshless integral method based on the regularized boundary integral equation is developed and applied to two-dimensional linear elasticity. The governing integral equation is obtained from the weak form of elasticity over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. The method is built on the LBIE method proposed by Atluri and co-workers, and several key improvements are introduced in this work that significantly enhance the accuracy and robustness of the method. The most critical improvement is the use of the subtraction technique to remove the strong singularity that results in a regularized governing integral equation. The technique is straightforward and efficient, and is much simpler and easier compared to other singularity removal techniques. A special numerical integration is employed for the calculation of integrals with weak singularity which further improves accuracy. The collocation method is employed to enforce the essential boundary conditions exactly, which is simple and computationally efficient. The natural boundary conditions are incorporated in the system governing equation and require no special handling. Numerical tests show that the meshless integral method is accurate and robust. The effects of weight function, support domain, sub-domain, and monomial basis are investigated and discussed.

On the stress-wave imaging of cavities in a semi-infinite solid

The problem of mapping underground cavities from surface seismic measurements is investigated within the framework of a regularized boundary integral equation (BIE) method. With the ground modeled as a uniform elastic half-space, the inverse analysis of elastic waves scattered by a three-dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry. For an accurate treatment of the gradient search technique employed to solve the inverse problem, sensitivities of the predictive BIE model with respect to cavity parameters are evaluated semi-analytically using an adjoint problem approach and a continuum kinematics description. Several key features of the formulation, including the rigorous treatment of the radiation condition for semi-infinite solids, modeling of an illuminating seismic wave field, and treatment of the prior information, are highlighted. A set of numerical examples with spherical and ellipsoidal cavity geometries is included to illustrate the performance of the method. It is shown that the featured adjoint problem approach reduces the computational requirements by an order of magnitude relative to conventional finite-difference estimates, thus rendering the three-dimensional elastic-wave imaging of solids tractable for engineering applications.

Minimum-order regular boundary integral equations for three-dimensional eddy-current problem

This paper provides regular boundary integral equations for determining the electromagnetic field for the three-dimensional eddy-current problem. The Mayergoyz approach enables us to split the problem into a magnetic problem and an electric problem, which are solved in succession. The magnetic problem leads to a set of one vector and one scalar regular integral equations (three scalar unknown functions), while the electric problem is reduced to a scalar regular integral equation (a scalar unknown function). In both cases, existence theorems for the solutions are proven.

Analysis of Two-Dimensional Transient Heat Conduction in Anisotropic Solids by Boundary Element Method.

This study is concerned with a new boundary element method based on the time-stepping scheme and the direct scheme for transient heat conduction problems in anisotropic solids. Mathematical formulations of the boundary element method are presented in detail for two-dimensional transient heat conduction. The reduced differential equations by the authors are transformed into regularized boundary integral equations which involve no integrals to be evaluated in the sense of the Cauchy principal value. Boundary element analysis can be performed by using the standard boundary element method. Computation is started from the initial state, and advanced by a time-marching procedure until the final stage is realized. The proposed solution procedures are applied to some typical examples, and the usefulness of the proposed BEM is demonstrated through discussions of the results obtained.

Internal variables and their sensitivities 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. A new formulation for design sensitivities in three-dimensional linear elasticity, based on the BCM, has been recently presented in Ref. [12]. This challenging 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 equation into its boundary contour version. The focus of [12] is the boundary problem, i.e., evaluation of displacements, stresses and their sensitivities on the bounding surface of a body. The focus of the present paper is the corresponding internal problem, i.e., analogous calculations at points inside a body. Numerical results for internal variables and their sensitivities are presented here for selected examples.

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless method based on a regular local integral equation and the moving least-squares approximation is developed. The present method is a truly meshless one as it does not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No derivatives of the shape functions are needed in constructing the system stiffness matrix for the internal nodes, as well as for those boundary nodes with no essential-boundary-condition-prescribed sections on their local boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable, and the computational results for the unknown variable and its derivatives are very accurate. No special post-processing procedure is required to compute the derivatives of the unknown variable, as the original result, from the moving least-squares approximation, is smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.

A new meshless regular local boundary integral equation (MRLBIE) approach

A new meshless regular local boundary integral equation (MRLBIE) approach

Surface variables and their sensitivities 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 and curvatures, at suitable points on the boundary of a body, can be easily obtained from a post-processing step of the standard BCM. A new formulation for design sensitivities in three-dimensional linear elasticity, based on the BCM, is presented in this paper. This challenging 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 equation into its boundary contour version. Finally, numerical results for surface variables, as well as their sensitivities, are presented for selected illustrative examples.

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.

Boundary Element Analysis for Unsteady Elastodynamic Problems Based on the Laplace Transform.

In the present paper, the boundary element method(BEM) for unsteady elastodynamic problems based on the Laplace transform is discussed. In the Laplace transformed BEM, the accuracy of teh numerical results is generally governed by the numerical treatment of the inverse Laplace transformation for the transformed solutions. Two types of numerical inverse Laplace transformation(NILT) formula, namely Krings & Waller's method and Hosono's method are applied to the 2-dimensional BEM analysis employing regularized boundary integral equations. It is shown that there is a stability condition between the element size and the time discretization. The characteristics of two types of NILT methods are found out through 2-dimensional BEM analyses for the unsteady elastodynamic problems.

Evaluation of the Hypersingular and Regularized Boundary Integral Equations for the Boundary Potential Gradients in 2D Field.

This paper presents a new scheme for the evaluation of the hypersingular and regularized boundary integral equations for the boundary potential gradients. In the process of taking the limit of the exterior collocation point to an arbitrary point on the boundary, no interpolation function is employed and continuity of the potential gradients at the collocation point is taken into account strictly. The regularized boundary integral equation for the boundary potential gradients is also derived and applicability of the conventional C0 elements to the discretization of the hypersingular and regularized boundary integral equations is discussed. A new interpolation scheme that satisfies the C1 continuity of the potential at any point on the boundary is presented for the potential and the flux, and is effectiveness is demonstrated through a numerical example.

Regularized bie formulations for first- and second-order shape sensitivity of elastic fields

The subject of this paper is the formulation of boundary integral equations for first- and second-order shape sensitivities of boundary elastic fields in three-dimensional bodies. Here the direct differentiation approach is considered. It relies on the repeated application of the material derivative concept to the governing regularized (i.e. weakly singular) displacement boundary integral equation (RDBIE) for an elastostatic state on a given domain. As a result, governing BIEs, which are also weakly singular, are obtained for the elastic sensitivities up to the second order. They are formulated so as to allow a straightforward implementation; in particular no strongly singular integral is involved. It is shown that the actual computation of shape sensitivities using usual BEM discretization uses the already built and factored discrete integral operators and needs only to set up additional right-hand sides and additional backsubstitutions. Some relevant discretization aspects are discussed.

Sensitivity Analysis for 2-D Steady-State Elastodynamic Problems Using Boundary Element Method and Its Application.

Sensitivity analysis is important in inverse analysis and optimization. In such problems, an iterative procedure is employed to find the minimum value of the cost function, which requires accurate and efficient computation of sensitivities with respect to unknowns or design parameters. We consider an isotropic, homogeneous, linear elastic body which is subjected to time-harmonic excitations. Differentiating the regularized boundary integral equations for these elastodynamic problems, we obtain the relationships between the displacements, tractions and design sensitivities. The accuracy and usefulness of the proposed method are demonstrated through sensitivity analysis of some examples in the two-dimensional elastodynamic problem. Finally, the method is applied to optimal shape design for reduction of stress concentration.

Boundary element formulation using relative quantity for viscous flow

The regularizing using relative quantity is applied to the problems of incompressible viscous flow in two dimensions. The regularized boundary integral equation for flow rate is obtained by superposing a particular solution of uniform flow rate upon the usual equation in the case of both finite and infinite region. The relative quantity is also introduced into the internal integral equations for flow rate and pressure. Since the integral equations for flow rate from inside to boundary become continuous, the integral equation for surface pressure is obtained, which has hitherto been absent in usual formulations. As a result, numerical results both of flow rate and pressure become very accurate all over the domain. Through two-dimensional examples of finite and infinite region, the new equations are verified to be correct.

Optimum design of cooling lines in injection moulds by using boundary element design sensitivity analysis

In injection moulding, it is important to obtain the optimal arrangenment of cooling lines in the mould to maintain a uniform temperature distribution on its cavity surface for high quality manufacturing. In this paper, a boundary integral design sensitivity formulation for steady-state heat conduction problems is presented, and is applied to the design of cooling lines in injection moulds. A regularized boundary integral equation for steady-state heat conduction is differentiated directly with respect to a design variable, and a boundary integral equation relating the sensitivity coefficients of the boundary temperature and the heat flux is derived. The sensitivity boundary integral equation is used for computation of the gradients of the defined objective function. Some two-dimensional numerical examples are presented.

Coincident collocation of displacement and tangent derivative boundary integral equations in elasticity

AbstractThe regular boundary integral equations of elastostatics are combined with regularized versions of the tangent derivative equations and collocated at the same points to formulate the elasticity problem in terms of displacements, tractions and the tangential displacement gradients. Hermitian cubic polynomials are used for functional interpolation on certain elements to formulate the boundary element method in terms of displacements, tractions and their tangent derivatives. Commensurate accuracy of nodal values of these functions and the tangent derivatives is obtained and makes possible the accurate and immediate recovery of all stress components. An example problem demonstrates the accuracy and atility of the approach.

Computation of Dynamic Stress Intensity Factors Using the Boundary Element Method Based on Laplace Transform and Regularized Boundary Integral Equations

Numerical computation of dynamical SIF is carried out for a rectangular plate with a center crack under impact tension

Boundary stress calculation using regularized boundary integral equation for displacement gradients

AbstractAccurate computation of boundary stress components is important in stress‐base shape optimization. The boundary element method usually gives us more accurate solutions of them compared with those obtained by the finite element method. In the present paper we propose a new approach to calculate the boundary stress components by using the regularized boundary integral equation relating the boundary displacement gradients to the tractions. Although the original integral equation must be evaluated in the sense of the Cauchy principal value, it can be regularized by subtracting and adding the displacement gradients at the source point from and to the original integral equation. When the displacement gradients satisfy the Hölder continuity at the source point, the regularized integral equation becomes non‐singular and can be evaluated numerically by using the standard Gaussian quadrature formula. The numerical implementation of the proposed integral equation is presented and the effectiveness of the approach is also discussed through some numerical demonstrations.