Regularized Boundary Integral Equation Research Articles

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.

A Time-Stepping Boundary Element Method for Transient Heat Conduction Problems in Orthotropic Bodies.

The paper proposes a boundary element method for analysis of transient heat conduction problems in orthotropic bodies. Time derivative in the governing equation is approximated by the time-stepping scheme. The reduced differential equation is transformed into a regularized boundary integral equation which involves no integrals to be evaluated in the sense of the Cauchy principal value. Although the formulation is presented for two-dimensional problems, it is also applicable to three-dimensional problems if the 3-D fundamental solution is used. A new computer program is developed and applied to several sample problems. The computational aspects of the proposed time-stepping boundary element method are investigated, first for the case where the principal axes of orthotropy coincide with the coordinates, and then for other more general cases. It is concluded that there is a suitable time-step and a suitable element size for each problem to be investigated, and that the optimal computational conditions for an arbitrary case can be estimated from the numerical results obtained for some of the sample problems for which the exact solutions are known.

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

This paper presents a computational method of dynamic stress intensity factors(DSIF) in two-dimensional problems. In order to obtain accurate numerical results of DSIF, the boundary element method using the Laplace transform based on regularized boundary integral equations is applied to the computation of transient elastodynamic responses. A computer program id newly developed for two-dimensional elastodynamics. Numerical computation is carried out for computation of DSIF for a rectangular plate with a center crack under impact tension. Accuracy of the results is investigated from the viewpoint of computational conditions such as the sampling number of the inverse Laplace transform and the number of boundary elements.

Continuity requirements for density functions in the boundary integral equation method

Two methods of forming regular or hypersingular boundary integral equations starting from an interior integral representations are discussed. One method involves direct treatment of the singularities such as Cauchy principal value and/or finite-part interpretation of the integrals and the other does not. By either approach, theory places the same restrictions on the smoothness of the density function for the integrals to exist, assuming sufficient smoothness of the geometrical boundary itself. Specifically, necessary conditions on the smoothness of the density function for meaningful boundary integral formulas to exist as required for the collocation procedure are established here. Cases for which such conditions may not be sufficient are also mentioned and it is understood that with Galerkin techniques, weaker smoothness requirements may pertain. Finally, the bearing of these issues on the choice of boundary elements, to numerically solve a hypersingular boundary integral equation, is explored and numerical examples in 2D are presented.

Regularized Boundary Integral Equation Formulations for Thin Elastic Plate Bending Analysis.

The boundary integral equations conventionally used for thin elastic plate bending analysis have the singularities of the Cauchy principal value order for which special care must be taken in numerical evaluation when the boundary is discretized by using the higher-order elements. In the present paper, the boundary integral equations are regularized up to an integrable order by using the subtracting and adding back technique. The obtained boundary integral equations both for deflection and rotation are weakly singular, and their discretized forms can be integrated accurately by the standard Gaussian quadrature formula. Numerical implementation of the resulting regularized integral equations is presented, and effectiveness of the proposed method is discussed through some numerical demonstrations.

Solution of potential problems using combinations of the regular and derivative boundary integral equations

The feasibility of using combinations of the boundary integral equation (BIE) and the normal derivative of the boundary integral equation (DBIE) is investigated for the two-dimensional Laplace equation. By using the combinations of these two equations it is possible to derive Fredholm integral equations of either the first or second kind regardless of the boundary conditions. Although the Fredholm equations of the second kind are well conditioned, in general they provide less accurate interior results than the Fredholm equations of the first kind and the classical direct boundary element method (DBEM). On the other hand, the Fredholm equations of the first kind can provide poor solutions in regions close to the boundary where large gradients exist in the boundary flux.

Direct differentiation method for shape design sensitivity analysis using boundary integral formulation

A direct differentiation method is proposed for shape design sensitivity analysis of elastic structures using the Boundary Integral Equation (BIE) formulation. A new BIE is derived, where unknowns are the design derivatives of the structural responses on the boundary, by applying the material derivative concept to the boundary integral identity. To remove the singularity problem arising in the new BIE, a regular BIE technique is employed by locating the collocation point outside the boundary. The accuracy of the results of the direct differentiation method is studied through the elliptic hole problem under tensile stress at infinity. A critical comparison is made for the sensitivity results of the direct differentiation and adjoint variable methods as well as for those of the finite difference method against the exact design sensitivity. Excellent accuracy is observed for the results of the direct differentiation method.

A regularized boundary integral equation method for elastodynamic crack problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.

On a regular boundary integral equation and a modified Trefftz method in Reissner's plate theory

On a regular boundary integral equation and a modified Trefftz method in Reissner's plate theory

On a regular boundary integral equation and a modified Trefftz method in Reissner's plate theory

If fundamental solutions are known it is convenient to formulate boundary value problems as boundary integral equations, usually locating the source points of the fundamental solutions on the deal boundary. This paper deals with Reissner's plate theory and presents ttwo solution techniques which avoid some inherent difficulties of the conventional approach, such as the need for accurate evaluation of singular integrals. If the points of source of the fundamental solutions are moved outside the domain of the problem they will never coincide with field points; this results in regular kernels and, thus, in regular integral equations. Use of the fundamental solutions as expansion functions, where the free expansion coefficients are determined by satisfying the prescribed conditions, permits integration to the avoided totally; this is a special formulation, a modification of the well-known Trefftz method. Several examples are presented to illustrate the merits and disadvantages of these two procedures.

An implementation of the boundary element method for zoned media with stress discontinuities

AbstractThe boundary element method as it applies to two‐dimensional elastostatic problems is implemented with quadratic variation in boundary variables and geometry. Additional parameters are associated with nodes at segment intersections to allow for discontinuous tractions. Supplementary equations consistent with linear elasticity are used to augment the regular boundary integral equations for certain mixed and displacement boundary conditions. Zoning or subregionalization capabilities are included to provide solutions to piecewise nonhomogeneous problems or problems on regions of irregular shape. Example problems are given to demonstrate the accuracy of the technique.