The analog equation method for large deflection analysis of heterogeneous orthotropic membranes: a boundary-only solution

In this paper, the analog equation method (AEM) is applied to nonlinear analysis of heterogeneous orthotropic membranes with arbitrary shape. In this case, the transverse deflections influence the in-plane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear with variable coefficients. The present formulation, being in terms of the three displacement components, permits the application of geometrical in-plane boundary conditions. The membrane may be prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation, the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on the BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method, and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of heterogeneous anisotropic membranes. The method has all the advantages of the pure BEM, since the discretization and integration are limited only to the boundary.     

Shape design sensitivity analysis for non-linear anisotropic heat conducting solids and shape optimization by the BEM

Shape design sensitivity analysis (SDSA) expressions have been derived for non-linear anisotropic heat conducting solid bodies by following the material derivative concept and adjoint variable method of optimal shape design given in the literature. The variation of a general integral functional has been described in terms of primary and adjoint quantities evaluated at the varying boundaries. As an example problem in shape optimization, optimal outer boundary profiles of an orthotropic solid body are obtained by the boundary element method (BEM), after reformulating the SDSA equations in a form which is most suitable for the BEM.     

Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity

In the direct formulation of the boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly boundary solution technique even when thermal effects are involved.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

Initial strain effects in multilayer composite laminates

A boundary integral formulation for the analysis of stress fields induced in composite laminates by initial strains, such as may be due to temperature changes and moisture absorption is presented. The study is formulated on the basis of the theory of generalized orthotropic thermo-elasticity and the governing integral equations are directly deduced through the generalized reciprocity theorem. A suitable expression of the problem fundamental solutions is given for use in computations. The resulting linear system of algebraic equations is obtained by the boundary element method and stress interlaminar distributions in the boundary-layer are calculated by using a boundary only discretization. The approach is general and it does not require a priori assumptions. Numerical results are presented to show the potential of the proposed approach.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Three-dimensional magnetostatic field calculation using boundaryelement method

The boundary-element calculation of three-dimensional magnetostatic field problems using the reduced and total magnetic scalar potential formulation is described. The method is based on a boundary integral equation that can be derived from Green's theorem. Two regions, a current-free iron region and an air region including the source domains, are considered. The material properties of the iron are assumed to be linear and either isotropic or anisotropic (orthotropic). Two examples are investigated: a C-shaped magnet and an iron cylinder of finite length immersed in the magnetic field of a cylinder coil     

A comparison of integral formulations for the analysis of low Reynolds number flows

Integral formulations for the analysis of low Reynolds number flows have been developed over the past 25 years. These formulations can typically be categorized as being either direct or indirect and either velocity integral equations or traction integral equations. Depending on the boundary conditions imposed for a given problem, the resulting integral formulation will result in either a Fredholm integral equation of the first kind, second kind or mixed. In general, Fredholm integral equations of the first kind lead to unstable numerical schemes based upon discretization and can result in low-order accuracy. For most practical problems, the direct velocity integral equation results in a Fredholm equation of the first kind. Nevertheless, many researchers have used this integral formulation with great success and any potential matrix ill-conditioning seems to have little influence on the accuracy of the numerical solutions. In this research, three integral formulations are compared, namely, the direct velocity integral equation, the indirect velocity integral equation, and the direct traction integral equation. Three benchmark problems are chosen for which there are analytic solutions. Although the discretized matrix condition numbers are in some cases orders of magnitude larger for the direct velocity integral formulation, there is little to distinguish between the three methods in terms of accuracy. In fact, the results for the direct velocity integral equation were slightly more accurate in most cases for the three benchmark problems considered in this research.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

The Galerkin boundary element method for exterior problems of 2-D Helmholtz equation with arbitrary wavenumber

Among many efforts put into the problems of eigenvalue for the Helmholtz equation with boundary integral equations, Kleinman proposed a scheme using the simultaneous equations of the Helmholtz integral equation with its boundary normal derivative equation. In this paper, the detailed formulation is given following Kleinman’s scheme. In order to solve the integral equation with hypersingularity, a Galerkin boundary element method is proposed and the idea of regularization in the sense of distributions is applied to transform the hypersingular integral to a weak one. At last, a least square method is applied to solve the overdetermined linear equation system. Several numerical examples testified that the scheme presented is practical and effective for the exterior problems of the 2-D Helmholtz equation with arbitrary wavenumber.     

Doubly periodic volume?surface integral equation formulation for modelling metamaterials

A generalised volume-surface integral equation is extended by way of the periodic Green's function to model arbitrarily complex designs of metamaterials consisting of high-contrast inhomogeneous anisotropic material regions as well as metallic inclusions. The unique aspect of the formulation is the integration of boundary and volume integral equations to increase modelling efficiency and capability. Specifically, the boundary integral approach with equivalent surface currents is adopted over regions consisting of piecewise homogeneous materials as well as metallic perfect electric/magnetic conductor inclusions, whereas the volume integral equation is employed only in inhomogeneous and/or anisotropic material regions. Because the periodic Green's function only needs to be evaluated for the equivalent surface currents enclosing an inhomogeneous and/or anisotropic region, matrix fill time is much less as compared to using a volume formulation. Furthermore, the incorporation of curvilinear finite elements allows for greater geometrical modelling flexibility for arbitrarily shaped high-contrast regions found in typical designs of engineered metamaterials     

An ancillary boundary integral equation for magnetostatic analysis

The boundary element method (BEM) has been established as an effective means for magnetostatic analysis. Direct BEM formulations for the magnetic vector potential have been developed over the past 20 years. There is a less well-known direct boundary integral equation (BIE) for the magnetic flux density which can be derived by taking the curl of the BIE for the magnetic vector potential and applying properties of the scalar triple product. On first inspection, the ancillary boundary integral equation for the magnetic flux density appears to be homogeneous, but it can be shown that the equation is well-posed and non-homogeneous using appropriate boundary conditions. In the current research, the use of the ancillary boundary integral equation for the magnetic flux density is investigated as a stand-alone equation and in tandem with the direct formulation for the magnetic vector potential.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

BIEM for 2D steady-state problems in cracked anisotropic materials

The two-dimensional ‘in-plane’ time-harmonic elasto-dynamic problem for anisotropic cracked solid is studied. The non-hypersingular traction boundary integral equation method (BIEM) is used in conjunction with closed form frequency dependent fundamental solution, obtained by Radon transform. Accuracy and convergence of the numerical solution for stress intensity factor (SIF) is studied by comparison with existing solutions in isotropic, transversely-isotropic and orthotropic cases. In addition a parametric study for the wave field sensitivity on wave, crack and anisotropic material parameters is presented.     

A robust formulation for solving fluid-structure interaction problems

 In this article, a coupled finite element and boundary element approach for the acoustic radiation and scattering from submerged elastic bodies of arbitrary shape is presented. An alternative to the direct boundary element method for acoustics is proposed. By taking an auxiliary source surface inside the radiating boundary and following the usual discretization and integration procedures employed in the boundary element method, both the singularities of the integrands and the nonuniqueness problems do not arise. In addition, the difficulty of slope discontinuity also can be overcome. This procedure is formulated in a similar fashion of wave superposition method, except that the direct boundary integral equations are adopted. The proposed formulation employ the surface Helmholtz integral equation and its normal gradient like that adopted in the Burton–Miller approach, but do not employ any coupling constant. Typical examples are presented that demonstrate the efficiency of the proposed technique. Received 9 April 2000     

Domain element local integral equation method for potential problems in anisotropic and functionally graded materials

An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material coefficients. The method is based on the local integral relationships (integral form of balance equation and/or integral equations utilizing fundamental solutions) and consistent approximation of field variable using standard domain-type elements. The accuracy and convergence of the proposed method is tested by several examples and compared with benchmark analytical solutions.     

Single edge-cracked orthotropic strip under three-point load

The problem of an orthotropic strip having a crack of unit length normal to one edge and subjected to a bending moment resulting from three-point loading is solved using integral transform method. The mixed boundary conditions lead to dual integral equations which are ultimately reduced to a Fredholm integral equation of second kind. The integral equation thus obtained is solved by the method developed by Fox and Goodwin. Numerical solutions for a fibre-reinforced composite material have been carried out to determine the stress intensity factor of an orthotropic medium. The same has been compared with the isotropic case.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.