Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

The 3‐D BEM implementation of a numerical Green's function for fracture mechanics applications

The use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed‐form expressions for Green's function components, however, have only been available for few simple 2‐D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2‐D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2‐D problems and the computational aspects of the 3‐D implementation of the numerical Green's function approach are now discussed, including examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407–26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.     

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

Three-dimensional fundamental solutions for transient heat transfer by conduction in an unbounded medium,half-space,slab and layered media

This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers.This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, António J, Simões N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.     

Scattering of acoustic waves by movable lightweight elastic screens

This paper computes the insertion loss provided by movable lightweight elastic screens, placed over an elastic half-space, when subjected to spatially sinusoidal harmonic line pressure sources. A gap between the acoustic screen and the elastic floor is allowed. The problem is formulated in the frequency domain via the boundary element method (BEM). The Green's functions used in the BEM formulation permit the solution to be obtained without the discretization of the flat solid–ground interface. Thus, only the boundary of the elastic screen is modeled, which allows the BEM to be efficient even for high frequencies of excitation. The formulation of the problem takes into account the full interaction between the fluid (air) and the solid elastic interfaces.The validation of the algorithm uses a BEM model, which incorporates the Green's functions for a full space, requiring the full discretization of the ground. The model developed is then used to simulate the wave propagation in the vicinity of lightweight elastic screens with different dimensions and geometries. Both frequency and insertion loss results are computed over a grid of receivers. These results are also compared with those obtained with a rigid barrier and an infinite elastic panel.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region

In this paper, numerical frequency domain formulations are developed to simulate the 2D acoustic wave propagation in the vicinity of an underwater configuration which combines two sub-regions: the first one consists of a wedge with rigid seabed and free surface, and the second one is assumed to have a rigid flat bottom and a free flat surface.The problem is solved using two different numerical methods: the Boundary Element Method (BEM) and the Method of Fundamental Solutions (MFS). Two models are developed by using a sub-region technique, where only the vertical interface between sub-regions of different geometries has to be discretized. These formulations incorporate Green's functions that take into account the presence of flat rigid and free surfaces and of a wedge. Green's functions are defined using two approaches: the image source method is used to model the rigid flat bottom and free flat interface, whereas the response provided by the wedge sub-region is based on a normal mode solution. Additionally, a MFS and a BEM model are also implemented which require the discretization of the sloping rigid seabed of the wedge, therefore making use of Green's functions for a rigid flat bottom and a free surface (using the image source method).A detailed discussion on the performance of these formulations is performed, with the aim of finding an efficient formulation to solve the problem. It is found that the model based on the MFS and on the sub-region technique has a significantly lower computational cost and is stable, therefore being the most suitable for the analysis of acoustic wave propagation in the studied configurations.     

Efficient evaluation of three-dimensional Green's functions in anisotropic elastostatic multilayered composites

The three-dimensional Green's functions in anisotropic elastostatic multilayered composites (MLCs) obtained within the framework of generalized Stroh formalism are expressed as two-dimensional integrals of Fourier inverse transform over an infinite plane. Their numerical evaluations involve tremendous computational efforts in particular in the presence of various singularities and near-singularities due to the presence of material mismatches across interfaces. The present paper derives the complete set of the Green's functions including displacement, stress and their derivatives with respect to source coordinates using a novel and computationally efficient approach. It is proposed for the first time that the Green's functions in the MLCs are expressed as a sum of a special solution and a general-part solution, with the former consisting of the first few terms of the trimaterial expansion solution around a source load. Since the zero-order term contains the singularity corresponding to the homogeneous full-space solution and can be evaluated analytically, and the other higher-order terms contain most of the near-singular behaviors and can be reduced to a line integral over a finite interval, the general-part solution becomes regular and the Green's functions overall can be evaluated efficiently. As an example, the Green's functions in a five-layered orthortropic plate are evaluated to demonstrate the efficiency of the proposed approach. Also, the detailed characteristics of these Green's functions are examined in both the transform- and physical-domains. These Green's functions are essential in developing the boundary-integral-equation formulation and numerical boundary element method for composite laminate problems involving regular and cracked geometries.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Open hole fatigue characteristics and damage growth of stitched plain weave carbon/epoxy laminates

Fatigue response of stitched plain weave carbon/epoxy laminates containing circular holes is experimentally investigated. Two carbon/epoxy laminates of cross-ply [(0/90)]₂₀ and quasi-isotropic [(±45)(0/90)₂(±45)₂(0/90)₂(±45)₂(0/90)]_s are reinforced using Kevlar-29® yarns in through-thickness direction. The laminates are drilled to produce a circular hole with diameter of 5.7 mm. Stitch configuration for cross-ply laminates is round stitch and parallel stitch, while that for quasi-isotropic laminates is parallel stitch only. For round stitch configuration, the hole is surrounded by circular stitch line of 7-mm diameter. For parallel stitch, the distance between two stitch lines (spacing) is 15 mm. In all, three independent cases are presented in this paper: Case 1 (cross-ply laminates, round stitch, tension–tension fatigue); Case 2 (cross-ply laminates, parallel stitch, tension–tension fatigue); Case 3 (quasi-isotropic laminates, parallel stitch, compression–compression fatigue). In each case, comparison with unstitched laminates is made. Case 1 shows that round stitch reduces tension fatigue curve of carbon/epoxy laminates. Round stitch seems to aggravate the damage, which is emanating from the hole rim of laminates. It gradually diverts the damage towards the edge of the specimen and causes premature fatigue failure. Case 2 shows that although parallel stitch generally does not influence the fatigue life of laminates, the damage growth due to parallel stitch is apparently unstable after 8 million cycles. As a result, laminates with parallel stitch eventually fail before reaching 10 million cycles. In contrast, unstitched laminates are able to sustain fatigue load for more than 10 million cycles. Case 3 shows that under compression fatigue load, fatigue limit of stitched plain weave laminates is better than that of the unstitched ones due to damage redistribution along the stitch lines.     

A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to boundary ones. This restores the BEM as a truly boundary solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.     

2D Green's functions of defective magnetoelectroelastic solids under thermal loading

Thermomagnetoelectroelastic problems for various defects embedded in an infinite matrix are considered in this paper. Using Stroh's formalism, conformal mapping, and perturbation technique, Green's functions are obtained in closed form for a defect in an infinite magnetoelectroelastic solid induced by the thermal analog of a line temperature discontinuity and a line heat source. The defect may be of an elliptic hole or a Griffith crack, a half-plane boundary, a bimaterial interface, or a rigid inclusion. These Green's functions satisfy the relevant boundary or interface conditions. The proposed Green's functions can be used to establish boundary element formulation and to analyzing fracture behaviour due to the defects mentioned above.     

A Green's function time‐domain boundary element method for the elastodynamic half‐plane

The transient Green's function of the 2‐D Lamb's problem for the general case where point source and receiver are situated beneath the traction‐free surface is derived. The derivations are based on Laplace‐transform methods, utilizing the Cagniard–de Hoop inversion. The Green's function is purely algebraic without any integrals and is presented in a numerically applicable form for the first time. It is used to develop a Green's function BEM in which surface discretizations on the traction‐free boundary can be saved. The time convolution is performed numerically in an abstract complex plane. Hence, the respective integrals are regularized and only a few evaluations of the Green's function are required. This fast procedure has been applied for the first time. The Green's function BEM developed proved to be very accurate and efficient in comparison with analogue BEMs that employ the fundamental solution. Copyright © 1999 John Wiley & Sons, Ltd.     

A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids

This paper presents a single-domain boundary element method (BEM) analysis of fracture mechanics in 2D anisotropic piezoelectric solids. In this analysis, the extended displacement (elastic displacement and electrical potential) and extended traction (elastic traction and electrical displacement) integral equations are collocated on the outside boundary (no-crack boundary) of the problem and on one side of the crack surface, respectively. The Green's functions for the anisotropic piezoelectric solids in an infinite plane, a half plane, and two joined dissimilar half-planes are also derived using the complex variable function method. The extrapolation of the extended relative crack displacement is employed to calculate the extended `stress intensity factors' (SIFs), i.e., K_I, K_II, K_III and K_IV. For a finite crack in an infinite anisotropic piezoelectric solid, the extended SIFs obtained with the current numerical formulation were found to be very close to the exact solutions. For a central and inclined crack in a finite and anisotropic piezoelectric solid, we found that both the coupled and uncoupled (i.e., the piezoelectric coefficient e_ijk=0) cases predict very similar stress intensity factors K_I and K_II when a uniform tension σ_yy is applied, and very similar electric displacement intensity factor K_IV when a uniform electrical displacement D_y is applied. However, the relative crack displacement and electrical potential along the crack surface are quite different for the coupled and uncoupled cases. Furthermore, for a inclined crack within a finite domain, we found that while a uniform σ_yy (=1 N m⁻²) induces only a very small electrical displacement intensity factor (in the unit of Cm^−3/2), a uniform D_y (=1 C m⁻²) can produce very large stress intensity factors (in the unit of Nm^−3/2).     

An analysis of solidification problems by the boundary element method

The problem of interest in this paper is the calculation of the motion of the solid–liquid interface and the time-dependent temperature field during solidification of a pure metal. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time-dependent Green's functions and convolution integrals. The BEM approach requires discretization of only the surface of the solidifying body. Thus, the numerical method closely follows the physics of the problems and is intuitively very appealing. The formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems. Emphasis is given to two-dimensional problems. Comparison with existing semi-analytical solutions and other numerical solutions from the literature reveals that the method is fast, accurate and without major time step limitations.     

Mixed Mode I,II and III analysis of multiple cracks in plane anisotropic solids by the BEM: a dislocation and point force approach

A direct boundary element method (BEM) for plane anisotropic elasticity is formulated for the generalized plane strain. It deals with the general case when the in-plane and out-of-plane deformations are coupled, including the special case when they are decoupled. The formulation is based on the distributions of point forces and dislocation dipoles following the physical interpretation of Somigliana's identity. We adopt Lekhnitskii-Eshelby-Stroh formalism for anisotropic elasticity and represent the point force and the dislocation, their dipoles, and continuous distributions systematically; the duality relations between the point force and the dislocation solutions are fully exploited. The analytical formulas for the displacement and the traction BEM are applied to the mixed mode crack analysis for multiply cracked anisotropic bodies. We extend the physical interpretation of Somigliana's identity to cracked bodies and represent the crack by the continuous distribution of dislocation dipoles. The mixed mode stress intensity factors (K_I, K_II and K_III) are determined accurately with the help of the conservation integrals of anisotropic elasticity.     

A simple estimation method of stress intensity factors for through-cracks in angle-ply laminates

In the present paper a simple method of estimation of stress intensity factors for through-cracks in angle-ply laminates is developed. In this procedure, Savin's elasticity solution for an elliptical hole in two-dimensional infinite plate is used as a basic solution for the stress distribution in each ply of laminate. The present method is applied to the problems of through cracks in a $\text{(}\text{0°}\text{90}\text{)}$_s laminate and a $\text{(}\text{?45°}\text{+ 45°}\text{)}$_s laminate. Comparison with existing numerical solutions obtained by three-dimensional finite element analysis shows good agreement. The simplicity of the present method gives the design engineer a useful tool for estimating stress concentration due to the presence of a hole or a crack.     

Three‐dimensional time‐harmonic Green's functions for a triclinic full‐space using a symbolic computation system

Time‐harmonic Green's functions for a triclinic anisotropic full‐space are evaluated through the use of a symbolic computation system.This procedure allows evaluation of the Green's functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions. Copyright © 2001 John Wiley & Sons, Ltd.     

Experimental determination of residual stresses in composite laminates [02/θ2]s

The present work aims to determine the residual stresses in carbon fiber/epoxy composite laminates, by means of the incremental hole-drilling method. Based on mechanical theories of composite laminates and an elastic plate with a circular hole, the relationship between the relaxed strains on the surface of laminates and the residual stresses in laminates was established. This newly deduced theoretical formula was adapted into the incremental hole-drilling method and allowed us to further study the residual stresses in the through-thickness direction for various composite laminates. Related numerical modeling of composite laminate with a hole was built to calibrate the coefficients within the formula. Experiments were conducted and the residual stresses in composite laminates [0₂/θ₂]_s are presented. The proposed approach was validated with the consistence between our results for cross-ply laminates and those in literature.