A new fast multipole boundary element method for two dimensional acoustic problems

A new fast multipole boundary element method (BEM) is presented in this paper for solving large-scale two dimensional (2D) acoustic problems based on the improved Burton–Miller formulation. This algorithm has several important improvements. The fast multipole BEM employs the improved Burton–Miller formulation, and successfully overcomes the non-uniqueness difficulty associated with the conventional BEM for exterior acoustic problems. The improved Burton–Miller formulation contains only weakly singular integrals, and avoids the numerical difficulties associated to the evaluation of the hypersingular integral, it leads to the numerical implementations more efficient and straightforward. Furthermore, the fast multipole method (FMM) and the approximate inverse preconditioned generalized minimum residual method (GMRES) iterative solver are adopted to greatly improve the overall computational efficiency. The numerical examples with Neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole BEM for solving large-scale 2D acoustic problems in a wide range of frequencies.     

Analysis of thin piezoelectric solids by the boundary element method

The piezoelectric boundary integral equation (BIE) formulation is applied to analyze thin piezoelectric solids, such as thin piezoelectric films and coatings, using the boundary element method (BEM). The nearly singular integrals existing in the piezoelectric BIE as applied to thin piezoelectric solids are addressed for the 2-D case. An efficient analytical method to deal with the nearly singular integrals in the piezoelectric BIE is developed to accurately compute these integrals in the piezoelectric BEM, no matter how close the source point is to the element of integration. Promising BEM results with only a small number of elements are obtained for thin films and coatings with the thickness-to-length ratio as small as 10⁻⁶, which is sufficient for modeling many thin piezoelectric films as used in smart materials and micro-electro-mechanical systems.     

Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems

A new completely analytical integral algorithm is proposed and applied to the evaluation of nearly singular integrals in boundary element method (BEM) for two-dimensional anisotropic potential problems. The boundary layer effect and thin body effect are dealt with. The completely analytical integral formulas are suitable for the linear and non-isoparametric quadratic elements. The present algorithm applies the analytical formulas to treat nearly singular integrals. The potentials and fluxes at the interior points very close to boundary are evaluated. The unknown potentials and fluxes at boundary nodes for thin body problems with the thickness-to-length ratios from 1E−1 to 1E−8 are accurately calculated by the present algorithm. Numerical examples on heat conduction demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body effect in BEM. Furthermore, the present linear BEM is especially accurate and efficient for the numerical analysis of thin body problems.     

Elasto-plastic buckling of curved webs

This paper presents an investigation on elastic buckling strength of curved girder webs subjected to uniform shears or bending stresses at the edges.An elastic 20 degrees of freedom finite element model was used to formulate the eigenvalue problem and a Gauss-Seidel iterative procedure was employed to yield the lowest critical edge loads.In case of pure bending, the investigation is extended into the plastic range. The deformation theory of plasticity in conjunction with a new formulation of the secant modulus is used to derive the elasto-plastic buckling equations. The same Gauss-Seidel iterative procedure was used to find the critical load for each assumed stress level. Further iterations with incremental stress were done to match the elasto-plastic buckling stresses. The material is assumed to be elastic-perfectly plastic and incompressible.In order to aid design professions, the dimensions of the web panel studied are within the practical ranges of curved plate girders. Four boundary conditions that represent various constrain conditions from flanges to stiffeners of plate girder designs, were considered.The results are presented in graphical forms. Interaction curves relating to various dimensionless parameters are constructed. Comparisons and convergence studies were made with existing available data. It is found that boundary conditions and aspect ratio influence the buckling stresses greatly. However, curvature effect is relatively insignificant over the range of practical application.     

Bending and stress responses of the hybrid axisymmetric system via state-space method and 3D-elasticity theory

This research presents bending responses of hybrid laminated nanocomposite reinforced axisymmetric circular/annular plates (HLNRACP/ HLNRAAP) within the framework of non-polynomial under mechanical loading and various type of initially stresses via the three-dimensional elasticity theory. The current structure is on the Pasternak type of elastic foundation and torsional interaction. The state-space approach and differential quadrature method (SS-DQM) are studied to present the bending characteristics of the current structure by considering various boundary conditions. To predict the material properties of the bulk, the role of mixture and Halpin–Tsai equations are studied. For modeling the circular plate, a singular point is studied. Finally, a parametric study investigates the impacts of various types of distribution of laminated layers, stacking sequence on the stress/strain information of the HLNRACP/ HLNRAAP. Results reveal that the system's static stability and bending behavior improve due to increasing the value of Winkler and Pasternak factors, and the stress distribution becomes more uniform.

     

A combined boundary element and finite element solution of slab and slab-on-girder bridges

The boundary element method (BEM) has been shown by many researchers to be an efficient numerical tool with which to analyse various engineering structures. In particular, the method has been extensively applied to a large number of plate bending problems. The distinct advantage of the method is in the reduction in dimensionality of the problem and, as a consequence of this, both computer time and data preparation are significantly reduced. In this paper, the suitability of a new method combining the advantages of both the BEM and the finite element method (FEM) is studied. The method is first applied to investigate the conventional plate bending problems. After the validity of the method is established it is then extended to analyse the more complicated problems of slabs and slab-on-girder bridges. Through a series of slab-on-girder bridge calculations, it is demonstrated that the method is not only accurate and fast converging but its ease of application and data preparation is not attainable by the ordinary FEM.     

A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods

The nearly singular integrals occur in the boundary integral equations when the source point is close to an integration element (as compared to its size) but not on the element. In this paper, the concept of a relative distance from a source point to the boundary element is introduced to describe possible influence of the singularity of the integrals. Then a semi-analytical algorithm is proposed for evaluating the nearly strongly singular and hypersingular integrals in the three-dimensional BEM. By using integration by parts, the nearly singular surface integrals on the elements are transformed to a series of line integrals along the contour of the element. The singular behavior, which appears as factor, is separated from remaining regular integrals. Consequently standard numerical quadrature can provide very accurate evaluation of the resulting line integrals. The semi-analytical algorithm is applied to analyzing the three-dimensional elasticity problems, such as very thin-walled structures. Meanwhile, the displacements and stresses at the interior points very close to its bounding surface are also determined efficiently. The results of the numerical investigation demonstrate the accuracy and effectiveness of the algorithm.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

Use of Trefftz functions in non-linear BEM/FEM

Simulation of many practical problems requires to use non-linear formulations with large displacements, large strains and large rotations. It is well known that the use of Trefftz (T-) functions (i. e. the functions satisfying the governing equations inside the domain) as weighting, or interpolation functions leads to more efficient formulations than those obtained by classical methods. In this paper we will show the use of T-functions and especially T-polynomials, Kelvin, or Kupradze and Boussinesq functions (Green functions with singularity points defined outside of the domain) and their combination in connection with the total Lagrangian formulation for multi-domain BEM (reciprocity based FEM) analysis of displacements and for the post-processing phase in the analysis (evaluation of both gradient of displacements and stress fields). The formulation results in non-singular boundary integrals which has numerical advantages over other formulations using singular boundary integral equations.     

The relationship amongst coefficient matrices in symmetric Galerkin BEM for 2D elastic problems

Based on the assumption that solutions from different methods are the same, the relationship amongst weakly singular, strongly singular and hypersingular matrices associated with symmetric Galerkin BEM is derived for 2D elastic problems. Hypersingularity is avoided through matrix manipulations so that only weakly and strongly singularities need to be solved. Compared with the advantages brought about by symmetry, the additional computation caused by matrix manipulations is not so important in many cases, especially for time domain dynamic problems or when one wants to couple BEM with other symmetric schemes. Both static and dynamic problems have been studied, and three numerical examples are included to show the effectiveness and accuracy of the present formulation.     

A boundary element analysis of symmetric laminated composite shallow shells

This paper presents a boundary element formulation for the analysis of symmetric laminated composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Fundamental solutions for elastostatic formulations are used and body forces are written as a sum of approximation functions multiplied by unknown coefficients. Two approximation functions are used. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Results for the approximation functions are compared and the accuracy of the proposed formulation is assessed by results from literature. It was shown that results obtained with the approximation function called augmented thin plate spline present very good agreement with literature even for shells that are not so shallow.     

A high precision coupled bending–extension triangular finite element for laminated plates

It is well recognized that the estimation of interlaminar stresses and strain energy release rates is important in designing laminated composite panels. Generally coupled bending–extension finite elements are necessary to study laminates to include the effects of coupling and/or combined transverse and extensional loads. Such elements are normally formulated adapting the classical theory of bending and extension. While the classical laminated plate theory of bending has provision to obtain interlaminar stresses due to transverse loading, it is necessary to include certain higher order terms in the extensional theory in order to obtain the interlaminar stresses due to inplane loads. A high precision triangular element based on a theory which includes both the bending and extension with necessary higher order terms is presented in this paper. The performance of this element is validated with the aid of examples. Numerical results for displacements in symmetric and unsymmetric laminates under bending loads have been given. Numerical results for interlaminar stresses in symmetric and unsymmetric laminates have been given for the well-known benchmark problem of a coupon with free edges. Strain energy release rate components at the delamination tip in coupons with unsymmetric sublaminates have been given. The effects of delamination length and location on the components of the strain energy release rate have been studied. Results indicated that with the use of this element, the interlaminar stresses can be estimated reasonably accurately, over a major part of the laminate except in a small local region close to the free edge. Global–local analysis with three-dimensional elements in the local region, is suggested to obtain local stresses more accurately. Interlaminar stresses at the boundary of a hole in a perforated plate under extension have been obtained to illustrate the use of the present element in a global–local analysis strategy.     

Studies on dynamic behaviour of a cantilever square plate with varying thickness

The effects of different types of variations in profile and thickness on the amplitude and the dynamic bending stresses of a square cantilever plate excited by a point harmonic load resonance has been investigated. A four-noded plate bending element has been used for the analysis. The response has been determined for the first three modes of vibration. In each case the results obtained for different thickness variations are compared with those of the uniform thickness plate. It is observed that considerable reductions in amplitude and/or bending stresses can be achieved by the proper selection of thickness variation.     

Boundary formulations for sensitivities of three-dimensional stress invariants

The direct, singular, boundary element analysis (BEA) formulation has been shown to provide a basis for a computationally efficient and accurate shape structural design sensitivity analysis (DSA) approach for three-dimensional solid objects. Within the boundary element analysis context, the theoretical formulation for sensitivities of important stress-related quantities including principal and deviatoric stresses, von Mises, maximum shear, and other stress invariants are presented, both for the surface as well as the interior of a continuum structure. Numerical results are given to demonstrate the accuracy of this approach.     

A powerful finite element for plate bending

A powerful finite element formulation for plate bending has been developed using a modified version of the variational method of Trefftz. The notion of a boundary has been generalized to include the interelement boundary. All boundary conditions and the interelement continuity requirements (displacements, slopes, internal forces) have been obtained as natural conditions on the generalized boundary. Coordinate functions have been constructed to satisfy the nonhomogeneous Lagrange equation locally within the elements. Singularities due to isolated loads have been properly taken into account. For practical use a general quadrilateral element has been developed and its accuracy illustrated on several numerical examples. Work is in progress to extend the formulation to anisotropic and moderately thick plates and to vibration analysis.     

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.     

Analysis of plates by finite strip method

New finite strips are developed for the analysis of plates. Based on Reissner's plate theory, the effect of shear deformation is included in the formulation. To eliminate artificial hardening, the shape functions for the strips are so chosen that there is no mismatched term along the interpolation functions for the interpolation parameters. Numerical examples are reported to demonstrate that the strips can work equally well in thick as well as thin plates.     

Solution of non-uniform torsion of bars by an integral equation method

In this paper, a boundary element method (BEM) is developed for the non-uniform torsion of simply or multiply connected cylindrical bars of arbitrary cross-section. The bar is subjected to an arbitrarily distributed twisting moment while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, besides the Saint-Venant torsional shear stresses, the warping normal stresses are also computed. Two boundary value problems with respect to the variable along the beam angle of twist and to the warping function are formulated and solved employing a BEM approach. Both the warping and the torsion constants are computed by employing an effective Gaussian integration over the domains of arbitrary shape. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The contribution of the normal stresses due to a restrained warping is investigated, by numerical examples, with great practical interest.     

Approximate solution of dual integral equations using Chebyshev polynomials

The aim of the present work is to introduce solution of special dual integral equations by the orthogonal polynomials. We consider a system of dual integral equations with trigonometric kernels which appear in formulation of the potential distribution of an electrified plate with mixed boundary conditions and convert them to Cauchy-type singular integral equations. We use the Chebyshev orthogonal polynomials to construct approximate solution for Cauchy-type singular integral equations which will solve the main dual integral equations. Numerical results demonstrate effectiveness of this method.