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In this paper, a boundary integral formulation is presented for obtaining the standard eigenvalue problem for the two-dimensional (2-D) Helmholtz equation. The formulation is derived by using the series expansions of zero-order Bessel functions for the fundamental solution to the Helmholtz equation. The proposed approach leads to a series of new fundamental functions which are independent of the wave number k of the Helmholtz equation. The coefficient matrix of the resulting homogeneous system of boundary element equations is of the form of a polynomial matrix in k which allows a much faster search for the eigenvalues by scanning k over an interval of interest or the standard eigenvalue problem to be formulated for directly solving for the eigenvalues without resort to iterative methods. The proposed technique was used to solve some known problems with available analytical solutions: 2-D domains with circular and rectangular geometries under Dirichlet and/or Neumann boundary conditions. The outcomes demonstrate that the proposed approach is computationally efficient and highly accurate.

This paper studies the stress intensity factors of square cracks horizontally placed in one-layered transversely isotropic halfspaces using the dual boundary element method (DBEM). This DBEM uses the point load fundamental solution of a transversely isotropic bi-material fullspace. On the uncracked boundary, the infinite boundary elements are used to consider the influence of the domain far from the crack. On the crack surface, special crack-front elements are introduced to capture the crack-tip behavior. The relative crack opening displacements on the crack surface are treated as unknown quantities. The crack opening displacements near the crack tip are used to calculate the stress intensity factors. Numerical results illustrate that the proposed method can obtain highly accurate results of the three stress intensity factors for the crack problems in one-layered transversely isotropic halfspaces. The square cracks in homogeneous or bi-material halfspaces under internal expansion or tensile traction are analyzed in detail. Results show that the material anisotropy of transversely isotropic materials exerts a strong influence on the stress intensity factors.

Made available in DSpace on 2022-05-01T06:02:17Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-10-01 Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) In this work, a non-linear Boundary Element (BE) formulation with damage model is extended for numerical simulation of structural masonry walls. The original formulation was presented by other researchers to investigate the concrete material in 2D stress analysis. The formulation proposal is written in terms of domain strain variables and it is based on the consistent tangent operator. A simple damage model is used to represent the material behavior of masonry components and internal variables and cell discretization of the domain are considered. However, for brick masonry it is necessary to take into account the orthotropy in this type of structure. The orthotropy is considered by expressing a residual elastic tensor as the difference of the orthotropic and the isotropic elastic property tensors. The complete integral formulation, using isotropic fundamental solution, and algebraic procedure are developed. Department of Civil and Environmental Engineering – School of Engineering –UNESP Centre for Infrastructure Performance and Reliability – School of Engineering The University of Newcastle Department of Geotechnical and Structural Engineering School of Engineering University of São Paulo – EPUSP Department of Mathematics – Federal Institute of Minas Gerais – IFMG Department of Civil and Environmental Engineering – School of Engineering –UNESP

The present paper deals with the development of a Galerkin Boundary Element Method (Galerkin-BEM) approach applied to the numerical simulation of plane and half-plane quasi-static viscoelastic problems. This approach makes use of a half-plane viscoelastic fundamental solution, thus avoiding the full discretization of the half-plane and consequently reducing the computational requirements of the proposed model. Time-domain responses are obtained by means of the viscoelastic reciprocal theorem and its temporal integrations are evaluated by using the Stieltjes convolutions. Numerical simulations are carried out in order to assess the accuracy and the applicability of the proposed approach to viscoelastic problems by comparing its results with those provided by the numerical and analytical solutions reported in the literature.

The efficiency of modeling of the axisymmetric dynamics of a gas bubble near a curved rigid wall by the boundary element method using the fundamental solution of the Laplace equation for an unbounded domain is numerically studied. For this purpose, the problems of the collapse of a bubble near a flat wall and the expansion and subsequent collapse of a bubble near the concave and convex walls are considered. To assess the effectiveness, the results of calculations of these problems are compared with the known results of their calculations using their fundamental solutions for the areas bounded by those walls. The results show the dependence of the numerical solution on the radius of the computational domain on the wall, the number of cells when the domain is uniformly partitioned, and the number of cells when it is non-uniformly partitioned with condensation toward the axis of symmetry along a geometric progression.

A boundary element method formulation is developed and validated through the solution of problems governed by the diffusion-wave equation, for which the order of the time derivative, say α, ranges in the interval (1, 2). This fractional time derivative is defined as an integro-differential operator in the Caputo sense. For α = 2, one recovers the classical wave equation. From the weighted residual method point of view, with the Laplace equation fundamental solution playing the role of the weighting function, a D-BEM type formulation is generated, in which the integrand of the domain integral contains the fractional time derivative. The Houbolt scheme is adopted to represent the second order time derivative of the variable of interest, say u, which appears in the Caputo operator. In all the examples, the proposed formulation provided accurate results for α as small as 1.05. Indeed, the analyses were carried out for α = 1.05, 1.2, 1.5, 1.8, 2.0, with the aim of verifying the influence of the order of the time derivative in the results.

In this paper, numerical computation using coupled finite element–boundary element (FE-BE) technique has been carried out for determination of elastic settlement of annular raft resting on elastic half-space. In the present study, annular raft has been idealised as a flexible plate and is discretised into plate finite element. However, the foundation soil has been considered as homogeneous isotropic elastic half-space and is discretised into soil boundary element. Plate stiffness matrix has been obtained using the finite element method based on Mindlin’s thick plate theory. Mindlin’s half-space solution has also been used as a fundamental solution to find the soil flexibility matrix by the boundary element method. Coupling of soil-foundation system has been performed by transforming boundary element matrix and then adding it to plate stiffness matrix using equilibrium equation and compatibility condition. In the present study, influence of various factors on elastic settlement of the raft has been analysed in terms of ring size, depth of embedment, raft thickness and soil’s Poisson’s ratio. A generic equation has also been developed incorporating these results for determination of elastic settlement. Sensitivity analysis of the proposed equation has also been done using various statistical methods.

The present work is concerned with a two-dimensional (2D) Stokes flow through a channel bounded by two parallel solid walls. The distance between the walls may be arbitrary, and the surface of one of the walls can be arbitrarily rough. The main objective of this work consists in homogenizing the heterogeneous interface between the rough wall and fluid so as to obtain an equivalent smooth slippery fluid/solid interface characterized by an effective slip length. To solve the corresponding problem, two efficient numerical approaches are elaborated on the basis of the method of fundamental solution (MFS) and the boundary element methods (BEMs). They are applied to different cases where the fluid/solid interface is periodically or randomly rough. The results obtained by the proposed two methods are compared with those given by the finite element method and some relevant ones reported in the literature. This comparison shows that the two proposed methods are particularly efficient and accurate.

The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane. Subsequently, based on Betti reciprocal theorem, by adopting the time dependent fundamental solutions in terms of displacement, traction and equivalent stress, the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem are established. The establishment procedures for the displacement and the stress boundary integral equations, together with the stress equation at boundary points, are presented in details, while the standard discretization both in time and space under the frame of time domain boundary element method (TD-BEM) and the solution of the algebraic equations are also briefly stated. Two verification examples are presented from different viewpoints, for elastic and elastoplastic analysis, for 1-D and 2-D geometries, and for finite and infinite domains. The TD-BEM formulation for dynamic elastoplastic analysis is presented for the plane strain problem as an example, where the formulation is also applicable for the plane stress problem by properly transforming the elastic constants and adopting the corresponding fundamental solutions. The equivalent stress fundamental solution for the elastoplastic dynamic plane strain problem is proposed to transform the virtual work in the third direction to the plane. Subsequently, the equivalent stress and the boundary integral equations for dynamic elastoplastic analysis for the plane strain problem, where a cavity under an uniformly distributed impact in an infinite medium, are established in the frame of the time domain boundary element method (TD-BEM).

The paper reports the new forms of the fundamental solutions for 3D magnetoelectroelasticity equations. The state-of-art presented in the paper shows that the fundamental solutions for magnetoelectroelasticity equations take complex forms, especially considering their applications in the boundary element method. In the paper, the magnetoelectroelastic continuum serves as a homogenized model of the piezoelectric-piezomagnetic composites. The model is obtained by the Mori-Tanaka micromechanical approach. The so-called quasi-static approximation of dynamics is applied to obtain the set of the partial differential equations consists of the hyperbolic equation of motion and two elliptic equations for the conservation laws of the electric and magnetic charge. The motivation example shows the analogy between the impulse response of the linear system of ordinary differential equations, known from the classical linear control theory, and the fundamental solution of the linear system of partial differential equations. The spatial Fourier transform changes the coupled system of hyperbolic-elliptic equations of magnetoelectroelasticity into differential-algebraic equations in the k-space and the time domain. Instead of the classic approach eliminating the constraint equations for the electromagnetic static potentials, the semi-state vector is introduced and the descriptor system with the Kronecker index equals 1 is obtained. The structural analysis is performed for the resultant system and the regular and singular matrix pencils were analyzed. The slowness surfaces of the mainly acoustic modes in the quasi-static approximation are constructed for the 3-phase composite. Further, the shifting method is applied and the general solution of the semi-state equations is obtained as an analogon of the impulse response solution. The obtained result serves as a base for obtaining the new form of the fundamental solution of dynamic magnetoelectroelasticity in 3-dimensional space and also for the step response and steady-state fundamental solution for magnetoelectroelastic continuum.