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Three kinds of Fragile Points Methods based on Petrov-Galerkin weak-forms (PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow-up of the previous study on the original FPM based on a symmetric Galerkin weak-form. The trial function is piecewise-continuous, written as local Taylor expansions at the Fragile Points. A modified Radial Basis Function-based Differential Quadrature (RBF-DQ) method is employed for establishing the local approximation. The Dirac delta function, Heaviside step function, and the local fundamental solution of the governing equation are alternatively used as test functions. Vanishing or pure contour integral formulation in subdomains or on local boundaries can be obtained. Extensive numerical examples in 2D and 3D are provided as validations. The collocation method (PG-FPM-1) is superior in transient analysis with arbitrary point distribution and domain partition. The finite volume method (PG-FPM-2) shows the best efficiency, saving 25% to 50% computational time comparing with the Galerkin FPM. The singular solution method (PG-FPM-3) is highly efficient in steady-state analysis. The anisotropy and nonhomogeneity give rise to no difficulties in all the methods. The proposed PG-FPM approaches represent an improvement to the original Galerkin FPM, as well as to other meshless methods in earlier literature.

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear non-local equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an $H^{5/2}$-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parameterized circular configuration, then global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as $t\rightarrow +\infty$. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions.

In this sequel to the article , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to HA¶rmander's criteria, see . It implies that the temperate fundamental solution of the operator [∂,∂]2+1 is not regular. Furthermore, a formula for the convolution of two O(p,q)†invariant distributions is presented, and, finally, L. Schwartz' question on the surjectivity of linear partial differential operators with constant coefficients on the space OM is completely answered in the case of O(p,q)†invariant differential operators.

We prove the existence of a global fundamental solution Γ(x;y) (with pole x) for any Hormander operator L=∑i=1mXi2 on Rn which is δλ-homogeneous of degree 2. Here homogeneity is meant with respect to a family of non-isotropic diagonal maps δλ of the form δλ(x)=(λσ1x1,…,λσnxn), with 1=σ1⩽⋯⩽σn. Due to a global lifting method for homogeneous operators proved by Folland [Comm. Partial Differential Equations 2 (1977) 161–207], there exists a Carnot group G and a polynomial surjective map π:G→Rn such that L is π-related to a sub-Laplacian LG on G. We show that it is always possible to perform a (global) change of variable on G such that the lifting map π becomes the projection of G≡Rn×Rp onto Rn. We prove that an integration argument over the (non-compact) fibers of π provides a fundamental solution for L. Indeed, if ΓG(x,x′;y,y′) (x,y∈Rn; x′,y′∈Rp) is the fundamental solution of LG, we show that ΓG(x,0;y,y′) is always integrable with respect to y′∈Rp, and its y′-integral is a fundamental solution for L.

Summary The singular boundary method (SBM) is a recent strong-form boundary collocation method that uses a linear combination of the fundamental solution of the governing equation to approximate the field variables. Because of its full interpolation matrix, the SBM solution encounters the high computational complexity and storage requirement that limit its applications to large-scale engineering problems. This paper presents a way to overcome this drawback by introducing the diagonal form fast multipole method. A diagonal form fast multipole singular boundary method is then developed to reduce the computational operations of the SBM with direct solvers from O(N3) to O(NlogN), where N is the number of unknowns. The proposed method works well for acoustic radiation and scattering with nondimensional wave number kD

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

In the present work, the scattering problem of an elastic wave by a penetrable thermoelastic body in an isotropic and homogeneous elastic medium is considered. The corresponding scattering problem is formulated in a suitable compact form, and taking into account the physical parameters of the thermoelastic body and integral representations for the total exterior elastic and the total interior thermoelastic field are presented. Using asymptotic analysis of the fundamental solution of the Navier equation, expressions of the far-field patterns are obtained, and reciprocity theorems for plane and spherical wave incidence are established. Finally, a general scattering theorem for plane wave incidence is also presented. Copyright © 2016 John Wiley & Sons, Ltd.

The two-dimensional transient problem that is studied concerns a semi-infinite crack in an isotropic solid comprising an infinite strip and a half-plane joined together and having the same elastic constants. The crack propagates along the interface at constant speed subject to time-independent loading. By means of the Laplace and Fourier transforms the problem is formulated as a vector Riemann-Hilbert problem. When the distance from the crack to the boundary grows to infinity the problem admits a closed-form solution. In the general case, a method of partial matrix factorization is proposed. In addition to factorizing some scalar functions it requires solving a certain system of integral equations whose numerical solution is computed by the collocation method. The stress intensity factors and the associated weight functions are derived. Numerical results for the weight functions are reported and the boundary effects are discussed. The weight functions are employed to describe propagation of a semi-infinite crack beneath the half-plane boundary at piecewise constant speed.

The site contains a Mathematica code for computing the displacement and stress induced by injection of fluid in a poroelastic layer bounded by impermeable elastic half-spaces

G. R. Franssens In this paper, we present an explicit construction for the fundamental solution of the heat operator, the Schrodinger operator, and related first-order parabolic Dirac operators on a class of some conformally flat non-orientable orbifolds. More concretely, we treat a class of projective cylinders and tori where we can study parabolic monogenic sections with values in different pin bundles. We present integral representation formulas together with some elementary tools of harmonic analysis that enable us to solve boundary value problems on these orbifolds. Copyright © 2015 John Wiley & Sons, Ltd.

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We present a boundary element method for computing numerical solutions of the reaction-diffusion telegraph equation in unbounded domains. This technique does not need artificial boundary conditions at the computational domain and uses a new algorithm to compute the Fourier transform, the convolution theorem, and the fact that the exact solution of the telegraph equation can be written as an integral transform in terms of the fundamental solution. We use the logistic growth model to find how the population of an organism evolves according to its growth rate. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 326–335, 2015

SUMMARY The scaled boundary finite-element method, a semi-analytical computational scheme primarily developed for dynamic stiffness of unbounded domains, is applied to the analysis of unsteady seepage flow problems. This method is based on the finite-element technology and gains the advantages of the boundary element method as well. Only boundary of the domain is discretized, no fundamental solution is required and singularity problems can be modeled rigorously. Anisotropic and non-homogeneous materials satisfying similarity are modeled with no additional efforts. In this study, firstly, formulation of the method for the transient seepage flow problems is derived followed by its solution procedures. The accuracy, simplicity and applicability of the method are demonstrated via four numerical examples of transient seepage flow – three of them are available in the literature. Homogenous, non-homogenous, isotropic and anisotropic material properties are considered to show the versatility of the technique. Excellent agreement with the finite-element method is observed. The method out-performs the finite-element method in modeling singularity points. Copyright © 2014 John Wiley & Sons, Ltd.

The anti-plane strain elastodynamic problem for a continuously inhomogeneous half-plane with free-surface relief subjected to time-harmonic SH-wave is studied. The computational tool is a boundary integral equation method (BIEM) based on analytically derived Green’s function for a quadratically inhomogeneous in depth half-plane. To show the versatility of the proposed BIE method, it is considered SH-wave propagation in an inhomogeneous half-plane with free surface relief presented by a semi-circle, semi-elliptic and triangle canyon. The inhomogeneous in depth half-plane is modeled in two different ways: (i) the material properties vary continuously in depth and BIEM based on Green’s function is used; (ii) the material properties vary in a discrete way and the half-plane is presented by a set of homogeneous layers with horizontal interfaces and a hybrid technique based on wave number integration method (WNIM) and BIEM is applied. The equivalence of these two different models is shown. The simulations reveal a marked dependence of the wave field on the material inhomogeneity and the potential of the BIEM based on the Green’s function for half-plane to produce highly accurate results by using strongly reduced discretization mesh in comparison with the conventional boundary element technique using fundamental solution for the full plane. c

SUMMARY In this paper, a coupling method between finite element and analytical layer-elements is utilized to analyze the time-dependent behavior of a plate of any shape and finite rigidity resting on layered saturated soils. Based on the integral transform techniques together with the aid of an order reduction method, an analytical layer-element solution is derived from the governing equations for three-dimensional Biot consolidation with respect to a Cartesian coordinate system and then extended to be the fundamental solution for the layered saturated soil under a point load. The Mindlin plate is modeled by eight-noded isoparametric elements. The governing equations of the interaction between soil and plate in the Laplace-Fourier transformed domain are deduced by referring to the coupling theory of FEM/BEM, and the final solution is obtained by applying numerical inversion. Numerical examples concerned with the time-dependent response of a plate are performed to demonstrate the influence of soil and plate properties on the interaction process. Copyright © 2014 John Wiley & Sons, Ltd.

SUMMARY In this study, the dynamic response of a poroelastic half-space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re-formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full-space fundamental singular solution for a point sink is also derived using potential methods. The mirror-image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half-space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.

Fracture and fatigue assessment of structures weakened by multiple site damage, such as two or more interacting cracks, is currently a very challenging problem. The main objective of this paper is to develop a mathematical model and an approach to investigate fatigue crack closure behaviour of two through-the-thickness collinear cracks of equal length in a plate of arbitrary thickness under remote tensile cyclic loading. The developed mathematical model of the problem under consideration is based on the Dugdale strip yield model and plasticity-induced crack closure concept. The approach utilises the fundamental solution for an edge dislocation in a plate of finite thickness and the distributed dislocation technique to obtain an effective and accurate solution to the system of governing equations. The obtained results show a very good agreement with the previously published analytical solutions for limiting cases. In particular, the new results confirm that the crack closure behaviour and the opening stress variation in the case of two collinear cracks are significantly dependent on the separation gap between two cracks as well as the plate thickness.

We develop a point source method (PSM) to obtain flow field reconstructions from remote measurements. The PSM belongs to the class of decomposition methods in inverse scattering because it solves the nonlinear and ill-posed inverse shape reconstruction problem by a decomposition into a linear ill-posed problem and a nonlinear well-posed problem. As a model problem, we investigate the reconstruction of the flow field of two-dimensional stationary Oseen equation, display math which is obtained by linearizing the Navier–Stokes equation with kinematic viscosity μ > 0 around the constant velocity u0. In contrast to acoustics or electromagnetics, the use of the PSM in fluid dynamics leads to a number of challenges in terms of the analysis and the proper setup of the scheme, in particular, because the null-spaces of the integral operators under consideration are no longer trivial and the fundamental solution is not symmetric in its spatial coordinate. We provide a suitable formulation of the method and prove convergence of flow reconstructions by the PSM. For the realization of the reconstruction when the inclusions are not known, we employ domain sampling. We will demonstrate the feasibility of the method for reconstructing one or several objects by numerical examples.

The aim of this paper is to study some classes of second-order divergence-form partial differential operators L of sub-Riemannian type. Our main assumption is the C^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(x, y) for L. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the L-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to M_r; the role of M_r in solving the homogeneous Dirichlet problem related to L in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Omega_r(x), superlevel sets of Gamma(x, ·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for L using M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).

The present paper is devoted to the two-dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the LaTeX2ϵ mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.

Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1, and their (2k+ 1)th-order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy-deformation” approach, where the limit in the first equation in (1) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non-self-adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [1]. Various other nonlinear operator and numerical methods for (1) are also applied. As a key alternative, the “super-nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in (1), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=pl(n)}l≥0 of the absorption exponents is discussed.

In the current article, we investigate the RBF solution of second-order two-space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary-only and integration-free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

A transversely isotropic linear elastic half-space with depth wise axis of material symmetry containing a cylindrical cavity of finite length is considered to be under the effect of a time-harmonic torsion force applied on a ring at an arbitrary depth on the surface of the cylindrical cavity. With the aid of cosine transforms, the boundary value problem for the fundamental solution is reduced to a generalized Cauchy singular integral equation. The Cauchy integral equation involved in this paper is analytically investigated and the final equation is numerically solved with an in-depth attention. Integral representation of the stress and displacement are obtained, and is shown that their degenerated form to the static problem of isotropic material is coincide with existing solutions in the literature. To investigate the effect of material anisotropy, the results are numerically evaluated and illustrated.

SUMMARY This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two-dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low-frequency FMM and the high-frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low-frequency FMM and the quadrature order for the high-frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite-size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.

Rock and rock mass are non-homogeneous medium with complex micro- and meso-scopic structures. Dispersion effect is caused by the interaction between the elastic wave and these micro- and meso-defects in rocks. A double-crack model is used to study the effect of microscopic structures on the dispersion of elastic waves. In the model, the interaction between these two cracks is considered to analyze the effect of multiple scattering. Among these double-crack systems, a linear superposition method is adopted to analyze the localization of the effect of defects in rocks. Mathematically, based on the fundamental solution of wave propagation equation with Green function method, and combined with the boundary integral method, cracks are treated as inner boundaries and dispersion equation is obtained. Furthermore, the difference between these two ways of interaction of cracks is analyzed, and the influence of the micro discontinuity parameters of the double-crack system, pore fluid pressure and unloading on the dispersion of rocks is discussed.

SUMMARY We consider the magnetohydrodynamic flow that is laminar and steady of a viscous, incompressible, and electrically conducting fluid in a semi-infinite duct under an externally applied magnetic field. The flow is driven by the current produced by a pressure gradient. The applied magnetic field is perpendicular to the semi-infinite walls that are kept at the same magnetic field value in magnitude but opposite in sign. The wall that connects the two semi-infinite walls is partly non-conducting and partly conducting (in the middle). A BEM solution was obtained using a fundamental solution that enables to treat the magnetohydrodynamic equations in coupled form with general wall conductivities. The inhomogeneity in the equations due to the pressure gradient was tackled, obtaining a particular solution, and the BEM was applied with a fundamental solution of coupled homogeneous convection–diffusion type partial differential equations. Constant elements were used for the discretization of the boundaries (y = 0, −a ⩽ x ⩽ a) and semi-infinite walls at x = ±a, by keeping them as finite since the boundary integral equations are restricted to these boundaries due to the regularity conditions as y ∞ . The solution is presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number (M), conducting length (l), and non-conducting wall conditions (k). The effect of the parameters on the solution is studied. Flow rates are also calculated for these values of parameters. Copyright © 2011 John Wiley & Sons, Ltd.

SUMMARY Couple-stresses are introduced to account for the microstructure of a material within the framework of continuum mechanics. Linear isotropic versions of such materials possess a characteristic material length l that becomes increasingly important as problem dimensions shrink to that level (e.g., as the radius a of a critical hole reduces to a size comparable to l). Consequently, this size-dependent elastic theory is essential to understand the behavior at micro- and nano-scales and to bridge the atomistic and classical continuum theories. Here we develop an integral representation for two-dimensional boundary value problems in the newly established fully determinate theory of isotropic couple stress elastic media. The resulting boundary-only formulation involves displacements, rotations, force-tractions and moment-tractions as primary variables. Details on the corresponding numerical implementation within a boundary element method are then provided, with emphasis on kernel singularities and numerical quadrature. Afterwards the new formulation is applied to several computational examples to validate the approach and to explore the consequences of size-dependent couple stress elasticity. Copyright © 2011 John Wiley & Sons, Ltd.

In this study, we use the addition theorem and superposition technique to solve the scattering problem with multiple circular cylinders arising from point sound sources. Using the superposition technique, the problem can be decomposed into two individual parts. One is the free-space fundamental solution. The other is a typical boundary value problem (BVP) with specified boundary conditions derived from the addition theorem by translating the fundamental solution. Following the success of null-field boundary integral formulation to solve the typical BVP of the Helmholtz equation with Fourier densities, the second-part solution is easily obtained after collocating the observation point exactly on the real boundary and matching the boundary condition. The total solution is obtained by superimposing the two parts which are the fundamental solution and the semianalytical solution of the Helmholtz problem. An example was demonstrated to validate the present approach. The parameter study of size and spacing between cylinders are addressed. The results are well compared with the available theoretical solutions and experimental data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011

SUMMARY The scaled boundary finite-element method (SBFEM), a novel semi-analytical technique, is applied to the analysis of the confined and unconfined seepage flow. This method combines the advantages of the finite-element method and the boundary element method. In this method, only the boundary of the domain is discretized; no fundamental solution is required, and singularity problems can be modeled rigorously. Anisotropic and nonhomogeneous materials satisfying similarity are modeled without additional efforts. In this paper, SBFE equations and solution procedures for the analysis of seepage flow are outlined. The accuracy of the proposed method in modeling singularity problems is demonstrated by analyzing seepage flow under a concrete dam with a cutoff at heel. As only the boundary is discretized, the variable mesh technique is advisable for modeling unconfined seepage analyses. The accuracy, effectiveness, and efficiency of the method are demonstrated by modeling several unconfined seepage flow problems. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper we develop an alternative boundary element method (BEM) formulation for the analysis of anisotropic three-dimensional (3D) elastic solids. Our implementation is based on the derivation of explicit expressions for the fundamental solution displacements and tractions, of general validity for any class of anisotropic materials, by means of Stroh formalism and Cauchy's residue theory. The resulting fundamental solution remains valid for mathematical degenerate cases when Stroh's eigenvalues are coincident, meanwhile it does not exhibit numerical instabilities for quasi-degenerate cases when Stroh's eigenvalues are nearly equal. A multiple pole residue approach is followed, leading to general explicit expressions to evaluate the traction fundamental solution for poles of m-multiplicity. Despite the existence of general displacement solutions in the literature, and for the sake of completeness, the same approach as for the traction solution is considered to derive the displacement fundamental solution as well. Based on these solutions, an explicit BEM approach for the numerical solution of 3D linear elastic problems for solids with general anisotropic behavior is presented. The analysis of cracked anisotropic solids is also considered. Details on the numerical implementation and its validation for degenerate cases are discussed. Copyright © 2010 John Wiley & Sons, Ltd.

This work addresses the evaluation of the stress and electric field concentrations around a circular hole in a functionally graded piezoelectric plane subjected to anti-plane elastic SH-wave and in-plane time-harmonic electric load. All material parameters vary exponentially along a line of arbitrary orientation in the plane of the piezoelectric material under consideration. The computational tool is a non-hypersingular traction based boundary integral equation method (BIEM). The kernel functions used in the BIEM are exact fundamental solutions that have been derived in previous work by the authors. Numerical solutions for the stress and electric field concentration factors (SCF and EFCF, respectively) around the perimeter of the hole are obtained. The simulation demonstrates the efficiency of the computational approach and its potential to reveal in an adequate way the dynamic stress and electric field distribution around the hole. Presented are results showing their dependence on various system parameters as e.g. the electro-mechanical coupling, the type of the dynamic load and its characteristics, the wave-hole and wave-material interaction and the magnitude and direction of the material inhomogeneity.

The method of fundamental solution is used to calculate the creeping flow around a spherical solid particle close to a porous membrane. The equations for the flow in the porous medium and conditions at the interface are satisfied automatically with the use of a Green function calculated by Elasmi and Feuillebois [11]. Singularities, i.e. the Green function and some of its derivatives, are distributed inside the particle and their positions and intensities are optimized by minimizing the difference between the approximate velocity and the exact one on the particle surface in the sense of least squares. It is proved that this procedure also minimizes the dissipated energy of the error velocity, that is of the difference between the approximation and the exact value. For the example cases treated here of a sphere moving normal to a wall, the singularities are stokeslets and stokeslet quadrupoles (or source doublets). In the particular case of an impermeable wall, the method is the same as that of Zhou and Pozrikidis [ 19]. Their results are recovered and extended to a higher number of singularities. The method is then applied to the case of a solid sphere moving normal to a thin porous slab. By comparison with results obtained by Elasmi and Feuillebois [11] with the boundary integral method, it is found that a good approximation is obtained here with only a few singularities. A comparison is also made with Goren [13] who treated analytically a related problem in the case of a low porosity.

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed-form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.

SUMMARY This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier-Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function-vorticity-magnetic induction-current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson-type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time-dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward-facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper a new and innovative method for computation of longitudinal dynamic characteristics of multi-cracked bars is proposed. Cracks are modeled by equivalent axial springs with specified flexibility. Making use of the Heaviside step function and Dirac's delta distribution, a single governing equation for the whole bar is developed. The governing equation is an ordinary differential equation. With the help of Laplace Transform, a general analytical solution in terms of several unknown coefficients is determined. Boundary conditions are then used to determine the analytical solution for specific problems. Making use of the proposed governing equation a new finite element formulation is derived. In this formulation the effect of cracks is considered by adding an equivalent mass matrix to the element mass matrix. Through numerical study, the accuracy, efficiency and robustness of the work is verified. Copyright © 2010 John Wiley & Sons, Ltd.

The author presents a Boundary Element Method (BEM) implementation for SH harmonic waves in a class of inhomogeneous anisotropic media. The inhomogeneity is assumed to be the same not only for the stiffnesses, but also for the density. The implementation is based on a closed form fundamental solution for SH waves derived by Daros [C. H. Daros, A fundamental solution for SH‐waves in class of inhomogeneous anisotropic media, Int. J. Eng. Science 46 (2008) 809–817]. He shows numerical results obtained by the traditional boundary integral equation. Moreover, the non‐hypersingular traction based BEM is also implemented, allowing the modelling of cracks in inhomogeneous anisotropic media. The author obtains numerical results for the stress intensity factors (SIF) which are compared to previous published results.

In this paper an abstract condition is given yielding universal series defined by sequences a = {a(j)}infinity j=1 in boolean AND(p > 1)l(p) but not in l(1). We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann zeta-function, or a fundamental solution of a given elliptic operator in R-nu with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in R-nu simultaneously with respect to all sigma-finite Borel measures in R-nu. Stronger results are obtained by using universal Dirichlet series.

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed.

In this paper, exponential basis functions (EBFs) are used in a boundary collocation style to solve engineering problems whose governing partial differential equations (PDEs) are of constant coefficient type. Complex-valued exponents are considered for the EBFs. Two-dimensional elasto-static and time harmonic elasto-dynamic problems are chosen in this paper. The solution procedure begins with first finding a set of appropriate EBFs and then considering the solution as a summation of such EBFs with unknown coefficients. The unknown coefficients are determined by the satisfaction of the boundary conditions through a collocation method with the aid of a consistent and complex discrete transformation technique. The basis and various forms of the transformation have been addressed and discussed. We shall propose several strategies for selection of EBFs with the aid of the basis explained for the transformation. While using the transformation, the number of EBFs should not necessarily be equal to (or less than) the number of boundary information data. A library of EBFs has also been presented for further use. The effect of body forces is included in the solution via construction of particular solution by the use of the discrete transformation and another series of EBFs. A number of sample problems are solved to demonstrate the capabilities of the method. It has been shown that the time harmonic problems with high wave number can be solved without much effort. The method, categorized in meshless methods, can be applied to many other problems in engineering mechanics and general physics since EBFs can easily be found for almost all problems with constant coefficient PDEs.