integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

a multi level fast multipole bem for 3 d elastodynamics in the frequency domain to reduce computational complexity and memory requirement for 3 d elastodynamics using the boundary element method bem a multi level fast multipole bem fm bem is proposed the diagonal form for the expansion of the elastodynamic fundamental solution is used with a truncation parameter adjusted to the subdivision level a feature necessary for achieving optimal computational efficiency both the single level and mul ti level forms of the elastodynamic fm bem are considered with emphasis on the latter crucial implementation issues including the truncation of the multipole expansion the optimal number of levels the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments a complexity analysis for both the single level and multi level versions is conducted the correctness and computational performances of the proposed elastodynamic fmm are demonstrated on numerical examples featuring up to o 10 6 dofs run on a single processor pc and including the diffraction of an incident p plane wave by a semi spherical or semi ellipsoidal canyon representative of topographic site effects

acoustic coupled fluid structure interactions using a unified fast multipole boundary element method this paper presents a numerical model for the acoustic coupled fluid structure interaction fsi of a submerged finite elastic body using the fast multipole boundary element method fmbem the helmholtz and elastodynamic boundary integral equations bies are respectively employed to model the exterior fluid and interior solid domains and the pressure and displacement unknowns are coupled between conforming meshes at the shared boundary interface to achieve the acoustic fsi the low frequency fmbem is applied to both bies to reduce the algorithmic complexity of the iterative solution from o n2 to o n1 5 operations per matrix vector product for n boundary unknowns numerical examples are presented to demonstrate the algorithmic and memory complexity of the method which are shown to be in good agreement with the theoretical estimates while the solution accuracy is comparable to that achieved by a conventional finite element boundary element fsi model

improved finite element methods for elastic waves abstract in this work we develop finite element methods for exterior problems of time harmonic elastic waves we employ dtn exterior boundary conditions on an artificial boundary to obtain a well posed equivalent problem in a bounded region that is suitable for domain based computation galerkin galerkin least squares and galerkin gradient least squares finite element methods are presented in acoustic wave problems the galerkin least squares method counters the numerical difficulties that result from employing the galerkin method but proves to be insufficient for elastic waves a specialization of the galerkin gradient least squares method to problems of elastic waves yields good results numerical tests validate these conclusions

regularized integral equation methods for elastic scattering problems in three dimensions abstract this paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three dimensional space the proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high order singular integration methods 13 introduced recently for the acoustic case in order to obtain formulations leading to iterative solvers gmres which converge in small numbers of iterations we investigate theoretically and computationally the character of the spectra of various operators associated with the elastic wave calderon relation including some of their possible compositions and combinations in particular by relying on the fact that the eigenvalues of the composite operator ns are bounded away from zero and infinity new uniquely solvable low gmres iteration integral formulation for the closed surface case are presented the introduction of corresponding low gmres iteration equations for the open surface equations additionally requires for both spectral quality as well as accuracy and efficiency use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges several numerical examples demonstrate the accuracy and efficiency of the proposed methodology

random parking euclidean functionals and rubber elasticity we study subadditive functions of the random parking model previously analyzed by the second author in particular we consider local functions s of subsets of mathbb r d and of point sets that are almost subadditive in their first variable denoting by the random parking measure in mathbb r d and by r the random parking measure in the cube qr r r d we show under some natural assumptions on s that there exists a constant overline s in mathbb r such that lim r to infty frac s q r xi q r lim r to infty frac s q r xi r q r overline s almost surely if zeta mapsto s q r zeta is the counting measure of zeta in qr then we retrieve the result by the second author on the existence of the jamming limit the present work generalizes this result to a wide class of almost subadditive functions in particular classical euclidean optimization problems as well as the discrete model for rubber previously studied by alicandro cicalese and the first author enter this class of functions in the case of rubber elasticity this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit as well as a generalization to stochastic networks generated on bounded sets

boundary integral equation methods for the elastic and thermoelastic waves in three dimensions in this paper we consider the boundary integral equation bie methods for solving the exterior neumann boundary value problems of elastic and thermoelastic waves in three dimensions based on the fredholm integral equations of the first kind the innovative contribution of this work lies in the proposal of the new regularized formulations for the hyper singular boundary integral operators bio associated with the time harmonic elastic and thermoelastic wave equations with the help of the new regularized formulations we only need to compute the integrals with weak singularities at most in the corresponding variational forms of the boundary integral equations the accuracy of the regularized formulations is demonstrated through numerical examples using the galerkin boundary element method bem

reconstruction of a constitutive law for rubber from in silico experiments using ogden s laws this article deals with the following data assimilation problem construct an analytical approximation of a numerical constitutive law in three dimensional nonlinear elasticity more precisely we are concerned with a micro macro model for rubber macroscopic quantities of interest such as the piola kirchhoff stress tensor can be approximated for any value of the strain gradient by numerically solving a nonlinear pde this procedure is however computationally demanding hence although conceptually satisfactory this physically based model is of no direct practical use the aim of this article is to circumvent this difficulty by proposing a numerical strategy to reconstruct from in silico experiments an accurate analytical proxy for the micro macro constitutive law

random parking sequential adsorption and the jamming limit identical cars are dropped sequentially from above into a large parking lot each car is positioned uniformly at random subject to non overlap with its predecessors until jamming occurs there have been many studies of the limiting mean coverage as the parking lot becomes large but no complete proof that such a limit exists until now

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

precorrected fft accelerated bem for large scale transient elastodynamic analysis using frequency domain approach summary a precorrected fast fourier transform pfft accelerated boundary element method bem for large scale transient elastodynamic analysis is developed and described in this paper the frequency domain approach is used to overcome the wrap around problem associated with the discrete fourier transform the exponential window method ewm is employed and incorporated in the frequency domain bem an improved implementation scheme of the pfft method based on polynomial interpolation technique is developed and applied to accelerate the elastodynamic bem this new scheme reduces the memory required to save the convolution matrix by a factor of 8 to further improve the efficiency of the code a newly developed linear system solver based on the induced dimension reduction method is employed its performance is investigated and compared with that of the well known gmres the accuracy and computational efficiency of the method are evaluated and demonstrated by three examples a classical benchmark a plate subject to an impact loading and a porous cube with nearly half million dofs subject to a step traction loading both analytical and experimental results are employed to validate the method it has been found that the ewm can effectively resolve the wrap around problem and accurate time responses for an arbitrarily chosen time period can be obtained copyright 2011 john wiley sons ltd

boundary integral equations for thin bodies summary a boundary integral equation formulation for thin bodies which uses cbie conventional bie only is well known to be degenerate a mixed formulation for a thin rigid scatterer which combines cbie and hbie hypersingular bie is motivated by examining the discretized form of the integral equations and this formulation is shown to be nondegenerate for thin non rigid inclusion problems a near singular integration procedure useful for singular integrals as well is presented finally numerical examples for acoustic wave scattering from rigid and soft scatterers are presented

efficiency of higher order finite elements for the analysis of seismic wave propagation the analysis of wave propagation problems in linear damped media must take into account both propagation features and attenuation process to perform accurate numerical investigations by the finite differences or finite element method one must consider a specific problem known as the numerical dispersion of waves numerical dispersion may increase the numerical error during the propagation process as the wave velocity phase and group depends on the features of the numerical model in this paper the numerical modelling of wave propagation by the finite element method is thus analyzed and dis cussed for linear constitutive laws numerical dispersion is analyzed herein through 1d computations investigating the accuracy of higher order 15 node finite elements towards numerical dispersion concerning the numerical analy sis of wave attenuation a rheological interpretation of the classical rayleigh assumption has for instance been previously proposed in this journal

rearrangement invariant spaces of functions on lca groups in this paper the rearrangement invariant hulls and kernels of the classes f f lp and f f l1loc on a noncompact nondiscrete locally compact abelian group are characterized the spacel1loc s of tempered distributions serves as the extended domain of the fourier transform instead of the classical domainl1 l2 we also introduce and study a new function spacep0which arises as the rearrangement invariant hull of some of the classes under consideration

an accurate boundary element method for the exterior elastic scattering problem in two dimensions this paper is concerned with a galerkin boundary element method solving the two dimensional exterior elastic wave scattering problem the original problem is first reduced to the so called burtonmiller 1 boundary integral formulation and essential mathematical features of its variational form are discussed in numerical implementations a newly derived and analytically accurate regularization formula 2 is employed for the numerical evaluation of hyper singular boundary integral operator a new computational approach is employed based on the series expansions of hankel functions for the computation of weakly singular boundary integral operators during the reduction of corresponding galerkin equations into a discrete linear system the effectiveness of proposed numerical methods is demonstrated using several numerical examples a new and accurate regularization formula which replaces the weak form of hypersingular boundary integral operator by a coupling of several weakly singular boundary integrals is given based on the series expansions of hankel functions several special integrals all weakly singular boundary integrals are evaluated exactly

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

the variational theory of complex rays an answer to the resolution of mid frequency 3d engineering problems the variational theory of complex rays vtrc is an approach to the simulation of mid frequency phenomena whose wavelengths are relatively small compared to the dimensions of the domain this is a wave based computational technique which involves a nonclassical variational formulation this paper focuses on the development of the approach for 3d engineering problems and shows that it is a mature technique illustrations are given to show the capabilities of this method

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

a wideband fast multipole accelerated boundary integral equation method for time harmonic elastodynamics in two dimensions 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

a fast and high order method for the three dimensional elastic wave scattering problem in this paper we present a fast and high order boundary integral equation method for the elastic scattering by three dimensional large penetrable obstacles the algorithm extends the method introduced in 5 for the acoustic surface scattering to the fully elastic case in our algorithm high order accuracy is achieved through the use of the partition of unity and a semi classical treatment of relevant singular integrals the computational efficiency associated with the nonsingular integrals is attained by the method of equivalent source representations on a cartesian grid and fast fourier transform fft the resulting algorithm computes one matrix vector product associated with the discretization of the integral equation with o n 4 3 log n operations and it shows algebraic convergence several numerical experiments are provided to demonstrate the efficiency of the method

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

a generalized calderon formula for open arc diffraction problems theoretical considerations we deal with the general problem of scattering by open arcs in two dimensional space we show that this problem can be solved by means of certain second kind integral equations of the form ns where n and s are first kind integral operators whose composition gives rise to a generalized calderon formula of the form ns j 0 k in a weighted periodized sobolev space here j 0 is a continuous and continuously invertible operator and k is a compact operator the ns formulation provides for the first time a second kind integral equation for the open arc scattering problem with neumann boundary conditions numerical experiments show that for both the dirichlet and neumann boundary conditions our second kind integral equations have spectra that are bounded away from zero and infinity as k to the authors knowledge these are the first integral equations for these problems that possess this desirable property this situation is in stark contrast with that arising from the related classical open surface hypersingular and single layer operators n and s whose composition ns maps for example the function 1 into a function that is not even square integrable our proofs rely on three main elements algebraic manipulations enabled by the presence of integral weights use of the classical result of continuity of the cesaro operator and explicit characterization of the point spectrum of j 0 which interestingly can be decomposed into the union of a countable set and an open set both of which are tightly clustered around 1 4 as shown in a separate contribution the new approach can be used to construct simple spectrally accurate numerical solvers and when used in conjunction with krylov subspace iterative solvers such as the generalized minimal residual method it gives rise to a dramatic reduction in the number of iterations compared with those required by other approaches

rearrangements of functions on a locally compact abelian group and integrability of the fourier transform abstract we find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group these classes consist of all functions for which the fourier transform belongs to a given lorentz space on the dual group different special cases of the problems considered in this paper have been originally studied by hardy littlewood hewitt ross cereteli and the author

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

an efficient and high order frequency domain approach for transient acoustic structure interactions in three dimensions this paper is concerned with numerical solution of the transient acoustic structure interaction problems in three dimensions an efficient and higher order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency domain the exponential window applied to the acoustic structure system yields an artificial damping to the system which eliminates the wrap around errors brought by the discrete fourier transform the frequency domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast fourier transform numerical studies are presented to demonstrate the accuracy and efficiency of the method copyright 2016 john wiley sons ltd

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

an advanced computational method for radiation and scattering of acoustic waves in three dimensions the method proposed in this paper provides a computational method for implementing the helmholtz integral formula for acoustic radiation and scattering problems associated with arbitrary shaped three dimensional bodies in particular an isoparametric element formulation is used in which both the surface geometry and the acoustic variables on the surface of the body are represented by second order shape functions within the local coordinate system a general formula for the surface velocity potential and the exterior field is derived this result is applicable to nonsmooth bodies i e it includes the case where the surface may have a nonunique normal e g at the edge of a cube test cases are shown involving spherical cylindrical and cubical geometry

nystrom method for elastic wave scattering by three dimensional obstacles nystrom method is developed to solve for boundary integral equations bie s for elastic wave scattering by three dimensional obstacles to generate the matrix equation from a bie nystrom method applies a quadrature rule to the integrations of smooth integrands over a discretized element directly and chooses the values of the unknown function at quadrature points as the system s unknowns to be solved this leads to a simple procedure to form the off diagonal entries of matrix by simply evaluating the integrands without numerical integrations for the diagonal or near diagonal entries corresponding to the integrals over a singular or near singular element where the kernels are singular or near singular we develop a systematic singularity treatment technique known as the local correction scheme based on the linear approximation of elements the scheme differs from the singularity regularization or subtraction technique used in the boundary element method bem it applies the series expansion of scalar green s function to the kernels and derives analytical solutions for the strongly singular integrals under the cauchy principal value like cpv like sense since the approach avoids the need for reformulating the bie for singularity removal in bem and solves for the somigliana s equation directly it is easy to implement and efficient in calculation numerical examples are used to demonstrate its robustness

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

a fast hierarchical dual boundary element method for three dimensional elastodynamic crack problems in this work a fast solver for large scale three dimensional elastodynamic crack problems is presented implemented and tested the dual boundary element method in the laplace transform domain is used for the accurate dynamic analysis of cracked bodies the fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the laplace parameter an aca adaptive cross approximation algorithm is used for the population of the low rank blocks and its performance at varying laplace parameters is investigated a preconditioned gmres is used for the solution of the resulting algebraic system of equations the preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format an original strategy based on the computation of some local preconditioners only is presented and tested to further speed up the overall analysis the reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

error minimization of multipole expansion in this paper we focus on the truncation error of the multipole expansion for the fast multipole method and the multilevel fast multipole algorithm when the buffer size is large enough the error can be controlled and minimized by using the conventional selection rules on the other hand if the buffer size is small the conventional selection rules no longer hold and the new approach which we have recently proposed is needed however this method is still not sufficient to minimize the error for small buffer cases we clarify this fact and show that the information about the placement of true worst case interaction is needed a novel algorithm to minimize the truncation error is proposed

numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method in this paper we investigate a numerical approach based on the partition of unity finite element method for the time harmonic elastic wave equations the aim of the proposed work is to accurately model two dimensional elastic wave problems with fewer elements capable of containing many wavelengths per nodal spacing and without refining the mesh at each frequency the approximation of the displacement field is performed via the standard finite element shape functions enriched by superimposing pressure and shear plane wave basis which incorporate knowledge of the wave propagation a variational framework able to handle mixed boundary conditions is described numerical examples dealing with the radiation and the scattering of elastic waves by a circular body are presented the results show the performance of the proposed method in both accuracy and efficiency copyright 2008 john wiley sons ltd

a high order integral solver for scalar problems of diffraction by screens and apertures in three dimensional space we present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three dimensional space our approach relies on new integral formulations as well as associated high order quadrature rules the new integral formulations involve weighted versions of the classical integral operators related to the thin screen dirichlet and neumann problems as well as a generalization to the open surface problem of the classical calderon formulae the high order quadrature rules we introduce for these operators in turn resolve the multiple green function and edge singularities which occur at arbitrarily close distances from each other and which include weakly singular as well as hypersingular kernels and thus give rise to super algebraically fast convergence as the discretization sizes are increased when used in conjunction with krylov subspace linear algebra solvers such as gmres the resulting solvers produce results of high accuracy in small numbers of iterations for low and high frequencies alike we demonstrate our methodology with a variety of numerical results for screen and aperture problems at high frequencies including simulation of classical experiments such as the diffraction by a circular disc featuring in particular the famous poisson spot evaluation of interference fringes resulting from diffraction across two nearby circular apertures as well as solution of problems of scattering by more complex geometries consisting of multiple scatterers and cavities

nonconforming finite element methods for the simulation of waves in viscoelastic solids the propagation of waves in two and three dimensional bounded viscoelastic media is described in the space frequency domain leading to a helmholtz type boundary value problem which is noncoercive non hermitian and complex valued first order absorbing boundary conditions are derived and used to minimize spurious reflections from the artificial boundaries the paper consists of two parts in part i we describe the global procedures for the approximate solution of the problem simplicial and rectangular nonconforming finite element methods are employed for the spatial discretization optimal error estimate in a broken energy and l2 norms are derived using a bootstrapping argument of schatz also a hybridization of these procedures is analyzed in part ii we define and analyze nonoverlapping domain decomposition iterative methods convergence results are derived and numerical experiments showing the potential applicability in seismology are presented

numerical resolution of the boundary integral equations for elastic scattering by a plane crack the problem of wave scattering by a plane crack is solved either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method a collocation method is often used to solve that equation but here we will use a variational method first writing the problem of fourier variables and then writing the associated integrals in the sesquilinear form with weak singularity kernels this representation is used in the numerical approach made with a finite element method in the surface of the crack numerical tests were made with circular and elliptical cracks but this method can be extended to other shapes with the same convergence profiles extensive results are given concerning the crack opening displacement the scattering cross section the back scattered amplitude and far field patterns

a high order spectral algorithm for elastic obstacle scattering in three dimensions in this paper we describe a high order spectral algorithm for solving the time harmonic navier equations in the exterior of a bounded obstacle in three space dimensions with dirichlet or neumann boundary conditions our approach is based on combined field boundary integral equation cfie reformulations of the navier equations we extend the spectral method developed by ganesh and hawkins 20 for solving second kind boundary integral equations in electromagnetism to linear elasticity for solving cfies that commonly involve integral operators with a strongly singular or hypersingular kernel the numerical scheme applies to boundaries which are globally parametrised by spherical coordinates the algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods the computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three dimensional convex and non convex smooth obstacles

elastic wave identification of penetrable obstacles using shape material sensitivity framework this study deals with elastic wave identification of discrete heterogeneities inclusions in an otherwise homogeneous reference solid from limited aperture waveform measurements taken on its surface on adopting the boundary integral equation bie framework for elastodynamic scattering the inverse query is cast as a minimization problem involving experimental observations and their simulations for a trial inclusion that is defined through its boundary elastic moduli and mass density for an optimal performance of the gradient based search methods suited to solve the problem explicit expressions for the shape i e boundary and material sensitivities of the misfit functional are obtained via the adjoint field approach and direct differentiation of the governing bies making use of the message passing interface the proposed sensitivity formulas are implemented in a data parallel code and integrated into a nonlinear optimization framework based on the direct bie method and an augmented lagrangian whose inequality constraints are employed to avoid solving forward scattering problems for physically inadmissible or overly distorted trial inclusion configurations numerical results for the reconstruction of an ellipsoidal defect in a semi infinite solid show the effectiveness of the proposed shape material sensitivity formulation which constitutes an essential computational component of the defect identification algorithm

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

on time harmonic elastic wave analysis by the boundary element method for moderate to high frequencies abstract certain inherent difficulties in obtaining accurate boundary element solutions to problems of elastodynamics at moderate to high frequencies are examined some procedures for overcoming or compensating for these difficulties are described in addition results demonstrating the effectiveness of these procedures are presented finally the use of these procedures in solving transient problems via fourier transform inversion is illustrated

hypersingular boundary integral equations for radiation and scattering of elastic waves in three dimensions abstract a weakly singular form of the hypersingular boundary integral equation bie traction equation for 3 d elastic wave problems is developed in this paper all integrals involved are at most weakly singular and except for a stronger smoothness requirement on boundary elements regular quadrature and collocation procedures used for conventional bies are sufficient for the discretization of the original hypersingular bie this weakly singular form of the hypersingular bie is applied to the composite bie formulation which uses a linear combination of the conventional bie and the hypersingular bie to remove the fictitious eigenfrequencies existing in the conventional bie formulation for elastic wave problems numerical examples employing different types of boundary elements clearly demonstrate the effectiveness and efficiency of the developed formulation

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

a boundary integral equation method for radiation and scattering of elastic waves in three dimensions etude du rayonnement et de la diffusion d ondes elastiques par des obstacles de forme arbitraire

a mixed form fast multipole algorithm the fast multipole algorithm manifests in two very different forms at low frequencies and at mid frequencies each can operate in their respective regimes but are not tenable in the other regimes the paper reports on a way to factorize the green s function for fast algorithm using a mixed form the low frequency fast multipole algorithm lf fma will be used at low frequencies or the long wavelength regime and the multilevel fast multipole algorithm mlfma will be used for the mid frequencies or the shorter wavelength regime for object modeling where both long wavelength and wave physics are important we propose a mixed form fast multipole algorithm mf fma this algorithm has no low frequency break down and it can work seamlessly from static where circuit physics is important to dynamic where wave physics is important

coupled finite element hierarchical boundary element methods for dynamic soil structure interaction in the frequency domain summary this paper discusses the coupling of finite element and fast boundary element methods for the solution of dynamic soil structure interaction problems in the frequency domain the application of hierarchical matrices in the boundary element formulation allows considering much larger problems compared to classical methods three coupling methodologies are presented and their computational performance is assessed through numerical examples it is demonstrated that the use of hierarchical matrices renders a direct coupling approach the least efficient as it requires the assembly of a dynamic soil stiffness matrix iterative solution procedures are presented as well and it is shown that the application of such schemes to dynamic soil structure interaction problems in the frequency domain is not trivial as convergence can hardly be achieved if no relaxation procedure is incorporated aitken s 2 method is therefore employed in sequential iterative schemes for the calculation of an optimized interface relaxation parameter while a novel relaxation technique is proposed for parallel iterative algorithms it is demonstrated that the efficiency of these algorithms strongly depends on the boundary conditions applied to each subdomain the fastest convergence is observed if neumann boundary conditions are imposed on the stiffest subdomain the use of a dedicated solver for each subdomain hence results in a reduced computational effort a monolithic coupling strategy often used for the solution of fluid structure interaction problems is also introduced the governing equations are simultaneously solved in this approach while the assembly of a dynamic soil stiffness matrix is avoided copyright 2013 john wiley sons ltd

a new fast multipole formulation for the elastodynamic half space green s tensor in this article a version of the frequency domain elastodynamic fast multipole boundary element method fm bem for semi infinite media based on the half space green s tensor and hence avoiding any discretization of the planar traction free surface is presented the half space green s tensor is often used in non multipole form until now for computing elastic wave propagation in the context of soil structure interaction with applications to seismology or civil engineering however unlike the full space green s tensor the elastodynamic half space green s tensor cannot be expressed using derivatives of the helmholtz fundamental solution as a result multipole expansions of that tensor cannot be obtained directly from known expansions and are instead derived here by means of a partial fourier transform with respect to the spatial coordinates parallel to the free surface the obtained formulation critically requires an efficient quadrature for the fourier integral whose integrand is both singular and oscillatory under these conditions classical gaussian quadratures would perform poorly fail or require a large number of points instead a version custom tailored for the present needs of a methodology proposed by rokhlin and coauthors which generates generalized gaussian quadrature rules for specific types of integrals has been implemented the accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single layer elastodynamic potentials involving up to about n 6 i 10 5 degrees of freedom in particular a complexity significantly lower than that of the non multipole version is shown to be achieved the 3d elastodynamic half space green s tensor is expressed as a 2d fourier integral it possesses the needed separated variable form allowing multipole like acceleration the 2d fourier integral is computed by a dedicated generalized gaussian quadrature efficiency and accuracy are assessed on comprehensive numerical tests a complexity significantly lower than using non multipole version is achieved

boundary integral equation methods for the two dimensional fluid solid interaction problem this paper is concerned with boundary integral equation methods for solving the two dimensional fluid solid interaction problem we reduce the problem to three different systems of boundary integral equations via direct and indirect approaches existence and uniqueness results for variational solutions of boundary integral equations are established since in all these variational formulations the hypersingular boundary integral operator associated with the time harmonic navier equation is a dominating integral operator we also include a new regularization formulation for this hypersingular operator which allows us to treat the hypersingular kernel by a weakly singular kernel numerical examples are presented to verify and validate the theoretical results

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

multilevel fast multipole algorithm for elastic wave scattering by large three dimensional objects multilevel fast multipole algorithm mlfma is developed for solving elastic wave scattering by large three dimensional 3d objects since the governing set of boundary integral equations bie for the problem includes both compressional and shear waves with different wave numbers in one medium the double tree structure for each medium is used in the mlfma implementation when both the object and surrounding media are elastic four wave numbers in total and thus four fma trees are involved we employ nystrom method to discretize the bie and generate the corresponding matrix equation the mlfma is used to accelerate the solution process by reducing the complexity of matrix vector product from o n 2 to o nlogn in iterative solvers the multiple tree structure differs from the single tree frame in electromagnetics em and acoustics and greatly complicates the mlfma implementation due to the different definitions for well separated groups in different fma trees our nystrom method has made use of the cancellation of leading terms in the series expansion of integral kernels to handle hyper singularities in near terms this feature is kept in the mlfma by seeking the common near patches in different fma trees and treating the involved near terms synergistically due to the high cost of the multiple tree structure our numerical examples show that we can only solve the elastic wave scattering problems with 0 3 0 4 millions of unknowns on our dell precision 690 workstation using one core

dynamics of structures coupled with elastic media a review of numerical models and methods abstract this paper reviews issues and developments in the field of structure environment interaction problems in which the environment is an elastic body possibly unbounded it covers in particular the fields of soil structure interaction ground borne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium the general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled and the domain decomposition technique classically used for its description is put up extensions for the cases of nonlinear structure and uncertain environment are then discussed finally the ongoing research in the fields of unbounded interface moving interface and multiple interfaces are reviewed and summarized