a galerkin boundary element method for high frequency scattering by convex polygons in this paper we consider the problem of time harmonic acoustic scattering in two dimensions by convex polygons standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave here we present a novel galerkin boundary element method which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon we prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency numerical results demonstrate the same logarithmic dependence on the frequency for the galerkin method solution our boundary element method is a discretization of a well known second kind combined layer potential integral equation we provide a proof that this equation and its adjoint are well posed and equivalent to the boundary value problem in a sobolev space setting for general lipschitz domains

an effective method for solving the hyper singular integral equations in 3 d acoustics the application of the boundary integral methods to the problem of acoustics exterior to a three dimensional surface suffers in general from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem the formulation that is most suitable for numerical implementation still appears to be that proposed by burton and miller proc r soc london ser a 323 201 210 1971 however the hypersingular kernels present in such a formulation render it computationally unattractive previous attempts to regularize such hypersingular kernels involved the use of double surface integrals or implicit use of tangential operators or closed form evaluations of hypersingular integrals the method of double surface integrals is computationally highly inefficient even though it allows higher order interpolation schemes on the surface the other two approaches are more conducive to the assumption of a constant value for each of the variables pressure a

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

a comparison of approximate boundary conditions and infinite element methods for exterior helmholtz problems we compare the accuracy of a variety of approximate boundary conditions abcs and two versions of an infinite element method iem for approximating solutions of exterior helmholtz problems the infinite elements are of variable order in the radial direction and differ only in whether the test functions in the weak formulation are conjugated or not we find that the accuracy of the abcs decreases with increasing frequency and that at higher frequencies their accuracy is several orders of magnitude lower than that obtained with the iems we show that the convergence rate of the conjugated element is slower than the rate for the unconjugated element and that the latter element fails to converge in the far field

on the efficiency of nested gmres preconditioners for 3d acoustic and elastodynamic h matrix accelerated boundary element methods abstract this article is concerned with the derivation of fast boundary element methods for 3d acoustic and elastodynamic problems in particular we are interested in the acceleration of hierarchical matrix h matrix based iterative solvers while h matrix representations allow to reduce the storage requirements and the cost of a matrix vector product the number of iterations for an iterative solver as the frequency or the problem size increases remains an issue we consider an inner outer preconditioning strategy i e the preconditioner is applied through an iterative solver at the inner level the preconditioner is defined as a h matrix representation of the system matrix with a given accuracy we investigate the influence of various parameters of the preconditioner i e the h matrix accuracy the gmres threshold and the maximum number of iterations of the inner solver different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system finally we propose a way to define the optimal setting for this preconditioner

spectral controllability for 2d and 3d linear schrodinger equations we consider a quantum particle in an infinite square potential well of rnrn n 2 3n 2 3 subjected to a control which is a uniform in space electric field under the dipolar moment approximation the wave function solves a pde of schrodinger type we study the spectral controllability in finite time of the linearized system around the ground state we characterize one necessary condition for spectral controllability in finite time kal if is the bottom of the well then for every eigenvalue of view the mathml source d the projections of the dipolar moment onto every normalized eigenvector associated to are linearly independent in rnrn in 3d our main result states that spectral controllability in finite time never holds for one directional dipolar moment the proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions in 2d we first prove the existence of a minimal time tmin 0tmin 0 for spectral controllability i e if t tmin t tmin one has spectral controllability in time t if condition kal holds true for and if ttmin holds generically with respect to the domain the proof relies on shape differentiation and a careful study of dirichlet to neumann operators associated to certain helmholtz equations we also show that one can recover exact controllability in abstract spaces from this 2d spectral controllability by adapting a classical variational argument from control theory

the use of chief to obtain unique solutions for acoustic radiation using boundary integral equations this report is concerned with the problem of obtaining a unique solution for radiation problems at characteristic wavenumbers when a boundary integral equation formulation is used it is shown that the combined helmholtz integral equation formulation chief method works even when numerous interior points lie exactly on nodal surfaces of a mode corresponding to an eigenfrequency of the related interior problem when at least one interior point does not lie on a nodal surface the use of chief is illustrated with numerical radiation experiments for the sphere and the finite cylinder chief is compared to the helmholtz gradient formulation hgf for circumventing nonuniqueness and is found to yield a more accurate solution in at least one published example a procedure is used to indicate the solution error when using integral equation methods with or without a technique to circumvent nonuniqueness this procedure uses the velocity potential of an interior point as an indicator of solution error in all cases considered the interior potential correctly indicated a good or bad solution whereas the matrix condition number falsely indicated a bad solution in several instances work supported in part by onr

a numerical method for scattering from acoustically soft and hard thin bodies in two dimensions abstract this paper presents a numerical method for predicting the acoustic scattering from two dimensional 2 d thin bodies both the dirichlet and neumann problems are considered applying the thin body formulation leads to the boundary integral equations involving weakly singular and hypersingular kernels completely regularizing these kinds of singular kernels is thus the main concern of this paper the basic subtraction addition technique is adopted the purpose of incorporating a parametric representation of the boundary surface with the integral equations is two fold the first is to facilitate the numerical implementation for arbitrarily shaped bodies the second one is to facilitate the expansion of the unknown function into a series of chebyshev polynomials some of the resultant integrals are evaluated by using the gauss chebyshev integration rules after moving the series coefficients to the outside of the integral sign others are evaluated exactly including the modified hypersingular integral the numerical implementation basically includes only two parts one for evaluating the ordinary integrals and the other for solving a system of algebraic equations thus the current method is highly efficient and accurate because these two solution procedures are easy and straightforward numerical calculations consist of the acoustic scattering by flat and curved plates comparisons with analytical solutions for flat plates are made

boundary spectral method for acoustic scattering and radiation problems the spectral method is considered in solving boundary integral equations by subtracting known solutions from the helmholtz integral equation the singularities of the helmholtz integral equation are removed therefore any order of basis functions used in spectral methods that times nonsingular kernels in the modified helmholtz integral equation can be integrated easily and efficiently instead of solving the variables at collocation points using conventional methods the generalized fourier coefficients are solved by the spectral method which reduces the number of unknowns and the storage of matrix elements in computation scattering and radiation problems from a sphere are used to illustrate the technique in the present paper

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

a transformation approach for efficient evaluation of oscillatory surface integrals arising in three dimensional boundary element methods we propose a method for efficient evaluation of surface integrals arising in boundary element methods for three dimensional helmholtz problems with real positive wavenumber k modelling wave scattering and or radiation in homogeneous media to reduce the number of degrees of freedom required when k is large a common approach is to include in the approximation space oscillatory basis functions with support extending across many wavelengths a difficulty with this approach is that it leads to highly oscillatory surface integrals whose evaluation by standard quadrature would require at least o k2 quadrature points here we use equivalent contour integrals developed for aperture scattering in optics to reduce this requirement to o k and possible extensions to reduce it further to o 1 are identified the contour integral is derived for arbitrary shaped elements but its application is limited to planar elements in many cases in addition the transform regularises the singularity in the surface integrand caused by the green s function including for the hyper singular case under appropriate conditions an open source matlab code library is available to demonstrate our routines

efficient hypersingular line and surface integrals direct evaluation by complex variable differentiation method abstract we present an efficient numerical scheme to evaluate hypersingular integrals appeared in boundary element methods the hypersingular integrals are first separated into regular and singular parts in which the singular integrals are defined as limits around the singularity and their values are determined analytically by taking the finite part values the remaining regular integrals can be evaluated using rational interpolatory quadrature or complex variable differentiation cvdm for the regular function when machine precision like accuracy is required the proposed method is then generalised for evaluating hypersingular surface integrals in which the inner integral is treated as the hypersingular line integral via coordinate transformations the procedure is implemented into 8 node rectangular boundary element and 6 node triangular element for numerical evaluation finally several numerical examples are presented to demonstrate the efficiency of the present method to the best of our knowledge the proposed method is more accurate faster and more generalised than other methods available in the literature to evaluate hypersingular integrals

a boundary integral approach for acoustic radiation of axisymmetric bodies with arbitrary boundary conditions valid for all wave numbers a boundary integral method for solving the exterior acoustic radiation problem of axisymmetric bodies with arbitrary boundary conditions is developed the new formulation derives from the method proposed by burton and miller proc r soc london ser a 323 201 210 1971 which uses a linear combination of the helmholtz integral equation and its normal derivative to ensure the uniqueness of the numerical solution at all frequencies it is shown that the use of tangential operators enables one to easily compute the hypersingular integral without much extra effort by taking advantage of the properties of axisymmetric geometry and using the expansion of the boundary conditions and the surface distribution functions in fourier series with respect to the angle of revolution the surface integral is reduced to a line integral along the generator of the body and fourier integrals of the green s function and its derivatives over the circumferential angle the fourier integrals with singular kernels are eval

a weighted residual formulation for the chief method in acoustics in this article the combined helmholtz integral equation formulation chief proposed by schenck is greatly improved so that it will alleviate the annoying nodal surface problem the constraint provided by each chief equation and its first order derivatives is enforced in a weighted residual sense over a small interior region rather than only at a single point consequently chief blocks rather than chief points are used in the numerical implementation although the corresponding interior problem may have nodal surfaces at characteristic frequencies it has no such things as nodal blocks therefore the chief block chief b method is expected to work even if all of the chief blocks fall on the intersection of several nodal surfaces this new formulation is implemented in an isoparametric element environment numerical examples in spherical radiation and scattering at intermediate frequencies are given to test the new formulation

direct evaluation of hypersingular galerkin surface integrals a direct algorithm for evaluating hypersingular integrals arising in a three dimensional galerkin boundary integral analysis is presented the singular integrals are defined as limits to the boundary and by integrating two of the four dimensions analytically the coincident integral is shown to be divergent however the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals a single analytic integration is employed for the edge and vertex singular integrals this is sufficient to display the divergent term in the edge adjacent integral and to show that the vertex integral is finite by explicitly identifying the divergent quantities we can compute the hypersingular integral without recourse to stokes s theorem or the hadamard finite part the algorithms are developed in the context of a linear element approximation for the laplace equation but are expected to be generally applicable as an example the algorithms are applied to solve a thermal problem in an exponentially graded material

hypersingular integrals how smooth must the density be hypersingular integrals are guaranteed to exist at a point x only if the density function f in the integrand satises certain conditions in a neighbourhood of x it is well known that a su cient condition is that f has a holder continuous rst derivative this is a stringent condition especially when it is incorporated into boundary element methods for solving hypersingular in tegral equations this paper is concerned with nding weaker conditions for the existence of one dimensional hadamard nite part integrals it is shown that it is sucien t for the even part of f with respect to x to have a holder continuous rst derivative the odd part is al lowed to be discontinuous a similar condition is obtained for cauchy principal value integrals these simple results have non trivial consequences they are applied to the calculation of the tangential derivative of a single layer potential and to the normal derivative of a double layer potential particular attention is paid to discontinuous densities and to discontinuous boundary conditions also despite the weaker sucien t conditions it is rearmed that for hypersingular integral equations collocation at a point x at the junction between two standard conforming boundary elements is not permissible theoretically various modications to the denition of nite part integral are explored

on efficient preconditioners for iterative solution of a galerkin boundary element equation for the three dimensional exterior helmholtz problem the paper presents a galerkin numerical method for solving the hyper singular boundary integral equations for the exterior helmholtz problem in three dimensions with a neumann s boundary condition previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and galerkin methods are not available due to the presence of a hypersingular integral operator this paper proposes a high order galerkin method by using singularity subtraction technique to reduce the hyper singular operator to a weakly singular one moreover we show here how to extend the previous work j appl numer math 36 4 2001 475 489 on sparse preconditioners to the galerkin case leading to fast convergence of two iterative solvers the conjugate gradient normal method and the generalised minimal residual method a comparison with the collocation method is also presented for the helmholtz problem with several wavenumbers

iterative solution of high order boundary element method for acoustic impedance boundary value problems abstract a high order boundary element method for time harmonic acoustic impedance boundary value problems is presented the method is based on the galerkin type formulation of the burton miller integral equation bmie with high order polynomial basis and testing functions this formulation has several important advantages it is free of the interior resonance problem the hypersingular integral operator of the traditional bmie formulation can be avoided it leads to faster convergence in terms of the number of unknowns than the low order methods and it shows a good performance with iterative solvers to avoid the numerical difficulties associated to the implementation of the singular surface integral equations the singularity extraction technique is applied to evaluate the integrals in the singular and near singular cases the resulting matrix equation is solved iteratively with the generalized minimal residual method gmres and a simple preconditioner based on the incomplete lu factorization is applied to expedite the convergence numerical results indicate that the bmie formulation with galerkin method and high order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the gmres method is applied to solve the matrix equation iteratively this in turn indicates that the formulation is well suited for an efficient application of fast solution procedures such as the fast multipole method

application of the inverse fast multipole method as a preconditioner in a 3d helmholtz boundary element method abstract we investigate an efficient preconditioning of iterative methods such as gmres for solving dense linear systems a x b that follow from a boundary element method bem for the 3d helmholtz equation focusing on the low frequency regime while matrix vector products in gmres can be accelerated through the low frequency fast multipole method lffmm the bem often remains computationally expensive due to the large number of gmres iterations we propose the application of the inverse fast multipole method ifmm as a preconditioner to accelerate the convergence of gmres the ifmm is in essence an approximate direct solver that uses a multilevel hierarchical decomposition and low rank approximations the proposed ifmm based preconditioning has a tunable parameter e that balances the cost to construct a preconditioner m which is an approximation of a 1 and the cost to perform the iterative process by means of m namely using a small respectively large value of e takes a long respectively short time to construct m while the number of iterations can be small respectively large a comprehensive set of numerical examples involving various boundary value problems with complicated geometries and mixed boundary conditions is presented to validate the efficiency of the proposed method we show that the ifmm preconditioner with a nearly optimal e of 10 2 clearly outperforms some common preconditioners for the bem achieving 1 2 10 8 times speed up of the computations in particular when the scale of the underlying scatterer is about five wavelengths or more in addition the ifmm preconditioner is capable of solving complicated problems in a reasonable amount of time that bd preconditioner can not

a comparison of techniques for overcoming non uniqueness of boundary integral equations for the collocation partition of unity method in two dimensional acoustic scattering the partition of unity method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements the non uniqueness of solution obtained from the conventional boundary integral equation cbie is well known the cbie derived through green s identities suffers from a problem of non uniqueness at certain characteristic frequencies two of the standard methods of overcoming this problem are the so called combined helmholtz integral equation formulation chief method and that of burton and miller the latter method introduces a hypersingular integral which may be treated in various ways in this paper we present the collocation partition of unity boundary element method pubem for the helmholtz problem and compare the performance of chief against a burton miller formulation regularised using the approach of li and huang

a unified boundary element method for the analysis of sound and shell like structure interactions i formulation and verification a unified boundary element method bem is developed in this paper to model both the exterior acoustic field and the elastic shell like structure in a coupled analysis the conventional boundary integral equation bie for three dimensional 3d elastodynamics is applied to thin shell like structures which can have arbitrary shapes and small thicknesses the nearly singular integrals existing in the bie when applied to thin bodies are transformed to nonsingular line integrals and are evaluated accurately and efficiently for the exterior 3d acoustic domain the burton and miller composite bie formulation is employed to overcome the fictitious eigenfrequency difficulty fed and the thin shape breakdown tsb conforming c0 quadratic elements are employed in the discretization of the two sets of bies the developed bie formulations are valid for both radiation and scattering problems and for all wave numbers numerical examples using spherical and cylindrical shells including nonuniform thickness and nond

more on the complexity of slice functions abstract one of the hardest problems in complexity theory seems to be to find a proof that the circuit complexity of some explicitly defined boolean function is nonlinear in the number of inputs and outputs an interesting approach for such a bound is the investigation of slice functions since for these functions circuits over a complete basis monotone circuits and the so called set circuits are nearly equivalent in this paper the theory of slice functions is extended a monotone representation of each boolean function whose monotone complexity is at most a factor n larger than its circuit complexity is presented for some slice functions and some other functions new pseudocomplements with small monotone complexity are presented these results lead to new and intuitive problems whose solution would lead to nonlinear lower bounds on circuit complexity some connections between the complexity of n output and 1 output slices and between the complexity of the k slice and the k 1 slice of the same function are discussed

the complexity of central slice functions let f be a monotone boolean function over x x1 xn the k slice of f is the function fk f tkn vtk 1n where tkn is the kth threshold function berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of kk imply lower bounds of the same order on the combinational complexity of f and that if f has large combinational complexity then some slice of f must have large monotone complexity however this latter result does not specify any particular slice and it is known that some nontrivial slice functions of np complete predicates have linear complexity in this paper we consider the 12n slice functions and show that for three basic monotone boolean np complete functions this slice is also np complete in addition the 12n slice of some boolean matrix functions f is studied and is proved to be no easier to compute than f

solution of 3d laplace and helmholtz equations in exterior domains using hp infinite elements abstract this work is devoted to a convergence study for infinite element discretizations for laplace and helmholtz equations in exterior domains the proposed approximation applies to separable geometries only combining an hp fe discretization on the boundary of the domain with a spectral like representation resulting from the separation of variables in the radial direction the presentation includes a convergence proof for the laplace equation and a stability analysis for the variational formulation of the helmholtz equation in weighted sobolev spaces the theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain

adaptively preconditioned gmres algorithms the restarted gmres algorithm proposed by saad and schultz siam j sci statist comput 7 1986 pp 856 869 is one of the most popular iterative methods for the solution of large linear systems of equations ax b with a nonsymmetric and sparse matrix this algorithm is particularly attractive when a good preconditioner is available the present paper describes two new methods for determining preconditioners from spectral information gathered by the arnoldi process during iterations by the restarted gmres algorithm these methods seek to determine an invariant subspace of the matrix a associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved

a coupled fe and boundary integral equation method based on exterior domain decomposition for fluid structure interface problems abstract a coupled finite element and exterior domain decomposition based boundary integral formulations for the solutions of two or three dimensional time harmonic fluid structure interaction problems is described in this paper it is known that the memory limitation of computers has been one of the major obstacles for solving large scale high frequency fluid structure interface systems using various existing nonlocal finite element and boundary integral equation coupling techniques due to the fully populated resultant matrix generated from the boundary integral equation representation the essence of this study is to decompose through domain decomposition of the exterior region the original exterior problem into arbitrary subproblems with data sharing only at the interfaces by decomposing the exterior infinite domain into an appropriate number of infinite subdomains this method not only ensures the validity of the formulation for all frequencies but also leads to a diagonalized blockwise banded system of discretized equations the size of an individual submatrix i e a block that is associated with an exterior subdomain may be decided by the user and may be selected such that the restriction due to the memory limitation of a given computer may be accommodated in addition the method is suited for parallel processing since the data associated with each subdomain impedance matrices load vectors etc may be generated in parallel and the communication needed will be only for the interface values most significantly unlike the existing coupled finite element and boundary integral equation techniques that are valid for all frequencies our method avoids the use of both the hypersingular operator and the double integrals therefore reducing the computational complexity numerical experiments have been performed for elastic cylindrical shells subjected to a plane incident wave the results have demonstrated the accuracy of the method for wavenumbers ranging from 0 to 30 both directly on the shell and in the far field and have confirmed that the procedure is valid for all frequencies

acoustic scattering by a modified werner method a modified integral werner method is used to calculate pressure scattered by an axisymmetric body immersed in a perfect and compressible fluid subject to a harmonic acoustic field this integral representation is built as the sum of a potential of a simple layer and a potential of volume it is equivalent to the exterior helmholtz problem with neumann boundary condition for all real wave numbers of the incident acoustic field for elastic structure scattering problems the modified werner method is coupled with an elastodynamic integral formulation in order to account for the elastic contribution of the displacement field at the fluid structure interface the resulting system of integral equations is solved by the collocation method with a quadratic interpolation the introduction of a weighting factor in the modified werner method decreases the number of volume elements necessary for a good convergence of results this approach becomes very competitive when it is compared with other integral methods that are valid for all wave numbers a numerical comparison with an experiment on a tungsten carbide end capped cylinder allows a glimpse of the interesting possibilities for using the coupling of the modified werner method and the integral elastodynamic equation used in this research

evaluation of 2 d green s boundary formula and its normal derivative using legendre polynomials with an application to acoustic scattering problems this paper presents the non singular forms in a global sense of two dimensional green s boundary formula and its normal derivative the main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain that is the proposed element free method requires only nodal data the approach includes expressing the unknown function as a truncated fourier legendre series together with transforming the integration interval a b to 1 1 the series coefficients are thus to be determined the hypersingular integral interpreted in the hadamard finite part sense and some weakly singular integrals can be evaluated analytically the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point the effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions the regularization is further applied to acoustic scattering problems the well known burton miller method using a linear combination of the surface helmholtz integral equation and its normal derivative is adopted to overcome the non uniqueness problem a general non singular form of the composite equation is derived comparisons with analytical solutions for acoustically soft and hard circular cylinders are made copyright 2001 john wiley sons ltd

optimizing the evaluation of finite element matrices assembling stiffness matrices represents a significant cost in many finite element computations we address the question of optimizing the evaluation of these matrices by finding redundant computations we are able to significantly reduce the cost of building local stiffness matrices for the laplace operator and for the trilinear form for navier stokes operators for the laplace operator in two space dimensions we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices up to cubics we are able to build the stiffness matrix on any triangle in less than one multiply add pair per entry up to sixth degree we can do it in less than about two pairs preliminary low degree results for poisson and navier stokes operators in three dimensions are also promising

on monotone simulations of nonmonotone networks we consider the following problem given some n argument monotone boolean function f xn with formal arguments xn x1 xn compute f using the 2n 1 inputs xn f0 f1 fn here fk is the k slice of f i e the n argument monotone boolean function fk xn f tnk tnk 1 where tnk is the n argument monotone boolean function which takes the value 1 iff at least k of its arguments are 1 it is easy to see that if nonmonotone operations are permitted then o n gates are sufficient by using the relation f v nk 0 fk tnk 1 the properties of slice functions imply th efficient monotone solutions would allow superlinear lower bounds on the combinational complexity of f to be obtained from large enough lower bounds on the monotone complexity of f since negation is known to be superpolynomially powerful some monotone functions must have superpolynomial complexity even if all the slice functions are given as extra inputs however it is possible that efficient simulations using slice functions exist for restricted classes of monotone functions in this paper we examine a broad class of monotone boolean functions proving that for almost all of the functions in the class no such simulation exists and that in a very weak sense negation is exponentially powerful in contrast to this an example of an efficient construction is given again for a natural class of monotone boolean functions

topological optimization of the evaluation of finite element matrices we present a topological framework for finding low flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight sided affine domains this framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element specific geometric tensor we then present a new concept of complexity reducing relations that serve as distance relations between these reference element tensors this notion sets up a graph theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree we present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization

stability assessment of a unified variational boundary integral method applicable to thin scatterers and scatterers with corners abstract the objective of this study is to assess the stability especially at and near critical frequencies of a new unified stable symmetric domain finite element and boundary integral methodology for solving time harmonic interface problems for scatterers of arbitrary shape the validity of this energy based variational procedure for rigid and penetrable smooth scatterers including the existence uniqueness and optimal convergence of the corresponding numerical approximations has been proved rigorously in the context of fluid structure interaction problems the emphasis here is on investigating the applicability of this procedure to the case of scatterers with corners using square and thin rectangular two dimensional rigid obstacles as prototypes we also establish that the apparently distinct direct boundary integral formulation due to burton and miller and the combined single and double layer indirect formulation due to leis brakhage and werner and panich introduced to insure that boundary integral equations for scattering problems are uniquely solvable for all wavenumbers are entirely equivalent within our variational setting by examining the condition number of the matrix of coefficients of the discretized equations as well as the resulting solutions both directly on the surface of the scatterers and in the far field it is demonstrated that the new methodology is robust and completely insensitive to critical frequencies even for thin objects most significantly the precise singular behaviour of the pressure field at corners is also predicted quite accurately by using standard finite elements along the boundary of the scatterer without having to introduce special singularity functions which for general interface problems may not be known in advance

unified embedded parallel finite element computations via software based frechet differentiation computational analysis of systems governed by partial differential equations pdes requires not only the calculation of a solution but the extraction of additional information such as the sensitivity of that solution with respect to input parameters or the inversion of the system in an optimization or design loop moving beyond the automation of discretization of pdes by finite element methods we present a mathematical framework that unifies the discretization of pdes with these additional analysis requirements in particular frechet differentiation on a class of functionals together with a high performance finite element framework has led to a code called sundance that provides high level programming abstractions for the automatic efficient evaluation of finite variational forms together with the derived operators required by engineering analysis

tangential derivative of singular boundary integrals with respect to the position of collocation points this paper investigates the evaluation of the sensitivity with respect to tangential perturbations of the singular point of boundary integrals having either weak or strong singularity both scalar potential and elastic problems are considered a proper definition of the derivative of a strongly singular integral with respect to singular point perturbations should accommodate the concomitant perturbation of the vanishing exclusion neighbourhood involved in the limiting process used in the definition of the integral itself this is done here by esorting to a shape sensitivity approach considering a particular class of infinitesimal domain perturbations that move individual points and especially the singular point but leave the initial domain globally unchanged this somewhat indirect strategy provides a proper mathematical setting for the analysis moreover the resulting sensitivity expressions apply to arbitrary potential type integrals with densities only subjected to some regularity requirements at the singular point and thus are applicable to approximate as well as exact bem solutions quite remarkable is the fact that the analysis is applicable when the singular point is located on an edge and simply continuous elements are used the hypersingular bie residual function is found to be equal to the derivative of the strongly singular bie residual when the same values of the boundary variables are substituted in both sbie and hbie formulations with interesting consequences for some error indicator computation strategies 1998 john wiley sons ltd

a novel technique in boundary integral equations for analyzing acoustic radiation from axisymmetric bodies a novel technique is suggested to avoid singular integrals of boundary integral equations bie for exterior sound radiation problems the technique reproduction of diagonal terms rdt utilizes special solutions of some monopole sources which give the distribution of theoretical surface pressure and velocity to reproduce diagonal terms in the bie matrices it is proposed to employ three monopole sources in reproduction of diagonal terms in both pressure and velocity matrices although the interior points are applied to generate monopole solutions the concept is basically different from that of general interior helmholtz equations so the present technique is different from those traditional methods to improve bie method through added interior helmholtz equations several examples are presented to demonstrate the effectiveness of the proposed technique for closed axisymmetric surfaces numerical results show that the technique of rdt can improve the accuracy of the solution up to 10 times compared to the conventional techniques such as shf chief shf

efficient assembly of h mathrm div and h mathrm curl conforming finite elements in this paper we discuss how to efficiently evaluate and assemble general finite element variational forms on h mathrm div and h mathrm curl the proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh dependent geometry tensor two key points must then be considered the appropriate mapping of basis functions from a reference element and the orientation of geometrical entities to address these issues we extend here a previously presented representation theorem for affinely mapped elements to piola mapped elements we also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents the result is an automated efficient and easy to use implementation that allows a user to specify finite element variational forms on h mathrm div and h mathrm curl in close to mathematical notation

a simple method avoiding non uniqueness in the boundary element method for acoustic scattering problem the boundary element method bem is widely used for sound field analysis problems however it has a non uniqueness problem in the exterior domain various methods to avoid this problem have been developed however these are not easily applied to the bem in this paper a simple method called the ica ring inner cavity ringing method is proposed for avoiding the non uniqueness problem and this method is applied to the bem in both single and plural domains the concept of the ica ring method is that a scatterer in free space is hollowed as a shell and the volume is smaller the eigenfrequencies are shifted to a higher range next the mechanism of the non uniqueness problem in plural domains and a reason of the application of the ica ring method to the case of plural domains are explained finally some results calculated by the bem using the ica ring method are shown the calculational condition is that a cylinder with radius 0 125m floats in two dimensional free space in this case no calculationa

solving the hypersingular boundary integral equation in three dimensional acoustics using a regularization relationship regularization of the hypersingular integral in the normal derivative of the conventional helmholtz integral equation through a double surface integral method or regularization relationship has been studied by introducing the new concept of discretized operator matrix evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices such a treatment greatly improves the computational efficiency as the number of frequencies to be computed increases the computational cost of solving the composite helmholtz integral equation is comparable to that of solving the conventional helmholtz integral equation in this paper the detailed formulation of the proposed regularization method is presented the computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems the radiation of a pulsating sphere an oscillating sphere and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements it is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied

acoustic scattering by a hard or soft body across a wide frequency range by the helmholtz integral equation method in this paper the acoustic scattering by an obstacle across a wide frequency range of sound waves is investigated on the basis of the helmholtz integral formulation to overcome the nonuniqueness difficulties the methods proposed by burton npl report nac 30 jan 1973 and by burton and miller proc r soc london ser a 323 201 210 1971 are adopted for the dirichlet and neumann problems respectively the aim of this paper is twofold the first is to bring together completely regular formulations of the helmholtz integral equation and its normal derivative the second is to extend these formulations to treat the higher frequency problems the weakly singular integrals are regularized by subtracting out one term and adding it back depending on the problem concerned the additional integral can finally be expressed in an explicit form or results in solving a surface source distribution of the equipotential body the hypersingular kernels are regularized by the method of using some properties of the associated laplace equation originally proposed by chien et al j acoust soc am 88 918 937 1990 the completely regularized integral equations are amenable to computation by direct use of the standard quadrature methods to study the acoustic scattering due to higher frequency waves filon s quadrature method proc r soc edinburgh 49 38 47 1928 is extended to treat the rapidly oscillatory integrands numerical examples consist of acoustic scattering from a hard or soft sphere of radius a across a wide spectrum of wave numbers ka 20 comparisons of the numerical results with the exact solutions demonstrate the validity and efficiency of the implementation

a boundary integral equation method for two dimensional acoustic scattering problems this work formulates the singularity free integral equations to study 2 d acoustic scattering problems to avert the nonuniqueness difficulties burton s and burton and miller s methods are employed to solve the dirichlet and neumann problems respectively the surface helmholtz integral equations and their normal derivative equations in bounded form are derived the weakly singular integrals are desingularized by subtracting a term from the integrand and adding it back with an exact value depending on the relevant problem the additional integral can finally be either expressed in an explicit form or transformed to form a surface source distribution of the related equipotential body the hypersingular kernel is desingularized further using some properties of an interior laplace problem the new formulations are advantageous in that they can be computed by directly using standard quadrature formulas also discussed is the contour a unique feature of 2 d problems instead of dividing the boundary surface into several small elements a parametric representation of a 2 d boundary curve is further proposed to facilitate a global numerical implementation calculations consist of acoustic scattering by a hard and a soft circular cylinder respectively comparing the numerical results with the exact solutions demonstrates the proposed method s effectiveness

geometric optimization of the evaluation of finite element matrices this paper continues earlier work on mathematical techniques for generating optimized algorithms for computing finite element stiffness matrices these techniques start from representing the stiffness matrix for an affine element as a collection of contractions between reference tensors and an element dependent geometry tensor we go beyond the complexity reducing binary relations explored in r c kirby a logg l r scott and a r terrel siam j sci comput 28 2006 pp 224 240 to consider geometric relationships between three or more objects algorithms based on these relationships often have even fewer operations than those based on complexity reducing relations

hybrid galerkin boundary elements theory and implementation summary in this paper we present a new quadrature method for computing galerkin stiffness matrices arising from the discretisation of 3d boundary integral equations using continuous piecewise linear boundary elements this rule takes as points some subset of the nodes of the mesh and can be used for computing non singular galerkin integrals corresponding to pairs of basis functions with non intersecting supports when this new rule is combined with standard methods for the singular galerkin integrals we obtain a hybrid galerkin method which has the same stability and asymptotic convergence properties as the true galerkin method but a complexity more akin to that of a collocation or nystrom method the method can be applied to a wide range of singular and weakly singular first and second kind equations including many for which the classical nystrom method is not even defined the results apply to equations on piecewise smooth lipschitz boundaries and to non quasiuniform but shape regular meshes a by product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane numerical experiments demonstrate that the new method realises the performance expected from the theory

solving the hypersingular boundary integral equation for the burton and miller formulation this paper presents an easy numerical implementation of the burton and miller bm formulation where the hypersingular helmholtz integral is regularized by identities from the associated laplace equation and thus needing only the evaluation of weakly singular integrals the helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth the hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the calderon identities linking the integral operators for associated laplace problems the method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three dimensional bodies unlike other approaches using direct evaluation of hypersingular integrals collocation points still coincide with mesh nodes as is usual when using conforming elements using higher order shape functions with the boundary element method model size kept fixed reduces the overall numerical integration effort while increasing the solution accuracy to reduce the condition number of the resulting bm formulation at low frequencies a regularized version ik k2 of the classical bm coupling factor i k is proposed comparisons with the combined helmholtz integral equation formulation method of schenck are made for four example configurations two of them featuring non smooth surfaces

combinatorial optimization of matrix vector multiplication in finite element assembly it has been shown that combinatorial optimization of matrix vector multiplication can lead to faster evaluation of finite element stiffness matrices based on a graph model characterizing relationships between rows an efficient set of operations can be generated to perform matrix vector multiplication for this problem we improve the graph model by extending the set of binary row relationships and solve this combinatorial optimization problem optimally for the binary row relationships implemented yielding significantly improved results over previously published graph models we also extend the representation by using hypergraphs to model more complicated row relationships expressing a three row relationship with a three vertex hyperedge for example our initial greedy algorithm for this hypergraph model has yielded significantly better results than the graph model for many matrices

on the singularities of green s formula and its normal derivative with an application to surface wave body interaction problems the paper presents the non singular forms of green s formula and its normal derivative of exterior problems for three dimensional laplace s equation the main advantage of these modified formulations is that they are amenable to solution by directly using quadrature formulas thus the conventional boundary element approximation which locally regularizes the singularities in each element is not required the weak singularities are treated by both the gauss flux theorem and the property of the associated equipotential body the hypersingularities are treated by further using the boundary formula for the associated interior problems the efficacy of the modified formulations is examined by a sphere in an infinite domain subject to neumann and dirichlet conditions respectively the modified integral formulations are further applied to a practical problem i e surface wave body interactions using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem removing the irregular frequencies is performed by linearly combining the standard integral equation with its normal derivative computations are presented of the added mass and damping coefficients and wave exciting forces on a floating hemisphere comparing the numerical results with that by other approaches demonstrates the effectiveness of the method copyright 2000 john wiley sons ltd

boolean functions whose monotone complexity is of size n2 log n we construct a sequence of monotone boolean functions hn 0 1 n 0 1 n such that the monotone complexity of hn is of order n2 log n this result includes the largest known lower bound of this kind previously there were an n3 2 bound for the boolean matrix product an n5 3 bound for boolean sums and an n2 log2n bound of the author for the same functions hn this new lower bound is proved by new methods which probably will turn out to be useful also for other problems

automatic generation of fast discrete signal transforms this paper presents an algorithm that derives fast versions for a broad class of discrete signal transforms symbolically the class includes but is not limited to the discrete fourier and the discrete trigonometric transforms this is achieved by finding fast sparse matrix factorizations for the matrix representations of these transforms unlike previous methods the algorithm is entirely automatic and uses the defining matrix as its sole input the sparse matrix factorization algorithm consists of two steps first the symmetry of the matrix is computed in the form of a pair of group representations second the representations are stepwise decomposed giving rise to a sparse factorization of the original transform matrix we have successfully demonstrated the method by computing automatically efficient transforms in several important cases for the dft we obtain the cooley tukey 1965 fft for a class of transforms including the dct type ii the number of arithmetic operations for our fast transforms is the same as for the best known algorithms our approach provides new insights and interpretations for the structure of these signal transforms and the question of why fast algorithms exist the sparse matrix factorization algorithm is implemented within the software package arep

a summary of infinite element formulations for exterior helmholtz problems this work is devoted to a study and summary of different infinite element ie formulations for helmholtz problems in arbitrary exterior domains the theoretical setting for each of the different formulations is presented and related to the mathematical existence theory the influence of a bilinear or a sesquilinear formulation is discussed as well as possible extensions to other elements the implementation of the infinite element method iem incorporates the use of 2d and 3d hp finite elements and allows for hp adaptive refinements numerical results show the computational efficiency of the coupled finite infinite element methodology

an integral equation approach to three dimensional acoustic radiation and scattering problems this paper provides numerical implementation and tests for a boundary integral equation method proposed recently by krutitskii ima j appl math 64 259 269 2000 for evaluating three dimensional acoustic radiation and scattering fields in the exterior neumann problem for several obstacles auxiliary boundaries with certain prescribed boundary conditions are introduced inside the obstacles for the purpose of eliminating the fictitious characteristic frequencies applying the simple source formulation on all boundaries leads to a set of uniquely solvable integral equations with weakly singular kernels to facilitate the numerical implementation the modified integral equations are further transformed to ordinary ones that contain nonsingular kernels only test cases consist of the acoustic radiation and scattering by one sphere and a pair of equal spheres comparisons are made with analytical solutions and schenck s chief method numerical results demonstrate the efficiency robustness and accuracy of

hypersingular boundary integral equations for exterior acoustic problems a bounded form of the hypersingular helmholtz integral equation for 3 d acoustic problems is developed in this paper all integrals with singular kernels are regularized within each regular surface element in the sense of kellogg with twice continuous differentiability the standard gaussian quadrature formula is applied without any special treatment for all the computational points the collocation points are chosen to be the gauss legendre nodes and no interpolation function is assumed for acoustic variables

evaluation of the helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions abstract this paper presents a means of evaluating singular integrals in the helmholtz boundary integral equation and its normal and tangential derivatives in two dimensions the subtraction addition technique is applied to the singular integral equations to convert the singular integrals to either ordinary integrals with bounded integrands or modified singular integrals including hypersingular integrals with exact integration values this regularization is performed before any discretization the modified integral equations can be calculated by directly applying standard quadrature rules over the entire integration domain numerical computations involve evaluating the acoustic field associated with a radiating inverse elliptic cylinder the velocity potential is obtained by applying the burton miller method which linearly combines the helmholtz boundary integral equation with its normal derivative to treat the fictitious characteristic frequencies further substituting the velocity potential into the regularized tangential derivative yields the surface tangential velocity comparing the numerical results with the analytical solutions verifies the effectiveness of the presented approach

a special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions this paper presents a simplified model of the helmholtz integral equation for bodies with axisymmetric shape and boundary conditions by taking advantage of a body s axisymmetric properties the surface integral is reduced to a line integral along the generator of the body and an integral over the angle of revolution the integration over the angle is performed partly analytically in terms of elliptic integrals and partly numerically using a gaussian quadrature formula the analytical formulation leading to elliptic integrals deals directly with the singularities in the green s function and its normal derivative examples are shown for scattering from a single sphere and two spheres and radiation from a finite cylinder the results are compared with analytical approximate and other accepted solutions

smoothness relaxation strategies for singular and hypersingular integral equations three stages are involved in the formulation of a typical direct boundary element method derivation of an integral representation taking a limit to the boundary ltb so as to obtain an integral equation and discretization we examine the second and third stages focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions two such strategies are indicated therst is the introduction of various apparent or pseudo ltbs the second is relaxed regularization in which a regularized integral equation derived rigorously under certain smoothness assumptions is used when less smoothness is available both strategies are shown to be based on inconsistent reasoning nevertheless reasons are oered for having some condence in numerical results obtained with certain strategies our work is done in two physical contexts namely two dimensionalpotential theory using functions of a complex variable and three dimensionalelastostatics 1998 john wiley sons ltd

evaluation of the stress tensor in 3 d elastoplasticity by direct solving of hypersingular integrals a new method of direct numerical evaluation of hypersingular boundary integrals has been applied to the differentiated form of the somigliana identity hypersingular identity in 3d elastostatics through this method it is possible to evaluate the stress tensor on the boundary of a complex 3d structure in a very accurate manner by employing the direct boundary element method bem the geometry of the elements and their arrangements over the boundary of the structure are not subjected to any restrictions numerical examples illustrate the accuracy of the proposed method

a new fast multipole boundary element method for two dimensional acoustic problems abstract a new fast multipole boundary element method bem is presented in this paper for solving large scale two dimensional 2d acoustic problems based on the improved burton miller formulation this algorithm has several important improvements the fast multipole bem employs the improved burton miller formulation and successfully overcomes the non uniqueness difficulty associated with the conventional bem for exterior acoustic problems the improved burton miller formulation contains only weakly singular integrals and avoids the numerical difficulties associated to the evaluation of the hypersingular integral it leads to the numerical implementations more efficient and straightforward furthermore the fast multipole method fmm and the approximate inverse preconditioned generalized minimum residual method gmres iterative solver are adopted to greatly improve the overall computational efficiency the numerical examples with neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole bem for solving large scale 2d acoustic problems in a wide range of frequencies

a boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions this paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions the difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface this process gives rise to a set of uniquely solvable boundary integral equations distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation the modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only this implementation allows direct application of standard quadrature formulas over the entire integration domain that is the collocation points are exactly the positions at which the integration points are located selecting the interior surface is an easy task moreover only a few corresponding interior nodal points are sufficient for the computation numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders comparisons with analytical solutions are made numerical results demonstrate the efficiency and accuracy of the current solution method

on the complexity of slice functions abstract by a result of berkowitz 1982 the monotone circuit complexity of slice functions cannot be much larger than the circuit combinational complexity of these functions for arbitrary complete bases this result strengthens the importance of the theory of monotone circuits we show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits here disjunction and conjunction are replaced by set union and set intersection all the main methods known for proving lower bounds on the monotone complexity of boolean functions fail to work in their present form for slice functions furthermore we show that the canonical slice functions of the boolean convolution the nechiporuk boolean sums and the clique function can be computed with a linear number of gates

a boundary integral method for acoustic radiation and scattering in this article a computational method is proposed for computing the helmholtz integral equation for acoustic radiation and scattering problems with a three dimensional body shape unlike the previous boundary element method the integration of acoustic properties is based on the global body surface instead of local elements the approximation function of body geometry and the weighting function of numerical integration are independently chosen when the singular kernels of helmholtz equation are regularized as bounded discontinuous functions the order of integration polynomials for acoustic properties can be arbitrarily chosen therefore the accuracy and efficiency of computation can be increased

an n superscript 3 2 lower bound on the monotone network complexity of the boolean convolution in this paper a general lower bound on the monotone network complexity of semidisjoint bilinear forms is proved by this method an n3 2 lower bound for the boolean convolution is obtained up to now the best known lower bound for the boolean convolution was of size n4 3 blum 1981 ieee annual sympos found comput sci 22 101 108