a new numerical method for rough surface scattering calculations a new approach to solving the magnetic field integral equation mfie for the current induced on a infinite perfectly conducting rough surface is presented by splitting the propagator matrix into contributions from the left and from the right of the point of observation a second kind integral equation can be formed with a new born term and a new kernel following discretization of this new integral equation the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional lu decomposition where the time saving factor is roughly n 3 where n is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated while the new born term is usually adequate for scattered field calculations the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems results are presented for one dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40 which illustrate the robustness of the new born term

higher order wavelet reconstruction differentiation filters and gibbs phenomena an orthogonal wavelet basis is characterized by its approximation order which relates to the ability of the basis to represent general smooth functions on a given scale it is known though perhaps not widely known that there are ways of exceeding the approximation order i e achieving higher order error in the discretized wavelet transform and its inverse the focus here is on the development of a practical formulation to accomplish this first for 1d smooth functions then for 1d functions with discontinuities and then for multidimensional here 2d functions with discontinuities it is shown how to transcend both the wavelet approximation order and the 2d gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet basis grid

wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

coifman wavelets in 3 d scattering from very rough random surfaces the coifman wavelets are employed to perform galerkin s procedure in the method of moments mom the coiflets are continuous smooth overlapping yet orthogonal which permit significant reduction of the sampling rate and dramatic compression of the matrix size with respect to the pulse based mom in addition the wavelet based impedance matrix is sparse more importantly the vanishing moments of the coiflets of order l 4 provide us with high precision o h sup 5 one point quadrature which knocks down the computational effort in filling the matrix entries from o n sup 2 to o n our approach is different from the conventional wavelet technique which employs wavelets to sparsify an existing impedance matrix under similarity transformation with computational cost in o n sup 2 excellent agreement between this coifman wavelet method and the previous publications is observed

coifman wavelets in 3 d scattering from a calibration target consisting of doubly periodic sharp metal cones the magnetic field integral equation mfie is solved by coiflet based galerkin s procedure in conjunction with the floquet theorem for plane wave scattering from a doubly periodic array of sharp conducting circular cones to avoid using intervallic wavelets to handle the boundary truncations we split the unknown surface current into four components enforcing periodic boundary conditions so that the standard coiflets can be employed owing to the orthogonality compact support continuity and smoothness of the coiflets well conditioned impedance matrices are obtained at the resolution level 5 majority of the matrix entries are obtained in the spectral domain by one point quadrature with high precision in o h5 imposing the dirac property of the coiflets for the self elements and elements associated with the same elevation of the field and source points z z spatial domain computation is applied bypassing the slow convergence of the spectral summation of the non damping propagating modes the simulation results of induced current and far zone scattering pattern are compared with the solutions from an rwg mlfma based commercial software feko and excellent agreement is observed

singularities interfaces and cracks in dissimilar anisotropic media for a non pathological bimaterial in which an interface crack displays no oscillatory behaviour it is observed that apart possibly from the stress intensity factors the structure of the near tip field in each of the two blocks is independent of the elastic moduli of the other block collinear interface cracks are analysed under this non oscillatory condition and a simple rule is formulated that allows one to construct the complete solutions from mode iii solutions in an isotropic homogeneous medium the general interfacial crack tip field is found to consist of a two dimensional oscillatory singularity and a one dimensional square root singularity a complex and a real stress intensity factors are proposed to scale the two singularities respectively owing to anisotropy a peculiar fact is that the complex stress intensity factor scaling the oscillatory fields however defined does not recover the classical stress intensity factors as the bimaterial degenerates to be non pathological collinear crack problems are also formulated in this context and a strikingly simple mathematical structure is identified interactive solutions for singularity interface and singularity interface crack are obtained the general results are specialized to decoupled antiplane and in plane deformations for this important case it is found that if a material pair is non pathological for one set of relative orientations of the interface and the two solids it is non pathological for any set of orientations for bonded orthotropic materials an intuitive choice of the principal measures of elastic anisotropy and dissimilarity is rationalized a complex variable representation is presented for a class of degenerate orthotropic materials throughout the paper the equivalence of the lekhnitskii and stroh formalisms is emphasized the article concludes with a formal statement of interfacial fracture mechanics for anisotropic solids

wavelet based deconvolution algorithm for time domain near field isar imaging based on correlation receiver in communication theory the measured backscattering waveform of a calibration conducting sphere is adopted as the system impulse response this antenna target system response is then approximated by the derivatives of a gaussian function dgf as well as the coiflets to restore the impulse response of the target and to reconstruct the isar images both the dgf and coiflets are applied to the deconvolution and their results are compared the gaussian and coiflet function parameters are derived by a best fit to the measured time domain system response employing the parametric fits a lowpass filter has been automatically embedded which performs regularization of the conjugate gradient cg optimization and speeds up the cg convergence numerical results show that the image quality from the coiflet system is similar to or better than from the dgf system owing to the continuity smoothness orthogonality and compact support of the coiflets the size of the system matrix has been reduced from 1024 times 1024 for dgf to 261 times 261 and the deconvolution process is accelerated more than 22 fold without sacrificing image characters

on the use of coifman intervallic wavelets in the method of moments for fast construction of wavelet sparsified matrices orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained however in many practical problems the solution domain is confined in a bounded interval while the wavelets are originally defined on the entire real line to overcome this problem periodic wavelets have been described in the literature nonetheless the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions we present the intervallic coifman wavelets coiflets for the method of moments mom the intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets the intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis mra of other usual unbounded wavelets the coiflets possesses a special property that their scaling functions have many vanishing moments as a result the zero entries of the matrices are identified directly without using a truncation scheme with an artificially established threshold further the majority of matrix elements are evaluated directly without performing numerical integration procedures such as gaussian quadrature for an n spl times n matrix the number of actual numerical integrations is reduced from n sup 2 to the order of 3n 2l 1 when the coiflets of order l is employed the construction of intervallic wavelets is presented numerical examples of scattering problems are discussed and the relative error of this method is studied analytically

generalized gaussian quadrature rules for systems of arbitrary functions a numerical algorithm is presented for the construction of generalized gaussian quadrature rules originally introduced by s karlin and w studden over three decades ago the quadrature rules to be discussed possess most of the desirable properties of the classical gaussian integration formulae such as positivity of the weights rapid convergence mathematical elegance etc the algorithm is applicable to a wide class of functions including smooth functions not necessarily polynomials as well as functions with end point singularities such as those encountered in the solution of integral equations complex analysis potential theory and several other areas the performance of the algorithm is illustrated with several numerical examples

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

interface corners in linear anisotropic viscoelastic materials abstract in this study an extended stroh formalism for two dimensional linear anisotropic viscoelasticity is developed for the problems of interface corners between two dissimilar viscoelastic materials in this formalism the solutions for the displacements and stress functions in the time domain can be written in the form of a matrix function using complex variables the correspondence relations for viscoelastic analysis are then obtained and verified for material eigenvectors displacement and stress eigenfunctions singularity orders of stresses and stress intensity factors explicit solutions for the material eigenvector matrices in the laplace domain are also obtained for standard linear and isotropic linear viscoelastic solids to calculate the singularity orders and stress intensity factors of the interface corners four different approaches are proposed through numerical examples on cracks interface cracks and interface corners an approach using the path independent h integral in the laplace domain with an elastic near tip solution which takes the correspondence relations for singularity orders and stress intensity factors is demonstrated to be better than the other three approaches

numerical simulation of scattering from rough surfaces a wavelet based approach in this paper a preliminary study is carried out to demonstrate the application of wavelets for improving the computation time and reducing computational memory required for evaluating the statistics of the scattered field from rough surfaces using the method of moments mom in conjunction with a monte carlo simulation in specific haar and the first order b spline wavelet basis functions are applied to the mom formulation of one dimensional rough surfaces in order to compare the computation time and sparsity for wavelets in the same family but of higher order since the scattering coefficient the second moment of the backscatter field per unit area is a gentle function of the surface parameters and the radar attributes it is demonstrated that a relatively high thresholding level can be applied to the impedance matrix which leads to a sparser impedance matrix and faster computation time it is also shown that applying a high threshold level the coefficients of the high order wavelets would increase out of proportion however the effect of these current components averages out when computing the scattering coefficients the resulting sparse impedance matrices are solved efficiently using fast search routines such as the conjugate gradient method a systematic study is carried out to investigate the effect of different threshold levels on the accuracy versus computing speed criterion the computed scattering coefficients are compared to previous results computed using a conventional pulse basis function as well as the existing theoretical solutions for rough surfaces it is shown that wavelet basis functions provide substantial reductions in both memory requirements and computation time

on unified numerical algorithm for 3 d scattering from dielectric and pec random rough surfaces we present a unified fast scattering algorithm for dielectric random rough surfaces that asymptotically reduces to the perfect electric conductor pec case when the loss tangent grows extremely large the coifman wavelets are employed to implement galerkin s procedure in the method of moments mom the coiflets based surface integral equations ies consist of both the tangential and normal components of electromagnetic fields as unknowns the inherited mathematical superiority e g local multiresolution analysis and high regularity with holder index 1 449 in smoothness allows efficiently implementing both electric field ie and magnetic field ie due to the high precision one point quadrature the coiflets yield fast evaluations of the most off diagonal entries reducing the matrix fill effort from o n2 to o n the orthogonality and riesz basis of the coiflets generate well conditioned impedance matrix with rapid convergence for the conjugate gradient solver in addition a semianalytical expression of the tapered wave carried power is derived which speeds up computations of the normalization factor of scattering coefficients numerical results demonstrate that the reduced pec model does not suffer from ill posed problems namely matrix condition numbers are kept small and solutions are stable under extremely large loss tangent where normal components of h field and tangential e field have vanished compared with the previous publications and laboratory measurements good agreement is observed

explicit solutions for collinear interface crack problems abstract by combining stroh s formalism the method of analytical continuation and the translating technique introduced in this paper the explicit full domain solutions for the problems of collinear interface cracks between dissimilar anisotropic media have been presented to define a proper bimaterial stress intensity factor the full domain solution is applied to the near tip of an interface crack the results show that the effect of material properties is totally reflected through the oscillation index e and the bimaterial constants d 11 d 22 and d 33 which have inverse relations with young s moduli e 1 e 2 and shear modulus g 23 g 31 respectively for the purpose of illustration three typical examples are solved explicitly such as a semi infinite interface crack a finite interface crack and two collinear interface cracks

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

a unified definition for stress intensity factors of interface corners and cracks abstract based upon linear fracture mechanics it is well known that the singular order of stresses near the crack tip in homogeneous materials is a constant value 1 2 which is nothing to do with the material properties for the interface cracks between two dissimilar materials the near tip stresses are oscillatory due to the order of singularity being 1 2 i e and 1 2 the oscillation index e is a constant related to the elastic properties of both materials while for the general interface corners their singular orders depend on the corner angle as well as the elastic properties of the materials owing to the difference of the singular orders of homogeneous cracks interface cracks and interface corners their associated stress intensity factors are usually defined separately and even not compatibly since homogenous cracks and interface cracks are just special cases of interface corners in order to build a direct connection among them a unified definition for their stress intensity factors is proposed in this paper based upon the analytical solutions obtained previously for the multibonded anisotropic wedges the near tip solutions for the general interface corners have been divided into five different categories depending on whether the singular order is distinct or repeated real or complex to provide a stable and efficient computing approach for the general mixed mode stress intensity factors the path independent h integral based on reciprocal theorem of betti and rayleigh is established in this paper the complementary solutions needed for calculation of h integral are also provided in this paper to illustrate our results several different kinds of examples are shown such as cracks in homogenous isotropic or anisotropic materials central or edge notches in isotropic materials interface cracks and interface corners between two dissimilar materials

a local converse theorem for mathrm sp _ 2r in this paper we prove the local converse theorem for mathrm sp 2r f over a p adic field f more precisely given two irreducible supercuspidal representations of mathrm sp 2r f with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of mathrm gl n f for all 1 le n le r then these two representations must be isomorphic our proof is based on the local analysis of the local integrals which define local gamma factors a key ingredient of the proof is certain partial bessel function property developed by cogdell shahidi tsai recently the same method can give the local converse theorem for mathrm u r r

tapered wave with dominant polarization state for all angles of incidence typical applications of the method of moments mom to rough surface three dimensional 3 d electromagnetic scattering require a truncation of the surface considered and call for a tapered incident wave it is shown how such a wave can be constructed as a superposition of plane waves avoiding problems near both normal and grazing incidence and providing clean footprints and clear polarization at all angles of incidence the proposed special choice of polarization vectors removes an irregularity at the origin of the wavenumber space and leads to a least squared error property of the wave issues in the application to 3 d scattering from an object over a rough surface are discussed approximate 3 d scalar and vector tapered waves which can be evaluated without resorting to any numerical integrations are derived and important limitations to the accuracy and applicability of these approximations are pointed out