diffraction of elastic waves by a penny shaped crack consider an infinite elastic solid containing a penny shaped crack we suppose that time harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i in this paper we describe a new method for solving the corresponding linear boundary value problem for u i which we denote by s we begin by defining an elastic double layer we prove that any solution of s can be represented by an elastic double layer whose density satisfies certain conditions we then introduce various green functions and define a new crack green function g ij that is discontinuous across the crack next we use g ij to derive a fredholm integral equation of the second kind for the discontinuity in u i across the crack we prove that this equation always has a unique solution hence we are able to prove that the original boundary value problem s always possesses a unique solution and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind

adaptive hausdorff estimation of density level sets consider the problem of estimating the level set g x f x of an unknown d dimensional density function f based on n independent observations x 1 x n from the density this problem has been addressed under global error criteria related to the symmetric set difference however in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere the hausdorff error criterion provides this degree of uniformity and hence is more appropriate in such situations it is known that the minimax optimal rate of error convergence for the hausdorff metric is n logn 1 d 2 for level sets with boundaries that have a lipschitz functional form where the parameter characterizes the regularity of the density around the level of interest however the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter furthermore previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets e g the boundary fragment and star shaped sets that effectively reduce the set estimation problem to a function estimation problem this characterization precludes level sets with multiple connected components which are fundamental to many applications this paper presents a fully data driven procedure that is adaptive to unknown regularity conditions and achieves near minimax optimal hausdorff error control for a class of density level sets with very general shapes and multiple connected components

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

embedding formulae in diffraction theory embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables angles of incidence and observation into those dependent upon fewer angular variables in terms of facilitating rapid computations across considerable parameter regimes this is a considerable advantage our aim is to derive embedding formulae for scattering and diffraction problems in acoustics electromagnetism and elasticity here we construct a general approach to formulating and using embedding formulae we do this using complementary approaches overly singular states and a physical interpretation in terms of sources the crucial point we identify is the form of the auxiliary state used in the reciprocal theorem this is unphysically singular at the edge and is reminiscent of weight function methods used in fracture mechanics illustrative implementations of our approach are given using wiener hopf techniques for semi infinite model problems in both elasticity and acoustics we also demonstrate our approach using a numerical example from acoustics and we make connections with high frequency asymptotic methods

transient elastodynamic antiplane crack analysis of anisotropic solids abstract transient elastodynamic analysis of an antiplane crack in anisotropic solids is presented a time domain traction boundary integral equation bie method is applied for this purpose the traction bie is hypersingular and has the crack opening displacement as its fundamental unknown quantity unlike the usual time domain bie method the present formulation applies a convolution quadrature developed by lubich lubich c 1988a b convolution quadrature and discretized operational calculus numer math 52 129 145 part i 413 425 part ii which requires only the laplace domain instead of the time domain green s functions the spatial variation of the crack opening displacement is approximated by an infinite series of chebyshev polynomials which take the local behavior of the crack opening displacement at crack tips into account by using a galerkin method the time domain bie is converted into a system of linear algebraic equations which can be solved step by step special attention is devoted to the computation of dynamic stress intensity factors of an antiplane crack in generally anisotropic solids numerical results for isotropic solids are presented and compared with the well known analytical results of thau and lu thau s a lu t h 1970 diffraction of transient horizontal shear waves by a finite crack and a finite rigid ribbon int j enggn sci 8 857 874 to check the accuracy and efficiency of the present time domain bie method the effect of the material anisotropy on the dynamic stress intensity factors is analyzed via several numerical examples

on the solutions of a class of dual integral equations occurring in diffraction problems it is shown that there exists a category of two dimensional diffraction problems which can be put into a standard form of dual integral equations these diffraction problems include diffraction of electromagnetic waves by a finite strip a finite slit the diffraction of scalar or vector elastic waves by a rigid strip or crack etc a general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions the sufficient conditions for a given dual integral equations to be solvable in this manner are also determined hence the method forms a complement to the weiner hopf method to illustrate the method solutions are obtained for a bench mark problem the diffraction of light by a finite perfectly conducting strip or equivalently the diffraction of sh waves by a crack comparison with results obtained by others for low intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies both the near field and the far field are obtained the latter is shown to correspond to the fraunhoffer diffraction pattern for high frequency it is also shown that for the equivalent crack problem the stress intensity factor sif fluctuates rapidly with changes in the angle of incidence for high frequencies thus making the sif especially sensitive to angle of incidence at high frequencies

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

shear and torsional impact of cracked viscoelastic bodies a numerical integral equation transform approach abstract the transient elastodynamic stress intensity factor was determined for a cracked linearly viscoelastic body under impact loading two separate geometries along with associated loading conditions were considered in the first case the body is in the form of an infinite strip containing a central finite length crack and is subjected to anti plane shear tractions various strip heights are considered including the possibility of a body of nearly infinite extent in the second case the body is of infinite extent containing a finite length penny shaped crack and is subjected to radial shear and torsional twisting tractions the analytical parts of the solutions are either given by a previous analysis of h g georgiadis or are obtained from results by g c sih through the use of the correspondence principle the numerical procedure consists of solving integral equations and then inverting the laplace transformed solution by the dubner abate crump technique numerical results were given for the standard linear solid by considering several combinations of material constants

convergence rates for persistence diagram estimation in topological data analysis computational topology has recently seen an important development toward data analysis giving birth to the field of topological data analysis topological persistence or persistent homology appears as a fundamental tool in this field in this paper we study topological persistence in general metric spaces with a statistical approach we show that the use of persistent homology can be naturally considered in general statistical frameworks and that persistence diagrams can be used as statistics with interesting convergence properties some numerical experiments are performed in various contexts to illustrate our results

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

pseudo differential operators for embedding formulae a new method is proposed for deriving embedding formulae in 2d diffraction problems in contrast to the approach developed in craster and shanin 2005 7 which is based on a differential operator here a pseudo differential i e a non local operator is applied to the wave field using this non local operator a new embedding formula is derived for scattering by a single wedge the formula has uniform structure for all opening angles including angles irrational with respect to p the earlier theory craster and shanin 2005 7 was valid only for rational angles

scattering of elastic waves from thin shapes in three dimensions using the composite boundary integral equation formulation in this paper the composite boundary integral equation bie formulation is applied to scattering of elastic waves from thin shapes with small but finite thickness open cracks or thin voids thin inclusions thin layer interfaces etc which are modeled with two surfaces this composite bie formulation which is an extension of the burton and miller s formulation for acoustic waves uses a linear combination of the conventional bie and the hypersingular bie for thin shapes the conventional bie as well as the hypersingular bie will degenerate or nearly degenerate if they are applied individually on the two surfaces the composite bie formulation however will not degenerate for such problems as demonstrated in this paper nearly singular and hypersingular integrals which arise in problems involving thin shapes modeled with two surfaces are transformed into sums of weakly singular integrals and nonsingular line integrals thus no finer mesh is needed to compute these nearly singular integrals

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

scattering by two penny shaped cracks with spring boundary conditions the elastodynamic scattering by a penny shaped crack with spring boundary conditions is investigated the transition t matrix of the crack is determined and the usefulness of this is illustrated by considering also the scattering by two cracks the t matrix of a single crack is first determined by a direct integral equation method which gives the crack opening displacement and the integral representation which subsequently gives the scattered field expanded in spherical waves two cracks are considered by a multi centred t matrix approach where matrix inverses are expanded in neuman series rotation matrices are employed so that the cracks may have an arbitrary rotation the back scattered longitudinal far field amplitude is computed both in the frequency and time domain in a few cases and the effects due to multiple scattering are in particular explored

efficient estimation of monotone boundaries let g 0 1 0 1 be a monotone nondecreasing function and let g be the closure of the set x y is an element of 0 1 x 0 1 0 less than or equal to y less than or equal to g x we consider the problem of estimating the set g from a sample of i i d observations uniformly distributed in g the estimation error is measured in the hausdorff metric we propose the estimator which is asymptotically efficient in the minimax sense

the homotopy perturbation method applied for solving complex valued differential equations with strong cubic nonlinearity abstract in this paper the homotopy perturbation method is adopted for solving a complex valued second order strongly nonlinear differential equation homotopy with an imbedding parameter p 0 1 is constructed the perturbation procedure with parameter p transforms the strongly nonlinear differential equation into a system of linear complex valued differential equations whose solutions give the approximate solution of the initial differential equation to illustrate the effectiveness and convenience of the suggested procedure a duffing equation with strong cubic nonlinearity is considered the periodic solution in the first approximation is obtained the solution is compared with the exact one and shows good agreement

linearized perturbation method for stochastic analysis of a rill erosion model the linearization and correction method lcm proposed by he is a simple and effective perturbation technique to solve nonlinear equations to analyze the random properties of rill erosion model a new stochastic perturbation technique called linearized perturbation method is developed by combining the traditional stochastic perturbation method with the lcm comparisons between the numerical results obtained by the linearized perturbation method and those obtained by monte carlo method indicated an excellent agreement however the calculation efficiency of the linearized perturbation method is higher

a plug in approach to support estimation we suggest a new approach based on the use of density estimators for the problem of estimating the compact support of a multivariate density this subject motivated in terms of pattern analysis by grenander has interesting connections with detection and clustering a natural class of density based estimators is defined universal consistency results and convergence rates are established for these estimators with respect to the usual measure based metric d between sets further convergence rates with respect to both d and the hausdorff metric d h are also obtained under some fairly intuitive shape restrictions

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

on nonparametric estimation of density level sets let x 1 x n be independent identically distributed observations from an unknown probability density f consider the problem of estimating the level set g g f x r 2 f x from the sample x 1 x n under the assumption that the boundary of g has a certain smoothness we propose piecewise polynomial estimators of g based on the maximization of local empirical excess masses we show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities we find also the optimal convergence rates for estimation of convex level sets a generalization to the n dimensional case where n 2 is given

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

manifold estimation and singular deconvolution under hausdorff loss we find lower and upper bounds for the risk of estimating a manifold in hausdorff distance under several models we also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure

weinstein s diffraction problem embedding formula and spectral equation in parabolic approximation a short wave problem of reflection and radiation by an open end of a two dimensional planar waveguide is studied the incident mode is assumed to have frequency close to the cut off the problem is studied in the parabolic approximation a recently developed approach based on the embedding formula and the spectral equation for the directivity of an edge green s function is applied to the problem

exact transient full field analysis of a finite crack subjected to dynamic anti plane concentrated loadings in anisotropic materials in this study the elastodynamic full field response of a finite crack in an anisotropic material subjected to a dynamic anti plane concentrated loading with heaviside function time dependence is investigated a linear coordinate transformation is introduced to simplify the problem the linear coordinate transformation reduces the anisotropic finite crack problem to an equivalent isotropic problem an alternative methodology different from the conventional superposition method is developed to construct the reflected and diffracted wave fields the transient solutions are determined by superposition of two proposed fundamental solutions in the laplace transform domain the fundamental solutions to be used are the problems for applying exponentially distributed traction and displacement on the crack faces and along the crack tip line in the laplace transform domain respectively exact analytical transient solutions for dynamic shear stresses displacement and stress intensity factor are obtained by using the cagniard de hoop method of laplace inversion and are expressed in explicitly compact formulations the solutions have accounted for the contributions of all diffracted waves generated from two crack tips numerical results for the time history of shear stresses and stress intensity factors during the transient process are calculated based on analytical solutions and are discussed in detail the transient solutions of stresses have been shown to approach the corresponding static values after the first eight waves have passed the field point the dynamic stress intensity factor will reach a maximum value when the incident wave arrives at the crack tip and remain constant before the first diffracted wave generated from the other crack tip arrives and then will oscillate near the static value a simple explicit expression of the dynamic overshoot for stress intensity factors is derived as a function of the location for applied loadings the crack length and material constants

impact response of a transversely isotropic cylinder with a penny shaped crack abstract the axisymmetric dynamic response of a penny shaped crack in a transversely isotropic infinite cylinder under normal impact is analyzed laplace and hankel transforms are used to reduce the transient problem to a pair of dual integral equations in the laplace transform plane the solution is given in terms of a fredholm integral equation of the second kind a numerical laplace inversion routine is used to recover the time dependence of the solution the dynamic stress intensity factor is determined and numerical results for some practical materials are shown graphically to demonstrate the influence of transverse isotropy

a topological view of unsupervised learning from noisy data in this paper we take a topological view of unsupervised learning from this point of view clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise we construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions a special case of our construction is the mixture of gaussians where there is gaussian noise concentrated around a finite set of points the means more generally we consider gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it we show that if the variance of the gaussian noise is small in a certain sense then the homology can be learned with high confidence by an algorithm that has a weak linear dependence on the ambient dimension our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial laplacian of a suitable data derived simplicial complex

invariance of stresses under a change in elastic compliances a recent discovery by cherkaev lurie and milton shows that in plane elasticity involving a material with varying compliances it is possible to keep the stress state the same by compensating for a constant shift in the bulk compliance with the opposite shift in the shear compliance the present article demonstrates that depending on the connectivity of the domain and some side conditions the stress state may also be invariant upon a linear shift in the elastic compliances discontinuous compliances such as encountered in composite materials and various interface conditions as well as global compatibility are discussed in detail

transient analysis of mode iii fracture in anisotropic layered media the transient response of a semi infinite crack in an anisotropic layered medium is presented in this study the individual layers are anisotropic and homogeneous the number of reflections and diffractions of stress waves that are generated from the interfaces of the layered medium and the crack are infinite it will cause extreme difficulties in analyzing this problem a modified linear coordinate transformation is introduced to reduce the anisotropic layered medium problem to an equivalent isotropic problem with a similar geometry configuration besides a useful fundamental solution is proposed and the solution can be determined by superposition of the fundamental solution in the laplace transform domain the cagniard method of laplace inversion is used to obtain the analytical transient solution in the time domain the final results for the stress intensity factor are expressed in compact formulations and are valid for an infinite length of time numerical calculations are also evaluated and discussed in detail nomenclature b j slowness in anisotropic materials c j 44 c j 45 c j 55 elastic moduli k mode iii dynamic stress intensity factor t time coordinate w j antiplane displacement in corresponding isotropic layered media w j antiplane displacement in anisotropic layered media x y z rectangular coordinates in corresponding isotropic layered media x y z rectangular coordinates in anisotropic layered media j material mass density j xz j yz shear stresses in corresponding isotropic layered media j xz j yz shear stresses in anisotropic layered media

a penny shaped crack in a transversely isotropic material under non axisymmetric impact loads abstract a solution is given for problems involving non axisymmetric dynamic impact loading of a penny shaped crack in a transversely isotropic medium laplace and hankel transforms are used to reduce the equations of elasticity to integral equations and solutions are obtained for the three modes of fracture the stress intensity factors are determined for a penny shaped crack loaded by concentrated normal impact forces and concentrated radial shear impact forces the integral equations are solved by numerical methods and the results are plotted showing how the dynamic stress intensity factors are influenced by the asymmetric loading

minimax manifold estimation we find the minimax rate of convergence in hausdorff distance for estimating a manifold m of dimension d embedded in rd given a noisy sample from the manifold under certain conditions we show that the optimal rate of convergence is n 2 2 d thus the minimax rate depends only on the dimension of the manifold not on the dimension of the space in which m is embedded

the stress intensity factor history for a half plane crack in a transversely isotropic solid due to impact point loading on the crack faces abstract three dimensional analysis is performed for a transversely isotropic solid containing a half plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack edge the solution of this problem is treated as the superposition of two simpler problems one considers the transient wave in an elastic half space generated by an impact point loading on the surface the other problem is that which cancels out the surface displacement ahead of the crack edge induced by problem 1 a half space subjected to a distributed dislocation on the surface is constructed as the fundamental problem and solved by the use of integral transforms the wiener hopf technique and the cagniard de hoop method an exact expression is derived for the mode i stress intensity factor as a function of time and position along the crack edge some features of the solution are discussed through numerical results

diffraction of elastic waves by a penny shaped crack analytical and numerical results the diffraction of time harmonic stress waves by a penny shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials martin proc r soc lond a 378 263 1981 has proved that the corresponding linear boundary value problem has precisely one solution and that this solution can be constructed by solving a two dimensional fredholm integral equation of the second kind however this integral equation has a complicated matrix kernel and the components of its vector solution are coupled the main purpose of the present paper is to show how martin s integral equation can be explicitly solved in terms of a sequence of functions each of which satisfies a very simple scalar integral equation of the second kind this simplification may be made for any incident wave for an incident plane wave further simplifications are possible we show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence namely grazing incidence the resulting computational procedure is especially attractive if only the stressintensity factors or the far field displacements are required finally we present some numerical results for the scattering of a p wave at normal incidence and an sv wave at oblique incidence and compare these with those of other authors

time harmonic body force loading of a mode i penny shaped crack abstract the present paper is concerned with the problem of determining dynamic sif of a penny shaped crack in an infinite elastic medium which is subjected to the action of time harmonic axial body forces placed symmetrically with respect to the crack plane the solution of the problem is obtained by superposition of the solutions of two simpler problems the first of these problems is related to the unperturbed crackless space under the prescribed axial body forces while the second problem consists in finding the dynamic sif of the penny shaped crack whose faces are directly acted upon by some axial stresses the form of these axial stresses is determined from the solution of the first problem fourier and hankel transforms have been used to solve the first problem next by means of the hankel transform the second problem has been reduced to a pair of dual integral equations which have been subsequently transformed into a fredholm integral equation of the second kind via an auxiliary function the integral equation has been solved numerically in order to determine the variations of the dynamic stress intensity factor at the rim of the penny shaped crack for some particular body force loading cases

relationship of anisotropic and isotropic materials for antiplane problems the material symmetry is assumed to be such that the inplane and antiplane deformations are uncoupled a correspondence is obtained between the anisotropic and the isotropic problem through such a correspondence the relationship of the stresses and displacement for the anisotropic and the corresponding isotropic problem is established for both polar and cartesian coordinate systems