a non hypersingular time domain biem for 3 d transient elastodynamic crack analysis a three dimensional 3 d time domain boundary integral equation method biem is presented for transient elastodynamic crack analysis a non hypersingular traction bie formulation is used with the crack opening displacements and their derivatives as unknown quantities a collocation method in conjunction with a time stepping scheme is developed to solve the non hypersingular time domain bies to simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front a constant spatial shape function is applied for elements away from the crack front while a spatial square root crack tip shape function is adopted for elements near the crack front a linear temporal shape function is used in the time stepping scheme numerical calculations have been carried out for penny shaped and square cracks results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables for the test examples considered good agreement between the present results and those of other authors is obtained

a mass matrix formulation for three dimensional dynamic fracture mechanics in this paper the dual reciprocity boundary element method is developed for the analysis of three dimensional elastodynamic fracture mechanics mixed mode problems the dual boundary element method is used to calculate the unknowns of boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method dynamic stress intensity factors are determined by the crack opening displacement cod in the time domain several numerical examples are presented to demonstrate the accuracy of the method

on the use of quarter point boundary elements for stress intensity factor computations this communication studies a procedure for stress intensity factor computations using traction singular quarter point boundary elements opening mode stress intensity factors are computed from the tractions nodal values at the crack tip a comparison is made between the factors calculated using this procedure and those obtained by previously recommended methods which made use of the nodal values of the displacements the proposed procedure was seen to be less discretization sensitive than any other of the considered methods accurate results were obtained even in the case of coarse meshes

spectral properties of the cauchy process on half line and interval we study the spectral properties of the transition semigroup of the killed one dimensional cauchy process on the half line 0 a nd the interval 1 1 this process is related to the square root of one dimensional laplacian a d 2 dx 2 with a dirichlet exterior condition on a complement of a domain and to a mixed steklov problem in the half plane for the half line an explicit formula for generalized eigenfunctions of a is derived and then used to construct a spectral representation of a explicit formulas for the transition density of the killed cauchy process on the half line or the heat kernel of a in 0 and for the distribution of the first exit time from the half line follow the formula for is also used to construct approximations to eigenfunctions of a in the interval for the eigenvalues n of a in the interval the asymptotic formula n n 2 8 o 1 n is derived and all eigenvalues n are proved to be simple finally efficient numerical methods of estimation of eigenvalues n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point

estimates and structure of α harmonic functions we prove a uniform boundary harnack inequality for nonnegative harmonic functions of the fractional laplacian on arbitrary open set d this yields a unique representation of such functions as integrals against measures on dc satisfying an integrability condition the corresponding martin boundary of d is a subset of the euclidean boundary determined by an integral test

a frequency domain biem combining dbies and tbies for 3 d crack inclusion interaction analysis abstract the influence of a spherical elastic inclusion on a penny shaped crack embedded in an infinite elastic matrix subjected to a time harmonic crack face or incident wave loading is investigated a boundary integral equation method biem combining displacement boundary integral equations dbies on the matrix inclusion interface and traction boundary integral equations tbies on the crack surface is developed and applied for the numerical solution of the corresponding 3 d elastodynamic problem in the frequency domain the singularity subtraction and mapping techniques in conjunction with a collocation scheme are implemented for the regularization and the discretization of the bies by taking into account the local structure of the solution at the crack front as numerical examples the interaction of an elastic inclusion and a neighboring penny shaped crack subjected to a tensile crack surface loading or an incident plane longitudinal wave loading is investigated the effects of the inclusion are assessed by the analysis of mixed mode dynamic stress intensity factors dsifs in dependence on the wave number the material combination of the matrix and the inclusion and the crack inclusion orientation size and distance

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

universal constraints on the location of extrema of eigenfunctions of non local schrodinger operators abstract we derive a lower bound on the location of global extrema of eigenfunctions for a large class of non local schrodinger operators in convex domains under dirichlet exterior conditions featuring the symbol of the kinetic term the strength of the potential and the corresponding eigenvalue and involving a new universal constant we show a number of probabilistic and spectral geometric implications and derive a faber krahn type inequality for non local operators our study also extends to potentials with compact support and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential

eigenvalues of the fractional laplace operator in the interval abstract two term weyl type asymptotic law for the eigenvalues of the one dimensional fractional laplace operator 2 0 2 in the interval 1 1 is given the n th eigenvalue is equal to n 2 2 8 o 1 n simplicity of eigenvalues is proved for 1 2 l 2 and l properties of eigenfunctions are studied we also give precise numerical bounds for the first few eigenvalues

bayesian designs for maximizing information and outcome abstract this article deals with a novel utility function to design experiments in a bayesian framework this utility function is a linear combination of the gain in shannon information and of the total outcome of the experiment defined as the sum of observed values in the dependent variable of a linear model thus the expected posterior utility to be maximized is a combination of the bayes d optimality criterion and the posterior expectation of the total output earlier studies have shown that bayesian parallels of the classical d a and e optimal designs can be obtained by considering utility or loss functions concerned with efficient estimation of the parameters of interest a different requirement that might be desirable in applied problems is to combine the accuracy of parameter estimation with the maximization of experimental output the utility function considered here does this we look at the implications of using this utility in deriving designs in the context of hierarchical linear models

spectral properties of the massless relativistic harmonic oscillator abstract an explicit solution of the spectral problem of the non local schrodinger operator obtained as the sum of the square root of the laplacian and a quartic potential in one dimension is presented the eigenvalues are obtained as zeroes of special functions related to the fourth order airy function and closed formulae for the fourier transform of the eigenfunctions are derived these representations allow to derive further spectral properties such as estimates of spectral gaps heat trace and the asymptotic distribution of eigenvalues as well as a detailed analysis of the eigenfunctions a subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the fourier representation of the eigenfunctions and its derivative

solving fractional schrodinger type spectral problems cauchy oscillator and cauchy well this paper is a direct offspring of the work of garbaczewski and stephanovich levy flights and nonlocal quantum dynamics j math phys 54 072103 2013 where basic tenets of the nonlocally induced random and quantum dynamics were analyzed a number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so called fractional quantum mechanics spectral problems presently we give a decisive computer assisted proof for an exemplary finite and ultimately infinite cauchy well problem that spectral solutions proposed so far were plainly wrong as a constructive input we provide an explicit spectral solution of the finite cauchy well the infinite well emerges as a limiting case in a sequence of deepening finite wells the employed numerical methodology algorithm based on the strang splitting method has been tested for an exemplary cauchy oscillator problem whose analytic solution is available an impact of the inherent spatial nonlocality of motion generators upon computer assisted outcomes potentially defective in view of various cutoffs i e detailed eigenvalues and shapes of eigenfunctions has been analyzed

sample size determination for robust bayesian analysis this article considers a robust bayesian approach to the sample size determination problem we focus on global bayesian robustness that studies lower bound ln upper bound un and range rn of posterior quantities of interest obtained as the prior varies in a class of distributions specifically we are interested in the selection of an appropriate sample size that gives guarantees to the researcher of observing a small value of the range and depending on the problems either a sufficiently large lower bound or a sufficiently small upper bound toward this end we approach the problem as a design issue and provide new sample size determination criteria based on summaries of the predictive distributions of ln un and rn such as expectations and tail probabilities relationships and comparison to standard classical and nonrobust bayesian methods are discussed the proposed methods are studied for the normal model with conjugate priors and used for choosing the size of a clinical trial

higher order pdes and symmetric stable processes we describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations as an application we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set d the variational formula is more user friendly than the classical rayleigh ritz formula we illustrate this by obtaining upper bounds on the eigenvalues in terms of dirichlet eigenvalues of the laplacian on d these results generalize some work of banuelos and kulczycki on the cauchy process along the way we prove an operator inequality for the operators associated with the transition densities of brownian motion and the brownian motion killed upon leaving d

to hit or to pass it over remarkable transient behavior of first arrivals and passages for levy flights in finite domains the term levy flights was coined by benoit mandelbrot who thus poeticized stable levy random motion a markovian process with stationary independent increments distributed according to the stable levy probability law contrary to the brownian motion the trajectories of the stable levy motion are discontinous that is exhibit jumps this feature implies that the process of first passage through the boundary of a given space domain or the first escape is different from the process of first arrival hit at the boundary here we investigate the properties of first escapes and first arrivals for levy flights and explore how the asymptotic behavior of the corresponding passage and hit probabilities is sensitive to the size of the domain in particular we find that the survival probability to stay in a large enough finite domain has a universal sparre andersen temporal scaling which is transient and changes to an exponential non universal decay at longer times also the probability to arrive at a finite domain possesses a similar transient sparre andersen universality that turns into a non universal and slower power law decay in course of time finally we demonstrate that the probability density of the leapover length l over the boundary related to overshooting events has an intermediate asymptotics which is inherent for the escape from a semi infinite domain however for larger leapovers the probability density decays faster according to the law thus we find that the laws derived for the stable processes on the semi infinite domain manifest themselves as transients for levy flights on the finite domain

the cauchy process and the steklov problem let xt be a cauchy process in rd d 1 we investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain d we establish a connection between the semigroup of this process and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem using this we derive a variational characterization for the eigenvalues of the cauchy process in d this characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the cauchy process inspired by those for brownian motion our results are new even in the simplest geometric setting of the interval 1 1 where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions such results although trivial for the laplacian take considerable work to prove for the cauchy processes and remain open for general symmetric stable processes along the way we present other general properties of the eigenfunctions such as real analyticity which even though well known in the case of the laplacian are not available for more general symmetric stable processes

two sided eigenvalue estimates for subordinate processes in domains let x xt t 0 be a symmetric markov process in a state space e and d an open set of e denote by x d the subprocess of x killed upon leaving d let s st t 0 be a subordinator with laplace exponent that is independent of x the processes x xst t 0 and x d x dt t 0 are called the subordinate processes of x and x d respectively under some mild conditions we show that if n n 1 and n n 1 denote the eigenvalues of the generators of the subprocess of x killed upon leaving d and of the process x d respectively then

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

the boundary element method applied to time dependent problems in anisotropic materials in this paper a multi domain boundary element formulation is developed for the solution of two dimensional dynamic anisotropic problems with or without cracks the inertia effects are treated as body forces and the dual reciprocity formulation is used to transform the domain integral into a boundary integral in fracture problems traction singular quarter point elements are used in order to give a correct representation of displacement and traction fields around the crack tip the dynamic stress intensity factors are computed directly from the tractions at the crack tip node the results obtained demonstrate good agreement with other reported results and show strong dependence on the material anisotropy

trace estimates for stable processes in this paper we study the behaviour in time of the trace the partition function of the heat semigroup associated with symmetric stable processes in domains of rd in particular we show that for domains with the so called r smoothness property the second terms in the asymptotic as t 0 involves the surface area of the domain just as in the case of brownian motion

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

martingale and weak solutions for a stochastic nonlocal burgers equation on finite intervals abstract this work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal burgers equation on a one dimensional bounded domain the existence of a martingale solution is shown by using a galerkin approximation prokhorov s theorem and skorokhod s embedding theorem the same galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal burgers equation on a bounded domain

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

time domain bem analysis for the 2d scalar wave equation initial conditions contributions to space and time derivatives the complete expressions corresponding to space and time derivatives of volterra s integral representation of the time dependent scalar wave equation are presented in this paper the concept of finite part of an integral is used in order to compute boundary kernels contributions time integration of boundary kernels is performed analytically for linear and constant time approximations for the potential and its normal derivative respectively the new integral representations presented in this text are quite general as contributions due to the initial conditions have also been included two examples are presented in order to assess the accuracy of the formulation

uniqueness of radial solutions for the fractional laplacian we prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional laplacian s with s 0 1 for any space dimensions n 1 by extending a monotonicity formula found by cabre and sire we show that the linear equation s u vu 0 in r n has at most one radial and bounded solution vanishing at infinity provided that the potential v is radial and nondecreasing in particular this result implies that all radial eigenvalues of the corresponding fractional schrodinger operator h s v are simple furthermore by combining these findings on linear equations with topological bounds for a related problem on the upper half space r n 1 we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation s q q q q 0 in r n for arbitrary space dimensions n 1 and all admissible exponents 0 this generalizes the nondegeneracy and uniqueness result for dimension n 1 recently obtained by the first two authors and in particular the uniqueness result for solitary waves of the benjamin ono equation found by amick and toland

quadratic time domain bem formulation for 2d elastodynamic transient analysis abstract an accurate and efficient time domain bem for the 2d elastodynamic wave transient analysis method is presented emphasis is focused on developing time domain fundamental convoluted kernels and methodology for quadratic temporal solution procedures which have never been presented before in the presented bem method called the qc method the temporal variations of displacement and traction are assumed to be quadratic and constant respectively the spatial variations of displacement and traction fields are assumed to be quadratic also the lc method assuming linear temporal variation of displacement field and constant temporal variation of traction field is examined in the paper for the verification of the qc method rectangular bars subject to different end loads are studied the numerical results by the qc method are compared with analytical solutions and those by the lc method numerical study reveals that the qc method is more accurate and stable in the qc method a good numerical result can still be obtained even when 1 0 2 0 however 0 50 0 75 should be used in the lc method in order to obtain good numerical results this would provide evidence that the qc method has great advantages over the lc method some conclusions regarding the qc method are also made

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

fractional schrodinger equation for a particle moving in a potential well in this paper the fractional schrodinger equation that contains the quantum riesz fractional derivative instead of the laplace operator is revisited for the case of a particle moving in the infinite potential well in the recent papers m jeng s l y xu e hawkins and j m schwarz on the nonlocality of the fractional schrodinger equation j math phys 51 062102 2010 10 1063 1 3430552 and s s bayin on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 10 1063 1 4705268 published in this journal controversial opinions regarding solutions to the fractional schrodinger equation for a particle moving in the infinite potential well that were derived by laskin fractals and quantum mechanics chaos 10 780 790 2000 10 1063 1 1050284 have been given in this paper a thorough mathematical treatment of these matters is provided the problem under consideration is reformulated in terms of three integral equations with the power

miscellanea a note on bayesian design for the normal linear model with unknown error variance summary the bayesian theory of optimal experimental design for the normal linear model has been developed under the assumption of known variance the insensitivity of specific design criteria to prior assumptions on the variance distribution has been demonstrated in special cases but a general result showing the way in which bayesian optimal designs are affected by prior information on the variance is lacking this note proves that bayesian designs are insensitive to information about

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

alpha continuity properties of the symmetric alpha stable process let d be a domain of finite lebesgue measure in d and let x t d be the symmetric stable process killed upon exiting d each element of the set i i 1 of eigenvalues associated to x t d regarded as a function of a 0 2 is right continuous in addition if d is lipschitz and bounded then each i is continuous in and the set of associated eigenfunctions is precompact

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

the dual boundary element method effective implementation for crack problems summary the present paper is concerned with the effective numerical implementation of the two dimensional dual boundary element method for linear elastic crack problems the dual equations of the method are the displacement and the traction boundary integral equations when the displacement equation is applied on one of the crack surfaces and the traction equation on the other general mixed mode crack problems can be solved with a single region formulation both crack surfaces are discretized with discontinuous quadratic boundary elements this strategy not only automatically satisfies the necessary conditions for the existence of the finite part integrals which occur naturally but also circumvents the problem of collocation at crack tips crack kinks and crack edge corners examples of geometries with edge and embedded crack are analysed with the present method highly accurate results are obtained when the stress intensity factor is evaluated with the j integral technique the accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed mode conditions

spectral properties of the dirichlet operator i 1d i2 s on domains in d dimensional euclidean space in this article we investigate the distribution of eigenvalues of the dirichlet pseudo differential operator i 1d i2 s s 0 1 on an open and bounded subdomain rd and predict bounds on the sum of the first n eigenvalues the counting function the riesz means and the trace of the heat kernel moreover utilizing the connection of coherent states to the semi classical approach of quantum mechanics we determine the sum for moments of eigenvalues of the associated schrodinger operator

asymptotic estimate of eigenvalues of pseudo differential operators in an interval abstract we prove a two term weyl type asymptotic law with error term o 1 n for the eigenvalues of the operator in an interval with zero exterior condition for complete bernstein functions such that converges to infinity as this extends previous results obtained by the authors for the fractional laplace operator 2 and for the klein gordon square root operator 1 2 1 2 1 the formula for the eigenvalues in a a is of the form n n 2 o 1 n where n is the solution of n n 2 a 1 a n and 0 2 is given as an integral involving

eigenvalues and minimizers for a non standard growth non local operator abstract in this article we study eigenvalues and minimizers of a fractional non standard growth problem we prove several properties on these quantities and their corresponding eigenfunctions

two dimensional transient wave propagation problems by time domain bem abstract in this paper an advanced formulation of time domain boundary element method bem for linear elastodynamics is used to study a number of problems involving wave propagation through half space as well as multi layered soils the algorithm incorporates isoparametric quadratic elements which facilitate proper modelling of problem geometry and can represent the field variables in dynamic problems very accurately which are very often wavy in nature also higher order temporal variation of functions is introduced improved techniques are employed for the accurate evaluation of both the singular and non singular spatial integrals most importantly this formulation incorporates simpler and better behaved kernels compared to those that have appeared in the recent bem literature by the present and other previous researchers with all these new and efficient features the present formulation is superior to the existing ones and as such represents a very effective tool for solving 2d transient wave propagation problems especially in infinite and semi infinite domains where other numerical methods have considerable difficulty in producing accurate solutions

continuity of eigenvalues of subordinate processes in domains let x x t t 0 be a symmetric markov process in a state space e and d an open set of e let s n s n t t 0 be a subordinator with laplace exponent n and s s t t 0 a subordinator with laplace exponent suppose that x is independent of s and s n in this paper we consider the subordinate processes open image in new window and open image in new window and their subprocesses open image in new window and x d killed upon leaving d suppose that the spectra of the semigroups of open image in new window and x d are all discrete with open image in new window being the eigenvalues of the generator of open image in new window and open image in new window being the eigenvalues of the generator of x d we show that if lim n n for every 0 then

interior stress calculations in 2 d time domain transient bem analysis abstract an improved and accurate formulation of time domain bem for determining the interior stresses in a 2 d transient dynamic analysis is presented a new set of simpler and better behaved transient dynamic stress kernels is derived with these interior stress kernels it has been possible to overcome all the analytical complexities that have hindered the accurate determination of interior stresses in the past the accuracy and stability are established via a few numerical examples including those involving stress concentrations

spectral properties of the operator of bessel potential type abstract let be a convex bounded domain in r m having regular boundary in this paper we study integral operators b m on l 2 of bessel potential type if n denotes the number of eigenvalues of b m that are for 0 we find the asymptotics of the regularized eigenvalue distribution function n d when 0 as a consequence we find the regularized traces of these operators

cracks in three dimensions a dynamic dual boundary element analysis in this paper the dual boundary integral equation for three dimensional dynamic problems in laplace space is presented for the first time the dual boundary element method is used to calculate dynamic stress intensity factors for three dimensional cracked structures the application of laplace transforms to the time dependent equations of elasticity reduce the problem to a static one in laplace space the displacement and traction boundary integral equations for the transformed variables are established by somigliana s identity by applying the displacement integral equation on one of the crack surfaces and the traction boundary integral equation on the other a general mixed mode crack problem can be solved in a single region in the same way as a static problem the transformed mixed mode stress intensity factors are calculated from the transformed displacement discontinuities near the crack tip the dynamic stress intensity factors in the time domain are obtained by the use of durbin s inversion method the accuracy of this method is demonstrated by its application to embedded penny shaped and elliptical cracks and edge cracks

a single region time domain bem for dynamic crack problems abstract a time domain boundary element method which allows the analysis of dynamic crack problems by using a single region formulation is presented the present method generates the distinct set of boundary integral equations by applying the displacement equation to one of the crack surfaces and the traction equation to the other the boundary of the structure is divided into continuous semi discontinuous and discontinuous quadratic elements the temporal variation of boundary displacements and tractions is approximated by piecewise linear and constant functions respectively the dynamic stress intensity factors are calculated using the crack opening displacements and the path independent integral this method is used to study dynamic behaviour of stationary cracks in finite and infinite domains in two dimensional analysis the results for two examples are compared with other reported solutions showing good agreement

fast estimation of expected information gains for bayesian experimental designs based on laplace approximations shannon type expected information gain can be used to evaluate the relevance of a proposed experiment subjected to uncertainty the estimation of such gain however relies on a double loop integration moreover its numerical integration in multi dimensional cases e g when using monte carlo sampling methods is therefore computationally too expensive for realistic physical models especially for those involving the solution of partial differential equations in this work we present a new methodology based on the laplace approximation for the integration of the posterior probability density function pdf to accelerate the estimation of the expected information gains in the model parameters and predictive quantities of interest we obtain a closed form approximation of the inner integral and the corresponding dominant error term in the cases where parameters are determined by the experiment such that only a single loop integration is needed to carry out the estimation of the expected information gain to deal with the issue of dimensionality in a complex problem we use a sparse quadrature for the integration over the prior pdf we demonstrate the accuracy efficiency and robustness of the proposed method via several nonlinear numerical examples including the designs of the scalar parameter in a one dimensional cubic polynomial function the design of the same scalar in a modified function with two indistinguishable parameters the resolution width and measurement time for a blurred single peak spectrum and the boundary source locations for impedance tomography in a square domain

nonlocal elliptic equations involving measures abstract in this article we study the existence of solution for the problem u f u in u 0 in r n where 0 is a parameter 0 1 and is a radon measure a weak solution is obtained by using schauder s fixed point theorem in the case where is dirac measure the symmetry of the solution is obtained by using the moving plane method