application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

optimized double sweep schwarz method by complete radiation boundary conditions we present an optimized double sweep nonoverlapping schwarz method for solving the helmholtz equation in semi infinite waveguides the domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high order absorbing boundary conditions are solved forward and backward sequentially for communication between subdomains neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and dirichlet data are exploited in the backward direction the complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find neumann data without introducing errors that would be produced if finite difference formulas were used for computing neumann data the convergence of the double sweep schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory

approximation of pseudodifferential operators in absorbing boundary conditions for hyperbolic equations engquist and majda 3 proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations in the case of the wave equation this boundary condition is valid at all frequencies here least squares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions it is shown that for the wave equation this approach leads to kreiss well posed initial boundary value problems and that the expectation of the reflected energy is lower than in the case of taylor and pade approximations 3 4 numerical examples indicate that this method works even more effectively for hyperbolic systems the least squares approach may be used to generate the boundary conditions automatically

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

on the stability of boundary conditions for molecular dynamics we study the stability of boundary conditions for molecular dynamics simulations a general stability criterion is established we first consider a one dimensional model with nearest neighbor interaction and multiple neighbor interactions we then generalize the results to more realistic models in 3d with nonlinear atomic interaction

complete radiation boundary conditions for the helmholtz equation i waveguides we consider the use of complete radiation boundary conditions for the solution of the helmholtz equation in waveguides a general analysis of well posedness convergence and finite element approximation is given in addition methods for the optimization of the boundary condition parameters are considered the theoretical results are illustrated by some simple numerical experiments

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

a comparative study of characteristic based algorithms for the maxwell equations characteristic based finite difference and finite volume schemes have been developed for solving the three dimensional maxwell equations in the time domain a detailed eigenvector analysis for the maxwell equations in a general curvilinear coordinate has also been completed to provide a basic framework for future finite difference schemes although the basic concepts of the two algorithms are identical the detailed formulations are vastly different for achieving split flux vectors according to the sign of the eigenvalues a comparative study of these algorithms applied to an oscillating electric dipole is carried out to assess their relative merit for further development in spherical coordinates second order windward numerical simulations of the radiating phenomenon are closely comparable in terms of accuracy and efficiency these methods also demonstrate an ability to suppress reflected waves from the truncated boundary by a simple compatibility condition

an optimized hybrid convolutional perfectly matched layer for efficient absorption of electromagnetic waves abstract a hybrid technique is studied in order to improve the performance of convolutional perfectly matched layer cpml in the finite difference time domain fdtd medium this technique combines the first order of higdon s annihilation equation as absorbing boundary condition abc with cpml to vanish the perfect electric conductor pec effects at the end of the cpml region an optimization algorithm is required to find optimum parameters of the proposed absorber in this investigation the particle swarm optimization pso is utilized with two separate objective functions in order to minimize the average and peak value of relative error using a standard test the overall performance of the proposed absorber is compared to the original cpml the results clearly illustrate this method provides approximately 10 db enhancements in cpml absorption error the performance memory and time requirement of the novel absorber hybrid cpml h cpml was analyzed during 2d and 3d tests and compared to most reported pmls the h cpml requirement of computer resources is similar to cpml and can simply be implemented to truncate fdtd domains furthermore an optimized set of parameters are provided to generalize the hybrid method

iterative radiation boundary conditions for finite element solutions of the scalar wave equation abstract new families of radiation boundary operators are proposed for finite element simulations of wave propagation problems characterized by the scalar wave equation these operators have the form b u j 1 c u j j 1 2 j where b is the dirichlet neumann or the robin operator and c is an appropriately chosen local operator since b is a standard operator compatible with the wave equation and c acts on the known iterate u j the radiation operators can be easily incorporated into a typical finite element code the proposed approaches are shown to produce very good results for the test cases considered

an efficient algorithm for truncating spatial domain in modeling light scattering by finite difference technique the finite difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry in practice this method requires that the spatial domain for the computation of near field be truncated an absorbing boundary condition must be imposed in conjunction with this truncation the performance of this boundary condition is essential to the stability of numerical computations and the reliability of results in the present study a new boundary condition referred to as the mixed t algorithm has been developed which is a generalization of the transmitting boundary condition originally developed by liao and co workers the present algorithm does not require spatial interpolation for wave values at interior grid points in addition it produces two minima of spurious reflections at small and large incident angles allowing efficient absorption of the scattered waves at the boundary for large incident angles when the third order mixed t algorithm is used the reflection coefficient of the boundary is less than 1 for incident angles from 0 to about 70 we find that the numerical instability associated with the transmitting boundary condition is caused by the location dependent amplitude of outgoing waves in the vicinity of the boundary for this reason the mixed t algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation when the stabilized algorithm is applied the near field within the truncated domain can be computed by using single precision arithmetic without overflows for more than 105steps in the time marching iteration finally the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a tm wave excited by a sinusoidal point source simultaneous simulation of the wave propagation in small and large domains and the scattering of a tm wave by an infinite circular cylinder

instability of difference models for hyperbolic initial boundary value problems a th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary according to this theory instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity c 0 to make this point of view precise we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries from these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms which extend earlier results due to osher and to gustafsson kreiss and sundstrom in particular we investigate l2 instability with respect to both initial and boundary data and show how they are affected by a finite versus infinite reflection coefficients and b wave radiation with c 0 versus c 0

pathloss calculation using the transmission line matrix and finite difference time domain methods with coarse grids numerical simulations of the wave equation that are intended to provide accurate time domain solutions require a computational mesh with grid points separated by a distance less than the wavelength of the source term and initial data however calculations of radio signal pathloss generally do not require accurate time domain solutions this paper describes an approach for calculating pathloss by using the finite difference time domain and transmission line matrix models of wave propagation on a grid with points separated by distances much greater than the signal wavelength the calculated pathloss can be kept close to the true value for freespace propagation with an appropriate selection of initial conditions this method can also simulate diffraction with an error governed by the ratio of the signal wavelength to the grid spacing

optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients when the coefficients of a problem have jumps of several orders of magnitude and are anisotropic many preconditioners and domain decomposition methods ddm suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system one way to improve the preconditioner is to use a linear algebra technique called deflation or very similarly coarse grid corrections in both cases it is necessary to identify and compute at least approximately all the eigenvectors corresponding to the bad eigenvalues in the framework of ddm we propose a way to design interface conditions so that convergence is fast and does not have any plateau the method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix the eigenvectors are not used the method relies on van der sluis result on a quasi optimal diagonal preconditioner for a symmetric positive definite matrix it is then possible to design robin interface conditions using only one real parameter for the entire interface by adding a second real parameter and more general interface conditions it is possible to take into account highly heterogeneous and anisotropic media numerical results are given and compared with other approaches

stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations abstract implicit noniterative finite difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations when applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable a stable as applied in practice the algorithms often face a severe time step restriction a major source of the difficulty is the treatment of the numerical boundary conditions one conjecture has been that unconditional stability requires implicit numerical boundary conditions an apparent counterexample was the space time extrapolation considered by gustafsson kreiss and sundstrom in this paper we examine space implicit and space time explicit extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals the results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit

termination of many 4 dimensional log flips we prove that any sequence of 4 dimensional log flips that begins with a klt pair x d such that k x d is numerically equivalent to an effective divisor terminates this implies termination of flips that begin with a log fano pair and termination of flips in a relative birational setting we also prove termination of directed flips with big k x d as a consequence we prove existence of minimal models of 4 dimensional dlt pairs of general type existence of 5 dimensional log flips and rationality of kodaira energy in dimension 4

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

the stretched mesh huygens absorbing boundary condition sm habc the huygens absorbing boundary condition habc is a new implementation of the analytical abcs which were in common use before the introduction of the perfectly matched layer pml abc in contrast to the traditional implementation the habc allows analytical abcs to be combined easily with other abcs this opens the way to combinations of two abcs where the advantages of both are preserved while their drawbacks are removed this is the case with the stretched mesh habc which relies on an analytical abc for the absorption of traveling waves and a real stretch of coordinates for the absorption of evanescent waves this paper demonstrates that such a combination can achieve a high absorption over the whole frequency spectrum and can challenge the pml abc in wave structure interaction problems

control of singular problem via differentiation of a min max abstract for a smooth domain in r n with boundary 0 1 we are concerned with the wave equation y y s in q t 0 t with t at source term satisfying s s l 1 0 t l 2 a dirichlet condition is imposed on 0 and we consider an absorbing condition y n uy 0 in 0 t gl 1 where u is the control parameter we introduce the cost function j u 1 2 t 0 1 uy 2 d d t and using the min max formulation of j we by pasas the sensitivity analysis of u y and obtain the gradient of j with a usual adjoint problem we first present an abstract frame for this kind of problems using the differentiability results of a min max 1 2 which we very shortly deduce here we show that the well posedness of the adjoint equation implies differentiability of the cost function governed by a linear well posed problem

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

unified formulation of radiation conditions for the wave equation a family of radiation boundary conditions for the wave equation is derived by truncating a rational function approximation of the corresponding plane wave representation and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities governed by second order wave equations on the radiating surface several well established radiation boundary conditions appear as special cases corresponding to different choice of the coefficients in the rational approximation the relation between these choices is established and an explicit formulation in terms of selected directions with ideal transmission is presented a mechanical interpretation of the fictitious surface densities enables identification of suitable conditions at corners and boundaries of the radiating surface numerical examples illustrate excellent results with one or two fictitious layers with suitable corner and boundary conditions

scattering of flexural waves by cavities in a plate abstract in this paper the scattering of slow flexural waves by arbitrary shaped cavities in an infinite elastic plate is studied using a combined finite element and analytical method the problem is considered as consisting of two interacting systems a bounded interior region containing all material and geometric irregularities and an unbounded exterior region the interior region is modelled by using mindlin type plate bending elements wave function expansion is used to represent the exterior region continuity of displacements and tractions are enforced at the nodes on the finite element interface with the exterior region comparison of present results for circular cavity with the analytical solution shows excellent agreement finally scattering by triangular and square shaped cavities as well as a pair of circular cavities is considered

cauchy characteristic evolution and waveforms we investigate a new methodology for the computation of waves generated by isolated sources this approach consists of a global spacetime evolution algorithm based on a cauchy initial value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior we match the two schemes at their common interface the characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval so that our numerical solution extends to infinity and accurately models the free space problem the matching interface need not be situated far from the sources the wavefronts may have arbitrary nonspherical geometry and strong nonlinearity may be present in both the interior and the exterior regions stability and second order convergence of the algorithms to the exact solution of the infinite domain problem are established numerically in three space dimensions the matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature for linear problems matching outperformed the local radiation conditions chosen for testing and was about as accurate for the same grid resolution as the exact nonlocal conditions however since the computational cost of the nonlocal conditions is many times that of matching this algorithm may be used with higher grid resolutions yielding a significantly higher final accuracy for strongly nonlinear problems matching was significantly more accurate than all other methods tested this seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field while matching consistently takes nonlinearity into account in both interior and exterior regions

stability theory of difference approximations for multidimensional initial boundary value problems an attempt is made to generalize results for arbitrary multidimensional difference schemes severe obstacles are encountered however for dissipative difference schemes the problem may be resolved

absorbing boundary conditions for the discretization schemes of the one dimensional wave equation when computing a partial differential equation it is often necessary to introduce artificial boundaries here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension fitting to the discretization scheme and stable moreover we give error estimates on the reflected part

derivation of identities involving some special polynomials and numbers via generating functions with applications the current article focus on the ordinary bernoulli euler and genocchi numbers and polynomials it introduces a new approach to obtain identities involving these special polynomials and numbers via generating functions as an application of the new approach an easy proof for the main result in 6 is given relationships between the genocchi and the bernoulli polynomials and numbers are obtained some interesting identities are discovered

analysis of conformal microstrip antennas with the discrete mode matching method an extension of the discrete mode matching dmm method is used to analyze microstrip antennas conformed to cylinders the structure needs to be discretized only along the azimuthal and the axial directions by sampling the fields and current densities in the interfaces between dielectric layers the solution in the radial direction is obtained analytically with the dyadic green s function which is calculated using the full wave equivalent circuit fwec in the spectral domain the structure is truncated in the axial direction by employing absorbing boundary conditions abcs the mathematical formulation is given in this paper followed by its application to compute the main properties of conformal microstrip antennas such as resonant frequency radiation patterns and input impedance validation with the commercial software hfss has been done and good agreement between the results has been observed

ascending chain condition for log canonical thresholds and termination of log flips we prove that the ascending chain condition acc for log canonical lc thresholds in dimension d existence of log flips in dimension d and the log minimal model program lmmp in dimension d 1 imply termination of any sequence of log flips starting with a d dimensional effective lc pair and also imply termination of flops in dimension d in particular the latter terminations in dimension 4 follow from the alexeev borisov conjecture in dimension 3

differential equation of appell polynomials via the factorization method let pn x n 0 be a sequence of polynomials of degree n we define two sequences of differential operators n and n satisfying the following properties n pn x pn 1 x n pn x pn 1 x by constructing these two operators for appell polynomials we determine their differential equations via the factorization method introduced by infeld and hull rev mod phys 23 1951 21 the differential equations for both bernoulli and euler polynomials are given as special cases of the appell polynomials

an accuracy analysis of boundary conditions for the forced shallow water equations the accuracy of numerical boundary conditions for the linearized one dimensional shallow water equations is determined by extending the dispersion fourier analysis originally presented by c k chu and a sereny j comput phys 15 476 1974 accuracy is measured by calculating steady state amplitudes of incoming and outgoing numerical waves at a boundary as functions of a forcing frequency radiating closed and forced radiating boundary conditions are studied in combination with two numerical schemes the richardson sielecki explicit finite difference scheme and the galerkin finite element method with piecewise linear basis functions and crank nicolson time stepping contamination of the finite element solution with short waves is seen to vary with the choice of extraneous boundary conditions an example of l n trefethen s j comput phys 49 199 1983 interpretation of the b gustafsson h kreiss and a sundstrom math comput 26 649 1972 instability is also given for the finite element method

application of a composition of generating functions for obtaining explicit formulas of polynomials abstract using notions of composita and composition of generating functions we obtain explicit formulas for the chebyshev polynomials the legendre polynomials the gegenbauer polynomials the associated laguerre polynomials the stirling polynomials the abel polynomials the bernoulli polynomials of the second kind the generalized bernoulli polynomials the euler polynomials the peters polynomials and the narumi polynomials

multi objective parameter identification of euler bernoulli beams under axial load abstract identification of physical parameters of the partial differential equation describing transverse vibrations of an axially loaded euler bernoulli beam ebb is proposed via a multi objective optimization formulation and solved by a genetic algorithm conflicting objectives such as performance and stability are specifically formulated and optimized simultaneously stability is quantified in terms of the solution s time growth factor physical parameter sets in the resulting pareto front approximation represent best trade offs with respect to the multiple objectives to compute output error performance objectives the ebb equation is discretized via finite differences in space and time and reformulated to a state space system identifiability is verified by checking regularity of the so called fisher information matrix the identification methodology is capable of determining material parameters including damping as well as the axial load from few spatially concentrated measurements its features are demonstrated and successfully validated on specific simulation data and measurement data obtained from a laboratory testbed

application of a complete radiation boundary condition for the helmholtz equation in locally perturbed waveguides abstract this paper deals with an application of a complete radiation boundary condition crbc for the helmholtz equation in locally perturbed waveguides the crbc one of efficient high order absorbing boundary conditions has been analyzed in straight waveguides in hagstrom and kim 2019 in this paper we apply crbc to the helmholtz equation posed in locally perturbed waveguides and establish the well posedness of the problem and convergence of crbc approximate solutions the new crbc proposed in this paper improves the one studied in hagstrom and kim 2019 in two aspects the first one is that the new crbc involves more damping parameters with the same computational cost as that of crbc in hagstrom and kim 2019 which results in 50 smaller reflection errors the second one is that the new crbc takes a neumann terminal condition of three term recurrence relations of auxiliary variables instead of a dirichlet terminal condition used in hagstrom and kim 2019 so that it can treat cutoff modes effectively finally we present numerical experiments illustrating the convergence theory

comparative evaluation of absorbing boundary conditions using green s functions for layered media absorbing boundary conditions are comparatively studied using the green s functions of the vector and scalar potentials for multilayer geometries and general sources since the absorbing boundaries are introduced as additional layers with predefined reflection coefficients into the calculation of the green s functions this approach provides an absolute measure of the effectiveness of different absorbing boundaries the green s functions are calculated using different reflection coefficients corresponding to different absorbing boundaries and compared to those obtained with no absorbing boundary it is observed that the perfectly matched layer pml is by far the best among the other absorbing boundary conditions whose reflection coefficients are available