utd diffraction coefficients for higher order wedge diffracted fields an asymptotic expansion of a multiple integral for knife edge diffraction is derived the expansion expresses the field in terms of single edge diffraction by considering this expression a similar expansion for higher order utd diffracted fields is proposed by means of a set of transition functions the result removes some of the shortcomings of the original set out of the utd when the incident field is not a ray optical field

high frequency green s function for planar periodic phased arrays with skewed grid and polygonal contour the derivation of an uniform high frequency asymptotic representation of the array green s function agf for finite planar periodic phased arrays with arbitrary polygonal contour and skewed grid is presented this result generalizes those obtained in a series of recent papers in the case of rectangular phased arrays with rectangular grid for the treatment of the high frequency phenomena the actual finite array is rigorously decomposed in terms of canonical constituents i e infinite semi infinite and sectoral arrays the final result is a representation of the agf in terms of spatially truncated floquet waves fws and fw induced diffracted fields arising from the edges and vertices of the polygonal rim of the array consequently the number of field contributions necessary to reconstruct the total field is independent of the number of elements of the array leading to a very efficient algorithm a series of numerical results is provided to demonstrate the effectiveness of the high frequency representation

the stability in _ and _ of the projection onto finite element function spaces on etablit la stabilite de la l 2 projection sur quelques espaces a elements finis v h consideree comme une application dans l p et w p 1 1 p sous des hypotheses de regularite faibles

an improved heuristic utd solution for multiple edge transition zone diffraction this paper describes a new heuristic approach for multiple edge diffraction modeling based on the uniform theory of diffraction utd it incorporates the slope diffraction terms which can be realized as a second order diffraction process vital for accurate predictions when more than one edge exists the predictions of this approach are compared with the vogler 1982 solution and were found to be in very good agreement for many different geometrical arrangements with low computational complexity the solution is also extended to wedge diffraction

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

the regularized chapman enskog expansion for scalar conservation laws rosenao r has recently proposed a regularized version of the chapmanenskog expansion of hydrodynamics this regularized expansion resembles the usual navier stokes viscosity terms at low wave numbers but unlike the latter it has the advantage of being a bounded macroscopic approximation to the linearized collision operator in this paper we study the behavior of the rosenau regularization of the chapman enskog expansion r c e in the context of scalar conservation laws we show that this r c e model retains the essential properties of the usual viscosity approximation e g existence of travelling waves monotonicity upper lipschitz continuity etc and at the same time it sharpens the standard viscous shock layers we prove that the regularized r c e approximation converges to the underlying inviscid entropy solution as its mean free path e 0 and we estimate the convergence rate

an efficient algorithm for the computation of the utd t transition function we present an efficient algorithm for the numerical calculation of the t canonical transition function which is encountered in the uniform geometrical theory of diffraction utd such a transition function allows the uniform ray field description of various high frequency diffraction mechanisms such as double wedge or vertex diffraction the proposed algorithm is valid for both real and imaginary arguments as required to deal with the general case in utd applications

a utd solution for multiple rounded surfaces this paper describes the diffraction mechanism when a ray optical field is obstructed by a cascade of smooth convex obstructions a new formulation of the uniform theory of diffraction utd is presented that incorporates the slope diffraction term the solution follows a generic approach that can be applied to arbitrary placed convex structures it will be shown that the slope diffraction term provides the required accuracy in radio propagation predictions when more than one obstruction exists in the propagation path the proposed solution is then compared with measurements and other analytical models with excellent agreement it is also found that the presented slope utd solution seems to be superior in terms of computation time and complexity while achieving very accurate results in addition when the radius of the cylinder tends to be very small the solution approaches the utd solution for multiple knife edge predictions

a heuristic utd slope diffraction coefficient for rough lossy wedges heuristic wedge diffraction coefficients for computing propagation path loss over finitely conducting earth are extended to include slope diffraction with the assumption that propagation of energy through the wedge is negligible the slope diffraction terms for the lossy wedge are obtained in an analogous manner as for the perfectly conducting case except that special care must be taken with the factor multiplying the incident field for grazing incidence results given show that the slope diffraction term produces continuous results that behave reasonably when compared with results for perfectly conducting wedges

a uniform double diffraction coefficient for a pair of wedges in arbitrary configuration an analytic closed form solution is presented for the double diffraction at a pair of arbitrarily placed wedges that is suitable to be used in a uniform theory of diffraction utd ray description framework here the particular assumption present in all the past literature that the two diffracting edges are coplanar is removed the doubly diffracted dd field at a pair of wedges is constructed via spectral synthesis such a procedure provides a double spectral integral representation of the dd field that is asymptotically evaluated by resorting to transition functions typical in double diffraction problems that can be expressed in terms of generalized fresnel integrals the final expression is arranged in the typical utd fashion and contains a uniform double diffraction dyadic coefficient the dd field behavior is analyzed in its transition regions to show how the dd field smoothly compensates for the total field discontinuities occurring at shadow boundaries where a singly diffracted field from a wedge is shadowed by the other wedge numerical examples are provided to verify such analysis and to test the accuracy of our utd ray description against method of moments full wave results

a slope utd solution for a cascade of multishaped canonical objects the case of multiple diffraction from a cascade of two dimensional multi shaped canonical objects such as wedges knife edges and cylinders is investigated a heuristic slope uniform theory of diffraction utd solution is presented that can give accurate and uniform field predictions at any area point and for any kind of environment the solution is then compared against measurements taken in an anechoic chamber and a very good agreement is observed furthermore an algorithm for implementing the proposed utd solution to rural area field predictions is also presented based on a novel algorithm for optimum fit of canonical shapes to the terrain irregularities comparisons between the model and measurements taken in real terrain profile show a very good agreement and uniform field predictions the importance of the choice of the used canonical object to terrain modeling is also discussed since different diffraction patterns are observed for different combination of the canonical objects

experimental and theoretical validation for the incremental theory of diffraction experimental and theoretical approaches to verify the validity of the incremental theory of diffraction itd are considered after providing a simple recipe for the application of the itd three geometries are examined for its validation first the itd formulation of the diffraction from a perfect electric conductor pec straight wedge is compared with the uniform theory of diffraction utd and with measurement results second the itd formulation of the diffraction from a pec disc is compared with measurement results and with the exact solution of a boundary value problem involving oblate spheroidal functions third the itd formulation of the diffraction from a hole in a pec plane is compared with the exact solution of a boundary value problem involving oblate spheroidal functions in particular this is the first time that itd results for diffraction from the disc and hole in a plane geometries are validated using exact solutions computed at a caustic in all cases examined very good agreement is found

a utd triple diffraction coefficient for straight wedges in arbitrary configuration a new utd solution is presented for the description of the ray triply diffracted by straight wedges in arbitrary configuration illuminated by a spherical wavefront field the proposed utd coefficient permits to uniformly describe the field at any aspect in particular including the case of overlapping transition regions where the subsequent application of utd single wedge diffraction coefficient fails the triple diffracted ray field is expressed in an analytical form providing physical insight into the triple diffraction mechanism and a new effective engineering tool within the utd framework as required in a modern ray based code

algorithm for the computation of the generalized fresnel integral the generalized fresnel integral appears as a canonical function in the uniform ray field representation of several high frequency diffraction mechanisms such as double wedge or vertex diffraction here we propose an algorithm for its calculation which is valid both for real and imaginary arguments as required for treating the general case in the uniform geometrical theory of diffraction framework

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

a modified diffraction coefficient for imperfect conducting wedges and buildings with finite dimensions multiple building diffraction loss calculation is often based on the assumption that their dimensions are infinite in the direction perpendicular to the propagation path even though in reality this assumption is not valid in this paper a modified diffraction coefficient model for the calculation of a more realistic diffraction loss caused by a single building with finite dimensions is proposed the overall diffraction loss is calculated for a group of buildings in a row by using the proposed model in addition after carrying out the field measurements it is shown that for some particular cases where the transmitter and diffracting obstacles are not far away from each other i e microcells or picocells spherical wave modeling produces a better accuracy than the plane wave approximation it is also demonstrated that the theoretical results can further be improved by considering the higher order terms in the calculation of diffraction coefficients lastly excess path loss obtained for a row of buildings with spherical wave assumption is compared with our measurement data this comparison shows reasonable correspondence between our theoretical model and measurements therefore our proposed model can successfully be used for improving the theoretical methods for predicting path loss in urban environments where diffraction from a number of buildings has to be considered

wide angle shift map pe for a piecewise linear terrain a finite difference approach a wide angle parabolic wave equation solution is presented using shift map and finite difference techniques the corresponding split step fourier solution is well known the solution using finite difference technique where the standard parabolic wave equation is modified into the so called claerbout equation allowing propagation angles up to 45deg from the paraxial direction is also well known here we present an extension to that solution in which the shift map technique is incorporated into the finite difference scheme allowing a varying terrain to be considered the result is a solution that corresponds to the well known split step solution which is believed to perform well for terrain slopes up to 10deg 15deg and discontinuous slope changes on the order of 15deg 20deg this solution is a first order one with respect to the terrain slope however when using the finite difference technique it is also possible to find a second order solution with respect to the terrain slope this new solution performs well for slopes up to about 15deg and discontinuous slope changes up to about 30deg which is an improvement

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

uniform asymptotic evaluation of surface integrals with polygonal integration domains in terms of utd transition functions the field scattered by a scattering body or by an aperture in the free space or in an unbounded homogenous medium can be described in terms of a double integral in this paper we show how a canonical integral on a polygonal domain with a constant amplitude function and a quadratic phase variation can be exactly expressed in terms of special functions namely fresnel integrals and generalized fresnel integrals this exact reduction represents a paradigm for deriving a new asymptotic evaluation for a more general integral this new asymptotic uniform integral evaluation is expressed in the format of the uniform geometrical theory of diffraction which is convenient for numerical computations

efficient algorithm for the evaluation of the physical optics scattering by nurbs surfaces with relatively general boundary condition an adaptive integration algorithm is presented for the computation of the physical optics po electric and magnetic field scattered by electrically large objects modeled by non uniform rational b splines nurbs the algorithm is the customization of a more general purpose result that has been recently published by using a unique formulation both impenetrable e g impedance surfaces coated conductors as well as transparent thin sheet materials e g thin dielectric panels or frequency selective surfaces are treated via their fresnel reflection and transmission coefficients the po radiation integral is evaluated over the nurbs parametric domain since most of the computer aided geometric design cagd tools are based on nurbs the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description with no need of generating new geometrical models furthermore the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard nyquist based sampling integration algorithms such reduction of the sampling points is achieved by resorting to high frequency technique concepts and allows a significant reduction of the cpu computational burden therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real life electrically large objects

multiple time domain diffraction for uwb signals the time domain td solution of the two dimensional multiple diffraction case is investigated the proposed td solution is based on the representation of the inverse fourier transform of the corresponding frequency domain fd solution in closed form as it is given by the uniform theory of diffraction utd and it incorporates the td representation of the higher order diffraction coefficients an algorithm to predict the td diffracted field after an arbitrary number of objects is also presented in the proposed algorithm different types of objects along the propagation path can be applied as well such as absorbing knife edges and metallic or nonperfectly conducting wedges the comparison between the td solution and the numerical inverse fast fourier transform of the fd solution proves the validity of the proposed solution

stability and analyticity in maximum norm for simplicial lagrange finite element semidiscretizations of parabolic equations with dirichlet boundary conditions stability and analyticity estimates in maximum norm are shown for spatially discrete finite element approximations based on simplicial lagrange elements for the model heat equation with dirichlet boundary conditions the bounds are logarithm free and valid in arbitrary dimension and for arbitrary polynomial degree the work continues an earlier study by schatz et al 5 in which neumann boundary conditions were considered

an efficient ray tracing algorithm for multiple straight wedge diffraction we present an efficient algorithm for the tracing of multiply edge diffracted rays the algorithm assumes a given sequence of infinite edges and complete visibility among them the ray tracing problem is formulated as the minimization of the ray total path length since such a cost function is strictly convex except for coplanar edges in the plane wave far field regime then the problem admits a unique global minimum and allows the use of a modified newton search algorithm which exhibits a very high converging rate we also propose a convenient starting point to effectively initialize the minimization algorithm the proposed algorithm is tested by some numerical examples that show its efficiency and effectiveness

asymptotic high frequency green s function for a large rectangular planar periodic phased array of dipoles with weakly tapered excitation in two dimensions this paper deals with the derivation and physical interpretation of a uniform high frequency representation of the green s function for a large planar rectangular phased array of dipoles with weakly varying excitation thereby our earlier published results valid for equiamplitude excitation and those for tapered illumination in one dimension are extended to tapering along both dimensions including dipole amplitudes tending to zero at the edges as previously the field obtained by direct summation over the contributions from the individual radiators is restructured into a double spectral integral whose high frequency asymptotic reduction yields a series of propagating and evanescent floquet waves fws together with corresponding fw modulated diffracted fields which arise from fw scattering at the array edges and vertexes to accommodate the weak amplitude tapering new generalized periodicity modulated edge and vertex slope transition functions are introduced accompanied by a systematic procedure for their numerical evaluation special attention is given to the analysis and physical interpretation of the complex vertex diffracted ray fields a sample calculation is included to demonstrate the accuracy of the asymptotic algorithm the resulting array green s function forms the basic building block for the full wave analysis of planar weakly amplitude tapered phased array antennas and for the description of electromagnetic radiation and scattering from weakly amplitude tapered rectangular periodic structures

a general uniform double wedge diffraction coefficient the uniform theory of diffraction utd is extended to include double diffraction by an arbitrary configuration of two wedges a scalar double diffraction coefficient is defined which is uniform throughout space while reducing to the product of single diffraction coefficients outside overlapping transition regions the result is formulated in terms of double fresnel integrals which can be rapidly evaluated

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

a note on double edge diffraction for parallel wedges an extended spectral theory of the diffraction approach is used in the investigation of double edge diffraction in the near field region of a parallel wedge geometry illuminated by a cylindrical wave a high frequency approximation can be formulated in terms of a canonical double stationary phase integral a uniform asymptotic reduction of the latter using the method of d s jones 1971 results in a compact representation involving the generalized fresnel integral with certain advantages in numerical applications over past representations

generalization of utd double diffraction to the case of impenetrable wedges with relatively general boundary conditions in the framework of the uniform geometrical theory of diffraction utd a novel high frequency solution is provided which allows the description of the rays doubly diffracted by a pair of impenetrable straight wedges with relatively general boundary conditions under a spherical wave illumination such a utd solution is heuristically derived by imposing the continuity of the total field at the shadow boundaries of the first order i e geometrical optics utd wedge single diffracted ray field utd solution it is the extension to the case of double diffraction of the first order solutions proposed by luebbers and by burnside and burgener it allows a uniform description of the field at any observation point including those aspects where transition regions overlap and the cascading of utd single wedge diffraction coefficients fails the proposed solution does not consider surface wave effects and for the special case of two perfectly conducting wedges it reduces to the formulation present in the literature of which it can be considered a generalization indeed it requires the same special functions involved in utd double diffraction by two perfectly conducting wedges the double diffracted ray field is expressed in analytical closed form providing a deeper understanding of the physics involved in double diffraction and a new powerful tool to be employed in any modern ray based code

equations involving fractional laplacian operator compactness and application abstract in this paper we consider the following problem involving fractional laplacian operator 1 u u 2 2 e u u in u 0 on where is a smooth bounded domain in r n e 0 2 2 0 1 2 2 n n 2 and is either the spectral fractional laplacian or the restricted fractional laplacian we show for problem 1 with the spectral fractional laplacian that for any sequence of solutions u n of 1 corresponding to e n 0 2 2 satisfying u n h c in the sobolev space h defined in 1 2 u n converges strongly in h provided that n 6 and 0 the same argument can also be used to obtain the same result for the restricted fractional laplacian an application of this compactness result is that problem 1 possesses infinitely many solutions under the same assumptions

numerical analysis of a nonlinear time relaxation model of fluids abstract this report investigates a time relaxation model for the regularization of the strong solution of the navier stokes equations following the earlier works of adams stolz and kleiser 37 38 ervin layton and neda 11 layton and neda 26 the nonlinear term investigated herein aims to better control the generalized turbulent fluctuations thus providing better stabilization effect on the underlying model we study the numerical properties of the resulting model and prove that the addition of the nonlinear term does not affect the overall order of convergence to the strong solution of the navier stokes equation our theoretical results based on the finite element method are confirmed by the numerical tests performed in the last section of the paper

liouville type theorems for nonlinear elliptic equations on extended grushin manifolds abstract in this paper we study two liouville theorems for nonlinear boundary value problems on extended grushin manifolds these problems are closely related to nonlinear fractional grushin equations we will prove that these problems do not possess nontrivial positive solution under some assumptions on the nonlinear terms we use the moving plane method in an integral form to prove our results

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

a novel utd type diffraction coefficient for a straight edge in a curved screen a novel uniform theory of diffraction utd type diffraction coefficient for a straight edge in a cylindrically curved screen valid for both convex and concave side wave incidence is obtained via uniform asymptotic theory of diffraction uat formulations the proposed curved screen utd type coefficient csutd allows uniform description of the fields at all aspect angles including the grazing incidence case where utd coefficient based predictions deviate sharply from available measurements when it is applied to the problem of monostatic radar cross section rcs of cylindrically curved plate

the numerical steepest descent path method for calculating physical optics integrals on smooth conducting quadratic surfaces in this paper we use the numerical steepest descent path nsdp method to analyze the highly oscillatory physical optics po integral on smooth conducting parabolic surfaces including both monostatic and bistatic cases quadratic variations of the amplitude and phase functions are used to approximate the integrand of po integral then the surface po integral is reduced into several highly oscillatory line integrals by invoking the nsdp method these highly oscillatory po line integrals are defined on the corresponding nsdps furthermore the critical point contributions for the po integral are exactly extracted and represented based on the nsdps the proposed nsdp method for calculating the po integral on the smooth conducting surfaces is frequency independent and error controllable compared with the traditional asymptotic expansion approach the nsdp method significantly improves the po integral accuracy by around two digits when the working wave frequencies are not extremely large numerical results are given to validate the nsdp method

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

numerical diffraction of array guided surface waves on finite frequency selective surfaces in this paper an adaptation of hurst s method to take into account the presence of array guided surface waves in the calculation of the scattered field of finite frequency selective surfaces in free space is presented numerical approaches to determine the guided wave s wavenumber needed in this calculation namely prony s method and the discrete fourier transform dft are examined and compared the results show that the scattered fields obtained with the proposed method which is based on numerically extracted diffraction coefficients are in very good agreement with those obtained with the method of moments

calculation of higher order diffracted fields for multiple edge transition zone diffraction a well known problem in conventional uniform theory of diffraction utd is multiple edge transition zone diffraction here higher order diffracted fields have to be used in order to improve the result and these fields are added by means of a series unfortunately this series will converge slowly for many edges which means that a good enough result might require a large number of terms however by summing up the series properly the computation time can be kept reasonable which will be shown for knife edges the summation technique is though valid for wedges as well