fast factorization acceleration of mom compressive domain decomposition domain decomposition dd for integral equation can be achieved by aggregating standard basis functions into specialized basis functions on each sub domain this results in a strong compression of the mom matrix which allows an iteration free e g lu decomposition solution also for electrically large problems fast matrix vector product algorithms can be used in the matrix filling and compression process of the employed aggregate functions approach this hybrid approach has received considerable attention in recent literature in order to quantitatively assess the performance advantages and limitations of this class of methods we start by proposing and demonstrating the use of the adaptive integral method aim fast factorization to accelerate the synthetic function expansion sfx dd approach the method remains iteration free with a significant boost in memory and time performances with analytical predictions of complexity scalings confirmed by numerical results then we address the complexity scaling of both stand alone dd and its combined use with fast mom this is done analytically and discussed with respect to known literature accounts of various implementations of the dd paradigm with nonobvious results that highlight needs and limitations and yielding practical indications

the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

curve straightening and the palais smale condition this paper considers the negative gradient trajectories associated with the modified total squared curvature functional f k2 iv ds the focus is on the limiting behavior as m tends to zero from the positive side it is shown that when m 0 spaces of curves exist in which some trajectories converge and others diverge in one instance the collection of critical points splits into two subsets as m tends to zero the critical curves in the first subset tend to the critical points present when m 0 meanwhile all the critical points in the second subset have lengths that tend to infinity it is shown that this is the only way the palais smale condition fails in the present context the behavior of the second class of critical points supports the view that some of the trajectories are dragged all the way to infinity when the curves are rescaled to have constant length the euler figure eight emerges as a critical point at infinity it is discovered that a reflectional symmetry need not be preserved along the trajectories there are examples where the length of the curves along the same trajectory is not a monotone fiunction of the flow time it is shown how to determine the elliptic modulus of the critical curves in all the standard cases the modulus p must satisfy 2e p k p 1 gl l when the space is limited to curves of fixed length l and the endpoints are separated by the vector g

electromagnetic analysis of large scale periodic arrays using a two level cbfs method accelerated with fmm fft to realize efficient analysis of large scale planar periodic arrays a two level characteristic basis functions method accelerated with fast multipole method fast fourier transform fmm fft is developed in this paper the characteristic basis functions cbfs for each element are constructed in a two level framework in which the cbfs construction based on plane wave derivation for single element at child level and the cbfs construction based on local interaction between array elements at father level are combined the number of unknowns is reduced significantly by the two level cbfs further fmm fft is applied to expedite the computation of interaction between elements numerical results of large scale planar periodic arrays are given to demonstrate the validity and efficiency of the present method

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

investigation of scattering properties of large scale aperiodic tilings using a combination of the characteristic basis function and adaptive integral methods this paper presents a hybrid approach for efficient analysis of electromagnetic em scattering from large scale aperiodic structures e g aperiodic penrose and danzer tilings which integrates the characteristic basis function method cbfm and the adaptive integral method aim by performing a domain decomposition a series of characteristic basis functions cbfs that are defined on a macro block and comprised of a relatively large number of sub domain basis functions facilitate a substantial reduction in the method of moments mom matrix size enabling the use of a direct solver for large problems the aim is applied to accelerate the calculation of cbfm reduced mom matrices significantly decreasing the cpu time and memory required for solving large scale problems as the size of one block becomes electrically large the original cbfm combined with the aim is employed to generate the initial cbfs by solving a large problem with multiple excitations which results in efficiently constructing the final cbfs afforded by the singular value decomposition svd procedure this methodology produces a two level cbfm aim hybrid algorithm for efficiently characterizing large scale objects the numerical results presented demonstrate the accuracy and efficiency of the proposed hybrid algorithm then the developed solver is applied to investigate em scattering properties of large scale aperiodic tilings the numerical results show that penrose danzer tilings exhibit significantly improved grating lobe suppression as compared to their periodic counterparts

an efficient symplectic reverse time migration method using a local nearly analytic discrete operator in acoustic transversely isotropic media with a vertical symmetry axis abstractreverse time migration rtm has become an extremely important imaging method due to its ability of handling complicated velocity models in recent years there has been an increasing need to perform rtm with anisotropic velocity models such as transversely isotropic media with a vertical symmetry axis vti finite difference fd is one of the most widely used numerical methods in rtm however it often suffers from serious numerical dispersion resulting in significant loss of imaging resolution over the years a nearly analytic discrete operator has been developed to approximate the high order spatial partial differential operators which is effective in suppressing numerical dispersion using this operator we developed a nearly analytic symplectic nas method to numerically solve acoustic vti wave equations which preserves the symplectic structure we have investigated its properties including the stability condition numerical dispersion relation and computational efficiency also we a

a spectral domain integral equation method utilizing analytically derived characteristic basis functions for the scattering from large faceted objects a novel technique based on a spectral domain integral equation method with analytically derived characteristic basis functions is introduced in this paper it enables us to treat scattering problems from electrically large faceted bodies in a numerically rigorous and computationally efficient manner in terms of both time and memory the analytically derived characteristic basis functions include certain desirable features of the asymptotic schemes and are defined on subdomains that can be electrically large not being bound to the typical discretization size of the conventional method of moments by properly weighting through a galerkin procedure the resulting electric field integral equation the problem is reduced to a matrix equation having dimensions that do not depend on the size of the scatterer but only on its shape electrically large problems can be handled in a computationally efficient manner by using the proposed method since the associated matrix size is relatively small moreover all the reduced matrix elements are calculated in the spectral domain without evaluating any convolution products

combining the moment method with geometrical modelling by nurbs surfaces and bezier patches induced current distributions on conducting bodies of arbitrary shape modelled by nurbs non uniform rational b splines surfaces are obtained by using a moment method approach to solve an electric field integral equation efie the nurbs surfaces are expanded in terms of bezier patches by applying the cox de boor transformation algorithm this transformation is justified because bezier patches are numerically more stable than nurbs surfaces new basis functions have been developed which extend over pairs of bezier patches these basis functions can be considered as a generalization of rooftop functions the method is applied to obtain rcs values of several objects modelled with nurbs surfaces good agreement with results from other methods is observed the method is efficient and versatile because it uses geometrical modelling tools that are quite powerful

environmental bias and elastic curves on surfaces the behavior of an elastic curve bound to a surface will reflect the geometry of its environment this may occur in an obvious way the curve may deform freely along directions tangent to the surface but not along the surface normal however even if the energy itself is symmetric in the curve s geodesic and normal curvatures which control these modes very distinct roles are played by the two if the elastic curve binds preferentially on one side or is itself assembled on the surface not only would one expect the bending moduli associated with the two modes to differ binding along specific directions reflected in spontaneous values of these curvatures may be favored the shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of lagrange multipliers to the darboux frame associated with the curve the forces transmitted to the surface along the surface normal will be determined features associated with a number of different energies both of physical relevance and of mathematical interest are described the conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined

stable p wave modeling for reverse time migration in tilted ti media we present an approach for p wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis tti media suitable for anisotropic reverse time migration the proposed approach is based on wave equations derived from first principles the equations of motion and hooke s law under the acoustic ti approximation consequently no assumptions are made about the spatial variation of medium parameters a rotation of the stress and strain tensors to a local coordinate system aligned with the ti symmetry axis makes it possible to benefit from the simple and sparse form of the ti elastic tensor in that system the resulting wave equations can be formulated either as a set of five first order or as a set of two second order partial differential equations for the constant density case the second order tti wave equations involve mixed and nonmixed second order spatial derivatives with respect to global nonrotated coordinates we propose a numerical implementation of these equations using high order centered finite differences to minimize modeling artifacts related to the use of centered first derivative operators we use discrete second derivative operators for the nonmixed second order spatial derivatives and repeated discrete first derivative operators for the mixed derivatives such a combination of finite difference operators leads to a stable wave propagator provided that the operators are designed properly in practice stability is achieved by slightly weighting down terms that contain mixed derivatives this has a minor practically negligible effect on the kinematics of wave propagation the stability of the presented scheme in inhomogeneous tti models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples

linear embedding via green s operators a modeling technique for finite electromagnetic band gap structures we propose a modular electromagnetic modeling procedure for large finite electromagnetic band gap ebg structures called linear embedding via green s operators it is a diakoptic method based on the huygens schelkunoff principle involving equivalent boundary current sources that electromagnetically characterize the enclosed domain of arbitrary shapes as if it were a multiport system in a cascade of embedding steps separate reusable domains are combined to form larger domains device design often involves tuning local medium properties in a compact designated domain with a large environment through an additional embedding step the equivalent sources describing the environment can be transferred to the boundary of the designated domain rendering subsequent design steps very fast this two stage optimization process is applied in the design of an ebg power splitter

fast analysis of a dual band reflectarray using two different numerical approaches based on the moment method this communication presents two efficient approaches for the electromagnetic field analysis of complex 3d bodies the first method can provide a fast and accurate analysis of arbitrary metallic and dielectric magnetic structures and combines the moment method mm the multilevel fast multipole algorithm mlfma physics optics po and the characteristic basis function method cbfm to solve very large scattering problems the second technique is based on a domain decomposition approach that divides the geometry into several parts to minimize the vast computational resources required when applying the mm a parallelization process was carried out by using the message passing interface mpi algorithm to minimize the memory and time requirements for each simulation a dual band reflectarray was successfully analyzed to compare the performance of both codes

a survey on symplectic and multi symplectic algorithms we simply review symplectic and multi symplectic algorithms in the paper these algorithms have many superiorities to general non symplectic algorithms such as they have long time behavior they can preserve the symplectic geometry structure of the hamiltonian system exactly and they can simulate original physical phenomena well some commonly used methods to construct symplectic and multi symplectic schemes are simply surveyed numerical examples are also enumerated to illustrate their performances

a moment method approach using frequency independent parametric meshes a moment method mm formulation using nonuniform rational b splines nurbs parametric surfaces is introduced to analyze radiation and scattering from arbitrarily shaped conducting bodies a feature of the technique presented is the independence of the geometrical model from the frequency which avoids the regeneration of the geometry model when the frequency changes this feature is obtained by defining new sets of basis and testing functions that extend over subpatches in each nurbs the proposed approach is accurate flexible and efficient

application of the characteristic basis function method utilizing a class of basis and testing functions defined on nurbs patches a novel implementation of the characteristic basis function method cbfm is given in which the high level basis functions called characteristic basis functions cbfs are represented in terms of curved rooftops generated from nonuniform rational b splines nurbs in the parametric u v domain the associated macro testing functions are defined by using curved razor blade functions corresponding to each rooftop the underlying objective of the cbfm is the reduction of the number of unknowns that arise from the discretization process when applying the conventional method of moments mom the result is therefore an approach which can handle many complex cases via direct solvers without suffering from convergence problems as many of the iterative techniques are known to do when dealing with ill conditioned matrices as a result of the combination of the cbfm with the special class of low level basis and testing functions directly located over nurbs surfaces complex and realistic geometries can be efficiently analyzed to yield accurate results while reducing the cpu time as well as required memory resources

confinement of semiflexible polymers a variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface as an application the confinement of a closed polymer loop of fixed length 2 r within a spherical cavity of smaller radius r 0 is considered it is shown that an infinite number of distinct periodic completely attached equilibrium states exist labeled by two integers n 2 3 4 and p 1 2 3 the number of periods of the polar and azimuthal angles respectively small loops oscillate about a geodesic circle n 2 p 1 is the stable ground state states with higher n exhibit instabilities if r 2r 0 new states appear as oscillations about a doubly covered geodesic circle the state n 3 p 2 replaces the twofold as the ground state in a finite band of values of r with increasing r loop states make a transition from oscillatory and orbital behavior on crossing the poles returning to oscillation upon collapse to a multiple cover of a geodesic circle signaled respectively by an increase in p and an increase in n the force transmitted to the surface does not increase monotonically with loop size but does asymptotically it behaves discontinuously where n changes the contribution to energy from geodesic curvature is bounded in large loops the energy becomes dominated by a state independent contribution proportional to the loop size the energy gap between the ground state and excited states disappears

efficient analysis of large scatterers by physical optics driven method of moments a new iterative procedure is presented that enables method of moment mom solution of scattered field from electrically large and complex perfectly conducting bodies using significantly reduced number of unknown coefficients in each iteration the body is excited by a plane wave and by the currents which are obtained as an approximate solution in the previous iteration the physical optics po and modified po techniques are used to determine the po and the correctional po currents which are expressed in terms of original mom basis functions and grouped into macro basis functions mbfs weighting coefficients of all mbfs are determined from the condition that mean square value of residuum of original mom matrix equation is minimized the iterative procedure finishes when the residuum decreases below the maximum allowed value the accuracy and efficiency of the proposed method are illustrated on two examples cube scatterer and airplane scatterer since the construction of mbfs by po and modified po techniques ensures fast convergence to the original mom solution the method is named po driven mom

elastic circles in 2 spheres by solving the euler lagrange equations we determine the elastic curves in the two dimensional sphere which are circular at rest we characterize the family of closed critical curves by a rational condition and the existence of closed elastic curves for the different possible values of their curvature at rest is proved in the final part of the paper we utilize a numerical approach to get a better understanding of the space of closed critical curves in this manner we analyse their shape uniqueness and minimizing properties

an iterative solution for electrically large problems combining the characteristic basis function method and the multilevel fast multipole algorithm an iterative scheme for the rigorous computation of electrically large problems is presented the approach is based on a combination of the characteristic basis function method cbfm and the multilevel fast multipole algorithm mlfma that can deal with very large problems that require an iterative solution process even considering that the application of the cbfm entails an important reduction of the number of unknowns when compared to method of moments approaches based on subdomain functions this reduction is due to the fact that the number of macro basis functions called characteristic basis functions cbfs is lesser than the number of low level subsectional functions used to sample the geometry in addition the use of the mlfma avoids the need to calculate and store the coupling terms in the reduced matrix that are not on or close to the diagonal thereby optimizing the cpu time and the memory storage requirements non uniform rational b splines nurbs surfaces are employed for the representation of the geometry and the cbfs are described in terms of curved rooftops generated in the parametric space the associated macro testing functions are defined as aggregations of curved razor blade functions

analysis of large complex structures with the synthetic functions approach an innovative procedure is presented that allows the method of moments mom analysis of large and complex antenna and scattering problems at a reduced memory and cpu cost bounded within the resources provided by a standard 32 bit personal computer the method is based on the separation of the overall geometry into smaller portions called blocks and on the degrees of freedom of the field the blocks need not be electrically unconnected on each block basis functions are generated with support on the entire block that are subsequently used as basis functions for the analysis of the complete structure only a small number of these functions is required to obtain an accurate solution therefore the overall number of unknowns is drastically reduced with consequent impact on storage and solution time these entire domain basis functions are called synthetic functions they are generated from the solution of the electromagnetic problem for the block in isolation under excitation by suitably defined sources the synthetic functions are obtained from the responses to all sources via a procedure based on the singular value decomposition because of the strong reduction of the global number of unknowns one can store the mom matrix and afford a direct solution the method is kernel free and can be implemented on existing mom codes

wave field interaction with complex structures using equivalence principle algorithm a domain decomposition scheme based on the equivalence principle similar to huygens principle for integral equation solvers and the method of moments is introduced the equivalence principle allows the replacement of unknown currents distributed in a volume in space by equivalence currents residing on the surface that bounds the volume it also allows the dissociation of the solution of one region from that of another region in this manner problems of high complexity can be encapsulated by surfaces of simpler shapes using less unknowns it can aid in parallel algorithms reusability of solutions as well as improving the condition number of a matrix system when disparate mesh or adaptive mesh are needed the challenge arises when an equivalence surface intercepts a current carrying conductor because the breakup of the current into separate pieces gives rise to charge singularity a junction basis can be used to mitigate this singularity however a better solution is to introduce a tap basis to model the current that intercepts with the equivalence surfaces using this scheme the current continuity is conserved and the singularity of the charges is avoided the solution is shown to be accurate