the bymoment method for two dimensional electromagnetic scattering the bymoment method is presented for analyzing two dimensional electromagnetic wave scattering in an unbounded region the method introduces a conforming surface to geometrically decouple the interior region containing the scatterer from the exterior region extending to infinity the solution in the interior is generated using the standard finite element solution of an interior dirichlet boundary value problem the interior solution is then coupled to the exterior using green s theorem in the unbounded exterior region the validity of the method and the associated computer program is demonstrated by comparing the results to those of other methods for scattering by various cylindrical geometries

caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

nonconformal feti dp domain decomposition methods for fe bi mlfma the nonconformal domain decomposition methods ddms of the hybrid finite element boundary integral multilevel fast multipole algorithm fe bi mlfma are presented for computing large electromagnetic scattering radiation problems the numerical performance of different transmission conditions between finite element method fem subdomains and fem and bi domains is studied in this paper the dirichlet transmission condition dtc employed at the interface between fem and bi domains has faster convergence than the robin type transmission condition and this finding is different from the conclusions for the fem ddms published in the literature furthermore the cement element ce based dual primal finite element tearing and interconnecting feti dp ddm has faster convergence than the lagrange multiplier based feti dp method under the fe bi system and is also different from that under the system of the fem with the absorbing boundary condition fem abc it shows that the ddms of fe bi mlfma have some essential differences from those of fem abc the analysis of these numerical phenomena is presented in this paper further comparison between the ddms of fe bi mlfma and fem abc demonstrates that the ddms of fe bi mlmfa have essential advantages over those of fem abc various numerical experiments confirm the accuracy efficiency and flexibility of the nonconformal ddm of fe bi mlfma which is the ce based feti dp ddm with dtc connecting fem and bi

marching on in degree solver of time domain finite element boundary integral method for transient electromagnetic analysis a marching on in degree solver of time domain finite element boundary integral method is proposed for the analysis of transient electromagnetic scattering from inhomogeneous objects weighted laguerre polynomials are employed as temporal basis functions and testing functions in the proposed algorithm which leads to a recursive matrix equation without time variable and it can be solved degree by degree then the time domain response can be obtained by post processing furthermore a hybrid algorithm using both direct method and iterative method is introduced to accelerate the convergence in the solving of matrix equation numerical results are presented to demonstrate the accuracy stability and effectiveness of the proposed method

implementation and experiments of a hybrid algorithm of the mlfma enhanced fe bi method for open region inhomogeneous electromagnetic problems although the computational complexity of the finite element boundary integral fe bi method is significantly reduced by the multilevel fast multipole algorithm mlfma this mlfma enhanced fe bi solution experiences a very slow convergence for some complex inhomogeneous problems a hybrid algorithm combining direct methods with iterative methods is designed to speed up the rate of convergence of this mlfma enhanced fe bi solution this hybrid algorithm is efficiently implemented with the aid of a newly developed package superlu of the lu decomposition solver numerical experiments are performed for scattering by a coated northrop wing to demonstrate the efficiency of this hybrid algorithm more importantly the thorough investigation of the numerical experiments clearly shows the better accuracy stability and robustness of this hybrid algorithm over the conventional algorithms

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

asymptotic dynamics of the dual billiard transformation given a strictly convex plane curve the dual billiard transformation is the transformation of its exterior defined as follows given a point x outside the curve draw a support line to it from the point and reflect x at the support point we show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a hamiltonian flow associated with the curve

a hybrid finite element method for 3 d scattering using nodal and edge elements a hybrid finite element method for three dimensional scattering is presented and numerical examples shown this approach which couples finite element discretization with the method of moments is particularly well suited for monostatic radar cross section calculations the method is based on a scalar and vector potential formulation of maxwell s equations the use of nodal elements and a highly efficient body of revolution implementation of the method of moments combined nodal and edge elements are employed to accurately model fields around corners and edges a curvature based sampling criterion is derived and shown to ensure accurate answers for highly curved scatterers numerical results and cray computer timings are compared with published results for an edge element code using radiation boundary conditions

a flexible and efficient higher order fe bi mlfma for scattering by a large body with deep cavities the hybrid finite element boundary integral multilevel fast multipole algorithm fe bi mlfma is applied to solve the challenge problem of scattering by a large body with deep cavities in this paper the hierarchical higher order curvilinear vector finite element hcvfe is employed to reduce the numerical dispersion error in the fem and to efficiently model the geometry of the cavity the coupling approaches are investigated at the interface between the fe region and the bi region for handling the problem of the different order basis functions and meshes used in the fe and bi region the problems encountered with the previous decomposition algorithm of fe bi mlfma are pointed out and analyzed in this paper a special algorithm of fe bi mlfma is designed based on the distinct geometry characteristics of deep cavity numerical results are presented to demonstrate the accuracy efficiency and flexibility of the higher order fe bi mlfma for scattering by a large body with big and deep cavities

inward looking and outward looking formulations for scattering from penetrable objects a number of approaches for solving the problem of scattering from two dimensional penetrable objects using the finite element method or other differential equation based solutions are reviewed different ways of joining the numerical solution on a finite domain to a field representation that satisfies the sommerfield radiation condition in the infinite exterior domain are also reviewed within the paradigm of the equivalence principle of electromagnetics depending on the region where the solution is deduced first in an inherently sequential process these approaches can often be classified as either inward looking or outward looking although inward looking formulations may exhibit computational advantages outward looking formulations appear preferable for electrically large domains because of internal resonance difficulties that may arise when solving the isolated interior problem first in an inward looking formulation

integral equation procedures for eddy current problems abstract two methods are presented for the solution of a class of two dimensional eddy current problems the specific problem considered is that of a wire carrying a periodic current parallel to a uniform conducting cylinder both methods yield integral equations over the boundary of the conductor the first method is exact while the second is an asymptotic procedure valid for a certain parameter range and reflecting the skin effect numerical implementation procedures are given for both methods and some numerical results are presented

coupling finite element and integral equation solutions using decoupled boundary meshes electromagnetic scattering a method is outlined for calculating scattered fields from inhomogeneous penetrable objects using a coupled finite element integral equation solution the finite element equation can efficiently model fields in penetrable and inhomogeneous regions while the integral equation exactly models fields on the finite element mesh boundary and in the exterior region by decoupling the interior finite element and exterior integral equation meshes considerable flexibility is found in both the number of field expansion points as well as their density only the nonmetal portions of the object need be modeled using a finite element expansion exterior perfect conducting surfaces are modeled using an integral equation with a single unknown field since e sub tan is identically zero on these surfaces numerical convergence accuracy and stability at interior resonant frequencies are studied in detail

a highly effective preconditioner for solving the finite element boundary integral matrix equation of 3 d scattering a highly effective preconditioner is presented for solving the system of equations obtained from the application of the hybrid finite element boundary integral fe bi method to three dimensional 3 d electromagnetic scattering problems different from widely used algebraic preconditioners the proposed one is based on a physical approximation and is constructed from the finite element method fem using an absorbing boundary condition abc on the truncation boundary it is shown that the large eigenvalues of the finite element fe abc system are similar to those of the fe bi system hence the preconditioned system has a spectrum distribution clustered around 1 in the complex plane consequently when a krylov subspace based method is employed to solve the preconditioned system the convergence can be greatly accelerated numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems

on the formulation of hybrid finite element and boundary integral methods for 3 d scattering this paper studies in detail a variety of formulations for the hybrid finite element and boundary integral fe bi method for three dimensional 3 d electromagnetic scattering by inhomogeneous objects it is shown that the efficiency and accuracy of the fe bi method depends highly on the formulation and discretization of the boundary integral equation bie used a simple analysis of the matrix condition number identifies the efficiency of the different fe bi formulations and an analysis of weighting functions shows that the traditional fe bi formulations cannot produce accurate solutions a new formulation is then proposed and numerical results show that the resulting solution has a good efficiency and accuracy and is completely immune to the problem of interior resonance finally the multilevel fast multipole algorithm mlfma is employed to significantly reduce the memory requirement and computational complexity of the proposed fe bi method

dual polygonal billiards and necklace dynamics we study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics our main result is that for a certain class of tables called the quasi rational polygons the dual billiard orbits are bounded this implies that for the subset of rational tables i e polygons with rational vertices the dual billiard orbits are periodic

a finite element boundary integral formulation for scattering by three dimensional cavity backed apertures a novel numerical technique is proposed for the electromagnetic characterization of the scattering by a three dimensional cavity backed aperture in an infinite ground plane the technique combines the finite element and boundary integral methods to formulate a system of equations for the solution of the aperture fields and those inside the cavity specifically the finite element method is used to formulate the fields in the cavity region and the boundary integral approach is used in conjunction with the equivalence principle to represent the fields above the ground plane unlike traditional approaches the proposed technique does not require a knowledge of the cavity s green s function and is therefore applicable to arbitrary shape depressions and material fillings furthermore the proposed formulation leads to a system having a partly full and partly sparse as well as symmetric and banded matrix which can be solved efficiently using special algorithms

finite element analysis of a scattering problem a finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed the method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition the space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics

a note on hybrid finite element method for solving scattering problems the hybrid finite element formulation hfem originated by p silvester and m s hsieh 1971 is modified in such a way that it results in a sparse or uniformly banded matrix rather than a partly full and partly sparse nonuniform matrix the modification is accomplished by changing the sequence of matrix substitutions and substantially improves the computational efficiency and enhances the capability of the method which is demonstrated by numerical examples a comparison with other numerical techniques is presented

an amplitude finite element formulation for electromagnetic radiation and scattering electromagnetic radiation and scattering in an exterior domain within the context of a finite element method has traditionally involved imposing a suitable absorbing boundary condition on the truncation boundary of the numerical domain to inhibit reflection from it in this work based on the wilcox asymptotic expansion of the electric far field we propose an amplitude formulation within the framework of the nodal finite element method whereby the highly oscillatory radial part of the field is separated out a priori so that the standard lagrange interpolation functions that are used have to capture a relatively gently varying function since these elements can be used in the immediate vicinity of the radiator or scatterer with few exceptions which we enumerate it is more effective compared to methods that impose absorbing boundary conditions at the truncation boundary especially for high frequency problems the proposed method is based on the standard galerkin finite element formulation and uses standard lagrange interpolation functions standard gaussian quadrature and the same degrees of freedom as a conventional formulation we show the effectiveness of the proposed formulation on a wide variety of radiation and scattering problems involving both conducting and dielectric bodies and involving both convex and non convex domains with sharp corners

numerically derived boundary conditions on artificial boundaries abstract we consider partial differential equations in an infinite domain in which an artificial boundary b is introduced in order to restrict the computational domain to the region bounded by b the nonlocal boundary condition on b is determined for equations of the type 2 u k 2 u 0 in a separable coordinate system and compared with two methods in which the boundary condition is approximated one method uses the free space green s function directly and does not involve the evaluation of surface integrals the other method in which the boundary condition is derived from the solution of the dirichlet problem in the domain exterior to b is considered by several authors in the literature using laplace s equation in two dimensions numerical results show that green s function approach is accurate and that the boundary condition can be computed readily with standard numerical packages

global numerical boundary condition based pde solution techniques for open region electromagnetic field problems the underlying concept and algorithm for global numerical boundary condition gnbc based pde solution techniques for open region emf problems are presented this numerical procedure has been applied with success for various boundary value problems in scattering waveguide and radiation analysis showing distinct advantages in generality flexibility and simplicity all the gnbcs treated are of the first kind i e eigenfunction based gnbcs

john von neumann s analysis of gaussian elimination and the origins of modern numerical analysis just when modern computers digital electronic and programmable were being invented john von neumann and herman goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers their foresight and the congruence of historical events made their work the first modern paper in numerical analysis von neumann once remarked that to found a mathematical theory one had to prove the first theorem which he and goldstine did for the accuracy of mechanized gaussian elimination but their paper was about more than that von neumann and goldstine described what they surmized would be the significant questions once computers became available for computational science and they suggested enduring ways to answer them

modeling radiation with an efficient hybrid finite element integral equation waveguide mode matching technique a method is presented to model electromagnetic radiation combining the finite element technique in the penetrating portion of a three dimensional 3 d radiating structure with an integral equation on the outer surface of the computational domain and with a waveguide mode matching technique on a cross section of the feeding waveguide the antenna can be of general shape provided a portion of the surrounding medium up to a surface of revolution is chosen as part of the computational domain this truncation scheme and the accurate source representation are more general than those existing in the literature for radiation modeling an open ended waveguide and a waveguide with a choke ring are analyzed and their radiation patterns are obtained and compared with available measurements

polygonal invariant curves for a planar piecewise isometry we investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones for this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components the union of all periodic components is an invariant pentagon with unequal sides transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange the restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region the proofs are based on exact calculations in a rational cyclotomic field we use the system to shed some light on a conjecture that pwis can possess transitive invariant curves that are not smooth

finite element method for unbounded field problems and application to two dimensional taper treatment of the finite element method for an unbounded field problem was proposed by mcdonald and wexler in 1972 their method is superior to others because it can exclude the singularities of green s functions this paper explains the treatment of the method in our 1979 letter which had some revisions of mcdonald and wexler s and calculated the time harmonic field problems examples presented are electromagnetic fields of two dimensional tapers which are open ended electromagnetic waves propagate in the taper and radiate from the taper to free space in this case the exact solutions for radiation from tapers are not available because of the complicated shape and so the finite element method is useful in solving these problems electromagnetic fields of tapers involving dielectric slabs are also calculated as examples of inhomogeneous problems

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

universality in two dimensional cellular structures evolving by cell division and disappearance the dynamics of two dimensional cellular networks is written in terms of coupled population equations which describe how the population of s sided cells is affected by cell division and disappearance in these equations the effect of the rest of the foam on the disappearing or dividing cell is treated as a local mean field under not too restrictive conditions the equilibrium distribution p s of cells satisfies a linear difference equation of order two or higher the population equations are asymptotically integrable the asymptotic integrability implies a universal distribution p s approximately cs kzs for large values of s which is also the boltzmann distribution associated with the maximum entropy inference asymptotic integrability of the population equations is absent in a global mean field approximation the importance of short range topological information to control the evolution of foams is thus confirmed

dynamics of kaleidoscopic maps abstract first we introduce a certain class of piecewise affine elliptic rotation maps on t 2 called the kaleidoscopic maps and describe its importance and then we concentrate our efforts on a special case when the rotation angle of a kaleidoscopic map is n n z for the special case we answer the conjectures regarding the periodicity and the singularity structure of such kaleidoscopic dynamics in the process we prove the partial riddling of the regular orbits that gives rise to the classification of periodic sets and estimate the hausdorff dimension of the singular set finally we study the dynamics of such kaleidoscopic maps restricted within the singular set and answer conjectures concerning the chaos the local chaos and the ergodicity with respect to the normalized hausdorff measure of the singular set

on variational electromagnetics theory and application a methodology to he named variational electromagnetics vem is introduced for handling time harmonic linear electromagnetic field problems first a desired variational formulation is derived from a fundamental variational principle fvp and then is solved by the hybrid element method along with the frontal solution algorithm the main advantages of such an approach are the unified derivation of variational equations for both interior and exterior field problems and their efficient numerical solution especially for problems with inhomogeneons media also included is a numerical example in which the scattering from an inhomogeneously coated cylinder is analyzed

parallel domain decomposition based algorithm of hybrid fe bi mlfma method for 3 d scattering by large inhomogeneous objects the hybrid method of the finite element boundary integral multilevel fast multipole algorithm fe bi mlfma has been recognized as one of the most powerful numerical methods for analyzing large inhomogeneous radiation scattering problems a domain decomposition algorithm dda of fe bi mlfma is presented in this paper by using the finite element tearing and interconnecting method feti the formulation of dda fe bi mlfma is presented and analyzed in detail the numerical performance of dda fe bi mlfma is investigated by numerical experiments from many aspects it includes the convergence speed versus types of domain decomposition number of subdomains types and inhomogeneity of dielectrics involving in solved problems and the scalability of dda fe bi mlfma the comparison of dda and previous algorithms of fe bi mlmfma is also carried out finally the capability of dda fe bi mlfma is shown for large inhomogeneous problems

modeling three dimensional scatterers using a coupled finite element integral equation formulation finite element modeling has proven useful for accurately simulating scattered or radiated fields from complex three dimensional objects whose geometry varies on the scale of a fraction of a wavelength to practically compute a solution to exterior problems the domain must be truncated at some finite surface where the sommerfeld radiation condition is enforced either approximately or exactly this paper outlines a method that couples three dimensional finite element solutions interior to a bounding surface with an efficient integral equation solution that exactly enforces the sommerfeld radiation condition the general formulation and the main features of the discretized problem are first briefly outlined results for far and near fields are presented for geometries where an analytic solution exists and compared with exact solutions to establish the accuracy of the model results are also presented for objects that do not allow an analytic solution and are compared with other calculations and or measurements

global stability of a class of discontinuous dual billiards an infinite parameter family of discontinuous area preserving maps is studied using geometrical methods necessary and sufficient conditions are determined for the existence of some bounding invariant sets which guarantee global stability it is shown that under some additional constraints all orbits become periodic most of them lyapounov stable and with a maximal period in any bounded domain of phase space this yields a class of maps acting on a discrete phase space

an effective domain decomposition based preconditioner for the fe bi mlfma method for 3d scattering problems a domain decomposition based preconditioner ddp is presented for the hybrid finite element boundary integral multilevel fast multipole algorithm fe bi mlfma the formulation of fe bi is first approximated by the finite element method fem with absorbing boundary condition abc then this approximate fem abc formulation is solved by using the finite element tearing and interconnecting method feti finally an inner outer iterative algorithm with variable convergence threshold is designed for fe bi mlfma because of good numerical scalability of feti and good approximation of fem abc this ddp fe bi mlfma performs well even for large lossless objects

fast solution methods in electromagnetics various methods for efficiently solving electromagnetic problems are presented electromagnetic scattering problems can be roughly classified into surface and volume problems while fast methods are either differential or integral equation based the resultant systems of linear equations are either solved directly or iteratively a review of various differential equation solvers their complexities and memory requirements is given the issues of grid dispersion and hybridization with integral equation solvers are discussed several fast integral equation solvers for surface and volume scatterers are presented these solvers have reduced computational complexities and memory requirements

scattering from a large body with cracks and cavities by the fast and accurate finite element boundary integral method a large body with cracks and cavities is a typical structure widely existing in realistic targets in this paper a newly developed fast and accurate finite element boundary integral fa fe bi method is applied to compute scattering by this kind of scatterer a thorough analysis on this fa fe bi numerical technique is presented clearly demonstrating that this technique has computational complexity o n log n and memory requirement o n n is the total number of surface unknowns an inward looking approach is employed as a preconditioner to speed up the rate of convergence of iterative solvers for this structure under these techniques a powerful code is developed for this kind of scatterer whose accuracy efficiency and capability is well confirmed by various numerical results

a novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems a novel hybridization of the finite element fe and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems the proposed method derives an adaptive numerical absorbing boundary condition abc for the finite element solution based on boundary integral equations unlike the standard finite element boundary integral approach the proposed method is free of interior resonance and produces a purely sparse system matrix which can be solved very efficiently unlike the traditional finite element absorbing boundary condition approach the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer radiator to minimize the computational domain and more importantly the method produces a solution that converges to the true solution of the problem to demonstrate its great potential the proposed method is implemented using higher order curvilinear vector elements a mixed functional is designed to yield both electric and magnetic fields on an integration surface without numerical differentiation to be used in the calculation of the adaptive abc the required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm which greatly reduces both the memory requirement and cpu time the finite element equations are solved efficiently using the multifrontal algorithm a mathematical analysis is conducted to study the convergence of the method finally a number of numerical examples are given to illustrate its accuracy and efficiency

a fast algorithm based on partial basic solution vectors domain decomposition method for scattering analysis of electrically large cylinders an efficient domain decomposition method ddm based on the partial basic solution vectors pbsv is presented for the electromagnetic scattering analysis of electrically large two dimensional objects the original computation domain is partitioned into nonoverlapping subdomains the pbsv of each subdomain are evaluated independently then the field on the interfaces between subdomains can easily be obtained by an iterative vector summation procedure and the final solution on each subdomain is solved independently and efficiently to improve the algorithm further two techniques expanding the pbsv by roof top basis functions and an under relaxed iteration method are also studied compared with the traditional ddm the proposed method can greatly reduce the computational complexity and the memory requirement moreover it can be implemented totally independently on both sequential and parallel computational platform which is distinct from the others the validity of this algorithm is verified by numerical examples

computing scattering amplitudes for arbitrary cylinders under incident plane waves a numerical method to compute scattering amplitudes for time harmonic waves scattered from infinite cylinders with arbitrary uniform cross section is described a nonlocal boundary condition is used to develop a variational formulation of the scattering problem and the finite element method is applied to determine approximations to the near field scattering amplitudes are then determined by means of an integral representation obtained from green s formula and properties of the nonlocal boundary operator computational results are presented to illustrate the method s application

invariant tori in hamiltonian systems with impacts it is shown that a large class of solutions in two degree of freedom hamiltonian systems of billiard type can be described by slowly varying one degree of freedom hamiltonian systems under some non degeneracy conditions such systems are found to possess a large set of quasiperiodic solutions filling out two dimensional tori which correspond to caustics in the classical billiard this provides a unified proof of existence of quasiperiodic solutions in convex billiards and other systems with impacts including classical billiard in electric and magnetic fields dual billiard and fermi ulam systems