topology optimization of metallic antennas we introduce an approach to carry out layout optimization of metallic antenna parts an optimization technique first developed for the optimization of load bearing elastic structures is adapted for the purpose of metallic antenna design the local conductivity values in a given region are used as design variables and are iteratively updated by a gradient based optimization algorithm given a set of time domain signals from exterior sources the design objective is here to maximize the energy received by the antenna and transmitted to a coaxial cable the optimization proceeds through a sequence of coarsely defined lossy designs with successively increasing details and less losses as the iterations proceed the objective function gradient is derived based on the fdtd discretization of maxwell s equations and is expressed in terms of field solutions of the original antenna problem and an adjoint field problem the same fdtd code but with different wave sources is used for both the original antenna problem and the adjoint problem for any number of design variables the gradient is evaluated on the basis of only two fdtd simulations one for the original antenna problem and another for the adjoint field problem we demonstrate the capability of the method by optimizing the radiating patch of both uwb monopole and microstrip antennas the uwb monopole is designed to radiate over a wide frequency band 1 10 ghz while the microstrip patch is designed for single and dual frequency band operation in these examples there are more than 20 000 design variables and the algorithm typically converges in less than 150 iterations the optimization results show a promising use of the proposed approach as a general method for conceptual design of near resonance metallic antennas

modified brinkman equation for a free flow problem at the interface of porous surfaces the cantor taylor brush configuration case 1 the free flow problem above at the surface interface and inside a cantor taylor brush configuration ctb simulating a porous medium was studied particle image velocimetry piv measurements confirm that the microscale stokes equation provides an accurate solution to the ctb microscale flow problem a comparison between the results of the averaged microscale stokes equation and that of the brinkman equation using an apparent viscosity shows that the concept of apparent viscosity did not provide a satisfactory agreement between the two approaches in order to develop a description of the average velocity profile across the interface flow region theoretical and numerical analyses were performed an averaging procedure of the navier stokes equations provided a set of three equations which were used to predict the average velocity in the fluid phase u f this set of equations is the suggested modified brinkman equation mbe the comparison between the results of the stokes equation and the mbe shows that optimizing the size of the averaging representative volume provides a good fit between the flow problem and the solution of the modified brinkman equation

the geometry of super conformal quantum mechanics n particle quantum mechanics described by a sigma model with an n dimensional target space with torsion is considered it is shown that an sl 2 conformal symmetry exists if and only if the geometry admits a homothetic killing vector d a a whose associated one form d a dx a is closed further the sl 2 can always be extended to osp 1 2 superconformal symmetry with a suitable choice of torsion by the addition of n real fermions extension to su 1 1 1 requires a complex structure i and a holomorphic u 1 isometry d a i a b b conditions for extension to the superconformal group d 2 1 which involve a triplet of complex structures and su 2 su 2 isometries are derived examples are given

level set based topology optimization for electromagnetic dipole antenna design this paper presents a level set framework for a typical electromagnetic design problem of dipole antenna in this study the geometrical configuration of an antenna is represented by the zero level contour of a higher dimensional level set function the governing equation for the induced current flow on a metal surface is the electric field integral equation efie which takes into account the electric component of the incident wave the design objective is formulated in terms of the surface current and incident electric field the normal velocity of the level set model which reflects the sensitivity of the objective function is derived from the adjoint variable method and shape derivative by optimizing the objective function the area with the highest current density to which the voltage feeding should be applied can be reshaped the advantages of adopting the level set technique for electromagnetic design lie in its capacity for capturing sophisticated topological changes and facilitation in mathematical representation of the design configuration the demonstrative examples of dipole antenna design show that the level set method results in a fairly smooth optimization process where the vacuum metal interface gradually attains its optimal configuration a series of design cases with self adjoint and non self adjoint sensitivity analyses are studied and compared to the benchmarking problems in dipole antenna

a finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics a new method for solving the time domain integral equations of electromagnetic scattering from conductors is introduced this method called finite difference delay modeling appears to be completely stable and accurate when applied to arbitrary structures the temporal discretization used is based on finite differences specifically based on a mapping from the laplace domain to the z transform domain first and second order unconditionally stable methods are derived spatial convergence is achieved using the higher order divergence conforming vector bases of graglia et al low frequency instability problems are avoided with the loop tree decomposition approach numerical results will illustrate the accuracy and stability of the technique

lie groups and twistor spaces on considere un espace symetrique de riemann simplement connexe m la connexion de levi civita et le fibre torseur metrique 7b f m j m dont les fibres 7b f x m pour x m est constitue de structures complexes orthogonales

rapid solution of the wave equation in unbounded domains in this paper we propose and analyze a new fast method for the numerical solution of time domain boundary integral formulations of the wave equation we employ lubich s convolution quadrature method for the time discretization and a galerkin boundary element method for the spatial discretization the coefficient matrix of the arising system of linear equations is a triangular block toeplitz matrix possible choices for solving the linear system arising from the above discretization include the use of fast fourier transform fft techniques and the use of data sparse approximations by using fft techniques the computational complexity can be reduced substantially while the storage cost remains unchanged and is typically high using data sparse approximations the gain is reversed i e the computational cost is approximately unchanged while the storage cost is substantially reduced the method proposed in this paper combines the advantages of these two approaches first the discrete convolution related to the block toeplitz system is transformed into the discrete fourier image thereby arriving at a decoupled system of discretized helmholtz equations with complex wave numbers a fast data sparse e g fast multipole or panel clustering method can then be applied to the transformed system additionally significant savings can be achieved if the boundary data are smooth and time limited in this case the right hand sides of many of the helmholtz problems are almost zero and hence can be disregarded finally the proposed method is inherently parallel we analyze the stability and convergence of these methods thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature we also present numerical results which illustrate the predicted convergence behavior

hybrid cross approximation of integral operators the efficient treatment of dense matrices arising e g from the finite element discretisation of integral operators requires special compression techniques in this article we use the matrix representation that approximates the dense stiffness matrix in admissible blocks corresponding to subdomains where the underlying kernel function is smooth by low rank matrices the low rank matrices are assembled by a new hybrid algorithm hca that has the same proven convergence as standard interpolation but also the same efficiency as the heuristic adaptive cross approximation aca

stable electric field tdie solvers via quasi exact evaluation of mot matrix elements prior theoretical studies and experience confirm that the stability of marching on in time mot solvers pertinent to the analysis of scattering from free standing three dimensional perfect electrically conducting surfaces hinges on the accurate evaluation of mot matrix elements resulting from a galerkin discretization of the underlying time domain integral equation tdie unfortunately the accurate evaluation of the four dimensional spatial integrals involved in the expressions for these matrix elements is prohibitively expensive when performed by computational means here a method that permits the quasi exact evaluation of mot matrix elements is presented specifically the proposed method permits the analytical evaluation of three out of the four spatial integrations leaving only one integral to be evaluated numerically since the latter has finite range and a piecewise smooth integrand it can be evaluated to very high accuracy using standard quadrature rules as a result the proposed method permits the fast evaluation of mot matrix elements with arbitrary user specified accuracy extensive numerical experiments show that an mot solver for the electric field tdie that uses the proposed quasi exact method is stable for a very wide range of time step sizes and yields solutions that decay exponentially after the excitation vanishes

a fast higher order time domain finite element boundary integral method for 3 d electromagnetic scattering analysis a novel hybrid time domain finite element boundary integral method for analyzing three dimensional 3 d electromagnetic scattering phenomena is presented the method couples finite element and boundary integral field representations in a way that results in a sparse system matrix and solutions that are devoid of spurious modes to accurately represent the unknown fields the scheme employs higher order vector basis functions defined on curvilinear tetrahedral elements to handle problems involving electrically large objects the multilevel plane wave time domain algorithm is used to accelerate the evaluation of the boundary integrals numerical results demonstrate the accuracy and versatility of the proposed scheme

structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique dielectric resonator antennas dras are a relatively new class of antenna that utilizes the radiation phenomena of dielectric resonators in open space since mainly analytical approaches have been applied to investigate dra designs the current dra forms are limited to simple geometric shapes and high performance dras with complex shapes have not yet been developed topology optimization is capable of yielding high performance structures and has been extensively applied to a variety of structural optimization problems applying it to the task of dra design may be extremely useful for the design of high performance antennas on the other hand the finite difference time domain fdtd method has been used to numerically evaluate general antenna performance since it is numerically robust during time domain analyses and can handle complex models thus the integration of topology optimization with the fdtd method has the potential to enable innovative designs of advanced antennas that offer exceptional performance in this research we propose a new topology optimization method for the design of dras that aim to operate with enhanced bandwidths using the fdtd method first the concept of topology optimization is briefly discussed and a way to integrate topology optimization with the fdtd method is proposed next design requirements are clarified and the corresponding objective functions and the optimization problem are formulated an optimization algorithm is constructed based on these formulations finally several dra design examples are presented to confirm the usefulness of the proposed method

time domain adaptive integral method for surface integral equations an efficient marching on in time mot scheme is presented for solving electric magnetic and combined field integral equations pertinent to the analysis of transient electromagnetic scattering from perfectly conducting surfaces residing in an unbounded homogenous medium the proposed scheme is the extension of the frequency domain adaptive integral pre corrected fast fourier transform fft method to the time domain fields on the scatterer that are produced by space time sources residing on its surface are computed 1 by locally projecting for each time step all sources onto a uniform auxiliary grid that encases the scatterer 2 by computing everywhere on this grid the transient fields produced by the resulting auxiliary sources via global multilevel blocked space time ffts 3 by locally interpolating these fields back onto the scatterer surface as this procedure is inaccurate when source and observer points reside close to each other and 4 near fields are computed classically albeit pre corrected for errors introduced through the use of global ffts the proposed scheme has a computational complexity and memory requirement of o n sub t n sub s log sup 2 n sub s and o n sub s sup 3 2 when applied to quasiplanar structures and of o n sub t n sub s sup 3 2 log sup 2 n sub s and o n sub s sup 2 when used to analyze scattering from general surfaces here n sub s and n sub t denote the number of spatial and temporal degrees of freedom of the surface current density these computational cost and memory requirements are contrasted to those of classical mot solvers which scale as o n sub t n sub s sup 2 and o n sub s sup 2 respectively a parallel implementation of the scheme on a distributed memory computer cluster that uses the message passing interface is described simulation results demonstrate the accuracy efficiency and the parallel performance of the implementation

simultaneous structural topology optimization of electromagnetic sources and ferromagnetic materials abstract for the design of electric machines such as actuators sensors and motors the topology optimization for em sources as well as ferromagnetic materials fms is in great request generally permanent magnets pms and coils serve as electromagnetic em sources in electric machines however most of researches in relation with topology optimization in electric machinery are focused mostly on the design of ferromagnetic materials such as yokes and stators in this paper the topology optimization methodology for the simultaneous design of pms and fms is developed and it is expanded to the design of fms with coils the proposed method makes it possible to obtain a clear topology with little gray density elements in the design domain and shows the stable convergence history the proposed method is applied to the optimization problem of a c core actuator and a magnetostrictive sensor the sensitivity analysis and the update of design variables are accomplished by the adjoint variable method and the sequential linear programming respectively

a multilevel cartesian non uniform grid time domain algorithm a multilevel cartesian non uniform grid time domain algorithm cngtda is introduced to rapidly compute transient wave fields radiated by time dependent three dimensional source constellations cngtda leverages the observation that transient wave fields generated by temporally bandlimited and spatially confined source constellations can be recovered via interpolation from appropriately delay and amplitude compensated field samples this property is used in conjunction with a multilevel scheme in which the computational domain is hierarchically decomposed into subdomains with sparse non uniform grids used to obtain the fields for both surface and volumetric source distributions the computational cost of cngtda to compute the transient field at n s observation locations from n s collocated sources for n t discrete time instances scales as o n tn slogn s and o n tn slog 2n s in the low and high frequency regimes respectively coupled with marching on in time mot time domain integral equations cngtda can facilitate efficient analysis of large scale time domain electromagnetic and acoustic problems

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

structural optimization of ferromagnetic materials based on the magnetic reluctivity for magnetic field problems abstract topology optimization of ferromagnetic materials in magnetic fields using the homogenization method and the density method has been attempted from an angle of the interpolation scheme based on the magnetic permeability however optimization results based on such methods may have poor convergence history in comparison with the structural topology optimization of an elastic problem this study intends to verify that the interpolation scheme using the magnetic permeability may cause problems in the sensitivity analysis for the structural optimization in magnetic fields and suggests a novel interpolation scheme based on the magnetic reluctivity numerical examples are focused on magnetic actuators to maximize the magnetic force attracting an armature this study is focused to maximize force itself using the maxwell stress method and the sensitivity analysis by the adjoint variable method whereas previous work to maximize force are based on the magnetic energy computation mostly without considering the actuating direction it is confirmed that optimal models by the proposed method have improved performance and better robust iteration history compared with the previous method

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

a pde sensitivity equation method for optimal aerodynamic design the use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design a typical procedure uses a simulation scheme to evaluate the objective function from the approximate states and its gradient then passes this information to an optimization algorithm once the simulation scheme cfd flow solver has been selected and used to provide approximate function evaluations there are several possible approaches to the problem of computing gradients one popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients although this black box approach has many advantages in shape optimization problems one must compute mesh sensitivities in order to compute the design sensitivity in this paper we present an alternative approach using the pde sensitivity equation to develop algorithms for computing gradients this approach has the advantage that mesh sensitivities need not be computed moreover when it is possible to use the cfd scheme for both the forward problem and the sensitivity equation then there are computational advantages an apparent disadvantage of this approach is that it does not always produce consistent derivatives however for a proper combination of discretization schemes one can showasymptotic consistencyunder mesh refinement which is often sufficient to guarantee convergence of the optimal design algorithm in particular we show that when asymptotically consistent schemes are combined with a trust region optimization algorithm the resulting optimal design method converges we denote this approach as thesensitivity equation method the sensitivity equation method is presented convergence results are given and the approach is illustrated on two optimal design problems involving shocks

on the multistep time discretization of linear initial boundary value problems and their boundary integral equations convergence estimates in terms of the data are shown for multistep methods applied to non homogeneous linear initial boundary value problems similar error bounds are derived for a new class of time discrete and fully discrete approximation schemes for boundary integral equations of such problems e g for the single layer potential equation of the wave equation in both cases the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions these estimates which are also proved here depend on bounds of the laplace transform of the distributional convolution kernel outside the stability region scaled by the time stepsize and on the smoothness of the data

combination of topology optimization and optimal control method this paper presents the combination of topology optimization and optimal control method to find the optimal match between the material topology and control in the presented method the material topology is determined using the simp solid isotropic material with penalization method which has been popularly used in topology optimization in the simp method the design variable is relaxed and bounded in the interval 0 1 and the evolution of the design variable is usually implemented by the method of moving asymptotes mma which can be used to deal with optimization problem with multiple integral constraints and bound constraint of the design variable in the combination of topology optimization and optimal control method the control variable appears along with the design variable in order to evolve the control variable and design variable using mma simultaneously the control variable is regularized using a bound constraint and the corresponding bound constraint is projected onto the interval 0 1 which is the same as the bound constraint of the design variable the optimization problem is analyzed using the adjoint method to obtain the adjoint sensitivity during the optimization procedure the design and control variables are filtered by the helmholtz filters to ensure the smoothness of the distribution to ensure the minimum scale length and remove the gray area in the material topology the filtered design variable is projected by the threshold method the feasibility and robustness of the combination of these two methods are demonstrated by several test problems including heat transfer fluid flow and compliance minimization

a consistent grayscale free topology optimization method using the level set method and zero level boundary tracking mesh summary this paper proposes a level set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming because the boundary tracking mesh is always conformed to the structural boundary good approximation to the boundary is maintained during optimization therefore structural design problems are solved completely without grayscale material previously we introduced the boundary tracking mesh generating method into level set based topology optimization and updated the design variables by solving the level set equation in order to adapt our previous method to general structural optimization frameworks the incorporation of the method with nonlinear programming is investigated in this paper to successfully incorporate nonlinear programming the optimization problem is regularized using a double well potential furthermore the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming we expect the investigation to open up a new class of grayscale free topology optimization the usefulness of the proposed method is demonstrated using several numerical examples targeting two dimensional compliant mechanism and metallic waveguide design problems copyright 2014 john wiley sons ltd

topology optimization of unsteady incompressible navier stokes flows this paper discusses the topology optimization of unsteady incompressible navier stokes flows an optimization problem is formulated by adding the artificial darcy frictional force into the incompressible navier stokes equations the optimization procedure is implemented using the continuous adjoint method and the finite element method the effects of dynamic inflow reynolds number and target flux on specified boundaries for the optimal topology of unsteady navier stokes flows are presented numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady navier stokes flows

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

runge kutta convolution quadrature for operators arising in wave propagation an error analysis of runge kutta convolution quadrature is presented for a class of non sectorial operators whose laplace transform satisfies besides the standard assumptions of analyticity in a half plane re s 0 and a polynomial bound operatorname o s mu 1 there the stronger polynomial bound operatorname o s mu 2 in convex sectors of the form operatorname arg s leq pi 2 theta for 0 the order of convergence of the runge kutta convolution quadrature is determined by 2 and the underlying runge kutta method but is independent of 1 time domain boundary integral operators for wave propagation problems have laplace transforms that satisfy bounds of the above type numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of runge kutta convolution quadrature for this class of applications our results show in particular that the full classical order of the runge kutta method is attained away from the scattering boundary

implicit runge kutta methods for the discretization of time domain integral equations implicit runge kutta based schemes are proposed for solving the time domain integral equations of electromagnetic theory the proposed technique maps a laplace domain equation to a domain equation using the butcher tableau of the implicit runge kutta scheme a discrete time domain system is recovered by computing the inverse transform numerically the resulting technique is capable of third or fifth order accuracy in time and is stable independent of time step numerical results illustrate the accuracy and stability of the technique

the fast multipole method i error analysis and asymptotic complexity this paper is concerned with the application of the fast multipole method fmm to the maxwell equations this application differs in many aspects from other applications such as the n body problem laplace equation and quantum chemistry etc the fmm leads to a significant speed up in cpu time with a major reduction in the amount of computer memory needed when performing matrix vector products this leads to faster resolution of scattering of harmonic plane waves from perfectly conducting obstacles emphasis here is on a rigorous mathematical approach to the problem we focus on the estimation of the error introduced by the fmm and a rigorous analysis of the complexity o n log n of the algorithm we show that error estimates reported previously are not entirely satisfactory and provide sharper and more precise estimates

nonuniform grid time domain ngtd algorithm for fast evaluation of transient wave fields a novel algorithm to efficiently compute transient wave fields produced by known three dimensional source constellations is proposed the algorithm uses domain decomposition concepts and comprises two steps to be repeated for each subdomain considered in the first step delay and amplitude compensated fields produced by sources residing inside each subdomain are computed at a sparse set of points surrounding the observation domain in the second step total fields in the observer domain are evaluated by interpolation delay and amplitude restoration and aggregation of subdomain fields the proposed scheme is well suited to accelerate the solution of time domain integral equations by marching on in time to carry out time domain physical optics calculations and to realize near to far field transformations of transients moreover the scheme automatically adapts to and takes advantage of special geometrical features of the source observer constellation studied a key benefit when analyzing quasi planar configurations in addition it realizes a seamless transition from the dynamic to the quasi static regime thus facilitating a unified treatment of electrically large and small problems last but not least the scheme is remarkably simple to implement

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

space time topology optimization for one dimensional wave propagation abstract a space time extension of the topology optimization method is presented the formulation with design variables in both the spatial and temporal domains is used to create structures with an optimized distribution of material properties that can vary in time the method is outlined for one dimensional transient wave propagation in an elastic rod with time dependent young s modulus by two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber frequency shifts and lack of energy conservation the optimization method s potential for creating structures with novel dynamic behavior is illustrated by a simple example it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure optimized designs in form of spatio temporal laminates and checkerboards are generated and discussed the example lays the foundation for creating designs with more advanced functionalities in future work

a galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions we consider retarded boundary integral formulations of the three dimensional wave equation in unbounded domains our goal is to apply a galerkin method in space and time in order to solve these problems numerically in this approach the computation of the system matrix entries is the major bottleneck we will propose new types of finite dimensional spaces for the time discretization they allow variable time stepping variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially the reason is that the basis functions of these spaces are globally smooth and compactly supported in order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere in this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

convolution quadrature method based symmetric galerkin boundary element method for 3 d elastodynamics the boundary integral equations in 3 d elastodynamics contain convolution integrals with respect to the time they can be performed analytically or with the convolution quadrature method the latter time stepping procedure s benefit is the usage of the laplace transformed fundamental solution therefore it is possible to apply this method also to problems where analytical time dependent fundamental solutions might not be known to obtain a symmetric formulation the second boundary integral equation has to be used which unfortunately requires special care in the numerical implementation since it involves hypersingular kernel functions therefore a regularization for closed surfaces of the laplace transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one supplementarily a weakly singular formulation of the laplace transformed elastodynamic double layer potential is presented this results in a time domain boundary element formulation involving at least only weakly singular integral kernels finally numerical studies validate this approach with respect to different spatial and time discretizations further a comparison with the wider used collocation method is presented it is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods copyright 2008 john wiley sons ltd

transparent boundary conditions for wave propagation in fractal trees convolution quadrature approach in this work we propose high order transparent boundary conditions for the weighted wave equation on a fractal tree with an application to the modeling of sound propagation in a human lung this article follows the recent work 29 dedicated to the mathematical analysis of the corresponding problem and the construction of low order absorbing boundary conditions the method proposed in this article consists in constructing the exact trans parent boundary conditions for the semi discretized problem in the spirit of the convolution quadrature method developed by ch lubich we analyze the stability and convergence of the method and propose an efficient algorithm for its implementation the exposition is concluded with numerical experiments

multistep and multistage convolution quadrature for the wave equation algorithms and experiments we describe how a time discretized wave equation in a homogeneous medium can be solved by boundary integral methods the time discretization can be a multistep runge kutta or a more general multistep multistage method the resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of lubich the aim of this work is twofold it describes an efficient robust and easily parallelizable method for solving the semidiscretized system the resulting algorithm has the main advantages of time stepping methods and of fourier synthesis at each time step a system of linear equations with the same system matrix needs to be solved yet computations can easily be done in parallel the computational cost is almost linear in the number of time steps and only the laplace transform of the time domain fundamental solution is needed the new aspect of the algorithm is that all this is possible without ever explicitly constructing the weights of the convolution quadrature this approach also readily allows the use of modern data sparse techniques to efficiently perform computation in space we investigate theoretically and numerically to which extent hierarchical matrix mathcal h matrix techniques can be used to speed up the space computation the second aim of this article is to perform a series of large scale 3d experiments with a range of multistep and multistage time discretization methods the backward difference formula of order 2 bdf2 the trapezoid rule and the 3 stage radau iia methods are investigated in detail one of the conclusions of the experiments is that the radau iia method often performs overwhelmingly better than the linear multistep methods especially for problems with many reflections yet in connection with hyperbolic problems bdfs have so far been predominant in the literature on convolution quadrature

a galerkin bem for transient acoustic scattering by an absorbing obstacle this paper deals with the numerical resolution of the so called time domain boundary integral equations the scattering problem by an absorbing obstacle serves as the model for this discussion a new system of retarded potential boundary integral equations rpbie is set up to solve the problem using an energy identity we are able to prove the unconditional stability for standard conforming galerkin approximation of the rpbie the actual space time boundary elements are described in some detail and their implementation has confirmed that stability in a wide range of situations and for large run times never reached before

wavelets and convolution quadrature for the efficient solution of a 2d space time bie for the wave equation abstract we consider a wave propagation problem in 2d reformulated in terms of a boundary integral equation bie in the space time domain for its solution we propose a numerical scheme based on a convolution quadrature formula by lubich for the discretization in time and on a galerkin method in space it is known that the main advantage of lubich s formulas is the use of the fft algorithm to retrieve discrete time integral operators with a computational complexity of order r log r r being twice the total number of time steps performed since the discretization in space leads in general to a quadratic complexity the global computational complexity is of order m 2 r log r and the working storage required is m 2 r 2 where m is the number of grid points on the domain boundary to reduce the complexity in space we consider here approximant functions of wavelet type by virtue of the properties of wavelet bases the discrete integral operators have a rapid decay to zero with respect to time and the overwhelming majority of the associated matrix entries assume negligible values based on an a priori estimate of the decaying behaviour in time of the matrix entries we devise a time downsampling strategy that allows to compute only those elements which are significant with respect to a prescribed tolerance such an approach allows to retrieve the temporal history of each entry e corresponding to a fixed couple of wavelet basis functions via a fast fourier transform with computational complexity of order r e log r e the parameter r e depends on the two basis functions and it satisfies r e r for a relevant percentage of matrix entries percentage which increases significantly as time and or space discretization are refined globally the numerical tests show that the computational complexity and memory storage of the overall procedure are linear in space and time for small velocities of the wave propagation and even sub linear for high velocities

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

transient elastic wave analysis of 3 d large scale cavities by fast multipole bem using implicit runge kutta convolution quadrature abstract existing boundary element methods bems have been proven to be efficient numerical techniques particularly in modeling wave propagation problems but still remain limitations in solving large scale problems the main disadvantages may be caused by several problems including high cost and large amount of computer memory for the computation a time domain bem in which a collocation method is used for the time discretization also shows numerical instability for small cfl courant friedrichs lewy number in this work we propose a new three dimensional 3 d approach for solving transient elastic wave propagation problems of large scale spherical and spheroidal cavities by the convolution quadrature bem accelerated by fast multipole method fmm using an implicit runge kutta scheme in abbreviation we name it as irk based cq fmbem two special techniques pertaining to the fmm including the scaling of modified spherical bessel functions and the truncation method are used to enhance the efficiency and stability of our proposed method this approach is found to be particularly suitable for 3 d large scale problems as its high accuracy stability of time marching process and computational efficiency these properties are subsequently illustrated through numerical examples dealing with large scale wave problems of 3 d single spherical and spheroidal cavities and multiple equally and irregularly spherical cavities the accuracy of the present formulation is verified by comparing the obtained results with available reference solutions in the literature some numerical aspects of time increments sizes and topological shapes of cavities and their influences on the deformations and the waveforms are investigated also detailed investigation of the computational efficiency of our proposed method such as cpu time required memory etc is presented all the implementation tasks are carried out using the tokyo tech supercomputer which is called tsubame 2 5 42

fast convolution quadrature for the wave equation in three dimensions this work addresses the numerical solution of time domain boundary integral equations arising from acoustic and electromagnetic scattering in three dimensions the semidiscretization of the time domain boundary integral equations by runge kutta convolution quadrature leads to a lower triangular toeplitz system of size n this system can be solved recursively in an almost linear time o n log 2 n but requires the construction of o n dense spatial discretizations of the single layer boundary operator for the helmholtz equation this work introduces an improvement of this algorithm that allows to solve the scattering problem in an almost linear time the new approach is based on two main ingredients the near field reuse and the application of data sparse techniques exponential decay of runge kutta convolution weights w n h d outside of a neighborhood of d n h where h is a time step allows to avoid constructing the near field i e singular and near singular integrals for most of the discretizations of the single layer boundary operators near field reuse the far field of these matrices is compressed with the help of data sparse techniques namely h matrices and the high frequency fast multipole method numerical experiments indicate the efficiency of the proposed approach compared to the conventional runge kutta convolution quadrature algorithm

topology optimization on two dimensional manifolds abstract this paper presents topology optimization on general two dimensional manifolds for phenomena described by second order partial differential equations where the material interpolation is implemented by using the material distribution method when a physical field is defined on a two dimensional manifold the material interpolation is implemented on a material parameter in the partial differential equation used to describe the distribution of the physical field when the physical field is defined on a three dimensional domain with its boundary conditions defined on a two dimensional manifold corresponding a surface or an interface of this three dimensional domain the material density is used to formulate a mixed boundary condition of the partial differential equation for the physical field and implement the penalization between two different boundary types based on the homeomorphic property of two dimensional manifolds typical two dimensional manifolds e g sphere torus mobius strip and klein bottle are included in the numerical tests which are used to demonstrate this topology optimization approach for the design problems of fluidic mechanics heat transfer and electromagnetics

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

topology optimization for three dimensional electromagnetic waves using an edge element based finite element method this paper develops a topology optimization procedure for three dimensional electromagnetic waves with an edge element based finite element method in contrast to the two dimensional case three dimensional electromagnetic waves must include an additional divergence free condition for the field variables the edge element based finite element method is used to both discretize the wave equations and enforce the divergence free condition for wave propagation described in terms of the magnetic field in the widely used class of non magnetic materials the divergence free condition is imposed on the magnetic field this naturally leads to a nodal topology optimization method when wave propagation is described using the electric field the divergence free condition must be imposed on the electric displacement in this case the material in the design domain is assumed to be piecewise homogeneous to impose the divergence free condition on the electric field this results in an element wise topology optimization algorithm the topology optimization problems are regularized using a helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method in order to ensure the applicability of the filter in the element wise topology optimization version a regularization method is presented to project the nodal into an element wise physical density variable

designing broad band patch antennas using the sequential quadratic programming method the utility of numerical codes is greatly enhanced if they can be used in design a situation that typically involves iterative optimization algorithms an attractive way is to use gradient based algorithms developed for solving nonlinear programming nlp problems we examine the performance of a general sequential quadratic programming sqp optimization algorithm for designing patch antennas in conjunction with a finite element boundary integral code