double sweep preconditioner for optimized schwarz methods applied to the helmholtz problem this paper presents a preconditioner for non overlapping schwarz methods applied to the helmholtz problem starting from a simple analytic example we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain the preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains and can be interpreted as a coarse grid for the domain decomposition method the resulting algorithm is shown to converge very fast independently of the number of subdomains and frequency the preconditioner has the advantage that like the original schwarz algorithm it can be implemented as a matrix free routine with no additional preprocessing

asymptotic formulas for the dual radon transform and applications caracterisation de la transformation de radon duale d une fonction indefiniment differentiable formules asymptotiques et applications

domain decomposition method for harmonic wave propagation a general presentation in this paper we give a general presentation of non overlapping domain decomposition methods for harmonic wave propagation models our abstract framework leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions it also includes applications to acoustic electromagnetic or elastic waves as well as the treatment of space discretization

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

perfectly matched transmission problem with absorbing layers application to anisotropic acoustics in convex polygonal domains this paper presents an original approach to design perfectly matched layers pml for anisotropic scalar wave equations this approach is based first on the introduction of a modified wave equation and second on the formulation of general perfectly matched transmission conditions for this equation the stability of the transmission problem is discussed by way of the adaptation of a high frequency stability necessary condition and we apply our approach to construct pml suited for any convex domain with straight boundaries a new variational formulation of the problem including a lagrange multiplier at the interface between the physical and the absorbing domains is then set and numerical results are presented in 2d and 3d these results show the efficiency of our approach when using constant damping coefficients combined with high order elements

a schwarz generalized eigen oscillation spectral element method gesem for 2 d high frequency electromagnetic scattering in dispersive inhomogeneous media in this paper we propose a parallel schwarz generalized eigen oscillation spectral element method gesem for 2 d complex helmholtz equations in high frequency wave scattering in dispersive inhomogeneous media this method is based on the spectral expansion of complex generalized eigen oscillations for the electromagnetic fields and the schwarz non overlapping domain decomposition iteration method the gesem takes advantages of a special real orthogonality property of the complex eigen oscillations and a new radiation interface condition for the system of equations for the spectral expansion coefficients numerical results validate the high resolution and the flexibility of the method for various materials

optimized schwarz methods without overlap for highly heterogeneous media in this paper an original variant of the schwarz domain decomposition method is introduced for heterogeneous media this method uses new optimized interface conditions specially designed to take into account the heterogeneity between the sub domains on each sides of the interfaces numerical experiments illustrate the dependency of the proposed method with respect to several parameters and confirm the robustness and efficiency of this method based on such optimized interface conditions several mesh partitions taking into account multiple cross points are considered in these experiments

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

phase field relaxation of topology optimization with local stress constraints we introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase field method in the basic formulation we have a pde constrained optimization problem where the finite element and design analysis are solved simultaneously the starting point of the relaxation is a reformulation of the material problem involving linear and 0 1 constraints only the 0 1 constraints are then relaxed and approximated by a cahn hilliard type penalty in the objective functional which yields convergence of minimizers to 0 1 designs as the penalty parameter decreases to zero a major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter the relaxation scheme yields a large scale optimization problem with a high number of linear inequality constraints we discretize the problem by finite elements and solve the arising finite dimensional programming problems by a primal dual interior point method numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach

null spaces and ranges for the classical and spherical radon transforms abstract let r be the classical radon transform that integrates a function over hyperplanes in r n and let sm be the transform that integrates a function over spheres containing the origin in r n we prove continuity results for both transforms and explicitly give the null space of r for a class of square integrable functions on the exterior of a ball in r n as well as the null space of sm for square integrable functions on a ball we show sm l 2 r n l 2 r n is one one and we characterize the range of sm on classes of smooth functions and square integrable functions by certain moment conditions if g x is a schwartz function on r n that is zero to infinite order at x 0 we prove moment conditions sufficient for g to be in the range of sm c r n we apply our results on sm to existence and uniqueness theorems for solutions to a characteristic initial value problem for the darboux partial differential equation

topology optimization using the lattice boltzmann method incorporating level set boundary expressions this paper presents a topology optimization method for fluid dynamics problems based on the level set method and using the lattice boltzmann method lbm in this optimization method the optimization problems are formulated based on the original boltzmann equation and the design sensitivities are precisely obtained without the time consuming numerical operations encountered when dealing with a large scale asymmetric matrix in contrast to previous research in which the lbm uses the lattice boltzmann equation lbe for the formulations of optimization problems and the derivation of their adjoint equations that is we newly derive sensitivity formulations from the original boltzmann equation not the lbe that can be said to be an approximated equation and these formulations yield strictly correct sensitivities that are error free based on the above formulations we construct a level set based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction furthermore two and three dimensional numerical examples are provided to confirm the validity and utility of the presented method

absorbing boundary conditions for the wave equation and parallel computing absorbing boundary conditions have been developed for truncating unbounded computational domains to bounded ones the authors investigate the use of absorbing boundary conditions in domain decomposition algorithms for the wave equation moreover they optimize the absorbing boundary conditions in order to obtain a good performance of the schwarz waveform relaxation algorithm the analysis given is confirmed by numerical experiments

non overlapping schwarz methods with optimized transmission conditions for the helmholtz equation abstract the development of efficient transmission conditions for the schwarz method without overlap is an important area of current research the techniques developed in recent years mainly consist of algebraic approximations of the discrete transparent operator or optimized approximations of the continuous transparent operator these optimized approximations are currently based on an analytical approach involving a minimization problem over all the available frequencies and using the l norm in this paper new numerical optimization techniques are investigated these original approaches allow selection of the frequency range for the minimization problem which is based on the l p norm new numerical experiments performed on a model problem demonstrate the performance and the robustness of the proposed optimization techniques in the particular cases p 1 and p applications to industrial automotive problems confirm the efficiency of these techniques when using arbitrary mesh partitions

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

a variational level set method for the topology optimization of steady state navier stokes flow the smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density based approaches though well established in structural designs inadequate this paper presents a level set method for topology optimization of steady state navier stokes flow subject to a specific fluid volume constraint the solid fluid interface is implicitly characterized by a zero level contour of a higher order scalar level set function and can be naturally transformed to other configurations as its host moves a variational form of the cost function is constructed based upon the adjoint variable and lagrangian multiplier techniques to satisfy the volume constraint effectively the lagrangian multiplier derived from the first order approximation of the cost function is amended by the bisection algorithm the procedure allows evolving initial design to an optimal shape and or topology by solving the hamilton jacobi equation two classes of benchmarking examples are presented in this paper 1 periodic microstructural material design for the maximum permeability and 2 topology optimization of flow channels for minimizing energy dissipation a number of 2d and 3d examples well demonstrated the feasibility and advantage of the level set method in solving fluid solid shape and topology optimization problems

non conformal domain decomposition method with mixed true second order transmission condition for solving large finite antenna arrays a non overlapping and non conformal domain decomposition method ddm is presented for modeling large finite antenna arrays two major benefits of the proposed ddm a a mixed true second order transmission condition sotc with corner edge penalty terms is developed to facilitate fast convergence in the ddm iterations numerical experiments indicate that the convergence of ddm with the aforementioned sotc is insensitive to the sizes of arrays and b the non conformal property of the proposed ddm permits the use of completely independent discretization for each of the sub domains we demonstrate the performance of the proposed approach through several large scale problems

new nonoverlapping domain decomposition methods for the harmonic maxwell system we study a nonoverlapping domain decomposition method for the harmonic maxwell equations with a new kind of interface condition using fourier analysis we derive suitable families of transmission conditions in r3 that involve second order tangential differential operators and that guarantee convergence for both propagative and evanescent modes such families depend upon parameters that are chosen to optimize the convergence rate of the corresponding iterative algorithm we then propose iterative solvers for the maxwell equations based on a domain decomposition procedure where such conditions are enforced on the interface some numerical results for a two domain decomposition show the effectiveness of the optimized interface conditions

non reflecting boundary conditions for waveguides new non reflecting boundary conditions are introduced for the solution of the helmholtz equation in a waveguide these boundary conditions are perfectly transparent for all propagating modes they do not require the determination of these propagating modes but only their propagation constants a quasi local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes related convergence properties to the exact solution and optimal error estimates are established

optimized schwarz method with complete radiation transmission conditions for the helmholtz equation in waveguides in this paper we present a nonoverlapping optimized schwarz algorithm with a high order transmission condition for the helmholtz equation posed in waveguides we introduce the new high order transmission conditions based on the complete radiation boundary conditions crbcs that have been developed for high order absorbing boundary conditions we obtain the convergence rate of the algorithm in terms of reflection coefficients of crbcs which decrease exponentially with the order of crbcs it will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the schwarz algorithm also this algorithm can be employed efficiently for a preconditioner in gmres implementations based on the substructured form as in m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 finally numerical examples confirming the theory will be presented

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

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

a problem of integral geometry abstract a new analytic method for finding a function from the knowledge of its integrals over the spheres passing through a fixed point is given this problem has been earlier studied by a cormack and e quinto

a quasi optimal domain decomposition algorithm for the time harmonic maxwell s equations this paper presents a new non overlapping domain decomposition method for the time harmonic maxwell s equations whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the magnetic to electric operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented

a domain decomposition method for discontinuous galerkin discretizations of helmholtz problems with plane waves and lagrange multipliers a nonoverlapping domain decomposition dd method is proposed for the iterative solution of systems of equations arising from the discretization of helmholtz problems by the discontinuous enrichment method this discretization method is a discontinuous galerkin finite element method with plane wave basis functions for approximating locally the solution and dual lagrange multipliers for weakly enforcing its continuity over the element interfaces the primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface lagrange multipliers only as in the feti h and feti dph dd methods for continuous galerkin discretizations this system of lagrange multipliers is iteratively solved by a krylov method equipped with both a local preconditioner based on subdomain data and a global one using a coarse space numerical experiments performed for two and three dimensional acoustic scattering problems suggest that the proposed dd based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains copyright 2009 john wiley sons ltd

a non overlapping optimized schwarz method for the heat equation with non linear boundary conditions and with applications to de icing abstract when simulating complex physical phenomena such as aircraft icing or de icing several dedicated solvers often need to be strongly coupled in this work a non overlapping schwarz method is constructed with the unsteady simulation of de icing as the targeted application to do so optimized coupling coefficients are first derived for the one dimensional unsteady heat equation with linear boundary conditions and for the steady heat equation with non linear boundary conditions the choice of these coefficients is shown to guarantee the convergence of the method using a linearization of the boundary conditions the method is then extended to the case of a general unsteady heat conduction problem the method is tested on simple cases and the convergence properties are assessed theoretically and numerically finally the method is applied to the simulation of an aircraft electrothermal de icing problem in two dimensions

superlinear convergence using block preconditioners for the real system formulation of complex helmholtz equations abstract complex valued helmholtz equations arise in various applications and a lot of research has been devoted to finding efficient preconditioners for the iterative solution of their discretizations in this paper we consider the helmholtz equation rewritten in real valued block form and use a preconditioner in a special two by two block form we show that the corresponding preconditioned krylov iteration converges at a mesh independent superlinear rate

a robin type domain decomposition method with red black partition in this paper we present a new robin type nonoverlapping domain decomposition dd preconditioner the unknown variables to be solved in this preconditioned algebraic system are the robin transmission data on the interface which are different from the well known dd methods like substructuring nonoverlapping dd method and feti method through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate for the second order elliptic problem we prove that the condition number of the preconditioned system is c 1 log frac h h 2 where h is the coarse mesh size and h is the fine mesh size moreover in this paper we always assume that there is a red black partition for the whole domain omega numerical results are given to illustrate the efficiency of our dd preconditioner

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

optimized schwarz methods for maxwell s equations over the last two decades classical schwarz methods have been extended to systems of hyperbolic partial differential equations using characteristic transmission conditions and it has been observed that the classical schwarz method can be convergent even without overlap in certain cases this is in strong contrast to the behavior of classical schwarz methods applied to elliptic problems for which overlap is essential for convergence more recently optimized schwarz methods have been developed for elliptic partial differential equations these methods use more effective transmission conditions between subdomains than the classical dirichlet conditions and optimized schwarz methods can be used both with and without overlap for elliptic problems we show here why the classical schwarz method applied to both the time harmonic and time discretized maxwell s equations converges without overlap the method has the same convergence factor as a simple optimized schwarz method for a scalar elliptic equation based on this insight we develop an entire new hierarchy of optimized overlapping and nonoverlapping schwarz methods for maxwell s equations with greatly enhanced performance compared to the classical schwarz method we also derive for each algorithm asymptotic formulas for the optimized transmission conditions which can easily be used in implementations of the algorithms for problems with variable coefficients we illustrate our findings with numerical experiments

a topology optimization method based on the level set method for the design of negative permeability dielectric metamaterials abstract this paper presents a level set based topology optimization method for the design of negative permeability dielectric metamaterials metamaterials are artificial materials that display extraordinary physical properties that are unavailable with natural materials the aim of the formulated optimization problem is to find optimized layouts of a dielectric material that achieve negative permeability the presence of grayscale areas in the optimized configurations critically affects the performance of metamaterials positively as well as negatively but configurations that contain grayscale areas are highly impractical from an engineering and manufacturing point of view therefore a topology optimization method that can obtain clear optimized configurations is desirable here a level set based topology optimization method incorporating a fictitious interface energy is applied to a negative permeability dielectric metamaterial design problem the optimization algorithm uses the finite element method fem for solving the equilibrium and adjoint equations and design problems are formulated for both two and three dimensional cases first the level set based topology optimization method is explained and the optimization problems for the design of metamaterials are then discussed several optimum design examples for the design of dielectric metamaterials that demonstrate negative effective permeability at prescribed frequencies are provided to confirm the utility and validity of the presented method

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

optimized schwarz methods for circular domain decompositions with overlap optimized schwarz methods are based on transmission conditions between subdo mains which are optimized for the problem class that is being solved such optimizations have been performed for many different types of partial differential equations but were almost exclusively based on the assumption of straight interfaces we study in this paper the influence of curvature on the optimization and we obtain four interesting new results first we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates second we develop an asymptotically accurate approximation technique based on turan type inequalities in our case to solve the much harder optimization problem on the curved interface and this approximation tech nique will also be applicable to currently too complex best approximation problems in the area of optimized schwarz methods third we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized schwarz method finally we find that in the case of curved interfaces optimized schwarz methods are not necessarily convergent for all admissible parameters our optimization leads however to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition we illustrate our analysis with numerical experiments

an adaptive dpg method for high frequency time harmonic wave propagation problems abstract the discontinuous petrov galerkin dpg method for high frequency wave propagation problems is discussed the dpg method with its attractive uniform mesh and wavenumber independent pre asymptotic stability property allows for a fully automatic adaptive h p algorithm that can be initiated from very coarse meshes moreover dpg always delivers a hermitian positive definite system suggesting the use of the conjugate gradient algorithm for its solution we present a new iterative solution scheme which capitalizes on these attractive properties of dpg this novel solver is integrated within the adaptive procedure by constructing a two grid like preconditioner for the conjugate gradient method that exploits information from previous meshes the construction of our preconditioner is discussed and its efficacy is illustrated with an example of a 2d acoustics problem our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber

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

near optimal perfectly matched layers for indefinite helmholtz problems a new construction of an absorbing boundary condition for indefinite helmholtz problems on unbounded domains is presented this construction is based on a near best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval designed with the help of a classical result by zolotarev using krein s interpretation of a stieltjes continued fraction this interpolant can be converted into a three term finite difference discretization of a perfectly matched layer pml which converges exponentially fast in the number of grid points the convergence rate is asymptotically optimal for both propagative and evanescent wave modes several numerical experiments and illustrations are included

optimal discrete transmission conditions for a nonoverlapping domain decomposition method for the helmholtz equation this paper is dedicated to recent developments of a two lagrange multipliers domain decomposition method for the helmholtz equation c farhat et al comput methods appl mech engrg 184 2000 pp 213 240 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 involving an additional augmented operator along the interface between the subdomains most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator b despres proceedings of the second international conference on mathematical and numerical aspects of wave propagation r kleinman et al eds siam philadelphia 1993 pp 197 206 j d benamou and b despres j comput phys 136 1997 pp 68 82 p chevalier and f nataf domain decomposition methods 10 ams providence ri 1998 pp 400 407 m j gander proceedings of the 12th international conference on domain decomposition methods chiba japan ddm org 2000 pp 247 254 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 at the discrete level the optimal operator can be proved to be equal to the schur complement of the outer domain this schur complement can be directly approximated using purely algebraic techniques like sparse approximate inverse methods or incomplete factorization the main advantage of such an algebraic approach is that it is much easier to implement in existing code without any information on the geometry of the interface and the finite element formulation used convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented

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

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

the optimized schwarz method with a coarse grid correction optimized schwarz methods osms use robin transmission conditions across the subdomain interfaces the robin parameter can then be optimized to obtain the fastest convergence a new formulation is presented with a coarse grid correction the optimal parameter is computed for a model problem on a cylinder together with the corresponding convergence factor which is smaller than that of classical schwarz methods a new coarse space is presented suitable for osm numerical experiments illustrating the effectiveness of osm with a coarse grid correction both as an iteration and as a preconditioner are reported

second order absorbing boundary conditions for the wave equation a solution for the corner problem the treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view a technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered a solution is proposed for the two dimensional case when second order conditions are used this solution consists of prescribing an adequate corner condition the problem thus obtained is analyzed theoretically and the condition is proved to be optimal the results obtained here are illustrated by numerical simulations some extensions to higher space dimensions and higher order conditions are proposed

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

convergence analysis for a second order schwarz method for nonoverlapping subdomains abstract a second order schwarz method for elliptic partial differential equations is developed and analyzed at a continuous level which can be parallel implemented with local communications it uses a second order boundary condition in terms of poincare steklov operators as its transmission condition on the interfaces the convergence of the algorithm in h 1 norm is obtained by using energy estimates and convergence results hold on arbitrary geometry of subdomains with no cross points

analysis and optimization of robin robin partitioned procedures in fluid structure interaction problems in the solution of fluid structure interaction fsi problems partitioned procedures are modular algorithms that involve separate fluid and structure solvers that interact in an iterative framework through the exchange of suitable transmission conditions at the fs interface in this work we study using fourier analysis the convergence of partitioned algorithms based on robin transmission conditions we derive for different models of the fluid and the structure a frequency dependent reduction factor at each iteration of the partitioned algorithm which is minimized by choosing optimal values of the coefficients in the robin transmission conditions two dimensional numerical results are also reported which highlight the effectiveness of the optimization procedure

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

fast alternating bidirectional preconditioner for the 2d high frequency lippmann schwinger equation this paper presents a fast iterative solver for the lippmann schwinger equation for high frequency waves scattered by a smooth medium with a compactly supported inhomogeneity the solver is based on the sparsifying preconditioner l ying multiscale model simul 13 2015 pp 644 660 and a domain decomposition approach similar to the method of polarized traces l zepeda nunez and l demanet j comput phys 308 2016 pp 347 388 the iterative solver has two levels the outer level in which a sparsifying preconditioner for the lippmann schwinger equation is constructed and the inner level in which the resulting sparsified system is solved quickly using an iterative solver preconditioned with a bidirectional matrix free variant of the method of polarized traces the complexity of the construction and application of the preconditioner is cal o n and cal o n log n respectively where n is the number of degrees of freedom numerical experiments in 2 dimensions indicate that t

asynchronous optimized schwarz methods with and without overlap an asynchronous version of the optimized schwarz method for the solution of differential equations on a parallel computational environment is studied in a one way subdivision of the computational domain with and without overlap the method is shown to converge when the optimal artificial interface conditions are used convergence is also proved for the laplacian operator under very mild conditions on the size of the subdomains when approximate non optimal interface conditions are utilized numerical results are presented on a large three dimensional problem on modern parallel clusters and supercomputers illustrating the efficiency of the asynchronous approach

convergence estimates for an higher order optimized schwarz method for domains with an arbitrary interface optimized schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations optimized schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence in the literature the analysis of optimized schwarz methods relies on fourier analysis and so the domains are restricted to be regular rectangular in this paper we express the interface operator of an optimized schwarz method in terms of poincare steklov operators this enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1 o h 1 4 on a class of general domains where h is the discretization parameter this is the predicted rate for a second order optimized schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

a level set method for vibration and multiple loads structural optimization abstract we extend the level set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads this method is based on a combination of the classical shape derivative and of the osher sethian level set algorithm for front propagation in two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations the shape derivative is used as an advection velocity in a hamilton jacobi equation for changing the shape this level set method is a low cost shape capturing algorithm working on a fixed eulerian mesh and it can easily handle topology changes

non overlapping domain decomposition algorithm based on modified transmission conditions for the helmholtz equation abstract a square root based transmission conditions domain decomposition method was recently introduced for the helmholtz equation it produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization we modify here these conditions in order to guarantee well posedness of local problems and further improve the efficiency of the whole method numerical results in particular in the three dimensional case show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm

a homographic best approximation problem with application to optimized schwarz waveform relaxation we present and study a homographic best approximation problem which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm we present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one the number of iterations to reach a certain accuracy can be orders of magnitudes smaller we illustrate our analysis with numerical experiments

band structure optimization of two dimensional photonic crystals in h polarization in the study of photonic crystals the question arises naturally which crystals produce the largest band gaps this question is investigated by means of an optimization based evolution algorithm which given two dielectric materials seeks to produce a material distribution within the fundamental cell which produces a maximal band gap at a given point in the spectrum the case of h polarization in two dimensions is considered several numerical examples are presented

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

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

a rapidly converging domain decomposition method for the helmholtz equation a new domain decomposition method is introduced for the heterogeneous 2 d and 3 d helmholtz equations transmission conditions based on the perfectly matched layer pml are derived that avoid artificial reflections and match incoming and outgoing waves at the subdomain interfaces we focus on a subdivision of the rectangular domain into many thin subdomains along one of the axes in combination with a certain ordering for solving the subdomain problems and a gmres outer iteration when combined with multifrontal methods the solver has near linear cost in examples due to very small iteration numbers that are essentially independent of problem size and number of subdomains it is to our knowledge only the second method with this property next to the moving pml sweeping method

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

algebraic multilevel preconditioner for the helmholtz equation in heterogeneous media an algebraic multilevel ml preconditioner is presented for the helmholtz equation in heterogeneous media it is based on a multilevel incomplete ldl t factorization and preserves the inherent complex symmetry of the helmholtz equation the ml preconditioner incorporates two key components for efficiency and numerical stability symmetric maximum weight matchings and an inverse based pivoting strategy the former increases the block diagonal dominance of the system whereas the latter controls l 1 for numerical stability when applied recursively their combined effect yields an algebraic coarsening strategy similar to algebraic multigrid methods even for highly indefinite matrices the ml preconditioner is combined with a krylov subspace method and applied as a black box solver to a series of challenging two and three dimensional test problems mainly from geophysical seismic imaging the numerical results demonstrate the robustness and efficiency of the ml preconditioner even at higher frequency regimes

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

a phase field method for shape optimization of incompressible flows abstract in this paper we present a phase field method applied to the fluid based shape optimization the fluid flow is governed by the incompressible navier stokes equations a phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized the shape sensitivity analysis is presented in terms of phase field variable which is the main contribution of this paper it saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes numerical results on some benchmark problems are reported and it is shown that the phase field approach for fluid shape optimization is efficient and robust

optimized schwarz methods for the time harmonic maxwell equations with damping in a previous paper two of the authors have proposed and analyzed an entire hierarchy of optimized schwarz methods for maxwell s equations in both the time harmonic and the time domain case the optimization process has been performed in a particular situation where the electric conductivity was neglected here we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case from the mathematical point of view the approach is different since the algorithm does not encounter the pathological situations of the zero conductivity case and thus the optimization problems are different we analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones we illustrate our analysis with numerical results

absorbing interface conditions for domain decomposition methods a general presentation the continuity conditions and the transmission conditions involved in domain decomposition methods are of major importance for the fast and robust convergence of these algorithms in this review paper we study step by step the methodology to derive various continuity conditions and compatible transmission conditions for two different types of problems i e the laplace equation and the helmholtz equation an original homogeneous formulation is also presented both in the continuous and in the discrete analysis numerical experiments show the relative efficiency of these continuity conditions and of these transmission conditions on academic problems

shape and topology optimization based on the phase field method and sensitivity analysis this paper discusses a structural optimization method that optimizes shape and topology based on the phase field method the proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions the advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required structural shapes are represented by the phase field function defined in the design domain and optimization of this function is performed by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis the proposed method is applied to two dimensional linear elastic and vibration optimization problems such as the minimum compliance problem a compliant mechanism design problem and the eigenfrequency maximization problem the numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations

a fem domain decomposition method for photonic and electromagnetic band gap structures this paper introduces an efficient domain decomposition algorithm for the solution of time harmonic electromagnetic fields arising in three dimensional finite size photonic band gap and electromagnetic band gap structures the method is based on the finite element approximation and a nonoverlapping domain decomposition method a set of cement unknowns on the inter domain interfaces has been explicitly introduced to enforce the appropriate field continuities the introduction of these extra unknowns allows for nonconforming nonmatching triangulations across domain eliminating the need for periodic mesh to ensure and improve the convergence of the outer iteration loop robin transmission condition is used to communicate information across interfaces the resulting system of equations is solved with a fast algorithm that loosely resembles the well known finite element tearing and interconnecting algorithm in this algorithm the method solves in the preprocessing step for the robin primal subdomain problems multiple times by exciting one dual unknown at a time this step generates an iteration matrix that is then used to update the dual unknowns in the outer loop iteration the present method becomes extremely efficient for problems with geometric repetitions such as photonic and electromagnetic band gap structures

analysis of the convected helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions abstract in this paper we propose complete radiation boundary conditions crbcs for solutions of the convected helmholtz equation with a uniform mean flow in a waveguide we first study crbcs for the helmholtz equation in a waveguide noting that the convected helmholtz equation is associated with the helmholtz equation via the prandtl glauert transformation crbcs for the convected helmholtz equation is derived from crbcs for the helmholtz equation we analyse well posedness and convergence of approximate solutions satisfying crbcs for the convected helmholtz equation in addition simple numerical experiments will be presented to confirm the theoretical results

how large a shift is needed in the shifted helmholtz preconditioner for its effective inversion by multigrid the shifted helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the helmholtz equation the idea is that if one uses a small complex shift the shifted helmholtz operator is still close to the original helmholtz operator and could thus be an effective preconditioner it was shown in m j gander i g graham and e a spence numer math 53 2015 pp 573 579 that the shift can be at most o k to prove rigorously wave number independent convergence of the preconditioned system solved with gmres provided the preconditioner is inverted exactly in practice however the preconditioner is inverted only approximately and if one shifts enough this can be done effectively by standard multigrid methods we show in this paper that for a finite element discretization the shift has to be at least o k 2 to be able to invert the shifted helmholtz preconditioner using multigrid there is therefore a gap between being a good precondi

short note shape and topology optimization for electrothermomechanical microactuators using level set methods in this short note a shape and topology optimization method is presented for multiphysics actuators including geometrically nonlinear modeling based on an implicit free boundary parameterization method a level set model is established to describe structural design boundary by embedding it into the zero level set of a higher dimensional level set function the compactly supported radial basis functions csrbf are introduced to parameterize the implicit level set surface with a high level of accuracy and smoothness the original more difficult shape and topology optimization driven by the hamilton jacobi partial differential equation pde is transferred into a relatively easier parametric size optimization to which many well founded optimization algorithms can be applied thus the structural optimization is transformed to a numerical process that describes the design as a sequence of motions of the design boundaries by updating the expansion coefficients of the size optimization two widely studied examples are chosen to demonstrate the effectiveness of the proposed method

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

3 d electromagnetic scattering computation in free space with the feti fdp2 method the electromagnetic dual primal finite element tearing and interconnecting feti dpem method is a nonoverlapping domain decomposition method developed for the finite element analysis of large scale electromagnetic problems where the corner edges are globally numbered this paper presents an extension of the feti dpem2 method named feti full dual primal feti fdp2 where more flexible robin type boundary conditions are imposed on the inner interfaces between subdomains as well as on the corner edges leading to a new interface problem its capacities are tested in the framework of a three dimensional 3 d free space scattering problem with a scattered field formulation and a computational domain truncated by perfectly mathed layers pml first we compare its accuracy with respect to other feti dpem2 methods and to a complete resolution of the fem problem thanks to a direct sparse solver we show that the convergence of iterative solvers is affected by the presence of the pml and can be accelerated by means of a more accurate approximation between adjacent subdomains of the dirichlet to neumann dtn operator the effectiveness of the iterative solvers are also considered for different test cases the advantages of the proposed feti fdp2 method combined with the associated dtn approximation is numerically demonstrated regardless the chosen working frequency or the iterative solvers

a source transfer domain decomposition method for helmholtz equations in unbounded domain we propose and study a domain decomposition method for solving the truncated perfectly matched layer pml approximation in bounded domain of helmholtz scattering problems the method is based on the decomposition of the domain into nonoverlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a pml method defined locally outside the last two layers the convergence of the method is proved for the case of constant wave number based on the analysis of the fundamental solution of the pml equation the method can be used as an efficient preconditioner in the preconditioned gmres method for solving discrete helmholtz equations with constant and heterogeneous wave numbers numerical examples are included