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 both in the time-harmonic and time-domain case. The optimization process has been perfomed 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

Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems.We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations

Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures

We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed. We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.Comment: 24 page

Comparison of a one and two parameter family of transmission conditions for Maxwell's equations with damping

Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell's equations, transmission conditions which lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell's equation, see [2, 3, 1], and also for first order formulations, see [7, 6]. These methods have well found their way into applications, see for example [9, 11, 10]. It turns out that good transmission conditions are approximations of transparent boundary conditions. For each form of approximation chosen, one can try to find the best remaining free parameters in the approximation by solving a min-max problem. Usually allowing more free parameters leads to a substantially better solution of the min-max problem, and thus to a much better algorithm. For a particular one parameter family of transmission conditions analyzed in [4], we investigate in this paper a two parameter counterpart. The analysis, which is substantially more complicated than in the one parameter case, reveals that in one particular asymptotic regime there is only negligible improvement possible using two parameters, compared to the one parameter results. This analysis settles an important open question for this family of transmission conditions, and also suggests a direction for systematically reducing the number of parameters in other optimized transmission conditions

Reconstructing Detailed Line Profiles of Lamellar Gratings from GISAXS Patterns with a Maxwell Solver

Laterally periodic nanostructures were investigated with grazing incidence small angle X-ray scattering (GISAXS) by using the diffraction patterns to reconstruct the surface shape. To model visible light scattering, rigorous calculations of the near and far field by numerically solving Maxwell's equations with a finite-element method are well established. The application of this technique to X-rays is still challenging, due to the discrepancy between incident wavelength and finite-element size. This drawback vanishes for GISAXS due to the small angles of incidence, the conical scattering geometry and the periodicity of the surface structures, which allows a rigorous computation of the diffraction efficiencies with sufficient numerical precision. To develop dimensional metrology tools based on GISAXS, lamellar gratings with line widths down to 55 nm were produced by state-of-the-art e-beam lithography and then etched into silicon. The high surface sensitivity of GISAXS in conjunction with a Maxwell solver allows a detailed reconstruction of the grating line shape also for thick, non-homogeneous substrates. The reconstructed geometrical line shape models are statistically validated by applying a Markov chain Monte Carlo (MCMC) sampling technique which reveals that GISAXS is able to reconstruct critical parameters like the widths of the lines with sub-nm uncertainty

Schwarz methods for second order Maxwell equations in 3D with coefficient jumps

We study non-overlapping Schwarz Methods for solving second order time-harmonic 3D Maxwell equations in heterogeneous media. Choosing the interfaces between the subdomains to be aligned with the discontinuities in the coefficients, we show for a model problem that while the classical Schwarz method is not convergent, optimized transmission conditions dependent on the discontinuities of the coefficients lead to convergent methods. We prove asymptotically that the resulting methods converge in certain cases independently of the mesh parameter, and convergence can even become better as the coefficient jumps increase

Optimized schwarz methods for Maxwell equations with discontinuous coefficients

We study non-overlapping Schwarz methods for solving time-harmonic Maxwell’s equations in heterogeneous media. We show that the classical Schwarz algorithm is always divergent when coefficient jumps are present along the interface. In the case of transverse magnetic or transverse electric two dimensional formulations, convergence can be achieved in specific configurations only. We then develop optimized Schwarz methods which can take coefficient jumps into account in their transmission conditions. These methods exhibit rapid convergence, and sometimes converge independently of the mesh parameter, even without overlap. We illustrate our analysis with numerical experiments

Scalable domain decomposition methods for time-harmonic wave propagation problems

The construction of efficient solvers for non self-adjoint problems, like Helmholtz equations is a challenging task. After the discretisation of the PDE by a finite element method, the resulting linear systems are large and because of their spectral properties, difficult to analyse theoretically and to solve by iterative methods. Domain decomposition methods are hybrid methods, as they use an iterative coupling of smaller problems which are solved in turn by direct methods. They rely on dividing the global problem into local subproblems on smaller subdomains. These methods can be used as iterative solvers but also as preconditioners in a Krylov method. Robustness with respect of the number of subdomains is important as this is related to the notion of scalability. We focus here on a configuration where scalability is achieved without the addition of a coarse-space correction. However, convergence can still be improved by modifying the transmission conditions imposed between the subdomains. In this manuscript, we start by giving an overview of the basic domain decomposition methods and their use as preconditioners. Then we consider these methods from an iterative point of view and we perform a study of convergence analysis of overlapping Schwarz methods with Dirichlet, Robin, zeroth and second order transmission conditions for many subdomains. We also present more sophisticated methods, which implement more effective transmission conditions depending on some optimised parameters. In our analysis, we focus on the Helmholtz problem and the magnetotelluric approximation of Maxwell’s equation for stripwise decompositions into many domains. Our theoretical findings are being demonstrated by the appropriate numerical evidence.The construction of efficient solvers for non self-adjoint problems, like Helmholtz equations is a challenging task. After the discretisation of the PDE by a finite element method, the resulting linear systems are large and because of their spectral properties, difficult to analyse theoretically and to solve by iterative methods. Domain decomposition methods are hybrid methods, as they use an iterative coupling of smaller problems which are solved in turn by direct methods. They rely on dividing the global problem into local subproblems on smaller subdomains. These methods can be used as iterative solvers but also as preconditioners in a Krylov method. Robustness with respect of the number of subdomains is important as this is related to the notion of scalability. We focus here on a configuration where scalability is achieved without the addition of a coarse-space correction. However, convergence can still be improved by modifying the transmission conditions imposed between the subdomains. In this manuscript, we start by giving an overview of the basic domain decomposition methods and their use as preconditioners. Then we consider these methods from an iterative point of view and we perform a study of convergence analysis of overlapping Schwarz methods with Dirichlet, Robin, zeroth and second order transmission conditions for many subdomains. We also present more sophisticated methods, which implement more effective transmission conditions depending on some optimised parameters. In our analysis, we focus on the Helmholtz problem and the magnetotelluric approximation of Maxwell’s equation for stripwise decompositions into many domains. Our theoretical findings are being demonstrated by the appropriate numerical evidence

Generalized harmonic modeling technique for 2D electromagnetic problems : applied to the design of a direct-drive active suspension system

The introduction of permanent magnets has significantly improved the performance and efficiency of advanced actuation systems. The demand for these systems in the industry is increasing and the specifications are becoming more challenging. Accurate and fast modeling of the electromagnetic phenomena is therefore required during the design stage to allow for multi-objective optimization of various topologies. This thesis presents a generalized technique to design and analyze 2D electromagnetic problems based on harmonic modeling. Therefore, the prior art is extended and unified to create a methodology which can be applied to almost any problem in the Cartesian, polar and axisymmetric coordinate system. This generalization allows for the automatic solving of complicated boundary value problems within a very short computation time. This method can be applied to a broad class of classical machines, however, more advanced and complex electromagnetic actuation systems can be designed or analyzed as well. The newly developed framework, based on the generalized harmonic modeling technique, is extensively demonstrated on slotted tubular permanent magnet actuators. As such, numerous tubular topologies, magnetization and winding configurations are analyzed. Additionally, force profiles, emf waveforms and synchronous inductances are accurately predicted. The results are within approximately 5 % of the non-linear finite element analysis including the slotted stator effects. A unique passive damping solution is integrated within the tubular permanent magnet actuator using eddy current damping. This is achieved by inserting conductive rings in the stator slot openings to provide a passive damping force without compromising the tubular actuator’s performance. This novel idea of integrating conductive rings is secured in a patent. A method to calculate the damping ratio due to these conductive rings is presented where the position, velocity and temperature dependencies are shown. The developed framework is applied to the design and optimization of a directdrive electromagnetic active suspension system for passenger cars. This innovative solution is an alternative for currently applied active hydraulic or pneumatic suspension systems for improvement of the comfort and handling of a vehicle. The electromagnetic system provides an improved bandwidth which is typically 20 times higher together with a power consumption which is approximately five times lower. As such, the proposed system eliminates two of the major drawbacks that prevented the widespread commercial breakthrough of active suspension systems. The direct-drive electromagnetic suspension system is composed of a coil spring in parallel with a tubular permanent magnet actuator with integrated eddy current damping. The coil spring supports the sprung mass while the tubular actuator either consumes, by applying direct-drive vertical forces, or regenerates energy. The applied tubular actuator is designed using a non-linear constrained optimization algorithm in combination with the developed analytical framework. This ensured the design with the highest force density together with low power consumption. In case of a power breakdown, the integrated eddy current damping in the slot openings of this tubular actuator, together with the passive coil spring, creates a passive suspension system to guarantee fail-safe operation. To validate the performance of the novel proof-of-concept electromagnetic suspension system, a prototype is constructed and a full-scale quarter car test setup is developed which mimics the vehicle corner of a BMW 530i. Consequently, controllers are designed for the active suspension strut for improvement of either comfort or handling. Finally, the suspension system is installed as a front suspension in a BMW 530i test vehicle. Both the extensive experimental laboratory and on-road tests prove the capability of the novel direct-drive electromagnetic active suspension system. Furthermore, it demonstrates the applicability of the developed modeling technique for design and optimization of electromagnetic actuators and devices