Calderón Preconditioned PMCHWT Equations for Analyzing Penetrable Objects in Layered Medium

published_or_final_versio

Benchmarking preconditioned boundary integral formulations for acoustics.

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves

Benchmarking preconditioned boundary integral formulations for acoustics

The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave scattering. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretisation of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave scattering at large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimised for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarking provide valuable information on the computational characteristics of several hundred different models for multiple scattering and transmission of acoustic wave fields

On a Calder\'on preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements

We present a Calder\'on preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calder\'on schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following shortcomings: 1) ill-conditioning when the frequency is low; 2) ill-conditioning when the discretization density is high; 3) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); 4) incorrect solution at low frequencies due to a loss of significant digits; and 5) the presence of spurious resonances. In this article, quasi-Helmholtz projectors are leveraged to obtain magnetic field integral equation (MFIE) that is immune to drawbacks 1)-4). Moreover, when this new MFIE is combined with a regularized electric field integral equation (EFIE), a new quasi-Helmholtz projector-combined field integral equation (CFIE) is obtained that also is immune to 5). The numerical results corroborate the theory and show the practical impact of the newly proposed formulations

Volume Integral Equation Methods for Forward and Inverse Bioelectromagnetic Approaches

L'abstract è presente nell'allegato / the abstract is in the attachmen

Solving Electrically Very Large Transient Electromagnetic Problems Using Plane-Wave Time-Domain Algorithms.

The marching-on-in-time (MOT)-based time domain integral equation solvers provide an appealing avenue for solving transient electromagnetic scattering/radiation problems. These state-of-the-art solvers are high-order accurate, rapidly converging and low-/high-frequency stable. Moreover, their computational efficiencies can be significantly improved by accelerators such as the multilevel plane-wave time-domain (PWTD) algorithm. However, practical transient electromagnetic problems involving millions of spatial unknowns and thousands of time steps were barely solved by PWTD-accelerated MOT solvers. This is due to the lack of (i) an efficient parallelization scheme for PWTD’s heterogeneous structure on modern computing platforms, and (ii) a temporal/angular/spatial adaptive PWTD that further improves the computational efficiency. The contributions of this work are as follows: First, a provably scalable parallelization scheme for the PWTD algorithm is developed. The proposed scheme scales well on thousands of CPU processors upon hierarchically partitioning the workloads in spatial, angular and temporal dimensions. The proposed scheme is adopted to time domain surface/volume integral equations (TD-SIE/TD-VIE) solvers for analyzing transient scattering from large and complex-shaped conducting/dielectric objects involving ten million/tens of millions of spatial unknowns. In addition, we developed a single/multiple graphics processing units (GPU) implementation of the PWTD algorithm that achieves at least one order of magnitude speedups compared to serial CPU implementations. Second, a wavelet compression scheme based on local cosine bases (LCBs) that exploits the sparsity in the temporal dimension is developed. All PWTD operations are performed in the wavelet domain with reduced computational complexity. The resultant wavelet-enhanced TD-SIE solver is capable of analyzing transient scattering from smooth quasi-planar conducting objects spanning well over one hundred wavelengths.PhDElectrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/113642/1/liuyangz_1.pd

Çeşitli sıradışı özelliklere sahip üç boyutlu metamalzemelerin hesaplamalı benzetimleri ve gerçeklenmeleri.

In this study, computational analysis and realization of three-dimensional metamaterial structures that induce negative and zero permittivity and/or permeability values in their host environment, as well as plasmonic nanoparticles that are used to design metamaterials at optical frequencies are presented. All these electromagnetic problems are challenging since effective material properties become negative/zero, while numerical solvers are commonly developed for ordinary positive parameters. In real life, three-dimensional metamaterial structures, involving split-ring resonators (SRR), thin wires, and similar subwavelength elements, are designed to exhibit single negativity (imaginary refractive index) and double negativity (negative refractive index) behaviors. However, metamaterial elements have small details with respect to wavelength and they operate when they resonate. Then, their numerical models lead to large matrix equations that are also ill-conditioned, making their solutions extremely difficult, if not impossible. If performed accurately, homogenization simplifies the analysis of metamaterials, while new challenges arise due to extreme parameters. For example, a combination of zero-index (ZI) and near-zero-index (NZI) materials with ordinary media (metals, free space, etc.) results in a high-contrast problem, and numerical instabilities occur particularly due to huge values of wavelength. Similar difficulties arise when considering the plasmonic effects of metals at optical frequencies since they must be modeled as penetrable bodies with negative real permittivity, leading to imaginary index values. Different surface-integral-equation (SIE) formulations and broadband multilevel fast multipole algorithm (MLFMA) implementations are extensively tested for accurate and efficient numerical solutions of ZI, NZI, imaginary-index, and negative-index materials. In addition to their computational simulations, metamaterial designs are fabricated with a low-cost inkjet-printing setup, which is based on using conventional printers that are modified and loaded with silver-based inks. Measurements demonstrate the feasibility of fabricating very low-cost three-dimensional metamaterials using simple inkjet printing.Thesis (M.S.) -- Graduate School of Natural and Applied Sciences. Electrical and Electronics Engineering