Finite element heterogeneous multiscale methods for the wave equation

Wave phenomena appear in a wide range of applications such as full-waveform seismic inversion, medical imaging, or composite materials. Often, they are modeled by the acoustic wave equation. It can be solved by standard numerical methods such as, e.g., the finite element (FE) or the finite difference method. However, if the wave propagation speed varies on a microscopic length scale denoted by epsilon, the computational cost becomes infeasible, since the medium must be resolved down to its finest scale. In this thesis we propose multiscale numerical methods which approximate the overall macroscopic behavior of the wave propagation with a substantially lower computational effort. We follow the design principles of the heterogeneous multiscale method (HMM), introduced in 2003 by E and Engquist. This method relies on a coarse discretization of an a priori unknown effective equation. The missing data, usually the parameters of the effective equation, are estimated on demand by solving microscale problems on small sampling domains. Hence, no precomputation of these effective parameters is needed. We choose FE methods to solve both the macroscopic and the microscopic problems. For limited time the overall behavior of the wave is well described by the homogenized wave equation. We prove that the FE-HMM method converges to the solution of the homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops. Neither the homogenized solution, nor the FE-HMM capture these dispersive effects. To capture them we need to modify the FE-HMM. Inspired by higher order homogenization techniques we additionally compute a correction term of order epsilon^2. Since its computation also relies on the solution of the same microscale problems as the original FE-HMM, the computational effort remains essentially unchanged. For this modified version we also prove convergence to the homogenized wave equation, but in contrast to the original FE-HMM the long-time dispersive behavior is recovered. The convergence proofs for the FE-HMM follow from new Strang-type results for the wave equation. The results are general enough such that the FE-HMM with and without the long-time correction fits into the setting, even if numerical quadrature is used to evaluate the arising L^2 inner product. In addition to these results we give alternative formulations of the FE-HMM, where the elliptic micro problems are replaced by hyperbolic ones. All the results are supported by numerical tests. The versatility of the method is demonstrated by various numerical examples

Beyond local effective material properties for metamaterials

To discuss the properties of metamaterials on physical grounds and to consider them in applications, effective material parameters are usually introduced and assigned to a given metamaterial. In most cases, only weak spatial dispersion is considered. It allows to assign local material properties, i.e. a permittivity and a permeability. However, this turned out to be insufficient. To solve this problem, we study here the effective properties of metamaterials with constitutive relations beyond a local response and take strong spatial dispersion into account. The isofrequency surfaces of the dispersion relation are investigated and compared to those of an actual metamaterial. The significant improvement provides evidence for the necessity to use nonlocal material laws in the effective description of metamaterials. The general formulation we choose here renders our approach applicable to a wide class of metamaterials

Interface conditions for a metamaterial with strong spatial dispersion

Local constitutive relations, i.e. a weak spatial dispersion, are usually considered in the effective description of metamaterials. However, they are often insufficient and effects due to a nonlocality, i.e. a strong spatial dispersion, are encountered. Recently (K.~Mnasri et al., arXiv:1705.10969), a generic form for a nonlocal constitutive relation has been introduced that could accurately describe the bulk properties of a metamaterial in terms of a dispersion relation. However, the description of functional devices made from such nonlocal metamaterials also requires the identification of suitable interface conditions. In this contribution, we derive the interface conditions for such nonlocal metamaterials

Finite element heterogeneous multiscale method for time-dependent Maxwell’s equation

We propose a Finite Element Heterogeneous Multiscale Method (FEHMM) for time dependent Maxwell’s equations in second-order formulation. This method can approximate the effective behavior of an electromagnetic wave traveling through a highly oscillatory material without the need to resolve the microscopic details of the material. To prove an a-priori error bound for the semi-discrete FE-HMM scheme, we need a new generalization of a Strang-type lemma for second-order hyperbolic equations. Finally, we present a numerical example that is in accordance with the theoretical results

Unified error analysis for non-conforming space discretizations ofwave-type equations

This paper provides a unified error analysis for non-conforming space discretizations of linear wave equations in time-domain. We propose a framework which studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite dimensional Hilbert spaces. A lift operator maps the semi-discrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and non-conforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation

A review and evaluation of the Langley Research Center's Scientific and Technical Information Program: Results of phase 6: The technical report. A survey and analysis

Current practice and usage using selected technical reports; literature relative to the sequential, language, and presentation components of technical reports; and NASA technical report publications standards are discussed. The effctiveness of the technical report as a product for information dissemination is considered

On the approximation of electromagnetic fields by edge finite elements. Part 2: A heterogeneous ultiscale method for Maxwell’s equations

In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these diffculties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell\u27s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell\u27s equations in a periodic setting and give numerical experiments in accordance to the stated results