Low Frequency Asymptotics and Electro-Magneto-Statics for Time-Harmonic Maxwell's Equations in Exterior Weak Lipschitz Domains with Mixed Boundary Conditions

Abstract We prove global and local versions of the so-called {\operatorname{div}} - {\operatorname{curl}} -lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

Download Full-text

A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains

Advances in Computational Mathematics ◽

10.1007/s10444-007-9027-6 ◽

2007 ◽

Vol 28 (4) ◽

pp. 355-382

Author(s):

Yang Liu ◽

Qiya Hu ◽

Dehao Yu

Keyword(s):

Domain Decomposition ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Low Frequency ◽

Unbounded Domains ◽

Time Harmonic ◽

Overlapping Domain Decomposition

Download Full-text

Impedance Functions for Surface Rigid Rectangular Foundations on Transversely Isotropic Multilayer Half-Spaces

Journal of Applied Mechanics ◽

10.1115/1.4023626 ◽

2013 ◽

Vol 80 (5) ◽

Cited By ~ 18

Author(s):

Amirhossein Amiri-Hezaveh ◽

Morteza Eskandari-Ghadi ◽

Mohammad Rahimian ◽

Amir K. Ghorbani-Tanha

Keyword(s):

Boundary Conditions ◽

Half Space ◽

Mixed Boundary ◽

Surface Displacement ◽

Transversely Isotropic ◽

Mixed Boundary Conditions ◽

Integral Transforms ◽

Material Symmetry ◽

Time Harmonic ◽

Impedance Functions

A horizontally multilayered Green elastic transversely isotropic half-space is considered as the domain of the boundary value problem involved in this paper, such that the axes of material symmetry of different layers are parallel to the axis of material symmetry of the lowest half-space, which is depthwise. The domain is assumed to be affected by an arbitrary time-harmonic forced vibration due to a rigid rectangular surface foundation. With the use of a potential function method and the Hankel integral transforms, the displacements and stresses Green's functions are determined in each layer. The unknown functions due to integrations in each layer are transformed to the unknown functions of the surface layer with the use of the concept of propagator matrix and the continuity conditions. The mixed boundary conditions at the surface of the whole domain are numerically satisfied with the assumption of piecewise constant distribution of tractions in the contact area. It is numerically shown that the surface displacement and stress boundary conditions are satisfied very well. The vertical and horizontal impedance functions of the rectangular foundation are determined, which may be used as lumped parameters in time-harmonic soil-structure interaction with transversely isotropic horizontally layered domain as the soil. It is shown that the impedance functions determined in this paper coincide with the same functions for the simpler case of isotropic homogeneous half-space as degenerations of this study.

Download Full-text