Flexible approximation schemes with numerical and ...

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COMPEL 30,2

Flexible approximation schemes with numerical and semi-analytical bases

552

Jianhua Dai Department of Electrical and Computer Engineering, The University of Akron, Akron, Ohio, USA

Helder Pinheiro and Jonathan P. Webb Department of Electrical and Computer Engineering, McGill University, Montreal, Canada, and

Igor Tsukerman Department of Electrical and Computer Engineering, The University of Akron, Akron, Ohio, USA Abstract Purpose – The purpose of this paper is to extend the generalized finite-difference calculus of flexible local approximation methods (FLAME) to problems where local analytical solutions are unavailable. Design/methodology/approach – FLAME uses accurate local approximations of the solution to generate difference schemes with small consistency errors. When local analytical approximations are too complicated, semi-analytical or numerical ones can be used instead. In the paper, this strategy is applied to electrostatic multi-particle simulations and to electromagnetic wave propagation and scattering. The FLAME basis is constructed by solving small local finite-element problems or, alternatively, by a local multipole-multicenter expansion. As yet another alternative, adaptive FLAME is applied to problems of wave propagation in electromagnetic (photonic) crystals. Findings – Numerical examples demonstrate the high rate of convergence of new five- and nine-point schemes in 2D and seven- and 19-point schemes in 3D. The accuracy of FLAME is much higher than that of the standard FD scheme. This paves the way for solving problems with a large number of particles on relatively coarse grids. FLAME with numerical bases has particular advantages for the multi-particle model of a random or quasi-random medium. Research limitations/implications – Irregular stencils produced by local refinement may adversely affect the accuracy. This drawback could be rectified by least squares FLAME, where the number of stencil nodes can be much greater than the number of basis functions, making the method more robust and less sensitive to the irregularities of the stencils. Originality/value – Previous applications of FLAME were limited to purely analytical basis functions. The present paper shows that numerical bases can be successfully used in FLAME when analytical ones are not available. Keywords Wave propagation, Wave properties, Numerical analysis, Approximation theory, Electrostatics Paper type Research paper COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 30 No. 2, 2011 pp. 552-573 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321641111101078

The authors are grateful to Professors Amir Boag and Yehuda Leviatan for discussions and for bringing several interesting references to their attention. They also thank all five anonymous reviewers of this paper for their kind and helpful comments.

1. Introduction 1.1 Flexible local approximation schemes and related methods Multi-body electrostatic, magnetostatic and electromagnetic wave scattering problems are important for the simulation of colloidal systems, polymers, macromolecules, magnetically driven assembly (Plaks et al., 2003; Yellen et al., 2005; Erb et al., 2009), electromagnetic/photonic crystals (Johnson and Joannopoulos, 2001; Sakoda, 2005; Tsukerman, 2007; Pinheiro et al., 2007) and in other applications. In such cases, conventional numerical methods have serious limitations: the finite-element method (FEM) requires geometrically conforming meshes that become extremely complex when the number of objects such as particles or scatterers is large; the fast multipole method (Cheng et al., 1999, and references therein) is not effective for inhomogeneous and especially for nonlinear problems. The generalized finite-difference (FD) calculus of flexible local approximation methods (FLAME) (Tsukerman, 2005, 2006, 2007, 2010) replaces the Taylor expansions of classical FD methods with more accurate approximating functions. This approach is particularly helpful when the Taylor expansion breaks down, for example at material interfaces. In the “Trefftz” version of FLAME, local basis functions satisfy the underlying differential equation: for example, in electro- or magnetostatic multiparticle problems, spherical harmonics near dielectric or magnetic particles can be used (Tsukerman, 2005, 2006; Sosonkina and Tsukerman, 2006). However, local analytical solutions could be either too complicated or unavailable; in multiparticle problems, this is the case when several particles are in close proximity to one another or when particles have complex shapes. In this situation, one option is to use local semi-analytic or numerical solutions as basis functions (Section 3). Another approach that has already been tried before is adaptive FLAME that locally reduces the grid size, thereby extending the effectiveness of relatively simple analytical bases (Dai and Tsukerman, 2008). FLAME is connected to several other classes of numerical methods such as variational Trefftz (Jirousek and Zielinski, 1997; Herrera, 2000), generalized FEM (Melenk and Babusˇka, 1996 and others), discontinuous Galerkin (Cockburn et al., 2000; Arnold et al., 2002 and many others), variational-difference schemes of Moskow et al. (1999), discontinuous enrichment and finite increment calculus (Hughes et al., 1998; Farhat et al., 2001; On˜ate, 2000). These connections were previously discussed in (Tsukerman 2006) (see in particular Figure 1, Sections 1 and 2, and references there) and are not revisited here to save space. However, a few additional contributions are pointed out below, partly at the suggestion of the anonymous reviewers. Special difference schemes that can now be viewed as natural particular cases of FLAME have been independently invented by various research groups. Mei et al. (1994) used the fundamental solutions of the Laplace equation to construct approximate absorbing conditions at the exterior boundary of the computational domain for unbounded problems. Similar ideas were put forward by Mittra and Ramahi (1989), Boag et al. (1994) and Boag and Mittra (1995). Hadley (2002a, b) derived difference schemes for the Helmholtz equation (with applications to electromagnetic waveguide analysis) from the Bessel function expansion in free space as well as at material boundaries and corners. Problems of three types are considered in this paper to fix ideas: electrostatics and magnetostatics in inhomogeneous media and wave propagation/scattering in electromagnetic (photonic) crystals. These ideas can be extended to various other

Flexible approximation schemes 553

COMPEL 30,2

5 Ω(i) 1

2

3

554 Figure 1. Patch Vði Þ (dashed line) intersects two nearby particles, which complicates the analytical approximation within this patch

4

problems, e.g. to the Poisson-Boltzmann equation in colloidal systems (Tsukerman, 2007, 2005, 2006). 1.2 Electrostatics The electrostatic potential u is governed by the well-known equation: Le u ; 27 · e 7u ¼ r

ð1Þ

where r is a given charge density and 1 is the dielectric permittivity that may in general be a function of coordinates. Standard conditions for the continuity of the potential and of the normal component of the electric flux density hold at particle boundaries; these conditions are equivalent to interpreting equation (1) in the sense of distributions (Vladimirov, 1984). If an external field with potential uext is applied, the Dirichlet boundary condition at infinity is: uðrÞ 2 uext ðrÞ ! 0;

r!1

ð2Þ

Alternatively, the equations may be rewritten in terms of the scattered potential us ¼ u 2 uext for which the condition at infinity is zero. This is often convenient, but in the present paper the scattered field will not be explicitly used. 1.3 Magnetostatics When formulated in terms of the magnetic scalar potential u, the problem for the magnetic field H ¼ 27u is mathematically analogous to the electrostatic problem: Lm u ; 27 · m7u ¼ 0 where m is the permeability and the boundary condition (2) applies. This formulation is valid for simply connected current-free regions (Bossavit, 1998) (these constraints can be removed by introducing cuts in multiply connected regions and “source fields” corresponding to given currents, but that is well beyond the scope of this paper).

1.4 Electromagnetic wave propagation and scattering Electromagnetic wave problems in this paper are limited to two dimensions; for electromagnetic vector problems in 3D, FLAME is still under development (Pinheiro and Webb, 2009). The 2D case arises if the material parameters and fields are independent of one coordinate (say, z). It is well known that electromagnetic waves can then be decomposed into two modes. In the E-mode (Transverse magnetic- or s-mode), the electric field has only one component E ¼ E z , whereas the magnetic field H has x and y components, but H z ¼ 0. Similarly, for the H-mode (Transverse electric- or p-mode) H ¼ H z , E z ¼ 0. These modes satisfy the familiar equations: 7 · m 21 7E þ v 2 1E ¼ 0

ð3Þ

7 · e 21 7H þ v 2 mH ¼ 0

ð4Þ

1.5 Multipole-multicenter expansions In the remainder of the paper, we shall frequently reference multipole-multicenter expansions. This approach is very well known in various applications, although its origin is not easy to pinpoint. It became popular in the 1980s (Stone and Alderton, 1985; Ballisti and Hafner, 1983), but the two-center expansion was already part of the DLVO theory in the 1940s (Verwey and Overbeek, 1948, pp. 145-50). The modern sources are (Chew, 1990; Mishchenko et al., 2002). The potential or field around a collection of particles is sought as a superposition of particle-centered cylindrical (2D) or spherical (3D) harmonics, either static ones or waves, depending on the type of the problem. To impose the interface boundary conditions on the surface of any given particle, one needs to translate the multipole expansions from all other particles to that particle; this is accomplished by the standard translation formulas (Cheng et al., 1999; Chew, 1990; Mishchenko et al., 2002). The details of this methodology are available in the literature and are not repeated here. In the context of FLAME, multipole-multicenter expansions play two different roles. First, they can be used to compute a global quasi-analytical solution for testing and verification. Second, they can help to generate local FLAME bases in situations where two or more particles happen to be in close proximity to one another (Section 3). 2. Trefftz-FLAME schemes The brief description of FLAME in this section has already appeared in a number of publications (Tsukerman, 2007, 2009, 2010) and is included here for convenience of the reader, to make the paper self-contained. Trefftz-FLAME is a generalized FD calculus that incorporates accurate local approximations of the solution into a difference scheme. Conceptually, the computational domain V is covered by a finite number of overlapping subdomains (“patches”) V(i ), V ¼