From Analytical to Numerical Methods and Back:

Trefftz Schemes, Whitney Forms, and Metamaterials

Igor Tsukerman

Department of Electrical and Computer Engineering,

The University of Akron, OH 44325-3904, USA

http://coel.ecgf.uakron.edu/igor/public_html

igor@uakron.edu

ETH, Monday July 4, 2011

Thanks to Christian Hafnerfor the invitation.

Also thanks toVadim Markel and John SchotlandC. T. Chan and W. C. ChewOszkar Bıro, Christian Magele, Kurt PreisSergey BozhevolnyiGreg RodinStéphane ClenetSascha Schnepp and Thomas WeilandGraeme MiltonDmitry GolovatyGuglielmo Rubinacci and his colleaguesPaolo Di BarbaBoris ShoykhetAlain Bossavit

Analytical methods

Numericalmethods

Ideas shared

Outline

From analytical to numerical methods: volume and boundary difference schemes with Trefftz approximations.

From numerical to analytical methods: metamaterial parameters via WNBK interpolation.

Examples.

Conclusion.

FROM ANALYTICAL TO NUMERICAL METHODS

What are the Biggest Problems in Finite Difference and Boundary Integral Methods?..

… and how to rectify them?

There will be some trade-offs.

Problems with FD & BIM

Finite Difference Methods:

the ―staircase effect‖.

Boundary Integral Methods:

Full matrices (but: use Fast MultipoleMethods, low-rank SVD or other acceleration).

Kernel singularities – arguably a tougher problem.

FLAME: Flexible Local Approximation MEthods

An automatic way to generate high-orderschemes (―compact schemes‖).

Replace Taylor expansions of standard FDwith local `Trefftz‘ approximations (basisfunctions satisfying the underlying differential equation).

Examples of basis functions: plane waves,exponentials, spherical / cylindricalharmonics, etc.

Accuracy often improves dramatically. `Staircase‘ approximation of curved and

slanted boundaries no longer an issue.

Highly Accurate Trefftz Approximations: Harmonic Polynomials

Harmonic polynomials provide an excellent approximation of harmonic functions.

J.M. Melenk, Numer. Math. 84:35–69, 1999.

I. Babuška and J.M. Melenk, Int. J. Numer. Meth. Eng. 40(4):727–758, 1997.

Flexible Local Approximation over Subdomains

(i) − local approximating

functions (e.g. sphericalharmonics, Bessel, planewaves, polynomials, …)

Trefftz methods: theapproximating functionssatisfy the underlyingdifferential equation exactly.

The Trefftz-FLAME Scheme:a simple null-space formula

The matrix of nodal values of basis functions

Approximating functions (i) are local solutions

of the underlying differential equation (spherical

harmonics, Bessel, plane waves, polynomials, …)

The difference scheme

I. Tsukerman, J Comput Physics, vol. 211, No. 2, 2006, pp. 659−699.

I. Tsukerman, IEEE Trans. Magn., vol. 41, No. 7, pp. 2206−2225, 2005.

I. Tsukerman, Computational Methods for Nanoscale Applications:

Particles, Plasmons and Waves, Springer, 2007.

Standard Taylor-based schemes

Control volume / flux balance schemes

Exact and super high-order schemes

Compact (―Mehrstellen‖) schemes

High-order schemes for particle interactions (electrostatic / Poisson-Boltzmann)

Perfectly Matched Layers

Wave scatteringTokamakplasma

Interconnects in printed circuit boards

FLAME Example: Super High Order

Schemes for the 1D Schrödinger Equation

Error vs. order of the FLAME scheme for the harmonic oscillator (5th

energy level as an example). The numerical error quickly approachesmachine precision even for a very small number of grid nodes.

u + V(x) u = 0

3-point schemes

A Singular Equation in 1D

G.W. Reddien and L.L. Schumaker, On a collocation method forsingular two-point boundary value problems, NumerischeMathematik, vol. 25, 1976, pp. 427—432.

Numerical solution at x = 0.5

Errors at x = 0.5

Scattering from Cylinder: Convergence

Waveguide Modes in Photonic Crystals (cont‘d)

Helder Pinheiro, Jon Webb, Igor Tsukerman, IEEE Trans. Magn., April 2007.

Photonic Band Structure Computation Using FLAME

Igor Tsukerman and František Čajko, IEEE Trans. Magn., Vol. 44, 2008.

Example of FLAME: Waves in Negative-Index Media

F. Čajko, I. Tsukerman, IEEE Trans. Magn. 44(6), 1378−1381, 2008.

FLAME on Regular and Irregular Stencils

Relative error in potential, regular grids. 19-point least-squares FLAME with 9 basis functions. mmc solution with 64 terms per particle taken as quasi-exact.

IT, Trefftz difference schemes on irregular stencils, J ComputPhys 229, 2010.

References on FLAME

Google tsukerman publications

Boundary-difference Methods

Classical Boundary Equation Methods

Introduce equivalent sources (charges or currents) residing on the boundaries.

Express the field as convolution of these sources with the respective Green functions.

Apply boundary conditions to derive a set of integral equations.

Classical Boundary Equation Methods: Capacitance

Conducting surface at a given

potential V with charge

distribution S.

Boundary integral equation

S * G = V, where G = 1/(4 r)

is the Green function.

Solve for S, then find

capacitance. [*]

Numerical solution: discretize,

apply the method of moments.

[*] A Fredholm equation of the first kind – not supposed to have a stable numerical solution but does.

Classical Boundary Integral Methods: Advantages

Lower dimensionality: 2D → 1D, 3D → 2D.

Natural treatment of unbounded problems (e.g. scattering and radiation), without the artificial domain truncation.

Classical Boundary Integral Methods: Disadvantages

Dense system matrices. Remedies: the Fast Multipole Method, wavelet transforms.

Singular kernels. Evaluation of singular integrals near surfaces may be especially problematic. Remedies?

A Singularity-Free Boundary Method

Drastic change: switch the sequence of operations.

The standard sequence: Differential formulation ⇒ Boundary integral

formulation ⇒ Discretization

The alternative sequence: Differential formulation ⇒ Discretization ⇒Boundary difference problem

A Singularity-Free Boundary Method (continued)

Discretization of the differential problem is performed on a regular grid and yields an FD scheme.

This scheme is converted into a boundary problem that involves discrete fundamental solutions (Green‘s functions) on the grid.

Discrete Green‘s functions, unlike their continuous counterparts, are always nonsingular.

FLAME schemes (with the underlying Trefftz approximations) are applied at the boundary.

Details on Wednesday, July 6.

Analytical methods

Numericalmethods

Ideas shared

METAMATERIALS AND PARAMETERS

IT, Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation, J Opt Soc Am B, March 2011.

Anders Pors, IT, Sergey I. Bozhevolnyi, Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation, Phys Rev E 83, 2011, accepted. arxiv:1104.2972

IT, Homogenization of metamaterials by dual interpolation of fields: a rigorous treatment of resonances and nonlocality, arxiv:1106.3227, submitted.

FROM ANALYTICAL TO NUMERICAL METHODS:

Thanks to Vadim Markel, Anders Pors, Boris Shoykhet, Dmitry Golovaty, Sergey Bozhevolnyi, Graeme Milton.

Metamaterials

Artificial periodic structures with geometric features smaller than the wavelength.

Controlling the flow of waves.

Effective parameters needed for design.

D.R. Smith et al., 2000

Nature, 1/25/07

Pendry, Schurig & Smith, Science 2006

Resonance Effects

http://staging.enthought.com

www.fen.bilkent.edu.tr/~aydin

radio.tkk.fi

Split rings ―LC‖ resonances magnetic dipoles

―Artificial Magnetism‖

Split rings ―LC‖ resonances magnetic dipoles.

Significance: artificial magnetism at high frequencies.

Conundrum: all materials are intrinsically nonmagnetic. The microscopic fields: b = h. How can the macroscopic (averaged) fields B and H differ?

Effective Parameters of Metamaterials: Partial Reference List

D. R. Smith and J. B. Pendry. JOSA B, 23(3):391–403, 2006.

C. R. Simovski. Optics and Spectroscopy, 107:726–753, 2009.

C. R. Simovski and S. A. Tretyakov. Photonics and Nanostructures –Fundamentals and Applications, 8:254–263, 2010.

D. Sjoberg et al. Multiscale Modeling & Sim, 4(1):149–171, 2005.

A. Bossavit et al, J. Math. Pures & Appl, 84(7): 819–850, 2005.

N. Wellander & G. Kristensson, SIAM J. Appl. Math. 64(1):170–195, 2003.

M. G. Silveirinha. Physical Review B, 75(11):115104, 2007.

A. K. Sarychev and V. M. Shalaev. Electrodynamics of Metamaterials. World Scientific, Singapore, 2007.

Sergei Tretyakov. Analytical Modeling in Applied Electromagnetics. Norwood, MA: Artech House, 2003.

C. Fietz & G. Shvets, Physica B: Cond Matter, 405(14): 2930–2934, 2010.

Some Pitfalls: zero cell size limit

Metamaterials: cell size smaller than the vacuum

wavelength but not vanishingly small. (Typical ratio

~0.1−0.3.)

This is a principal limitation, not just a fabrication

constraint (Merlin, PNAS 2009, IT, JOSA B, 2008;

Sjoberg et al. Multiscale Mod & Sim, 2005; Bossavit

et al, J. Math. Pures & Appl, 2005).

Cell size a 0: nontrivial physical effects

(e.g.“artificial magnetism”) disappear.

Some Pitfalls: Volume Averaging

The average field <b> differs from the field immediately at the boundary.

Nonphysical artifacts of volume averaging: spurious boundary sources.

The Defining Principles

The coarse-grained fields vary much less rapidly than the total fields but approximate (not necessarily in a point-wise fashion) the physical fields.

The coarse-grained fields must satisfy Maxwell‘s equations and interface boundary conditions.

Defining Principles (cont‘d)

The fast and coarse components must be semi-decoupled. (The fast components may depend on the coarse ones, but not vice versa.)

There exists a linear relationship between (E, H) and (D, B) that is independent of the incident waves (at least to a given level of approximation).

Div- and Curl-conforming Fields

e.g. Peter Monk, Finite Element Methods for Maxwell's Equations, 2003.

K. Urban, Math Comp 70, 2000.

The coarse-grained fields must satisfy Maxwell‘s equations and interface boundary conditions.

The ―Scaffolding‖ for Coarse-Grained Fields

E and H constructed as

WNBK-interpolants of edge

circulations; D and B – as

WNBK-interpolants of face

fluxes.

Hassler Whitney (1907–1989), a world

class mathematician and a skilled rock

climber. http://www128.pair.com

Jean-Claude Nédélec

Ecole Polytechnique, Palaiseau, France

Alain Bossavit, LGEP,

Gif-sur-Yvette, FranceRobert Kotiuga, Boston University

Edge Interpolation

Circulations have the Kronecker-delta property

A basis function with tangential

continuity across edges

J. van Welij, IEEE Trans Magn 21(6): 2239–2241, 1985.J.-C. Nedelec, Numer Math 35, 315–341, 1980.

Face Interpolation

Face fluxes have the Kronecker-delta property

A basis function with normal

continuity across faces

A. Bossavit, IEE Proc. 135A, 1988; Computational Electromagnetism, 1998.

A. Bossavit, IEE Proc. 135A, 1988; Computational Electromagnetism, 1998.

p = 0: p = 1:

Whitney Forms (very, very briefly)

Properties of WNBK Interpolants

Separation of Scales

The Coarse Scale

The coarse-grained fields satisfy Maxwell‘s equations exactly –even for nonlinear materials!

The Fine Scale

Solve the cell problem (optional):

Approximation of the Field: Basis Modes

―Linear Algebra‖The linear map L

is highly multidimensional.

Compress it by switching to a ―canonical‖ physical basis.

IT, Homogenization of metamaterials by dual interpolation of fields: a rigorous

treatment of resonances and nonlocality, arxiv:1106.3227 , submitted.

―Linear Algebra‖ (cont‘d)

Switching to a new basis (mean values + variations):

IT, Homogenization of metamaterials by dual interpolation of fields: a rigorous

treatment of resonances and nonlocality, arxiv:1106.3227 , submitted.

―Linear Algebra‖ (cont‘d)

In the new (―canonical‖) basis

IT, Homogenization of metamaterials by dual interpolation of fields: a rigorous

treatment of resonances and nonlocality, arxiv:1106.3227 , submitted.

=

quantifying ―spatial dispersion‖

VERIFICATION AND EXAMPLES

Verification: Empty Cell

A trivial ―sanity check‖. Still, some known procedures yield spurious Bloch factors.

By direct computation, one obtains unit parameters (exactly) for any plane wave in an empty cell if a < λ/2.

Since any field can be expanded into plane waves, unit parameters are obtained for any field. (Technically, for an infinite basis set.)

Verification: One-component static fields

One-component (z ) static field independent of z.

The div-conforming WNBK interpolant for d reduces just to a constant D0.

The WNBK-interpolant for E, generally a bilinear function of coordinates, also reduces to a constant E0 = E.

The new procedure yields the right result

Verification: Comparison with the Maxwell-Garnett formula

Dielectric inclusions in a host medium. Static case for simplicity.

M-G: ϵeff = (1+f χ)/(1−f χ), where χ is the

polarizability of inclusions in the host.

Two basis functions in the new method:

(the 2nd function is completely similar)

Verification: Comparison with M-G (cont‘d)

Verification: Comparison with M-G (cont‘d)

Cylindrical inclusion with a varying radius and permittivity

Verification: Bloch Bands, 2D array of cylinders, p-mode

rcyl = 0.33a, ϵcyl = 9.61

Markers: effective parameters, the new method

Solid lines: accurate numerical computation (FLAME)

X direction

Verification: effective parameters, 2D array of cylinders, p-mode

Real parts of ϵeff and μ eff

shown.

ΓX direction.

Double-negative

parameters between a/

~0.33 and 0.42. Agrees

very well with the band

diagrams and with the

published results.

Verification: Wave Propagation through an Array of Cylinders

rcyl = 0.33a, ϵcyl = 9.61

ωa/(2πc) = a/λ = 0.24959

(one of the data points in

previous simulations).

p-mode

60 5 cylinders in the

simulation.

FLAME simulation

(generalized FD)

Verification: Waves in an Array of Cylinders (cont‘d)

Incidence at /6

Verification: Slab with Resonating Inclusions

rcyl = 0.25a, ϵcyl = 200+5i

p-mode

70 3 or 70 5 cylinders

in the FLAME

simulation.

λ/a = 11

λ/a = 9

Slab with Resonating Inclusions (cont‘d)

rcyl = 0.25a,

ϵcyl = 200+5i

p-mode

Slab with Resonating Inclusions (cont‘d)

rcyl = 0.25a,

ϵcyl = 200+5i

p-mode

Inclusions offset to

(0.1a, 0.1a)

SRR material parametersarxiv:1104.2972, Phys Rev E, accepted

A. Pors, I.T., S. I. BozhevolnyiD: proposed ―direct‖ method;

S: retrieval from scattering parametersχ0: magnetoelectric coupling parameter

Complex reflection and transmission coefficients for a 1-μm

thick SRR-metamaterial slab. Incident wave: x-polarized,

propagating in the y-direction. r+ and r− correspond to +y and –y

propagation, respectively.

Transmission / reflection

D: proposed ―direct‖ method; S: retrieval from scattering

parameters‗Sim‘: full-wave FEM simulation

of a 5-layer SRR-slab.A. Pors, I.T., S. I. Bozhevolnyi

arxiv:1104.2972, Phys Rev E, accepted

Future Work

Applications: metamaterials, electrical machines (laminated cores), … Applications beyond electromagnetics.

Formal mathematical theory: discrete Hodge operators? (Hiptmair, Bossavit, Kettunen et al, Auchmann, …)

Parameter estimates and optimization (especially magnetic properties).

Nonlocality and chirality.

Energy and losses.

Conclusion on Homogenization

A new methodology is put forward for the evaluation of the effective parameters of electromagnetic and optical metamaterials.

The main underlying principle is that the coarse-grained E and H fields have to be curl-conforming, while the B and D fields have to be div-conforming.

Conclusion (cont‘d)

WNBK interpolation provides an excellent framework for constructing the coarse-grained fields: ensures not only the proper continuity condition, but also compatibility of the respective interpolants.

As a result, remarkably, Maxwell‘s equations for the coarse-grained fields are satisfied exactly.

Conclusion (cont‘d)

The field is approximated with a linear combination of basis functions (e.g. Bloch waves). Effective parameters devised to provide the most accurate linear relation between the WNBK interpolants of the basis functions.

Exact result in the limiting case of a vanishingly small cell size.

Nonlocality and ―spatial dispersion‖ demystified.