# GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size

## Abstract

Due to the recent explosion of interest in studying the electromagnetic behavior of large (truncated) periodic structures such as phased arrays, frequency-selective surfaces, and metamaterials, there has been a renewed interest in efficiently modeling such structures. Since straightforward numerical analyses of large, finite structures (i.e., explicitly meshing and computing interactions between all mesh elements of the entire structure) involve significant memory storage and computation times, much effort is currently being expended on developing techniques that minimize the high demand on computer resources. One such technique that belongs to the class of fast solvers for large periodic structures is the GIFFT algorithm (Green's function interpolation and FFT), which is first discussed in [1]. This method is a modification of the adaptive integral method (AIM) [2], a technique based on the projection of subdomain basis functions onto a rectangular grid. Like the methods presented in [3]-[4], the GIFFT algorithm is an extension of the AIM method in that it uses basis-function projections onto a rectangular grid through Lagrange interpolating polynomials. The use of a rectangular grid results in a matrix-vector product that is convolutional in form and can thus be evaluated using FFTs. Although our method differs from [3]-[6] in various respects,more »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 894343

- Report Number(s):
- UCRL-CONF-218954

TRN: US0700151

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Conference

- Resource Relation:
- Conference: Presented at: IEEE AP-S International Symposium, Albuquerque, NM, United States, Jul 09 - Jul 14, 2006

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGORITHMS; COMPUTERS; DEFECTS; DIPOLES; ELECTROMAGNETIC INTERACTIONS; EXCITATION; EXPLOSIONS; FABRICATION; INTERPOLATION; MATRICES; MODIFICATIONS; NUMERICAL ANALYSIS; POLYNOMIALS; SIMULATION; STORAGE

### Citation Formats

```
Capolino, F, Basilio, L, Fasenfest, B J, and Wilton, D R.
```*GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size*. United States: N. p., 2006.
Web. doi:10.1109/APS.2006.1711660.

```
Capolino, F, Basilio, L, Fasenfest, B J, & Wilton, D R.
```*GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size*. United States. doi:10.1109/APS.2006.1711660.

```
Capolino, F, Basilio, L, Fasenfest, B J, and Wilton, D R. Mon .
"GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size". United States.
doi:10.1109/APS.2006.1711660. https://www.osti.gov/servlets/purl/894343.
```

```
@article{osti_894343,
```

title = {GIFFT: A Fast Solver for Modeling Sources in a Metamaterial Environment of Finite Size},

author = {Capolino, F and Basilio, L and Fasenfest, B J and Wilton, D R},

abstractNote = {Due to the recent explosion of interest in studying the electromagnetic behavior of large (truncated) periodic structures such as phased arrays, frequency-selective surfaces, and metamaterials, there has been a renewed interest in efficiently modeling such structures. Since straightforward numerical analyses of large, finite structures (i.e., explicitly meshing and computing interactions between all mesh elements of the entire structure) involve significant memory storage and computation times, much effort is currently being expended on developing techniques that minimize the high demand on computer resources. One such technique that belongs to the class of fast solvers for large periodic structures is the GIFFT algorithm (Green's function interpolation and FFT), which is first discussed in [1]. This method is a modification of the adaptive integral method (AIM) [2], a technique based on the projection of subdomain basis functions onto a rectangular grid. Like the methods presented in [3]-[4], the GIFFT algorithm is an extension of the AIM method in that it uses basis-function projections onto a rectangular grid through Lagrange interpolating polynomials. The use of a rectangular grid results in a matrix-vector product that is convolutional in form and can thus be evaluated using FFTs. Although our method differs from [3]-[6] in various respects, the primary differences between the AIM approach [2] and the GIFFT method [1] is the latter's use of interpolation to represent the Green's function (GF) and its specialization to periodic structures by taking into account the reusability properties of matrices that arise from interactions between identical cell elements. The present work extends the GIFFT algorithm to allow for a complete numerical analysis of a periodic structure excited by dipole source, as shown in Fig 1. Although GIFFT [1] was originally developed to handle strictly periodic structures, the technique has now been extended to efficiently handle a small number of distinct element types. Thus, in addition to reducing the computational burden associated with large periodic structures, GIFFT now permits modeling these structures with source and defect elements. Relaxing the restriction to strictly identical periodic elements is, of course, useful for practical applications where, for example, a dipole excitation may be of interest or, as is often the case for metamaterials, defective elements are introduced in the structure's fabrication process. The main extensions of the GIFFT method compared to [1] are the following: (1) Both periodic ''background'' and ''source'' or ''defect'' elements are now separately defined in translatable unit cells so that, in the algorithm, mutual electromagnetic interactions can be computed. (2) The near-interaction block matrix must allow for the possibility of ''background-to-source'' or ''background-to-defect'' cell interactions. (3) Matrices representing projections of both ''background and source'' or ''background and defect'' subdomain bases onto the interpolation polynomials must be defined and appropriately selected in forming the matrix-vector product. It is important to note that, although here we consider a metamaterial layer with a dipole-antenna excitation, as per the extended GIFFT algorithm, ''defect'' elements could be considered as well.},

doi = {10.1109/APS.2006.1711660},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon Jan 23 00:00:00 EST 2006},

month = {Mon Jan 23 00:00:00 EST 2006}

}