Abstract
In this paper, we follow the Test-Driven Development (TDD) paradigm in the development of an in-house code to allow for the finite element analysis of finite periodic type electromagnetic structures (e.g., antenna arrays, metamaterials, and several relevant electromagnetic problems). We use unit and integration tests, system tests (using the Method of Manufactured Solutions—MMS), and application tests (smoke, performance, and validation tests) to increase the reliability of the code and to shorten its development cycle. We apply substructuring techniques based on the definition of a unit cell to benefit from the repeatability of the problem and speed up the computations. Specifically, we propose an approach to model the problem using only one type of Schur complement which has advantages concerning other substructuring techniques.
Highlights
The Finite Element Method (FEM) has been extensively used for Computational Electromagnetics [1] due to its robustness and versatility. It requires the use of volumetric meshes, which limit the size of the problems to be analyzed. To leverage this problem in FEM, some techniques have been proposed in the last years: Domain Decomposition Methods (DDM) [2,3,4] have increased the size of the problems that can be solved at the expense of a more difficult formulation and the use of iterative solvers; Model Order Reduction (MOR) techniques [5,6] reduce the number of simulations required for a given bandwidth using polynomial interpolation techniques; multigrid methods [7,8] are used as preconditioners for iterative solvers; and substructuring methods that are purely algebraic approaches based on the Schur complement [9,10]
The paper is structured as follows: In Section 2, we define the formulation used in the FEM code, the Schur complement-based substructuring methods that we use, and the types of validation procedures we have considered to introduce the technique; in Section 3, we show the unit and integration tests that we have considered, we use the Method of Manufactured Solutions (MMS) to show the accuracy of the implementation and to conduct error convergence tests, we define smoke tests and use a pyramidal horn antenna array as a real-world application, and we observe the performance of the introduced technique
We have followed the Test-Driven Development (TDD) paradigm in the development of an in-house code to allow for the finite element analysis of finite periodic type electromagnetic structures
Summary
The Finite Element Method (FEM) has been extensively used for Computational Electromagnetics [1] due to its robustness and versatility It requires the use of volumetric meshes, which limit the size of the problems to be analyzed. The infinitely periodic approximation is useful to model large structures, it neglects the border effects due to the finite size of the structures in practice. In this context, substructuring methods are useful as they provide an algebraically exact solution for the analysis of the finite periodic structure while alleviating the computational effort by working with a finite number of different Schur complements (being that number independent of the size of the structure). We propose an approach in which there is only one type of Schur complement involved in the computations with the corresponding advantages in terms of computational resources
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