Modeling of Permittivity Variations in Stochastic Computational Electromagnetics

Abstract

With the evolution of 5G systems offering high data rates, major changes are required in the design approach of the components of communication systems. Furthermore, building complex electromagnetic systems at the terahertz frequency range is of particular interest to the scientific community. Transmitting and receiving electromagnetic (EM) subsystems, including antennas, RF circuits & devices, RF filters, waveguides, etc are essential building blocks of these systems. It has been observed that there is significant uncertainty in the realization of these components due to fabrication tolerance, especially at millimeter wave frequencies and above. In addition to the variations in material properties due to these, the complex nature of antenna hosting environments, excitation function and point of source feeding, affect the performance characteristics of these devices. Incorporating these uncertainties in the EM design of the above advanced systems, both in terms of mathematical formulation and computational implementation is challenging. The tolerance in the fabrication process results in variations of dielectric material properties, which affects the system response. Therefore, a proper quantification of uncertainties using an efficient numerical stochastic EM solver help deliver a robust and optimal design. In this scenario, this thesis explores developing fast and efficient numerical stochastic EM solvers by considering parameters with a statistical variation. Various uncertainty modeling algorithms are formulated, implemented, and their performance is evaluated, validated, and compared by considering different practical stochastic EM problems. Both intrusive and non-intrusive finite element methods (FEM) for uncertainty quantification (UQ) in electromagnetics have been studied extensively in this work. For this analysis, FEM is used due to its versatility in handling complex EM structures with multiple dielectric domains. EM problems are unique due to the special boundary conditions employed, the possibility of resonances due to structural features and the broad frequency range of analysis required. A popular intrusive method for stochastic analysis is the polynomial chaos expansion (PCE) based stochastic spectral finite element (SSFEM) method. SSFEM can capture variations in an EM problem accurately and is shown to be computational efficient when compared with the Monte Carlo (MC) method. But SSFEM computational complexity scales with the number of random variables and results in a curse of dimensionality. Therefore the Neumann expansion (NE) is developed as an intrusive method for solving stochastic EM problems, wherein the matrix obtained by the discretization, can be split into the deterministic and stochastic parts. The Neumann series expansion after appropriate truncation is applied here to obtain the stochastic response. Unlike SSFEM, the computational complexity of NE method is shown to scale marginally with the number of stochastic regions, but is shown to have a limitation of capturing large variations. Another PCE based scheme for uncertainty modeling namely, least square polynomial chaos expansion (LSPCE ) is proposed here, as a non-intrusive method. A non-intrusive scheme is easier to implement and treats the EM solver as a black-box and therefore can be integrated with even commercial EM solvers. LSPCE minimizes the sum of squared error due to PCE truncation, through a system of algebraic equations, to solve the unknown PCE coefficients. This formulation is found to be computationally efficient compared with Monte Carlo and can efficiently handle EM problems with large stochastic dimensionality. Implementation aspects such as the initial number of samples for the proposed method is chosen by analyzing probability distance measures. The number of initial samples is found to be at least twice the number of orthonormal stochastic basis (over-determined system). Furthermore, the computational complexity of LSPCE can be reduced using fewer initial samples, but this results in an under-determined system, which is highly ill-conditioned. It has been shown that, such an ill-posed problem is solved using regularization methods such as regularized steepest descent. Fabrication tolerance can also result in spatial variations in material properties for EM structures and can be modeled as a random field using Karhunen Loeve (KL) expansion for a given covariance kernel and correlation length. KL expansion is truncated for a finite-dimensional representation and analyzed using intrusive methods such as SSFEM and NE. This implementation of stochastic modeling requires several random variables, which is difficult to solve using conventional stochastic algorithms. However, it has been shown that sparse algorithms can be utilized for solving these problems, as PCE coefficients are sparse in this case. Sparse algorithms, namely orthogonal matching pursuit and subspace pursuit, have been applied to enhance computational efficiency. Modern EM systems are expected to be operated over a broad frequency range, which increases the computation cost when frequency domain methods such as FEM is used. Large degree of freedom in complex EM problems increases this further. A formulation involving proper orthogonal decomposition (POD) is attempted, which forms a basis of low dimension and is shown to be efficient and accurate for a single frequency. Extending this to be operated over the frequency range, the intrusive UQ methods, SSFEM and NE are applied to this low dimensional POD basis. It is shown that the use of this modified POD-SSFEM and POD-NE formulations offers significant computational and memory advantages and can be analysed over a broad range of frequency. This new formulation is also shown to be effectively capturing the stochastic response for EM problems with large degree of freedom with limited computational resources. All the above numerical stochastic algorithms are implemented with in-house edge element FEM programs to solve stochastic electromagnetic problems involving variations in the permittivity of dielectric regions. Accuracy of these methods is evaluated by comparing with Monte Carlo simulations and performing statistical tests. Computational constraints have been discussed, and the resulting efficiencies are evaluated. These statistical formulations can be used by the EM designers for developing optimal models, which overcome the impact of fabrication tolerance.