Stochastic Methods in Time-Domain Electromagnetic Computations

Abstract

The world is advancing towards a highly connected, fast data rate, digital trans- formation through communication technologies like 5G and 6G, high-speed data centres, Terahertz imaging etc. High-frequency RF systems are playing a pivotal role in bringing about this revolution. Better-performing, budget-optimized designs are always in demand and have become challenging. A robust design should consider various uncertainties in real-world operating scenarios during the design and analysis phase. Many of these uncertainties are critical with the advent of microwave and millimetre wave systems and detrimental to the optimum performance of the sys- tem. Such uncertainties may be present in the material characteristics or geometry naturally, by ageing or by tolerances in its manufacturing process. External factors like temperature and pressure in its operating environment may also cause random fluctuations in the device’s performance. Despite all these, the omnipresence of RF systems can inadvertently introduce interference problems in other systems. Random electrical characteristics of the surrounding are to be considered in such scenarios, particularly in the design of life-critical implant devices and wearable electronics. This thesis attempt to develop polynomial chaos expansion (PCE) based computational methods to quantify the parametric uncertainties in the material property as well as geometry in microwave designs and its operating environments. A computational method to address the ‘curse of dimensionality’ in PCE-based uncertainty quantification methods is also proposed as a part of this work. A time domain computational technique based on the radial point interpolation method (RPIM) is adopted as the deterministic framework, owing to its accuracy in complex and curved geometries with a straightforward node-based formulation. Uncertainties are intrusively incorporated into this computational model through random variables ex- pressed as PCEs. Challenges involved in formulating such a stochastic mathematical framework are addressed through Euclidean geometry, linear algebra, analysis and numerical techniques. Random variations in material properties such as permittivity and conductivity are arising due to tolerances in its manufacturing methods. These variations usually follow a standard probability distribution and transform Maxwell’s equations that govern the electromagnetic interactions, to behave as a random process. In the computational perspective of RPIM, these are a set of numerical stochastic partial differential equations (SPDEs) in the time domain. A polynomial chaos expansion (PCE) is used to represent the random variables embedded in these SPDEs. Following a Galerkin procedure, this can be simplified to obtain a concise set of equations, called the stochastic RPIM (SRPIM), that is numerically solvable with good accuracy, highly reduced time complexity, and a marginally increased space complexity. The method is validated in various two-dimensional and three-dimensional microwave designs, and the results and advantages are compared with the Monte Carlo method and Stochastic Collocation. Despite numerous advantages of the PCE-based methods, computational complexity explodes in factorial order with the number of random variables involved. We propose a reduced order method, called reduced SRPIM (R-SRPIM), to partially address this problem. A two-stage order reduction strategy is adopted here, using the linearization property of orthogonal polynomials and the singular value de- composition (SVD). Significant computational gain is achieved with a reduced time complexity. A bio-electromagnetic problem of signal interference on a pacemaker probe with a large stochastic dimension is analyzed using the proposed method. The results are compared with SRPIM, the Monte Carlo method, and also with the un- certainty quantification toolbox in COMSOL Multiphysics v. 6.0. using adaptive Gaussian (AG) and adaptive sparse polynomial chaos (ASP). This kind of analysis is helpful in improvising futuristic wearable device designs, encompassing statistics of electrical parameters of tissues and a wide range of similar bio-electromagnetics problems. Random geometric variations in RF designs often result in undesired performance, particularly when high-frequency systems are involved. A computational framework is proposed in this thesis to analyze such effects on the time domain and wideband characteristics. The simplicity of the node-based formulation of RPIM is exploited to incorporate a PCE-based representation for the random variations of the coordinate locations without any mesh transformation. Challenges involved in inverting sub- domain stochastic matrices are addressed through block matrix inversion utilizing its symmetry and the Neumann approximation. The accuracy and complexity of the proposed method are validated through numerical examples by comparing the results with Monte Carlo methods implemented in RPIM and CST Studio Suit 2018