Stochastic Finite Element Modeling of Material and Geometric Uncertainties in Electromagnetics
Design methodologies for RF/Microwave systems require major changes to cope up with the evolution of faster, high data/rate wireless communication systems using 5G and futuristic 6G technologies. The research on Terahertz communication and imaging has gained the interest of electromagnetics (EM) community. Robust design of these components incorporating manufacturing process tolerances would enhance the fabrication yield and thereby reduce the overall production cost. Estimating these effects in the design phase is ideal, but the limitations of analytical methods both in EM and stochastic analysis call for numerical estimation of such uncertainties. Similar uncertainties are also encountered in the analysis of electromagnetic field interaction with biological samples as the EM properties of these samples may change with physiological or diurnal variations. Most popular computational electromagnetics (CEM) algorithms are deterministic and the impact of uncertainties such as these random variations in properties of the media or geometric variations is usually ignored due to the high computational time. This thesis develops efficient computational methods to estimate these stochastic variations using a spectral representation of these randomly varying parameters. First, a computational framework for analyzing variations in the dielectric constant of a region in an EM model is represented using random variables. The resultant electric field becomes a random field which is represented using polynomial chaos expansion (PCE). This representation has been extended further to develop algorithms for stochastic variation in multiple subdomains. The above random variable representation alone is insufficient when the material has spatial variations. Karhunen-Loève expansion (KLE) is used for representing these input variations with minimum stochastic dimension. KLE can be truncated at the desired accuracy level to obtain the stochastic response to the spatial variation of material properties. As KLE-SSFEM is extended to multiple subdomains the stochastic dimension may increase. Even though the scheme has a sparse matrix, its size may cross the capability of computers. To tackle this situation, a spatially averaged SSFEM (SA-SSFEM) is developed which limits the stochastic dimension to the number of stochastic subregions. This enables the stochastic analysis of large EM models using spectral stochastic FEM. Simulation outcomes like, transmission coefficient and reflection coefficient are analyzed to obtain their probability density estimation due to variation in permittivity and loss tangent in discrete dielectric sections inside a waveguide. The stochastic characterization of resonance frequency and transmission coefficients of a dielectric loaded microstrip line is also estimated where the dielectric properties of a pellet are allowed to be stochastic and spatially varying. These numerical results are found to match the Monte Carlo simulation (MCS) with 10,000 samples. Probability density function (PDF) obtained from the proposed method is systematically compared MCS by performing two variable KS-test. Based on the success of above method for analyzing material variations, a novel geometrical SSFEM (GSSFEM) is proposed to analyze dimensional uncertainty in 3-D microwave models. The most challenging part in the stochastic analysis of geometric variation in finite element model is handling of mesh modifications in an intrusive framework. Therefore, currently only non-intrusive methods which require re-meshing of the physical model in every sample execution, are employed for such analyses. On the other hand, this is handled in the proposed GSSFEM by using Piola transformation where the mesh movement in tetrahedral elements are captured to a reference element. The Jacobian of an element used for Piola mapping is represented as a function of stochastic variables to capture mesh level uncertainties. Polynomial chaos expansion is used for approximating the electric field as a random process. This is the first such formulation of geometric uncertainties in a full-wave 3-D model, using an intrusive approach. The proposed technique is validated by applying it to waveguide devices with uncertainty in geometric dimensions and the results are compared with the Monte Carlo simulations. The proposed scheme is faster than sparse grid stochastic collocation, even with the computational scaling for the same degrees of freedom and stochastic dimension. The above finite-element based stochastic formulations for electromagnetics employ edge elements and the resulting mathematical models result in sparse matrices thereby resulting in computationally efficient schemes. These stochastic methods can be employed by designers for analyzing the impact of fabrication tolerance of passive components at microwave frequencies and beyond.
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