An adaptive hierarchical sparse grid collocation method for stochastic scattering systems analysis

To quantify the impacts of random inputs on hybrid electromagnetics (EM)-circuit systems or EM scattering from objects, an adaptive hierarchical sparse grid collocation (ASGC) algorithm combined with discontinuous Galerkin time-domain (DGTD) method is presented in this work. As a stochastic polynomial chaos modality, the ASGC method approximates the interested stochastic observables using interpolation functions over a set of collocation points. Instead of employing a full-tensor product sense, the collocation points in ASGC method are hierarchically marched with interpolation level based on Smolyak´s algorithm. To further reduce the collocation points, the hierarchical surplus is used as the error indicator for each collocation point to achieve adaptivity. To handle different stochastic systems, both piecewise linear and Lagrange polynomial basis functions are applied. More specifically, the locally supported piecewise linear basis functions based on Newton-Cotes grid are particularly suitable to attack sharp variations and discontinuities in stochastic observables, while the Lagrange polynomial basis functions based on Clenshaw-Curtis grid are more favorable for smoothly stochastic observables due to its global property. With these strategies, the number of collocation points is significantly reduced with exponential convergence rate. To characterize the far-field scattering properties of objects, the radar-cross-section (RCS) of perfectly electrical conducting (PEC) and dielectric spheres are investigated under the influence of geometrical and material uncertainties such as radius R and relative electrical permittivity ϵ_r. With this stochastic simulation algorithm, statistical information including the mean values, variances, probability density functions (pdfs) and cumulative distribution functions (cdfs) can be acquired conveniently.