Uncertainty Quantification for Electromagnetic Systems Using ASGC and DGTD Method

In this paper, an adaptive hierarchical sparse grid collocation (ASGC) method combined with the discontinuous Galerkin time-domain method is leveraged to quantify the impacts of random parameters on the electromagnetics systems. The ASGC method approximates the stochastic observables of interest using interpolation functions over a set of collocation points determined by the Smolyak's algorithm integrated with an adaptive strategy. Instead of resorting to a full-tensor product sense, the Smolyak's algorithm constructs the collocation points in a hierarchical scheme with the interpolation level. Enhanced by an adaptive strategy, the Smolyak's algorithm will sample more points along important dimensions with sharp variations or discontinuities, resulting in a nonuniform sampling scheme. To flexibly handle different stochastic systems, either piecewise linear or Lagrange polynomial basis functions are applied. With these strategies, the number of collocation points is significantly reduced. The statistical knowledge of stochastic observables including the expected value, variance, probability density function, and cumulative distribution function are presented. The accuracy and robustness of the algorithm are demonstrated by various examples.

[1]  Dani Gamerman,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 1997 .

[2]  George E. Karniadakis,et al.  Multi-element probabilistic collocation method in high dimensions , 2010, J. Comput. Phys..

[3]  Li Jun Jiang,et al.  Cosimulation of Electromagnetics-Circuit Systems Exploiting DGTD and MNA , 2014, IEEE Transactions on Components, Packaging and Manufacturing Technology.

[4]  Minseok Choi,et al.  A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes , 2013, J. Comput. Phys..

[5]  Erich Novak,et al.  High dimensional polynomial interpolation on sparse grids , 2000, Adv. Comput. Math..

[6]  D. Xiu.  Efficient collocational approach for parametric uncertainty analysis , 2007 .

[7]  J. Hesthaven,et al.  Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty , 2011 .

[8]  B. Fox.  Strategies for Quasi-Monte Carlo , 1999, International Series in Operations Research & Management Science.

[9]  Li Jun Jiang,et al.  Integration of Arbitrary Lumped Multiport Circuit Networks Into the Discontinuous Galerkin Time-Domain Analysis , 2013, IEEE Transactions on Microwave Theory and Techniques.

[10]  Thomas Gerstner,et al.  Numerical integration using sparse grids , 2004, Numerical Algorithms.

[11]  A. Taflove,et al.  Single Realization Stochastic FDTD for Weak Scattering Waves in Biological Random Media , 2013, IEEE Transactions on Antennas and Propagation.

[12]  Hans-Joachim Bungartz,et al.  Multivariate Quadrature on Adaptive Sparse Grids , 2003, Computing.

[13]  Li Jun Jiang,et al.  Simulation of Electromagnetic Waves in the Magnetized Cold Plasma by a DGFETD Method , 2013, IEEE Antennas and Wireless Propagation Letters.

[14]  Li Jun Jiang,et al.  A Hybrid Time-Domain Discontinuous Galerkin-Boundary Integral Method for Electromagnetic Scattering Analysis , 2014, IEEE Transactions on Antennas and Propagation.

[15]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .

[16]  Eric Michielssen,et al.  An Adaptive Multi-Element Probabilistic Collocation Method for Statistical EMC/EMI Characterization , 2013, IEEE Transactions on Electromagnetic Compatibility.

[17]  G. Karniadakis,et al.  Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations , 2006 .

[18]  Jian-Ming Jin,et al.  A Symmetric Electromagnetic-Circuit Simulator Based on the Extended Time-Domain Finite Element Method , 2008, IEEE Transactions on Microwave Theory and Techniques.

[19]  George E. Karniadakis,et al.  The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications , 2008, J. Comput. Phys..

[20]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[21]  Xiang Ma,et al.  An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations , 2009, J. Comput. Phys..

[22]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[23]  Qing He,et al.  Fast Electromagnetics-Based Co-Simulation of Linear Network and Nonlinear Circuits for the Analysis of High-Speed Integrated Circuits , 2010, IEEE Transactions on Microwave Theory and Techniques.

[24]  E. Michielssen,et al.  A Fast Stroud-Based Collocation Method for Statistically Characterizing EMI/EMC Phenomena on Complex Platforms , 2009, IEEE Transactions on Electromagnetic Compatibility.

[25]  Yahya Rahmat-Samii,et al.  The field equivalence principle: illustration of the establishment of the non-intuitive null fields , 2000 .

[26]  Jan S. Hesthaven,et al.  Computational modeling of uncertainty in time-domain electromagnetics , 2005, Workshop on Computational Electromagnetics in Time-Domain, 2005. CEM-TD 2005..

[27]  Albert E. Ruehli,et al.  The modified nodal approach to network analysis , 1975 .

[28]  M. Mishchenko,et al.  T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database , 2004 .

[29]  Xiang Ma,et al.  An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations , 2010, J. Comput. Phys..

[30]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[31]  Nitin Agarwal,et al.  A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties , 2009, J. Comput. Phys..

[32]  Barbara I. Wohlmuth,et al.  Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB , 2005, TOMS.

[33]  Costas D. Sarris,et al.  Efficient Analysis of Geometrical Uncertainty in the FDTD Method Using Polynomial Chaos With Application to Microwave Circuits , 2013, IEEE Transactions on Microwave Theory and Techniques.

[34]  Uncertainty quantification of EM-circuit systems using stochastic polynomial chaos method , 2014, 2014 IEEE International Symposium on Electromagnetic Compatibility (EMC).