Sensitivity Analysis With Finite Element Method For Microwave Design .

Chapter 1 Introduction[6], [7] and S-parameters [8], [10]. Yet, the computation is based on Tellegen'stheory of circuit concepts, not field solutions.Recently, an adjoint-variable method has been proposed for the sensitivitycomputation in numerical electromagnetic (EM) problems both in the timedomain (the transmission line method, TLM [11 ], and the finite-difference intime-domain method, FDTD [12]) and in the frequency domain (the frequencydomain TLM [13], [14], the method of moments, MoM, and the finite-elementmethod, FEM [15]). For the frequency-domain application, this method uses thefinite-difference approximation of the system matrix derivatives, which eliminatesthe need for analytical pre-processing [15].The computational load issignificantly reduced without sacrificing the accuracy.However, all traditional AVM techniques still require one more full-waveanalysis (the adjoint system analysis) in addition to the original system analysis.Moreover, the excitation in the adjoint system analysis depends on the responseand its relation to the solution of the original problem. This may cause potentialdifficulties in the formulation of this additional analysis problem, and itsimplementation in the framework of a commercial high-frequency CAD package.In this thesis, a general approach, named self-adjoint sensitivity analysis(SASA), is proposed for the sensitivity analysis of microwave networkparameters, i.e., S-parameters. With this approach, we eliminate the additional(adjoint) system analysis and further improve the efficiency of the sensitivity2

Chapter 1 Introductioncomputation [16], [17], [18]. This possibility has been first discussed in Akel etal. [19], in the case of the FEM formulation based on the tetrahedral edgeelements.With a commercial high-frequency FEM solver, this standalonealgorithm runs independently of the underlying analysis algorithm.The onlyinformation it needs is the field solution at specific grid points. Thus, it is easilyintegrated into any kind of design automation process with a commercial solver.The most promising application of the SASA is in the area of gradientbased optimization [15], [20].The optimization process can be significantlyaccelerated with the Jacobian provided by the SASA, since the computation of theJacobian and/or Hessian is the bottleneck of the optimization efficiency with fullwave EM solvers.In this thesis, the advantage of the SASA is validated by its application ingradient based optimization.Notably, the efficiency of the SASA basedoptimization can be further improved by applying the Broyden update [21] at thelevel of the system matrix derivatives. As the overhead of the Broyden update isnegligible compared to the finite-difference approximation of the system matrix,the computational load for the Jacobian calculation is practically zero.The accumulation of inaccuracies during the iterative Broyden update ofthe system matrix derivatives may lead to wrong solutions, or even to divergenceof the optimization process. To guarantee robustness of the convergence, thecomputation of the system matrix derivatives should switch between the fast3

Chapter 1 Introductioniterative method and the robust finite-difference approximation.·Certainswitching criteria are suggested in this thesis [22]. The resulting algorithm isnamed the Broyden/finite-difference SASA (BIFD-SASA). The BIFD-SASA isvalidated by the optimization of a microwave filter and by the solution of aninverse imaging problem using various gradient based optimization algorithms.Chapter 2 introduces the basics of the AVM methodology. We review thebasic sensitivity expression of the AVM and discuss its implementation in thesensitivity analysis of complex linear systems. The computational overhead of thetraditional AVM is analyzed and compared with the overhead of the finitedifference approximation.Chapter 3 introduces the SASA with the finite-element method andexplores its application in S-parameter sensitivity computation for microwavestructures.The general SASA expression is derived from the 3-D FEMformulation. The algorithm is validated by a rectangular waveguide bend exampleand a dielectric coupling filter design example.Error estimation andcomputational overhead evaluation are conducted.Comparisons with thetraditional finite-difference approximation are provided.Chapter 4 integrates the SASA method with the Broyden update in theframework of gradient-based optimization.The B/FD-SASA method isimplemented through switching criteria which ensure the reliability of thesensitivity results.The BIFD-SASA method is validated through two4

Chapter 1 Introductionoptimization examples. The convergence of the optimization using this method iscompared with that using FD-SASA as well as the finite differences at theresponse leveLAn overall conclusion is made in Chapter 5 and suggestions for futuredevelopment are given.The contributions of this research can be summarized as follows:(1)Development of an efficient sensitivity computation algorithm with thefinite-element method, i.e., SASA [16], [18].(2)Implementation of the SASA algorithm with the commercial finiteelement solver FEMLAB [16], [18].(3)Validation of the efficiency of SASA through numerical designexamples [ 16].(4)Development of a Broyden-update-based sensitivity computationalgorithm for gradient based optimization [22].(5)Validation of the efficiency of the sensitivity computation algorithmwith different optimization algorithms [22].5

Chapter 1 IntroductionREFERENCES[1]D. G. Cacuci, Sensitivity & Uncertainty Analysis, Volume 1: Theory. BocaRaton, FL: Chapman & HalVCRC, 2003.[2]A. D. Belegundu and T.R. Chandrupatla, Optimization Concepts andApplications in Engineering. Upper Saddle River, NJ: Prentice Hall, 1999.[3]E. J. Haug, K.K. Choi and V. Komkov, Design Sensitivity Analysis ofStructural Systems. Orlando: Academic Press Inc., 1986.[4]S. W. Director and R. A. Rohrer, "The generalized adjoint network andnetwork sensitivities," IEEE Trans. Circuit Theory, vol. CT-16, pp.318323, Aug. 1969.[5]J. W. Bandler and R. E. Seviora, "Current trends in networkoptimization," IEEE Trans. Microwave Theory Tech., vol. MTT-18,pp.1159-1170, Dec. 1970.[6]V. A. Monaco and P. Tiberio, "Computer-aided analysis of microwavecircuits," IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp.249263, Mar. 1974.[7]J. W. Bandler, "Computer-aided circuit optimization," in Modern FilterTheory and Design, G. C. Ternes and S. K. Mitra, Eds. New York: Wiley,1973, ch. 6.[8]K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design ofMicrowave Circuits. Norwood, MA: Artech House, 1981.[9]J. Vlach and K. Singhal, Computer Methods for Circuit Analysis andDesign. New York: Van Nostrand, 1983.[10]J. W. Bandler and R. E. Seviora, "Wave sensitivities of networks," IEEETrans. Microwave Theory Tech., vol. MTT-20, pp. 138-147, Feb. 1972.[11]M. H. Bakr and N. K. Nikolova, "An adjoint variable method for timedomain TLM with wide-band Johns matrix boundaries," IEEE Trans.Microwave Theory Tech., vol. 52, pp. 678-685, Feb. 2004.6

Chapter 1 Introduction[12]N. K. Nikolova, H. W. Tam, and M. H. Bakr, "Sensitivity analysis withthe FDTD method on structured grids," IEEE Trans. Microwave TheoryTech., vol. 52, pp. 1207-1216, Apr. 2004.[13]M. H. Bakr and N. K. Nikolova, "An adjoint variable method forfrequency domain TLM problems with conducting boundaries," IEEEMicrowave and Wireless Component Letters, vol. 13, pp. 408-410, Sept.2003.[14]S. M. Ali, N. K. Nikolova, and M. H. Bakr, "Central adjoint variablemethod for sensitivity analysis with structured grid electromagneticsolvers," IEEE Trans. Magnetics, vol. 40, pp. 1969-1971, Jul. 2004.[15]N. K. Georgieva, S. Glavic, M. H. Bakr and J. W. Bandler, ''Feasibleadjoint sensitivity technique for EM design optimization," IEEE Trans.Microwave Theory Tech., vol. 50, pp. 2751-2758, Dec. 2002.[16]N. K. Niko1ova, J. Zhu, D. Li, M. Bakr, and J. Bandler, "Sensitivities ofnetwork parameters with electromagnetic frequency domain simulator,"IEEE Trans. Microwave Theory Tech. vol. 54, pp. 670-681, Feb, 2006.[17]N. K. Nikolova, J. Zhu, D. Li, et al., "Extracting the derivatives of networkparameters from frequency domain electromagnetic solutions," GeneralAssembly ofInti. Union ofRadio Science CDROM, Oct. 2005.[18]D. Li and N. K. Nikolova, "S-parameter sensitivity analysis of waveguidestructures with FEMLAB," COMSOL Multiphysics Conf 2005Proceedings, Cambridge, MA, Oct. 2005, pp. 267-271.[19]H. Akel and J. P. Webb, "Design sensitivities for scattering-matrixcalculation with tetrahedral edge elements," IEEE Trans. Magnetics, vol.36, pp. 1043-1046, Jul. 2000.[20] · N. K. Nikolova, et al., "Accelerated gradient based optimization usingadjoint sensitivities," IEEE Trans. Antenna Propagat., vol. 52, pp. 21472157, Aug. 2004.[21]C. G. Broyden, "A class of methods for solving nonlinear simultaneousequations," Mathematics of Computation, vol. 19, pp. 577-593, 1965.7

Chapter 1 Introduction[22]D. Li, N. K. Nikolova, and M. H. Bakr, "Optimization using Broydenupdate self-adjoint sensitivities," IEEE Antenna Propagat. Symposium2006, accepted.8

CHAPTER2METHODOLOGY OF THEADJOINT VARIABLE METHOD2.1INTRODUCTIONCommercial high-frequency electromagnetic (EM) solvers usually do notcompute sensitivity information, i.e., the Jacobian of the objective function, withrespect to the design parameters.For design purposes, when sensitivityinformation is required, a finite-difference approximation at the response level isusually performed as a simple although inefficient way to obtain the responsederivatives. The finite-difference approximation is highly inefficient in numericalcomputations, since it requires at least N 1 additional full system analyses for aproblem with N designable parameters [1]. With higher-order approximations,the number of analyses increases.ql,J stionableThe feasibility of this approach becomeswhen the design-variable space is large.The adjoint-variable method (AVM) is proved to be the most efficientmethod for sensitivity analysis [1], [2], [3], as it requires only one additional

Chapter 2 Methodology of the Adjoint-Variable methodsystem analysis to compute all sensitivities. The additional analysis is known asadjoint system analysis, with the adjoint system matrix being the transpose of thesystem matrix ofthe original problem [1], [3], [4]. Thus, the AVM improves theefficiency of the sensitivity computation by a factor of N in comparison with theforward or backward finite-difference approximations. The performance of theA VM has been validated in control theory [3], structural engineering [4], and incircuit and computational EM applications in electrical engineering [3], [5].In this chapter, we give a brief introduction into the methodology of theAVM, especially its applications with frequency-domain numerical EM solvers.Most of the discussion in this chapter and in the rest of the thesis focuses onapplications with the finite-element method (FEM).In Section 2.2, we present the concept of the frequency-domain AVM forlinear systems. Then, we give a general discussion on the sensitivity analysis ofcomplex linear systems in Section 2.3. Section 2.4 discusses the difficulties in theimplementation of the AVM with commercial solvers. The efficiency and therequired computational resources are discussed in Section 2.5 along with acomparison with the finite-difference approximation at the response level. Theadvantages and the drawbacks of the AVM are also discussed.10

Chapter 2 Methodology of the Adjoint-Variable method2.2FREQUENCY-DOMAIN ADJOINT-VARIABLEMETHODAfter proper discretization, a time-harmonic linear EM problem (by linear,we refer to the fact that the problem has linear material properties) can be writtenas a linear system of equations [6]:Ax b.(2.1)Here, A is the M by M system matrix, x is the 1 by M state variable vector, inFEM, known as the field solution vector, and b is the 1 by M excitation vector,which can be derived from the electromagnetic sources and the inhomogeneousboundary conditions according to the FEM formulation. The system matrix is afunction of the vector of design (shape or material) parameters p, i.e., A(p).Thus, the field solution vector x is an implicit function of p .For sensitivity analysis purposes, we need to determine the gradient of auser defined response function f(p,x(p)) with respect to p at the field solutionx of(2.1):V p/(p,x(p)) subject to AX b.(2.2)Here, the gradient of the response function f(p,x(p)) is defined as a row vector[4], [6]v,t [Z Z, . . :J11(2.3)

Chapter 2 Methodology of the Adjoint-Variable methodNote that the response function f(p,x(p)) is formulated so that it may have anexplicit dependence on the design variables in addition to its implicit dependenceon p through the field solution vector x . In some situations, both dependencesexist.We first constrain our problem as a real-number problem, i.e., both thesystem matrix A(p) and the response function f(p,x(p)) are real.Thesensitivity analysis with complex numbers is discussed in the next section.According to [7], an AVM sensitivity expression can be formulated as:v pf V f xr[v pb-V p(AX)].(2.4)Here, we divide the gradient of the response function into two parts:(2.5)V fstands for the explicit dependence of the response function f on the designvariables p , and V xf · V pX reflects the implicit dependence on p through thefield solution x. The vectorx is the adjoint solution, which is the solution of theadjoint system of equations [4], [8]:(2.6)The adjoint system excitation is(2.7)12

Chapter 2 Methodology of the Adjoint-Variable methodHere, we need to compute the original system solutionsystem solutionx,the adjointx, and the derivative of the system matrix with respect to eachdesign variable oAif)p;, i l, . ,N. Thus, with only two full-wave simulations,namely, the original system simulation and the adjoint system simulation, we cancompute the sensitivities.It is important to notice that we need to compute the system matrixderivative with respect to each design parameter oAif)p;, i l, . ,N. In some rarecases, the matrix derivatives may be analytically available [9], [10]. Then, thesensitivities are exact.According to [9], the time needed for the analyticalcomputation of one system matrix derivative is comparable with one systemmatrix fill.Thus, the sensitivity computation of a problem with N designparameters leads to an overhead of N matrix fills in addition to the original andadjoint system analyses.For most of the full-wave EM analysis methods, the system matrixderivatives are either not analytically available or too complicated to beanalytically derived for general design software. In these cases, the system matrixderivatives are computed by the finite-.difference approximation [ 11]:oA(p) A(p Ap;-e;)-A(p)Op;Here, e; is the unit vector whose ith element equals 1 and all others equal 0:13(2.8)

Chapter 2 Methodology of the Adjoint-Variable method0e; 1 ith element(2.9)0and /lp; is the finite-difference perturbation of the ith design variable.Thisapproximation also requires N additional matrix fills, similarly to the exactmethod. Our studies have shown that the accuracy for the sensitivity computationusing this system-matrix-level finite-difference approximation is satisfactory, witha relative error well below 1%, compared with the exact sensitivity computation[1].2.3SENSITIVITIES OF COMPLEX LINEAR SYSTEMSThe derivations in the above section apply to real-number problems only.However, in electromagnetic frequency-domain sensitivity analysis, the systemequations are complex, and, often, the responses are complex, too. It can beshown that the sensitivity formula in the complex case can be derived from thereal-number sensitivity formula.A complex linear system of equations in the form of (2.1) can bereformulated in a real-valued form [11]:[9U AHere, 9l and - AJ[mx] [mb]9U x b·(2.10)stand for the real and the imaginary parts of a matrix,respectively. We can re-write (2.1 0) as14

Chapter 2 Methodology of the Adjoint-Variable method(2.11)where(2.12)The size of the real-valued system of equations is twice the size of the originalone.With this real-valued system of equations, the AVM sensitivity expressionfor a real-valued response [becomes:(2.13)Here, Xr is the solution ofthe corresponding adjoint system of equations(2.14)The real-valued adjoint system excitationbris(2.15)The adjoint system of equations can b

Broyden Self-adjoint Sensitivity Analysis Broyden/Finite-Difference Self-Adjoint Sensitivity Analysis Broyden-Fletcher-Goldfarb-Shannon Electromagnetics Feasible Adjoint Sensitivity Technique Finite Difference Finite-Difference Time Domain Finite Element Method Method ofMoment Sequential Quadratic Programming Transmission-Line Method Trust Regions