Adjoint Sensitivity Analysis of High Frequency Structures with MATLAB®
2: Colorado School of Mines, Golden, CO, USA
3: Northern Illinois University, DeKalb, IL, USA
This book presents the theory of adjoint sensitivity analysis for high frequency applications through timedomain electromagnetic simulations in MATLAB®. This theory enables the efficient estimation of the sensitivities of an arbitrary response with respect to all parameters in the considered problem. These sensitivities are required in many applications including gradientbased optimization, surrogatebased modeling, statistical analysis, and yield analysis. Using the popular FDTD method, the book shows how wideband sensitivities can be efficiently estimated for different types of materials and structures, and includes plenty of well explained MATLAB ® examples to help readers absorb the content more easily and to make the theory more understandable to the broadest possible audience. Topics covered include a review of FDTD and an introduction to adjoint sensitivity analysis; the adjoint variable method for frequencyindependent constitutive parameters; sensitivity analysis for frequencydependent objective functions; transient adjoint sensitivity analysis; adjoint sensitivity analysis with dispersive materials; adjoint sensitivity analysis of anisotropic structures; nonlinear adjoint sensitivity analysis; secondorder adjoint sensitivities; and advanced topics.
Other keywords: adjoint variable method; high frequency structures; 2D FDTD solvers; transient responses; frequencydomain responses; electric circuits; highfrequency structures; FDTD numerical approach; 1D FDTD solvers; nonlinear materials; Matlab; 3D FDTD solvers; scalar objective functions; Sparameters; AVM techniques; adjoint sensitivity analysis
 Book DOI: 10.1049/SBEW525E
 Chapter DOI: 10.1049/SBEW525E
 ISBN: 9781613532317
 eISBN: 9781613532324
 Page count: 320
 Format: PDF

Front Matter
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1 Introduction to sensitivity analysis approaches
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In this chapter, we give an initial introduction to adjoint techniques. We first review the classical approaches for derivative estimation and illustrate their computational cost. Approaches such as forward finite differences (FFD), backward finite differences (BFD), and central finite differences (CFD) are addressed. These approaches are utilized in later chapters as a reference to compare the estimated adjoint sensitivities against. We then present an adjoint sensitivity formulation that is suitable for analyzing electric circuits. The adjoint SA of electrical circuits and conductor transmission lines was addressed by several researchers. This analysis serves as a smooth introduction to the basic concepts involved in adjoint analysis of highfrequency structures. The same theorem is extended to highfrequency electromagnetic simulations as will be illustrated in the following chapters. We illustrate the theory presented in this chapter with circuit examples.

2 Introduction to FDTD
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Computational electromagnetics (CEM) has evolved rapidly during the past decade to a point where now extremely accurate predictions can be given for a variety of electromagnetic problems, including the scattering crosssection of radar targets and the precise design of antennas and microwave devices. In general, commonly used CEM methods are based on the applications of Maxwell's equations and the appropriate boundary conditions associated with the problem to be solved.The basic formulation of the commonly used timedomain differential equation approach, namely the finitedifference timedomain (FDTD) method for CEM applications, is covered in this chapter. This formulation sets the stage for the adjoint variable methods covered in the subsequent chapters. Most of the materials in this chapter is extracted from the authors' recent book [1].

3 The adjoint variable method for frequencyindependent constitutive parameters
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In this chapter, we show how adjoint sensitivity analysis techniques can be applied to computational electromagnetics problems. We focus in this chapter on linear, isotropic, and nondispersive electromagnetic structures. In this case, the properties of all materials in the computational domain do not depend on the field magnitude, the excitation polarization, or the excited frequency band. We adopt a stepbystep approach for introducing the basic concepts. First, we address the 1D finite difference time domain (FDTD) case and show how adjoint sensitivities are estimated. This case is illustrated through a numerical example. We then address the 2D TMz case. A full derivation of the adjoint variable method is given. We show that using a similar formulation to the 1D case, the sensitivities of a general objective function with respect to all parameters are estimated using one extra simulation. The 2D transverse electric to z (TEz) is similar to the 2D transverse magnetic to z (TMz) and will not be addressed here. Two numerical examples are presented to illustrate the 2D case. Finally, we address the full 3D case and show that the same concept applies to full 3D problems.

4 Sensitivity analysis for frequencydependent objective functions
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In this chapter, we address the practical case of frequencydependent objective functions such as the Sparameters, the Zparameters, and the Vparameters. These parameters are usually estimated over a band of frequencies. The timedomain solver returns a vector of complex numbers representing the values of these responses at different frequencies. The AVM approach estimates the sensitivities of these responses over the desired frequency band using at most one extra simulation.

5 Transient adjoint sensitivity analysis
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In this chapter, we address the transient time domain adjoint sensitivity analysis problem. Without loss of generalization, we will consider the desired transient response to be an electric field component at a point r_{o} in the computational domain E(r_{o}, kΔt). We show in this chapter that we can evaluate the derivatives ∂E(r_{o}, kΔt)/ ∂pi, for all parameters, and for all time steps, using only one extra simulation. In other words, we predict how the complete transient response at the probe changes due to incremental changes in any of the parameters.

6 Adjoint sensitivity analysis with dispersive materials
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In all the previously discussed cases, we assumed that the materials utilized in the FDTD simulations are nondispersive. This means that all material properties (permittivity, permeability, and conductivity) do not change with frequency. In many interesting applications, however, this is not the case. For example, in the emerging area of metamaterials, the effective permittivity and permeability show strong dependency on frequency [1]. In the area of plasmonics, all metals have dispersive properties [25]. The same applies to modeling materials in the THz and infrared frequency regimes [6,7]. It is thus of prime importance to be able to estimate the sensitivities of different responses with respect to all geometrical and material parameters of structures with dispersive material properties. In this chapter, we develop a general theory for adjoint sensitivity analysis of high frequency dispersive structures. This formulation applies to materials with commonly used types of dispersion profiles such as Lorentz [8,9], Drude [1012], and Debye [13,14]. We show that only one dispersive adjoint simulation is required to estimate the sensitivities of the desired response with respect to all parameters. We illustrate this approach through one example.

7 Adjoint sensitivity analysis of anisotropic structures
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In this chapter, we present an algorithm for adjoint sensitivity analysis of anisotropic materials. We show that using only one extra adjoint simulation, the sensitivities of the objective function with respect to all parameters are estimated. The material property tensors of the adjoint problem are the transpose of those of the original problem. The computational cost of the adjoint problem is the same as that of the original simulation. The derivation considered in this chapter addresses the nondispersive case. The considered tensors are assumed to be independent of time. The formulation presented in this chapter is adapted. The anisotropic and dispersive case is still a subject of research.

8 Nonlinear adjoint sensitivity analysis
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In this chapter, we focus on the nonlinear case. We show that if the relative permittivity, conductivity, or both are polynomial functions of the electric field, we can extract the sensitivities of the considered objective function or response with respect to all nonlinearity parameters using one extra simulation [4,5]. This approach is illustrated for the isotropic and nondispersive case. However, the theory can be extended to nonlinear materials with arbitrary anisotropy and dispersion profiles. The theory of adjoint sensitivity analysis of nonlinear material in computational electromagnetics is still at its initial stages. We report here on some initial results using the finite difference time domain (FDTD) method

9 Secondorder adjoint sensitivities
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The authors show in this chapter, two different approaches for efficiently estimating the secondorder derivatives (Hessian matrix) of a given objective function. The cost of evaluating the Hessian using classical finite difference approach is O(n^{2}) where n is the number of parameters. The first adjoint approach reduces the cost of estimating all components of the Hessian matrix to only 2n extra simulations. This approach is simple, and it uses the algorithms developed in previous chapters. A second approach for estimating the complete Hessian is also presented. This approach is more complex than the first approach and requires extra memory storage. This approach requires only n + 1 extra simulations per Hessian evaluation. It follows that the computational cost is approximately one half of the first adjoint approach. This saving comes at the cost of a more complex algorithm and more extensive storage.

10 Advanced topics
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The authors discuss in this chapter approaches that can significantly reduce the memory storage associated with the AVM method. We show that the methods of spatial and spectral sampling can be applied to make the AVM approach more efficient. The authors also discuss how the AVM approach can be implemented using numerical techniques other than the FDTD method. The authors discuss how the approach applies to the transmission line modeling (TLM) method. The TLM method is another time domain technique that can be shown to be related to the FDTD. The authors also briefly discuss how the AVM technique can be applied using frequency domain solvers such as the finite element method (FEM) and the method of moments (MoM). The authors conclude this chapter by briefly illustrating some applications of adjoint sensitivity analysis.

Back Matter
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Supplementary material

Supplementary files for "Adjoint Sensitivity Analysis of High Frequency Structures with MATLAB"

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The zip file contains Matlab codes for chapters 1, 3, 4, 5, 6, 7, 8, 9.

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Click on the icon below to download Matlab files for this book.
The zip file contains Matlab codes for chapters 1, 3, 4, 5, 6, 7, 8, 9.