CHAPTER 3 THE MACRO-MODEL
3.3 Model Order Reduction
In order to improve the computational efficiency of the sub-gridded model, we utilize model order reduction (MOR). In essence, MOR projects the column- space of the full-order system to a compact (i.e. reduced-order) sub-space; this is accomplished by finding the strong orthogonal axes inside the full-
order space and placing them in a sub-space projection matrix V .
Various MOR techniques may be used [2–7] to decrease the degrees of freedom of the full-order system while maintaining sufficient engineering ac- curacy. The MOR algorithm of ENOR [5] is convenient because it uses the fast Cholesky factorization for computationally efficient solution of matrix equations to produce a symmetric reduced order model while providing adap- tive error control to derive the projection matrix. The orthogonal projection operation is essentially the Galerkin process, which takes a (differential or integral) system of equations with infinite degrees of freedom and transforms it to a system of equations with finite degrees of freedom. The projection matrix is obtained by moment matching which involves expansion about the frequency of interest, and it is applied to the full-order system to produce passive reciprocal reduced order models in the form of a symmetric positive semi-definite system transfer function matrix, so long as the original system transfer function matrix is also positive semi-definite. The positive semi- definite nature of the reduced order system ensures passivity, and therefore stability, of the reduced order model.
Both [5] and [6] used Cholesky factorization in the MOR algorithm for its utility in speeding up the solution of the projection matrix by providing upper and lower triangular matrices for use in back substitution; this choice eliminates the need for computationally costly row-echelon operations. The Cholesky factorization becomes especially important for reduction of very large matrices.
It is worth noting that Cholesky factorization works on positive definite matrices only. For the matrix equation to be Cholesky factorizable, an ad- ditional requirement is imposed on the choice of the expansion frequency s0; that is, s0 must be a real frequency if Cholesky factorization is indeed used. Following a similar procedure to [5], we define the matrix quantities C ∈ <M ×M, G ∈ <M ×M, B ∈ <M ×n, J ∈ <n×n, X ∈ <M ×n, and Γ ∈ <M ×M, where M, n are integers, as follows:
C = P (3.21)
Γ = DHPµ−1DE (3.23)
B = Qh (3.24)
Using (3.21) - (3.23), we may re-write the admittance transfer function matrix (3.9) as Y = Cs + G + 1 sΓ (3.25) The total system in Laplace (s-domain) is given by
Cs + G + 1 sΓ
X[s] = BJ [s] (3.26)
We set s = s0(1 − z), where the complex variable z = −1s0(s − s0). Also, we define the auxiliary variable
Y0[z] = X[z]
1 − z (3.27)
The expansion frequency can be chosen along any path on the complex Laplace plane, and was chosen to be on the real axis in [5,6]; however, because ENOR requires Cholesky factorization, and Cholesky factorization requires a positive-definite matrix, which in turn must comprise all real components, we choose to expand about a real frequency s0 = f0 Hz. Further discussion on expansion about real vs. imaginary frequency may be found at [44–47].
Next, we expand X[z], Y0[z], J [z] in powers of z about the frequency s0, and substitute these expansions into (3.26) and (3.27), equate coefficients of same powers of z, and after some algebraic manipulations obtain the following recurrence relation: Cs0+ G + 1 s0 Γ Xk = Cs0Xk−1− 1 s0 ΓYk−10 + QhJk (3.28) where the desired number of iterations (i.e., number of moments about s0) k is an integer, the relative tolerance tolrel is a real number signifying the eigenvalue noise floor of the system, and it is
Yk0 = Xk+ Yk−10 X−1 = Y−10 = 0
The yet unknown Xk are proportional to the (block) moments of the elec- tric field vector (i.e. system response) when expanded about the frequency s0.
We set the h-field source (i.e. system excitation) as follows:
Jk = In×n if k = 0, 0 if k 6= 0, (3.29)
and solve (3.28) for as many k-terms as desired.
The choice of setting J(k=0) = In×n, where In×n is the n × n identity matrix, is equivalent to having a discrete impulse (magnetic) source at each excitation node in Qhhb, resulting in the discrete impulse response of the system.
Once the solutions to Xkare obtained, we form V = [X0, X1, ..., Xq], where the projection matrix V ∈ <M ×m is comprised of the first q block moments, and the integer m M is the order of reduction.
We propose that the model order reduction technique of ENOR produces an approximation of the full-order Maxwell’s state-space system, which yields sufficient accuracy in representing fields of a broad-band nature. What con- stitutes “sufficient accuracy” is decided by the specific application and its given criteria for accuracy. The previous work in the literature shows that a finite set of dominant eigenvectors obtains a good-enough approximation of transient response of various systems (e.g., waveguides). ENOR achieves this by moment-matching (about s0) of the system transfer function, while employing an adaptive error control scheme to decide what minimal set of modes and moments is necessary to minimize the error between the time response of projected reduced-order and full-order models. ENOR provides the mechanism for this adaptive error control through parameters of relative tolerance tolrel and maximum number of iterations kiter.
The argument tolrel is used to admit into the projection matrix only those eigenvectors associated with eigenvalues above the noise floor (a level be-
low which the associated eigenvector is deemed to have insignificant relative contribution to the system response). The argument kiter is used to decide the maximum number of moments about the expansion frequency s0 that are necessary to represent the system with sufficient accuracy. This itera- tive process continues until the reduced-order and full-order system discrete impulse responses are matched to desired specification. Thus, the ENOR algorithm produces the necessary and sufficient vector basis for construction of the eigen-space which is used to represent the dominant eigen-modes of the original system transfer function, thereby leading to a projection matrix whose column space spans the eigen-space of the full-order system.
In the following chapter, the stochastic domains are assumed to be rela- tively small fine-featured sub-domains of the much larger coarse deterministic domains, in multi-scale structures. This implies that the stochastic macro- model region is electrically small and thus the spatial distribution of fields in the stochastic macro-model region exhibits small sensitivity over frequency. Thus, a relatively small set of moments obtains sufficiently accurate repre- sentation of the transient field profile in the stochastic macro-model region.
Next, we apply the above MOR technique of ENOR to project the full- order system to obtain a reduced order ATF.