Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

Georg A. Mensah; Philip E. Buschmann; Alessandro Orchini

doi:10.1177/17568277221084464

Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

Mensah, Georg A.; Buschmann, Philip E.; Orchini, Alessandro 2022-03-01 00:00:00 The spectrum of the thermoacoustic operator is governed by a nonlinear eigenvalue problem. A few different strategies have been proposed by the thermoacoustic community to tackle it and identify the frequencies and growth rates of ther- moacoustic eigenmodes. These strategies typically require the use of iterative algorithms, which need an initial guess and are not necessarily guaranteed to converge to an eigenvalue. A quantitative comparison between the convergence prop- erties of these methods has however never been addressed. By using adjoint-based sensitivity, in this study we derive an explicit formula that can be used to quantify the behaviour of an iterative method in the vicinity of an eigenvalue. In par- ticular, we employ Banach’s ﬁxed-point theorem to demonstrate that there exist thermoacoustic eigenvalues that cannot be identiﬁed by some of the iterative methods proposed in the literature, in particular ﬁxed-point iterations, regardless of the accuracy of the initial guess provided. We then introduce a family of iterative methods known as Householder’s meth- ods, of which Newton’s method is a special case. The coefﬁcients needed to use these methods are explicitly derived by means of high-order adjoint-based perturbation theory. We demonstrate how these methods are always guaranteed to converge to the closest eigenvalue, provided that the initial guess is accurate enough. We also show numerically how the basin of attraction of the eigenvalues varies with the order of the employed Householder’s method. Keywords Thermoacoustics, nonlinear eigenvalue problem, ITA, basin of attraction solve generalized linear eigenvalue problems, for which Introduction L(ω) = (X − ωY), solving an eigenvalue problem that is Thermoacoustic instabilities can be assessed by solving the 4–6 nonlinear in the eigenvalue ω is intrinsically harder. A inhomogeneous Helmholtz equation typical approach is to transform the NLEVP into a (series 2 2 2 2 −iωτ of) associated eigenvalue problems linear in the eigenvalue. ∇ · c ∇p ˆ + ω p ˆ = (c − c )n(x)e ∇p ˆ (1) 2 1 x ref For example: on a prescribed geometry with appropriate boundary con- ditions. In Eq. (1) ˆp indicates the complex-valued amp- � NLEVPs that are polynomial in ω with order K can be litude of the Fourier transform of the pressure ﬂuctuations recast into linear eigenvalue problems of dimension (the eigenfunction) and c the speed of sound, with the sub- KN ; scripts and indicating the regions upstream and down- � Solutions of NLEVPs can be found by iteratively 1 2 stream the ﬂame, respectively. The interaction index n, solving linear eigenvalue problems resulting from the which is non-zero only over an acoustically compact expansion of L(ω) (to any desired order) in the 2,3 volume, and the time delay τ model the ﬂame response. eigenvalue ; Equation (1) deﬁnes a nonlinear eigenvalue problem (NLEVP) in the complex frequency ω. Once discretized, Institute of Fluid Dynamics and Technical Acoustics, TU Berlin, DE e.g. by means of ﬁnite elements, the NLEVP can be Department of Energy and Process Engineering, NTNU Trondheim, NO expressed in compact form as Corresponding author: L(ω) p = 0, (2) Alessandro Orchini, Institute of Fluid Dynamics and Technical Acoustics, where L is an N × N large, sparse matrix depending non- TU Berlin, DE. linearly on ω. Although there exist efﬁcient algorithms to Email: a.orchini@tu-berlin.de Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access page (https://us.sagepub.com/en-us/nam/open-access-at-sage). Mensah et al. 31 � Contour integration methods reduce the NLEVP to a with respect to these solution methods, and how the linear eigenvalue problem possessing only the eigen- basins of attraction of each eigenvalue vary depending values of L inside a given contour in the complex on the order of the Householder’s method used. 7,8 plane. The thermoacoustic nonlinear eigenvalue These approaches exploit the fact that efﬁcient and robust problem methods (e.g. Arnoldi) exist for large scale linear eigenvalue 9,10 problems. A common feature of iterative methods is that The weak formulation and discretization of the thermoa- they all require an initial guess ω for the eigenvalue , which coustic Helmholtz equation (1) results in an N-dimensional is updated at every iteration. Eigenvalues of the thermoa- NLEVP that reads coustic operator are ﬁxed points of the mapping deﬁned L(ω)p = K + ωC + ω M + Q(ω) p = 0. (3) by the chosen iterative algorithm. According to Banach’s ﬁxed-point theorem p. 152, for The matrices in (3) arise from discretizing the operators in an iterative algorithm to be able to identify an eigenvalue, equation (1), viz. the stiffness operator ∇ · (c ∇(·)) 7! K, the mapping deﬁned by the algorithm needs to be contracting 2 2 the mass operator ω (·) 7! ω M, and the ﬂame operator in the vicinity of the eigenvalue. This means that, provided 2 2 −iωτ −(c − c )n(x)e ∇| (·) 7! Q(ω). The matrix C results 2 1 x that the initial guess is sufﬁciently close to an eigenvalue, ref from the discretization of the boundary conditions needed at each iteration the algorithm will move the guess closer to close the problem. They can be expressed in terms of to the eigenvalue, until a prescribed tolerance is reached. the acoustic impedance Z on all boundaries On the other hand, if the mapping is repelling in the vicinity of an eigenvalue, it will not be possible to identify this eigen- iωp ˆ + Z(ω)∇p ˆ · n ˆ = 0, (4) value with the chosen algorithm, since a sufﬁciently close guess will be pushed further away from the solution. An where n ˆ is a unit vector normal to the boundary. overlooked eigenvalue can have serious consequences in Equation (3) represents a nonlinear eigenvalue problem. thermoacoustics, as the reliable and accurate determination The challenge is to ﬁnd all its eigenvalues ω – and their cor- of all relevant eigenvalues is of paramount importance to responding eigenvectors p – in a prescribed portion of the ensure the safe operability of an engine. complex plane. The ﬁxed-point iteration method described in is a com- monly used algorithm for identifying eigenvalues of the 12–17 thermoacoustic Helmholtz operator – it is used in to Rijke tube test-case name a few. However, with the exception of a short discus- sion in, there is no reference in the literature investigating Throughout this study we will employ the classic Rijke tube the convergence properties of this method in relation to the conﬁguration to demonstrate our results. It consists of a spectrum of the thermoacoustic operator. The aim of this straight duct in which a ﬂame is located. Across the ﬂame work is to quantify the convergence properties of ﬁxed- the temperature, and hence the speed of sound, rise point iteration methods that are commonly used in thermo- abruptly. The axial ﬂame location x is chosen to be in acoustics. It will be shown that some thermoacoustic eigen- the middle of the tube. The parameters that determine the values cannot be found using a ﬁxed-point algorithm, acoustic response of the system are chosen to be identical regardless of the chosen initial guess. We will then intro- to those used in. By non-dimensionalizing all quantities duce an alternative adjoint-based iterative method that has using the tube length L as a characteristic length and the more robust convergence properties and is thus better speed of sound c in the cold section of the duct as a char- suited to tackle the thermoacoustic problem. acteristic velocity, the chosen Rijke tube’s parameter are This study is organized as follows. First the thermoa- x = 0.5, c(x) = 1 for x ≤ x and c(x) = 2 otherwise. The f f coustic problem and the ﬁxed-point iteration presented boundary conditions are chosen to be acoustically closed in are introduced. An explicit formula that quantiﬁes (∇ ˆp = 0) and opened ( ˆp = 0) at the inlet (x = 0) and the convergence properties of the ﬁxed-point mapping outlet (x = 1), respectively. The ﬂame model parameters is derived. The same procedure is conducted for a are chosen to be n = 1/3 and τ = 2. so-called Picard iteration, which is another form of ﬁxed- For the range of frequencies that we will consider all point iteration. Both methods are applied to a generic transverse modes are cut-off. The problem can therefore be thermoacoustic test case to demonstrate that some considered as one-dimensional, and the ﬂame to be a point eigenvalues are repellors with respect to these mappings. source in the Helmholtz equation (1), n(x) = nδ(x − x ). We will then introduce a class of adjoint-based solution The eigenvalues of this problem can be calculated semi- methods known as Householder, of which Newton’s analytically by using an acoustic network approach. By method is a special case. It will be discussed how all assuming one-dimensional acoustics, equation (1) can be eigenvalues of the thermoacoustic operator are attractors expressed in terms of the Riemann invariants f and g. 32 International Journal of Spray and Combustion Dynamics 14(1-2) Table 1. Eigenvalues of the Rijke tube network model (5) in the quadratic in ω, except for the heat release term. The ﬂame considered portion of the complex plane together with the operator can therefore be thought of as a perturbation of the associated acoustic (no ﬂame) and intrinsic (anechoic boundary underlying purely acoustic problem, which is obtained with conditions) eigenvalues from which they stem. Q = 0. This motivates the use of an iterative strategy, starting from the (known) purely acoustic solutions. ωω ω ac ITA The key idea is to recast the problem into a linear eigen- 1 2.396–0.262i 2.462+0i – value problem of doubled dimension, and then to iteratively 2 4.692+0.304i – 4.712+0i identify a thermoacoustic eigenvalue starting from an initial 3 6.283+0i 6.283+0i – guess. We shall indicate this linear eigenvalue problem as 4 7.874+0.304i – 7.854+0i L (ω ; ω) p = 0. It reads FP n With suitable matching conditions at the ﬂame zone and for 0 −I p I0 p =−ω . (10) the parameters introduced above, the acoustic-network K + Q(ω ) C p 0M p I I eigenvalue problem reads X Y FP FP g g 0 On the left hand side, the nonlinear dependency of the 1 1 L = L − L = , (5) 1D ac fl f f 0 problem on the eigenvalue has been removed by freezing 2 2 the evaluation of the ﬂame operator Q, i.e, by evaluating where L is the acoustic response in the absence of an active ac at ω , the eigenvalue guess. This guess is updated by itera- ﬂame tively replacing ω with the closest eigenvalue ω of (10), −iω −iω/2 which can be found by means of, e.g., the Arnoldi iteration e + 1 e − 1 L ≡ , (6) ac −iω −iω/2 method. The eigenvector in (10) has been extended into 1 − e 2e + 2 ˜p = [ p, ω p]. (11) and L provides the feedback between heat release rate ﬂuc- fl This can be seen from the ﬁrst matrix equation deﬁned by tuations and the acoustics (10), which reads p = ω p. By using this relation, the second matrix equation of (10) reduces to (3) when L ≡ . (7) fl −2iω −iω e e − 1 0 ω = ω. The algorithm terminates successfully if the differ- ence between the results of two successive iterations is less By imposing det(L ) = 0 the eigenvalues of the Rijke tube 1D than a prescribed tolerance tol , or is aborted if a predeﬁned can be obtained by using standard root-ﬁnder methods. We maximum number of iterations is reached. shall focus our analysis on the properties of the four thermo- acoustic eigenvalues found in the region S of the complex Algorithm 1 Nicoud’s algorithm plane delimited by 1: function ITERATE(ω , p , tol , maxiter, K, C, M, Q) 1 ω 2: ω ← ∞ S= { ω ∣ ℜ(ω) ∈ [0, 10] ∧ ℑ(ω) ∈ [ − 1, 1] }, (8) 0 3: n ← 1 which are reported in Table 1. Two of these eigenvalues can 4: p ← ω p be linked to acoustic modes – whose eigenvalues ω can be ac I0 calculated by setting n = 0, which implies L = 0 – while 5: Y ← fl FP 0M the other two can be linked to intrinsic thermoacoustic 6: while |ω − ω | > tol and n < maxiter do 20,21 n n−1 ω (ITA) modes – whose frequencies are given by 0 −I 7: X ← FP K + Q(ω ) C 2k + 1 1 c n ω = π − i log − 1 n . (9) ITA ˜ ˜ 8: ω , p ← EIGS −X , Y , ω , p n+1 FP FP n n+1 n τ τ c 9: n ← n + 1 10: end while 11: p ← p [1 : N] In the following, we will discretize by ﬁnite elements the n n 12: return ω , p , n thermoacoustic eigenproblem (3) and use various iterative n 13: end function solution techniques to identify its eigenvalues. The semi- analytical wave-based solutions presented in Table 1 will The pseudocode for the ﬁxed-point iterative scheme serve as benchmarks. The dimensions of the resulting is outlined in Alg. 1. We have denoted with matrix operator L do not allow for the use of determinant- EIGS(X, Y, ω , p ) the Arnoldi method that identiﬁes the based methods, and alternative iterative strategies are needed. eigenpair ω, p of the linear eigenvalue problem X p = ωY p closest to ω . An initial guess for the eigenvector State-of-the-art iterative solution method p , if known, may also be provided to the algorithm, in A ﬁxed-point iteration to solve the NLEVP (3) was proposed order to improve the convergence of the Arnoldi method. in. This algorithm exploits the fact that the problem is Also note that the matrix M is always positive-deﬁnite – as Mensah et al. 33 H ∗ H † it deﬁnes a mass matrix – and hence is Y . Thus, the eigen- FP deﬁned by (14) shows that p = (C + ω M ) p ,so that ˜ † value problem (X (ω ) − ωY ) p = 0 appearing at each FP n FP the extended adjoint eigenvector ˜p satisﬁes iteration, can be solved efﬁciently. † H ∗ H † † ˜p = C + ω M p , p . (15) Fixed-point iteration schemes in thermoacoustics The ﬁrst matrix equation in (14) deﬁnes the adjoint eigen- † † value problem L (ω) p = 0. A single iteration of the procedure described in Alg. 1 From adjoint-based sensitivity analysis, the following deﬁnes a mapping ω = f (ω ) for which ﬁxed-points ω n+1 n formula is known for evaluating the derivative of f with are sought. This ﬁxed-point map f can be explicitly 22,23 respect to ω : expressed by rewriting the eigenvalue problem (10) as L (f ; ω ) ˜p = X (ω ) + fY ˜p = 0. (12) FP n FP n FP H ∂L FP ˜p ˜p df ′ ∂ω A condition for the convergence of the mapping to an f = =− (16) H ∂L dω FP eigenvalue of the thermoacoustic operator is provided by ˜ ˜ p p ∂f Banach’s ﬁxed-point theorem, which we brieﬂy recall. Consider a bounded domain Γ ⊂ C that contains the ﬁxed- By combining the known information on the direct and point ω. Banach’s ﬁxed-point theorem guarantees that f has adjoint eigenvectors – Eqs. (11) and (15), respectively – one (unique) ﬁxed-point in Γ as long as f is a contraction as well as on the functional shape of L – Eqs. (12) and FP everywhere in Γ. (10) – we have For our purposes, we are only interested at the behaviour of the iterative map at the ﬁxed-point. In particular, to know H∂L H FP † † ′ †H ′ if an eigenvalue ω is an attractor of a given iterative method ˜p ˜p = ˜p X ˜p = p Q (ω) p (17a) FP ∂ω f it is sufﬁcient to verify that H∂L H FP † † †H ˜ ˜ ˜ ˜ p p = p Y p = p() C + 2ωM p (17b) |f (ω)| < 1. (13) FP ∂f This in fact implies that the mapping f represents a contrac- tion in the vicinity of ω. Provided that Eq. (13) is satisﬁed, This yields a closed-form expression for the sensitivity of the eigenvalue ω can be found if the initial guess lies sufﬁ- the mapping f as required to apply Banach’s theorem: ciently close to it. 3 H † ′ Nicoud recognized the importance of the contraction p Q (ω) p f (ω) = . (18) FP properties of f , but stated that “[ …] obtaining general †H p() C + 2ωM p results about the contracting properties of the operator f from physical arguments is out of reach of the current under- The mapping sensitivity (18) is generally expected to be standing of the thermoacoustic instabilities.” Signiﬁcant pro- non-zero. This implies that, if the ﬁxed-point iteration con- gress has been made in this direction by the thermoacoustic verges, |f | < 1, then it possesses a linear convergence rate – community in recent years, and tools are now available to see Eq. (21). quantify the behaviour of the mapping f from the thermoa- coustic equations. In particular, the contraction properties of the mapping f can be quantiﬁed by exploiting adjoint-based An alternative ﬁxed-point iteration scheme 22,23 sensitivity. In the following, we use adjoint methods to Thechoiceofa ﬁxed-point iteration method is not unique. derive an analytical expression that allows us to compute One of the disadvantages of the solution method proposed |f (ω)| and thus make explicit use of Banach’stheorem. in is that it casts the original problem in an eigenvalue problem having doubled dimensions, which is computa- tionally more expensive to solve. Alternatively, one can Contraction properties of the ﬁxed-point iteration perform an iteration in the ω term of Eq. (3), while the The adjoint equation associated with the mapping (12) reads: remaining occurrences of the eigenvalues are ﬁxed to ω . The square root of the eigenvalues of the resulting L (f ; ω) ˜p = FP (linear) eigenvalue problem are then an approximation of H H † 0K + Q (ω) I0 p the sought eigenvalues. This choice is convenient since ∗ I + f = 0, H H the matrix M that appears on the r.h.s. of the resulting gen- 0M −IC eralised eigenvalue problem is positive deﬁnite, allowing (14) for the use of robust linear eigenvalue solvers. We shall where f denotes the complex conjugate of f.Its ﬁxed-points refer to this solution scheme, described in Alg. 2, as ∗ ∗ are found when f = ω . The second matrix equation Picard iteration. p. 154 34 International Journal of Spray and Combustion Dynamics 14(1-2) Algorithm 2 Picard iteration Table 2. Eigenvalues identiﬁed by means of ﬁnite elements and values of the contracting operator |f |. Only the eigenvalues for 1: function ITERATE(ω , p , tol , maxiter, K, C, M, Q) 1 ω which |f | < 1 can be identiﬁed, provided that an accurate initial 2: ω ← ∞ guess is known. 3: n ← 1 4: while |ω − ω | > tol and n < maxiter do ′ ′ n n−1 ω ω |f||f | FP PC 5: X ← K + ω C + Q(ω ) n n 6: ω , p ← EIGS −X, M, ω , p 1 2.396−0.260i 0.519 0.519 n+1 n n+1 n 7: ω ← ω 2 4.696+0.297i 17.025 17.025 n+1 n+1 8: n ← n + 1 3 6.300−0.000i 0.022 0.022 9: end while 4 7.880+0.316i 12.204 12.204 10: return ω , p , n 11: end function The Picard iteration deﬁnes the mapping f given by ⎛ ⎞ ⎝ ⎠ L ( f ; ω ) p = K + ω C + Q(ω ) +f M p = 0. PC n n n X (ω ) PC n (19) By invoking the associated adjoint eigenvalue problem, and by using the adjoint sensitivity Eq. (16), one can show that the sensitivity of the Picard iteration reads † ′ p (Q (ω) + C) p Figure 1. Modeshapes of the four eigenvalues in the considered f (ω) = . (20) PC H portion of the complex plane. The colors orange, purple, blue p 2ωM p and green indicate, respectively, the thermoacoustic modes number 1, 2, 3 and 4, as listed in Table 2. Modes 1 and 3 are of Notably, the only difference with respect to the sensitivity 3 acoustic origin, 2 and 4 of ITA origin (see Table 1). of the ﬁxed-point iteration proposed by , Eq. (17), is that the matrix C is moved to the numerator. However, for the simple boundary conditions considered in this study, one ﬁnds that the matrix C has non-zero elements only at the iterative method, described in the next section. This is open-end boundary, where both p and p vanish, so that because, as we will now show, ﬁxed-point methods fail in the contribution of the term p Cp vanishes. identifying some eigenvalues. The modeshapes associated This is a general result. When considering non-trivial with the four eigenvalues are shown in Figure 1. boundary conditions, the convergence properties of the The sensitivities that the mapping f takes at these solu- two proposed ﬁxed-point solution methods are going to tions in Table 2 are then evaluated according to Eq. (18). differ, and one cannot draw general conclusions on which ′ By comparison with Table 1 one ﬁnds that |f | is less method will be able to identify more eigenvalues a priori. than one for thermoacoustic modes of acoustic origin, On the other hand, for trivial (open and/or closed) boundary whereas it is greater than one for modes of ITA origin. conditions one can show that p Cp = 0. Therefore, when This implies that modes of ITA origin cannot be identiﬁed considering geometries with simple open or closed bound- with this iterative scheme. Indeed, even if a good initial esti- ary conditions, the Picard iteration method should be pre- ′ mate is known for those, since |f | > 1 the iterative algo- ferred as a ﬁxed-point iteration scheme since (i) it is rithm will repel the initial guess from the actual solution. computationally cheaper and (ii) has the same contraction This fact might be partly responsible for the late discovery properties at the eigenvalues. of modes of ITA origin in thermoacoustics, in particular for 21,24 annular combustion chambers, for which larger non- linear eigenvalue problems need to be solved. Convergence properties of ﬁxed-point iterative Although a formal proof is not available, the difﬁculties methods that the ﬁxed-point iterations have in converging towards Table 2 highlights the contraction properties of the discussed some ITA modes can be intuitively explained by the follow- ﬁxed-point algorithm for the four eigenvalues of interest that ing argument. The matrix in the numerator of (18) is propor- the problem has when discretized using an equidistant linear tional to the ﬂame delay, exp (−iωτ). Modes of ITA origin ﬁnite elements mesh with 127 degrees of freedom. These tend to have largely negative growth rates, ℑ[ω] ≫ 0, eigenvalues have been calculated using a higher-order implying that | exp (−iωτ)| ≫ 1. Additionally, the Mensah et al. 35 eigenvectors of modes of ITA origin (and their adjoint) have iterations is exceeded – the algorithm does not converge a strong magnitude at the ﬂame – see Figure 1 – exactly the with this initial condition to the desired tolerance. The zone in which the ﬂame response Q is large, due to a strong results are shown in Fig. 2. The axes indicate the value gradient in the pressure modeshapes at the reference pos- taken by the initial guess, and the color scheme indicates ition. In combination, these two effects yield a large value to which eigenvalue the initial guess has converged and † ′ of the term p Q (ω) p, appearing in the numerator of how many steps it took. The actual solutions are highlighted (18). Thus, |f | ≫ 1 can be expected for ITA modes. This with white markers. statement is not expected to hold generally, but only Brute-force numerical calculations conﬁrm that the serves to explain the different convergence behaviours of modes of ITA origin, #2 and #4 in Table 1, cannot be modes of intrinsic and acoustic origin for this speciﬁc found by using ﬁxed-point iterative methods, even for case. Indeed, per Eq. (18), a different choice of boundary guesses that start very close to the actual solutions. The conditions – via complex-valued impedances in C or modes of acoustic origin, instead, can be identiﬁed. The more terms associated with different boundaries – can portions of the complex plane that, when used as an decrease the value of |f | below unity. When this is the initial guess, converge to modes #1 and #3, are highlighted case, then also intrinsic modes could be computed with a in orange and blue, respectively. Notably, convergence of ﬁxed-point iteration. mode #1 always requires a number of iterations much To demonstrate that the ﬁxed-point iterations cannot larger than those needed for mode #3 to converge. This identify modes for which |f | > 1, we perform a large is in line with the fact that the sensitivity of mode #1, number of brute-force calculations in the complex plane. although less than 1, is an order of magnitude larger We initialize the ﬁxed-point algorithm with initial than that of mode #3. More quantitatively, when guesses ω in the region S deﬁned in (8) sampled on a 0 < |f | < 1, in the vicinity of a solution the number of uniform 1001 × 201 grid. The eigenvectors for the steps N needed to improve the accuracy by an order of Krylov-subspace methods were initialized with the magnitude can be estimated by N =⌈1/ log (1/|f |)⌉. p = [1, 1, .. . , 1] . We stop the algorithm when a toler- From the values reported in Table 2, this yields N = 4 −10 ance tol ≡ |ω − ω | < 10 is reached – an eigen- for mode #1 and N = 1 for mode #3, i.e., the former ω n n−1 s value has converged – or when a maximum number of 16 requires approximately 4 times more iterations than the Figure 2. Convergence and basins of attractions for (top) the ﬁxed-point iteration described in , Alg. 1, and (bottom) the Picard iteration described in this study, Alg. 2. The areas in orange and blue indicate, respectively, the basins of attraction of the thermoacoustic modes #1 and #3, as listed in Table 2. The basin of attraction of mode #1 is hatched with black because, although the algorithm is slowly converging, the convergence rate is very low, and it takes more than 16 iterations to identify the eigenvalue to the −10 prescribed accuracy of 10 . Both methods are unable to identify modes number 2 and 4, of ITA origin. The lighter the color, the less steps are needed for convergence. Gray-black hatched surfaces indicate (slow) convergence towards an eigenvalue outside of the considered domain. Yellow indicates convergence to a (spurious) eigenvalue with ω = 0. 36 International Journal of Spray and Combustion Dynamics 14(1-2) latter to converge to a predeﬁned tolerance. Within the method to det[L(ω)] = 0 leads to required 16 iterations limit, the eigenvalue #1 can be iden- −6 tiﬁed with a tolerance of 10 . A larger number of itera- det[L(ω)] ω = ω − . (22) n+1 n tions is needed to converge to the prescribed tolerance d(det[L(ω)])/dω ω=ω −10 tol = 10 , emphasized by the presence of black stripes in Figure 2. Lastly, in yellow we have highlighted Evaluating the determinant is however a demanding and the portion of the complex plane that converges to the ω = error-prone operation. A determinant-free Newton scheme 0 eigenvalue. This is always a solution of the thermoa- was suggested in, which exploits Jacobi’s formula coustic weak form (3), but it represents a spurious solution that does not satisfy the boundary conditions, except when det[L(ω)] 1 = . (23) −1 all boundaries are acoustically closed. This spurious solu- d(det[L(ω)])/dω tr L (ω)dL(ω)/dω tion arises from manipulation of the impedance boundary condition equations when deriving the weak form (3). The Although this scheme is more robust than a direct evaluation ﬁxed-point algorithm proposed by and summarised in of the determinant, the evaluation of the inverse of the matrix Alg. 1 strongly promotes convergence towards this spuri- operator L becomes prohibitively expensive when L has hun- ous eigenvalue because it separates the action of the dreds of thousand of degrees of freedom, as is typical in real- boundary matrix, C, from that of the stiffness and ﬂame world thermoacoustic calculations. matrices, K and Q, as can be seen in Eq. (10). To bypass these issues, in the following section we will Together with the higher dimensions of the linearized discuss how adjoint-based methods allow us to use eigenvalue problem, the promotion of the convergence Newton’s method directly on the NLEVP equations. This towards a spurious eigenvalue is another disadvantage of will not require the evaluation of determinants nor matrix the method of in contrast to the Picard iteration (19). inverses, and provides more accurate information on the Nonetheless, the slow convergence of some modes and mapping sensitivity than ﬁnite difference approximations. the non-convergence of other modes for both ﬁxed-point iterations are inherently linked to their linear convergence rate. To overcome this limitations, we will now consider Adjoint-based Newton iteration methods with a super-linear convergence rate. By introducing an auxiliary variable λ, the NLEVP (2) can be written as L(ω) p = λY p. (24) Newton-like iteration methods A sequence generated from an iterative algorithm is said to Equation (24) can be thought of as a generalized eigenvalue have a rate of convergence m if there exists a bound G > 0 problem with eigenvalue λ depending on a parameter ω. such that From this viewpoint λ is an implicitly deﬁned function of ω, i.e. λ = λ(ω). If the parameter ω is chosen such that |ω − ω|≤ G|ω − ω| . (21) the eigenvalue problem (24) has an eigenvalue λ = 0, n+1 n then ω (and the corresponding eigenvector p) is a solution of the NLEVP (2). Since the implicitly restarted Arnoldi When the contraction condition (13) is satisﬁed, the iterative algorithm is used to solve the sparse eigenvalue problem ﬁxed-point algorithms described so far converge towards the (24), linear in λ, choosing the matrix Y to be semi-positive closest eigenvalue at a linear rate, m = 1. However, when deﬁnite has signiﬁcant advantages. Possible choices are one has |f (ω)|= 0, the rate of convergence becomes super- Y = I, the identity matrix, or Y = M, the mass matrix. In linear, with m > 1. Importantly, |f (ω)|= 0 always satisﬁes our analysis the latter will be used, since it naturally Banach’s condition (13), implying that super-linear conver- arises when discretizing the continuous thermoacoustic ging algorithms are guaranteed to converge to all eigenva- operator by means of ﬁnite elements. The following discus- lues. Super-linear convergence is a feature of Newton’s sion on the convergence of the method is however inde- method, which is therefore more robust than ﬁxed-point pendent from the speciﬁc form chosen for Y. methods (0th order), since it uses gradient information (1st To ﬁnd values of ω for which (24) has a zero eigenvalue order). Gradient information may be included also in a ﬁxed- in λ, we seek the roots of the implicit relation λ = λ(ω). point iteration by introducing a relaxation parameter Newton’s iteration on this relation reads Appenidx C.4. This procedure is however based on ﬁnite difference approximations and requires solving two eigen- ω = ω − λ(ω )/λ (ω ) = f (ω ), (25) n+1 n n n n value problems at each iteration. A direct application of Newton’s method is preferable. which requires the sensitivity of the eigenvalue with respect Since the eigenvalues of (3) are found when the deter- to the parameter, λ = dλ/dω. This sensitivity can be minant of L vanishes, a direct application of Newton’s obtained by means of adjoint methods. Starting from Eq. Mensah et al. 37 (24) it can be shown that the sensitivity of a linear eigen- m > 0 the iterative Householder step reads 22,23,27 value problem is {m−1} 1/λ (ω ) ω = ω + m . (27) † ′ n+1 n {m} p L (ω ) p 1/λ (ω ) λ (ω ) = . (26) n p Y p At ﬁrst order (m = 1), Eq. (27) corresponds to the Newton By using this expression, one can iteratively update the par- iteration (25). At second order (m = 2) one obtains ameter ω in (25) until an eigenvalue λ = 0 is found. 2λ(ω )λ (ω ) n n ω = ω − , (28) n+1 n Algorithm 3 Newton adjoint-based method ′ ′′ 2 λ (ω ) −λ(ω )λ (ω ) n n n 1: function ITERATE(ω , p , p , L, tol , maxiter) 1 ω 1 1 which is also known as Halley’s method. In order to apply 2: ω ← ∞ higher-order Householder’s methods, the evaluation of 3: n ← 1 {m} 4: while |ω − ω | > tol and n < maxiter do n n−1 ω high-order eigenvalue sensitivities λ is required. Once 5: λ , p ← EIGS L(ω ), M, 0, p n n n+1 n again, adjoint-based theory can be exploited in this ∗ H † 6: λ , p ← EIGS L (ω ), M, 0, p regard. By using perturbation theory, explicit formulas for n n n+1 n p L (ω ) p the calculation of arbitrarily high-order sensitivity have n+1 n+1 7: λ ← − n H been derived in, to which we refer the interested reader. p M p n+1 n+1 For the purpose of this study, we shall assume that a func- 8: ω ← ω − n+1 n ′ tion PERT that computes arbitrarily high-order sensitivities is 9: n ← n + 1 available. With this at hand, the pseudocode for the 10: end while Householder iterative method is given in Alg. 4, in which 11: λ ← EIGS L(ω ), M, 0, p ⊳ This line basically computes n n the function HOUSE computes the Householder update to the residual for possible later use ω at the desired order, according to Eq. (27). † n 12: return ω , p , p , λ n n n n n 13: end function Algorithm 4 Householder’s method 1: function ITERATE(ω , L, p , p , tol , maxiter, order) 1 ω 1 1 2: ω ← ∞ The algorithm for the adjoint-based Newton iteration 3: n ← 1 4: while |ω − ω | > tol and n < maxiter do is given in Alg. 3. Note that this method requires n n−1 ω 5: λ , p ← EIGS L(ω ), M, 0, p solving both the direct and the adjoint problem, which n n n+1 n ∗ H requires a numerical implementation of the adjoint 6: λ , p ← EIGS L (ω ), M, 0, p n n+1 n 7: λ ← PERT() L(ω )− λ M, order matrix L . This may cause difﬁcultiesincodeswhich derivs,n n n 8: ω ← ω + HOUSE λ , order n+1 n derivs,n are implemented in a matrix-free fashion, or in which the 9: n ← n + 1 direct and adjoint discrete equations are derived independently 10: end while from the continuous equations, and have different discretiza- 33 d 11: λ ← EIGS L(ω ), M, 0, p n n tions. In our implementation , the elements of the sparse 12: return ω , p , p, λ , n n n matrix L are explicitly stored. Furthermore, since we 13: end function employ a Bubnov-Galerkin ﬁnite elements discretization, the discretization of the continuous adjoint equation is equivalent to the adjoint of the discretized direct equation, and one can Convergence properties of Householder’s methods. By con- † H ′ show that L = L . As always for mappings obtained struction, all Houselder’smethods have f = 0at the ﬁxed from Newton’s method, the mapping deﬁned by (25) has points, implying that all eigenvalues are attractors and have super-linear convergence, i.e. f = 0. This guarantees that a non-empty basin of attraction. When considering simple all thermoacosutic eigenvalues are attractors for this iter- roots, the mth order Householder’s method has a conver- ation method, and, provided sufﬁciently good guesses are gence rate of m + 1, meaning that a lower number of itera- provided, they can always be identiﬁed. This is formally tions in generally required to identify a solution. Yet, these shown in the Appendix. methods are generally not applied to scalar equations because the increase in function evaluations per iteration outweighs the improved convergence rate when the order is increased. This limitation however does not straightfor- Householder’s methods wardly apply to large eigenvalue problems. This is because Newton’s method can be generalized by considering one needs to weight the computational effort needed for the higher-order expansions of the relation λ = λ(ω). The solution of a linear eigenvalue problem at each step – ruled resulting class of iterative root-ﬁnding methods are by the dimension of the problem only – against the one known as Householder’s methods. At an arbitrary order needed to perform all the matrix-vector products needed – 38 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 3. Convergence and basins of attractions for the Householder’s methods from ﬁrst order (top, also known as Newton’s method) to ﬁfth order (bottom). The areas in orange, purple, blue and green indicate, respectively, convergence to the thermoacoustic modes number 1, 2, 3 and 4, as listed in Table 2. Modes 1 and 3 are of acoustic origin, 2 and 4 of ITA origin. The lighter the color, the less steps are needed for convergence. Gray indicates convergence to an eigenvalue outside of the considered domain. Black indicates that the algorithm does not converge with these initial conditions. Mensah et al. 39 which depends on the problem size, but also signiﬁcantly steps are generally needed to converge to the desired increases at each perturbation order. Thus, for adjoint- accuracy, as can be seen from the fact that the higher is based perturbation theory applied on large matrices, compu- the perturbation order, the brighter is its ﬁgure. This tational optimality is ruled by a non-trivial trade-off does not necessarily mean a faster convergence time, but between the number of iterations needed to converge it demonstrates the increase in convergence rate, Eq. and the cost of the perturbation method at the chosen (21), with an increase in the order of the method m. order. Numerical experiments have shown (details not Another notable feature of the Householder’s methods is reported here) that the Householder’s method of order 3 that the size of the basins of attraction of the ITA modes has, on average, optimal performances (minimum time (purple and green) grows when the order of the method and memory used) for eigenvalue problems that have a is increased. This implies that, when using a (say) fourth few thousands degrees of freedom. order Householder scheme (fourth row in Figure 3), a coarse grid of initial guesses is sufﬁcient to identify all the eigenvalues of interest. On the other hand for the Degenerate eigenvalues. Special care needs to be taken if the Newton method (top row in Figure 3), although the eigenvalue searched for is a multiple root of the character- basins of attraction of modes of ITA origin are non-empty, istic function of L, i.e., if the eigenvalue is degenerate. For they have a small size. A ﬁner mesh of initial guesses is degenerate semi-simple eigenvalues perturbation theory 32,38 therefore required if the eigenvalues are randomly can still be used to compute the necessary derivatives. searched. Higher-order Householder’s scheme suffer, To retain the high convergence rate of Householder’s however, from the existence of larger portions of the methods, the scheme must be applied to the ath root of λ, complex plane within which the method does not con- where that a is the algebraic multiplicity of the eigenvalue. verge to any eigenvalues (in black). The mth order expansion reads All these considerations suggest that a third order {m−1} a Householder scheme is on average the optimal choice, as 1/λ (ω ) ω = ω + m . (29) it provides both faster computational times and possess n+1 n {m} 1/λ (ω ) good convergence properties. To conclude, we remark that the convergence maps This however requires apriori knowledge on the multiplicity show fractal patterns at the boundaries between different of an eigenvalue. In thermoacoustic applications, degenera- basins of attraction. The fractal nature of the boundaries cies with multiplicity 2 are typically induced by symmetries of attraction (sometimes referred to as Julia sets) is a of the system and can be often predicted with intuition known feature of Newton-like iteration maps already and/or the use of symmetry breaking criteria. In most when applied to scalar equations. However, we note cases, multiplicities can be removed from the problem by a how some of the boundaries in Figure 3 are non-fractal. proper symmetry reduction scheme, such as reduction to This can be attributed to the fact that we apply Newton/ Bloch-periodic unit cells. Lastly, in case of defective eigen- Householder’s methods to eigenvalue problems. Our values, the adjoint-based schemes presented here cannot be adjoint-based algorithms require the identiﬁcation of the applied, because defective eigenvalues are not analytic in smallest eigenvalue of a linear eigenvalues problem to their parameters. However, defective eigenvalues can only the initial guess (see Alg. 3 and 4). This choice becomes appear as isolated points in the parametrized spectrum of ambiguous when the closest eigenvalue of the linear eigen- L and are, thus, not considered generic. value problem (24) becomes degenerate. This is what happens at non-fractal boundaries, and the clear-cut between convergence to two different eigenvalues when Convergence examples using very close guesses is explained by the fact that our Figure 3 shows the eigenvalue convergence maps when solution method suddenly tracks a different eigenvalue using Householder’s methods from ﬁrst order (top) to branch of the linear eigenvalue problem. ﬁfth order (bottom). The tolerance for convergence is set −10 to tol = 10 for all cases. The convergence properties have signiﬁcantly improved compared to those of the Conclusions ﬁxed-point methods shown in Figure 2. Indeed, for all considered orders of the Houselder’s method, all the In this study, we have investigated the convergence proper- four thermoacoustic eigenvalues in the considered ties of the most common methods used to identify eigenva- portion of the complex plane can be found. In fact, as pre- lues in thermoacoustic systems by means of iterative dicted by Banach’s theorem for this mapping each mode methods. By exploiting adjoint-based sensitivity, we have possesses a non-empty basin of attraction in its vicinity derived explicit equations that can be used to assess the con- – recall that f = 0. Furthermore, the higher the order of traction properties of various algorithms in the vicinity of the considered Householder iteration, the fewer iteration eigenvalues. Thanks to Banach’s ﬁxed-point theorem, this 40 International Journal of Spray and Combustion Dynamics 14(1-2) interest, since these are often observed in experiments and for information is sufﬁcient to know if a given algorithm can or these the thermoacoustic Helmholtz equation (1) is valid. cannot converge to an eigenvalue. c. There might also be ω dependent boundary conditions, as dis- For ﬁxed-point iterations, which are commonly employed cussed in. This would not alter the following discussion on the in thermoacoustics, we found that not all eigenvalues are convergence properties of the method. attractors. In particular, numerical calculations on a Rijke d. The code is open-source and is implemented in the Julia package tube system showed that thermoacoustic modes of ITA WavesAndEigenvalues (https://github.com/JulHoltzDevelopers/ origin are repellors. This implies that these eigenvalues WavesAndEigenvalues.jl). can never be identiﬁed by ﬁxed-point algorithms, regard- less on the accuracy of the initial guess. This may be References linked to the late discovery of ITA modes by the thermoa- 1. Dowling AP and Stow SR. Acoustic Analysis of coustic community. Gas Turbine Combustors. J Propuls Power 2003; 19(5): We then discussed how Newton-like iterative methods 751–764. should always be preferred to ﬁxed-point iterations. This 2. Crocco L. 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Accounting for complex ﬂow-acoustic interactions in a 3D thermo-acoustic Helmholtz solver. PhD Thesis, Université By evaluating the sensitivity of the mapping f deﬁned in de Toulouse, 2017. Eq. (25) we can quantify the convergence properties of 26. Juniper MP. Sensitivity Analysis of Thermoacoustic Instability this method. We have with Adjoint Helmholtz Solvers. Physical Review Fluids ′ 2 ′′ ′ 2 2018; 3: 110509. (43 pages). DOI: 10.1103/PhysRevFluids.3. df (λ ) −λ λ (λ ) = 1 − = 1 − = 0, (30) ′ 2 ′ 2 dω (λ ) (λ ) 27. Greenbaum A, Li RC and Overton ML. First-order where in the second step we have used the fact that, for any Perturbation Theory for Eigenvalues and Eigenvectors. thermoacoustic solution, λ = 0. This proves that the Banach SIAM Rev 2020; 62(2): 463–482. condition (13) is satisﬁed for all thermoacoustic eigenva- 28. Devaney RL, Branner B, Keen L, Douady A, Blanchard P, lues. Thus, provided a sufﬁciently good guess, this Hubbard JH and Schleicher D. Complex Dynamical method converges towards the closest eigenvalue at a Systems: The Mathematics Behind the Mandelbrot and super-linear rate, and is able to identify all eigenvalues in Julia Sets. Providence, USA: American Mathematical Society, 1994. DOI: 10.1090/psapm/049. the spectrum of the thermoacoustic operator. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Spray and Combustion Dynamics SAGE http://www.deepdyve.com/lp/sage/iterative-solvers-for-the-thermoacoustic-nonlinear-eigenvalue-problem-70hpPq3j6h

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ISSN: 1756-8277
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DOI: 10.1177/17568277221084464
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Abstract

The spectrum of the thermoacoustic operator is governed by a nonlinear eigenvalue problem. A few different strategies have been proposed by the thermoacoustic community to tackle it and identify the frequencies and growth rates of ther- moacoustic eigenmodes. These strategies typically require the use of iterative algorithms, which need an initial guess and are not necessarily guaranteed to converge to an eigenvalue. A quantitative comparison between the convergence prop- erties of these methods has however never been addressed. By using adjoint-based sensitivity, in this study we derive an explicit formula that can be used to quantify the behaviour of an iterative method in the vicinity of an eigenvalue. In par- ticular, we employ Banach’s ﬁxed-point theorem to demonstrate that there exist thermoacoustic eigenvalues that cannot be identiﬁed by some of the iterative methods proposed in the literature, in particular ﬁxed-point iterations, regardless of the accuracy of the initial guess provided. We then introduce a family of iterative methods known as Householder’s meth- ods, of which Newton’s method is a special case. The coefﬁcients needed to use these methods are explicitly derived by means of high-order adjoint-based perturbation theory. We demonstrate how these methods are always guaranteed to converge to the closest eigenvalue, provided that the initial guess is accurate enough. We also show numerically how the basin of attraction of the eigenvalues varies with the order of the employed Householder’s method. Keywords Thermoacoustics, nonlinear eigenvalue problem, ITA, basin of attraction solve generalized linear eigenvalue problems, for which Introduction L(ω) = (X − ωY), solving an eigenvalue problem that is Thermoacoustic instabilities can be assessed by solving the 4–6 nonlinear in the eigenvalue ω is intrinsically harder. A inhomogeneous Helmholtz equation typical approach is to transform the NLEVP into a (series 2 2 2 2 −iωτ of) associated eigenvalue problems linear in the eigenvalue. ∇ · c ∇p ˆ + ω p ˆ = (c − c )n(x)e ∇p ˆ (1) 2 1 x ref For example: on a prescribed geometry with appropriate boundary con- ditions. In Eq. (1) ˆp indicates the complex-valued amp- � NLEVPs that are polynomial in ω with order K can be litude of the Fourier transform of the pressure ﬂuctuations recast into linear eigenvalue problems of dimension (the eigenfunction) and c the speed of sound, with the sub- KN ; scripts and indicating the regions upstream and down- � Solutions of NLEVPs can be found by iteratively 1 2 stream the ﬂame, respectively. The interaction index n, solving linear eigenvalue problems resulting from the which is non-zero only over an acoustically compact expansion of L(ω) (to any desired order) in the 2,3 volume, and the time delay τ model the ﬂame response. eigenvalue ; Equation (1) deﬁnes a nonlinear eigenvalue problem (NLEVP) in the complex frequency ω. Once discretized, Institute of Fluid Dynamics and Technical Acoustics, TU Berlin, DE e.g. by means of ﬁnite elements, the NLEVP can be Department of Energy and Process Engineering, NTNU Trondheim, NO expressed in compact form as Corresponding author: L(ω) p = 0, (2) Alessandro Orchini, Institute of Fluid Dynamics and Technical Acoustics, where L is an N × N large, sparse matrix depending non- TU Berlin, DE. linearly on ω. Although there exist efﬁcient algorithms to Email: a.orchini@tu-berlin.de Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https:// creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as speciﬁed on the SAGE and Open Access page (https://us.sagepub.com/en-us/nam/open-access-at-sage). Mensah et al. 31 � Contour integration methods reduce the NLEVP to a with respect to these solution methods, and how the linear eigenvalue problem possessing only the eigen- basins of attraction of each eigenvalue vary depending values of L inside a given contour in the complex on the order of the Householder’s method used. 7,8 plane. The thermoacoustic nonlinear eigenvalue These approaches exploit the fact that efﬁcient and robust problem methods (e.g. Arnoldi) exist for large scale linear eigenvalue 9,10 problems. A common feature of iterative methods is that The weak formulation and discretization of the thermoa- they all require an initial guess ω for the eigenvalue , which coustic Helmholtz equation (1) results in an N-dimensional is updated at every iteration. Eigenvalues of the thermoa- NLEVP that reads coustic operator are ﬁxed points of the mapping deﬁned L(ω)p = K + ωC + ω M + Q(ω) p = 0. (3) by the chosen iterative algorithm. According to Banach’s ﬁxed-point theorem p. 152, for The matrices in (3) arise from discretizing the operators in an iterative algorithm to be able to identify an eigenvalue, equation (1), viz. the stiffness operator ∇ · (c ∇(·)) 7! K, the mapping deﬁned by the algorithm needs to be contracting 2 2 the mass operator ω (·) 7! ω M, and the ﬂame operator in the vicinity of the eigenvalue. This means that, provided 2 2 −iωτ −(c − c )n(x)e ∇| (·) 7! Q(ω). The matrix C results 2 1 x that the initial guess is sufﬁciently close to an eigenvalue, ref from the discretization of the boundary conditions needed at each iteration the algorithm will move the guess closer to close the problem. They can be expressed in terms of to the eigenvalue, until a prescribed tolerance is reached. the acoustic impedance Z on all boundaries On the other hand, if the mapping is repelling in the vicinity of an eigenvalue, it will not be possible to identify this eigen- iωp ˆ + Z(ω)∇p ˆ · n ˆ = 0, (4) value with the chosen algorithm, since a sufﬁciently close guess will be pushed further away from the solution. An where n ˆ is a unit vector normal to the boundary. overlooked eigenvalue can have serious consequences in Equation (3) represents a nonlinear eigenvalue problem. thermoacoustics, as the reliable and accurate determination The challenge is to ﬁnd all its eigenvalues ω – and their cor- of all relevant eigenvalues is of paramount importance to responding eigenvectors p – in a prescribed portion of the ensure the safe operability of an engine. complex plane. The ﬁxed-point iteration method described in is a com- monly used algorithm for identifying eigenvalues of the 12–17 thermoacoustic Helmholtz operator – it is used in to Rijke tube test-case name a few. However, with the exception of a short discus- sion in, there is no reference in the literature investigating Throughout this study we will employ the classic Rijke tube the convergence properties of this method in relation to the conﬁguration to demonstrate our results. It consists of a spectrum of the thermoacoustic operator. The aim of this straight duct in which a ﬂame is located. Across the ﬂame work is to quantify the convergence properties of ﬁxed- the temperature, and hence the speed of sound, rise point iteration methods that are commonly used in thermo- abruptly. The axial ﬂame location x is chosen to be in acoustics. It will be shown that some thermoacoustic eigen- the middle of the tube. The parameters that determine the values cannot be found using a ﬁxed-point algorithm, acoustic response of the system are chosen to be identical regardless of the chosen initial guess. We will then intro- to those used in. By non-dimensionalizing all quantities duce an alternative adjoint-based iterative method that has using the tube length L as a characteristic length and the more robust convergence properties and is thus better speed of sound c in the cold section of the duct as a char- suited to tackle the thermoacoustic problem. acteristic velocity, the chosen Rijke tube’s parameter are This study is organized as follows. First the thermoa- x = 0.5, c(x) = 1 for x ≤ x and c(x) = 2 otherwise. The f f coustic problem and the ﬁxed-point iteration presented boundary conditions are chosen to be acoustically closed in are introduced. An explicit formula that quantiﬁes (∇ ˆp = 0) and opened ( ˆp = 0) at the inlet (x = 0) and the convergence properties of the ﬁxed-point mapping outlet (x = 1), respectively. The ﬂame model parameters is derived. The same procedure is conducted for a are chosen to be n = 1/3 and τ = 2. so-called Picard iteration, which is another form of ﬁxed- For the range of frequencies that we will consider all point iteration. Both methods are applied to a generic transverse modes are cut-off. The problem can therefore be thermoacoustic test case to demonstrate that some considered as one-dimensional, and the ﬂame to be a point eigenvalues are repellors with respect to these mappings. source in the Helmholtz equation (1), n(x) = nδ(x − x ). We will then introduce a class of adjoint-based solution The eigenvalues of this problem can be calculated semi- methods known as Householder, of which Newton’s analytically by using an acoustic network approach. By method is a special case. It will be discussed how all assuming one-dimensional acoustics, equation (1) can be eigenvalues of the thermoacoustic operator are attractors expressed in terms of the Riemann invariants f and g. 32 International Journal of Spray and Combustion Dynamics 14(1-2) Table 1. Eigenvalues of the Rijke tube network model (5) in the quadratic in ω, except for the heat release term. The ﬂame considered portion of the complex plane together with the operator can therefore be thought of as a perturbation of the associated acoustic (no ﬂame) and intrinsic (anechoic boundary underlying purely acoustic problem, which is obtained with conditions) eigenvalues from which they stem. Q = 0. This motivates the use of an iterative strategy, starting from the (known) purely acoustic solutions. ωω ω ac ITA The key idea is to recast the problem into a linear eigen- 1 2.396–0.262i 2.462+0i – value problem of doubled dimension, and then to iteratively 2 4.692+0.304i – 4.712+0i identify a thermoacoustic eigenvalue starting from an initial 3 6.283+0i 6.283+0i – guess. We shall indicate this linear eigenvalue problem as 4 7.874+0.304i – 7.854+0i L (ω ; ω) p = 0. It reads FP n With suitable matching conditions at the ﬂame zone and for 0 −I p I0 p =−ω . (10) the parameters introduced above, the acoustic-network K + Q(ω ) C p 0M p I I eigenvalue problem reads X Y FP FP g g 0 On the left hand side, the nonlinear dependency of the 1 1 L = L − L = , (5) 1D ac fl f f 0 problem on the eigenvalue has been removed by freezing 2 2 the evaluation of the ﬂame operator Q, i.e, by evaluating where L is the acoustic response in the absence of an active ac at ω , the eigenvalue guess. This guess is updated by itera- ﬂame tively replacing ω with the closest eigenvalue ω of (10), −iω −iω/2 which can be found by means of, e.g., the Arnoldi iteration e + 1 e − 1 L ≡ , (6) ac −iω −iω/2 method. The eigenvector in (10) has been extended into 1 − e 2e + 2 ˜p = [ p, ω p]. (11) and L provides the feedback between heat release rate ﬂuc- fl This can be seen from the ﬁrst matrix equation deﬁned by tuations and the acoustics (10), which reads p = ω p. By using this relation, the second matrix equation of (10) reduces to (3) when L ≡ . (7) fl −2iω −iω e e − 1 0 ω = ω. The algorithm terminates successfully if the differ- ence between the results of two successive iterations is less By imposing det(L ) = 0 the eigenvalues of the Rijke tube 1D than a prescribed tolerance tol , or is aborted if a predeﬁned can be obtained by using standard root-ﬁnder methods. We maximum number of iterations is reached. shall focus our analysis on the properties of the four thermo- acoustic eigenvalues found in the region S of the complex Algorithm 1 Nicoud’s algorithm plane delimited by 1: function ITERATE(ω , p , tol , maxiter, K, C, M, Q) 1 ω 2: ω ← ∞ S= { ω ∣ ℜ(ω) ∈ [0, 10] ∧ ℑ(ω) ∈ [ − 1, 1] }, (8) 0 3: n ← 1 which are reported in Table 1. Two of these eigenvalues can 4: p ← ω p be linked to acoustic modes – whose eigenvalues ω can be ac I0 calculated by setting n = 0, which implies L = 0 – while 5: Y ← fl FP 0M the other two can be linked to intrinsic thermoacoustic 6: while |ω − ω | > tol and n < maxiter do 20,21 n n−1 ω (ITA) modes – whose frequencies are given by 0 −I 7: X ← FP K + Q(ω ) C 2k + 1 1 c n ω = π − i log − 1 n . (9) ITA ˜ ˜ 8: ω , p ← EIGS −X , Y , ω , p n+1 FP FP n n+1 n τ τ c 9: n ← n + 1 10: end while 11: p ← p [1 : N] In the following, we will discretize by ﬁnite elements the n n 12: return ω , p , n thermoacoustic eigenproblem (3) and use various iterative n 13: end function solution techniques to identify its eigenvalues. The semi- analytical wave-based solutions presented in Table 1 will The pseudocode for the ﬁxed-point iterative scheme serve as benchmarks. The dimensions of the resulting is outlined in Alg. 1. We have denoted with matrix operator L do not allow for the use of determinant- EIGS(X, Y, ω , p ) the Arnoldi method that identiﬁes the based methods, and alternative iterative strategies are needed. eigenpair ω, p of the linear eigenvalue problem X p = ωY p closest to ω . An initial guess for the eigenvector State-of-the-art iterative solution method p , if known, may also be provided to the algorithm, in A ﬁxed-point iteration to solve the NLEVP (3) was proposed order to improve the convergence of the Arnoldi method. in. This algorithm exploits the fact that the problem is Also note that the matrix M is always positive-deﬁnite – as Mensah et al. 33 H ∗ H † it deﬁnes a mass matrix – and hence is Y . Thus, the eigen- FP deﬁned by (14) shows that p = (C + ω M ) p ,so that ˜ † value problem (X (ω ) − ωY ) p = 0 appearing at each FP n FP the extended adjoint eigenvector ˜p satisﬁes iteration, can be solved efﬁciently. † H ∗ H † † ˜p = C + ω M p , p . (15) Fixed-point iteration schemes in thermoacoustics The ﬁrst matrix equation in (14) deﬁnes the adjoint eigen- † † value problem L (ω) p = 0. A single iteration of the procedure described in Alg. 1 From adjoint-based sensitivity analysis, the following deﬁnes a mapping ω = f (ω ) for which ﬁxed-points ω n+1 n formula is known for evaluating the derivative of f with are sought. This ﬁxed-point map f can be explicitly 22,23 respect to ω : expressed by rewriting the eigenvalue problem (10) as L (f ; ω ) ˜p = X (ω ) + fY ˜p = 0. (12) FP n FP n FP H ∂L FP ˜p ˜p df ′ ∂ω A condition for the convergence of the mapping to an f = =− (16) H ∂L dω FP eigenvalue of the thermoacoustic operator is provided by ˜ ˜ p p ∂f Banach’s ﬁxed-point theorem, which we brieﬂy recall. Consider a bounded domain Γ ⊂ C that contains the ﬁxed- By combining the known information on the direct and point ω. Banach’s ﬁxed-point theorem guarantees that f has adjoint eigenvectors – Eqs. (11) and (15), respectively – one (unique) ﬁxed-point in Γ as long as f is a contraction as well as on the functional shape of L – Eqs. (12) and FP everywhere in Γ. (10) – we have For our purposes, we are only interested at the behaviour of the iterative map at the ﬁxed-point. In particular, to know H∂L H FP † † ′ †H ′ if an eigenvalue ω is an attractor of a given iterative method ˜p ˜p = ˜p X ˜p = p Q (ω) p (17a) FP ∂ω f it is sufﬁcient to verify that H∂L H FP † † †H ˜ ˜ ˜ ˜ p p = p Y p = p() C + 2ωM p (17b) |f (ω)| < 1. (13) FP ∂f This in fact implies that the mapping f represents a contrac- tion in the vicinity of ω. Provided that Eq. (13) is satisﬁed, This yields a closed-form expression for the sensitivity of the eigenvalue ω can be found if the initial guess lies sufﬁ- the mapping f as required to apply Banach’s theorem: ciently close to it. 3 H † ′ Nicoud recognized the importance of the contraction p Q (ω) p f (ω) = . (18) FP properties of f , but stated that “[ …] obtaining general †H p() C + 2ωM p results about the contracting properties of the operator f from physical arguments is out of reach of the current under- The mapping sensitivity (18) is generally expected to be standing of the thermoacoustic instabilities.” Signiﬁcant pro- non-zero. This implies that, if the ﬁxed-point iteration con- gress has been made in this direction by the thermoacoustic verges, |f | < 1, then it possesses a linear convergence rate – community in recent years, and tools are now available to see Eq. (21). quantify the behaviour of the mapping f from the thermoa- coustic equations. In particular, the contraction properties of the mapping f can be quantiﬁed by exploiting adjoint-based An alternative ﬁxed-point iteration scheme 22,23 sensitivity. In the following, we use adjoint methods to Thechoiceofa ﬁxed-point iteration method is not unique. derive an analytical expression that allows us to compute One of the disadvantages of the solution method proposed |f (ω)| and thus make explicit use of Banach’stheorem. in is that it casts the original problem in an eigenvalue problem having doubled dimensions, which is computa- tionally more expensive to solve. Alternatively, one can Contraction properties of the ﬁxed-point iteration perform an iteration in the ω term of Eq. (3), while the The adjoint equation associated with the mapping (12) reads: remaining occurrences of the eigenvalues are ﬁxed to ω . The square root of the eigenvalues of the resulting L (f ; ω) ˜p = FP (linear) eigenvalue problem are then an approximation of H H † 0K + Q (ω) I0 p the sought eigenvalues. This choice is convenient since ∗ I + f = 0, H H the matrix M that appears on the r.h.s. of the resulting gen- 0M −IC eralised eigenvalue problem is positive deﬁnite, allowing (14) for the use of robust linear eigenvalue solvers. We shall where f denotes the complex conjugate of f.Its ﬁxed-points refer to this solution scheme, described in Alg. 2, as ∗ ∗ are found when f = ω . The second matrix equation Picard iteration. p. 154 34 International Journal of Spray and Combustion Dynamics 14(1-2) Algorithm 2 Picard iteration Table 2. Eigenvalues identiﬁed by means of ﬁnite elements and values of the contracting operator |f |. Only the eigenvalues for 1: function ITERATE(ω , p , tol , maxiter, K, C, M, Q) 1 ω which |f | < 1 can be identiﬁed, provided that an accurate initial 2: ω ← ∞ guess is known. 3: n ← 1 4: while |ω − ω | > tol and n < maxiter do ′ ′ n n−1 ω ω |f||f | FP PC 5: X ← K + ω C + Q(ω ) n n 6: ω , p ← EIGS −X, M, ω , p 1 2.396−0.260i 0.519 0.519 n+1 n n+1 n 7: ω ← ω 2 4.696+0.297i 17.025 17.025 n+1 n+1 8: n ← n + 1 3 6.300−0.000i 0.022 0.022 9: end while 4 7.880+0.316i 12.204 12.204 10: return ω , p , n 11: end function The Picard iteration deﬁnes the mapping f given by ⎛ ⎞ ⎝ ⎠ L ( f ; ω ) p = K + ω C + Q(ω ) +f M p = 0. PC n n n X (ω ) PC n (19) By invoking the associated adjoint eigenvalue problem, and by using the adjoint sensitivity Eq. (16), one can show that the sensitivity of the Picard iteration reads † ′ p (Q (ω) + C) p Figure 1. Modeshapes of the four eigenvalues in the considered f (ω) = . (20) PC H portion of the complex plane. The colors orange, purple, blue p 2ωM p and green indicate, respectively, the thermoacoustic modes number 1, 2, 3 and 4, as listed in Table 2. Modes 1 and 3 are of Notably, the only difference with respect to the sensitivity 3 acoustic origin, 2 and 4 of ITA origin (see Table 1). of the ﬁxed-point iteration proposed by , Eq. (17), is that the matrix C is moved to the numerator. However, for the simple boundary conditions considered in this study, one ﬁnds that the matrix C has non-zero elements only at the iterative method, described in the next section. This is open-end boundary, where both p and p vanish, so that because, as we will now show, ﬁxed-point methods fail in the contribution of the term p Cp vanishes. identifying some eigenvalues. The modeshapes associated This is a general result. When considering non-trivial with the four eigenvalues are shown in Figure 1. boundary conditions, the convergence properties of the The sensitivities that the mapping f takes at these solu- two proposed ﬁxed-point solution methods are going to tions in Table 2 are then evaluated according to Eq. (18). differ, and one cannot draw general conclusions on which ′ By comparison with Table 1 one ﬁnds that |f | is less method will be able to identify more eigenvalues a priori. than one for thermoacoustic modes of acoustic origin, On the other hand, for trivial (open and/or closed) boundary whereas it is greater than one for modes of ITA origin. conditions one can show that p Cp = 0. Therefore, when This implies that modes of ITA origin cannot be identiﬁed considering geometries with simple open or closed bound- with this iterative scheme. Indeed, even if a good initial esti- ary conditions, the Picard iteration method should be pre- ′ mate is known for those, since |f | > 1 the iterative algo- ferred as a ﬁxed-point iteration scheme since (i) it is rithm will repel the initial guess from the actual solution. computationally cheaper and (ii) has the same contraction This fact might be partly responsible for the late discovery properties at the eigenvalues. of modes of ITA origin in thermoacoustics, in particular for 21,24 annular combustion chambers, for which larger non- linear eigenvalue problems need to be solved. Convergence properties of ﬁxed-point iterative Although a formal proof is not available, the difﬁculties methods that the ﬁxed-point iterations have in converging towards Table 2 highlights the contraction properties of the discussed some ITA modes can be intuitively explained by the follow- ﬁxed-point algorithm for the four eigenvalues of interest that ing argument. The matrix in the numerator of (18) is propor- the problem has when discretized using an equidistant linear tional to the ﬂame delay, exp (−iωτ). Modes of ITA origin ﬁnite elements mesh with 127 degrees of freedom. These tend to have largely negative growth rates, ℑ[ω] ≫ 0, eigenvalues have been calculated using a higher-order implying that | exp (−iωτ)| ≫ 1. Additionally, the Mensah et al. 35 eigenvectors of modes of ITA origin (and their adjoint) have iterations is exceeded – the algorithm does not converge a strong magnitude at the ﬂame – see Figure 1 – exactly the with this initial condition to the desired tolerance. The zone in which the ﬂame response Q is large, due to a strong results are shown in Fig. 2. The axes indicate the value gradient in the pressure modeshapes at the reference pos- taken by the initial guess, and the color scheme indicates ition. In combination, these two effects yield a large value to which eigenvalue the initial guess has converged and † ′ of the term p Q (ω) p, appearing in the numerator of how many steps it took. The actual solutions are highlighted (18). Thus, |f | ≫ 1 can be expected for ITA modes. This with white markers. statement is not expected to hold generally, but only Brute-force numerical calculations conﬁrm that the serves to explain the different convergence behaviours of modes of ITA origin, #2 and #4 in Table 1, cannot be modes of intrinsic and acoustic origin for this speciﬁc found by using ﬁxed-point iterative methods, even for case. Indeed, per Eq. (18), a different choice of boundary guesses that start very close to the actual solutions. The conditions – via complex-valued impedances in C or modes of acoustic origin, instead, can be identiﬁed. The more terms associated with different boundaries – can portions of the complex plane that, when used as an decrease the value of |f | below unity. When this is the initial guess, converge to modes #1 and #3, are highlighted case, then also intrinsic modes could be computed with a in orange and blue, respectively. Notably, convergence of ﬁxed-point iteration. mode #1 always requires a number of iterations much To demonstrate that the ﬁxed-point iterations cannot larger than those needed for mode #3 to converge. This identify modes for which |f | > 1, we perform a large is in line with the fact that the sensitivity of mode #1, number of brute-force calculations in the complex plane. although less than 1, is an order of magnitude larger We initialize the ﬁxed-point algorithm with initial than that of mode #3. More quantitatively, when guesses ω in the region S deﬁned in (8) sampled on a 0 < |f | < 1, in the vicinity of a solution the number of uniform 1001 × 201 grid. The eigenvectors for the steps N needed to improve the accuracy by an order of Krylov-subspace methods were initialized with the magnitude can be estimated by N =⌈1/ log (1/|f |)⌉. p = [1, 1, .. . , 1] . We stop the algorithm when a toler- From the values reported in Table 2, this yields N = 4 −10 ance tol ≡ |ω − ω | < 10 is reached – an eigen- for mode #1 and N = 1 for mode #3, i.e., the former ω n n−1 s value has converged – or when a maximum number of 16 requires approximately 4 times more iterations than the Figure 2. Convergence and basins of attractions for (top) the ﬁxed-point iteration described in , Alg. 1, and (bottom) the Picard iteration described in this study, Alg. 2. The areas in orange and blue indicate, respectively, the basins of attraction of the thermoacoustic modes #1 and #3, as listed in Table 2. The basin of attraction of mode #1 is hatched with black because, although the algorithm is slowly converging, the convergence rate is very low, and it takes more than 16 iterations to identify the eigenvalue to the −10 prescribed accuracy of 10 . Both methods are unable to identify modes number 2 and 4, of ITA origin. The lighter the color, the less steps are needed for convergence. Gray-black hatched surfaces indicate (slow) convergence towards an eigenvalue outside of the considered domain. Yellow indicates convergence to a (spurious) eigenvalue with ω = 0. 36 International Journal of Spray and Combustion Dynamics 14(1-2) latter to converge to a predeﬁned tolerance. Within the method to det[L(ω)] = 0 leads to required 16 iterations limit, the eigenvalue #1 can be iden- −6 tiﬁed with a tolerance of 10 . A larger number of itera- det[L(ω)] ω = ω − . (22) n+1 n tions is needed to converge to the prescribed tolerance d(det[L(ω)])/dω ω=ω −10 tol = 10 , emphasized by the presence of black stripes in Figure 2. Lastly, in yellow we have highlighted Evaluating the determinant is however a demanding and the portion of the complex plane that converges to the ω = error-prone operation. A determinant-free Newton scheme 0 eigenvalue. This is always a solution of the thermoa- was suggested in, which exploits Jacobi’s formula coustic weak form (3), but it represents a spurious solution that does not satisfy the boundary conditions, except when det[L(ω)] 1 = . (23) −1 all boundaries are acoustically closed. This spurious solu- d(det[L(ω)])/dω tr L (ω)dL(ω)/dω tion arises from manipulation of the impedance boundary condition equations when deriving the weak form (3). The Although this scheme is more robust than a direct evaluation ﬁxed-point algorithm proposed by and summarised in of the determinant, the evaluation of the inverse of the matrix Alg. 1 strongly promotes convergence towards this spuri- operator L becomes prohibitively expensive when L has hun- ous eigenvalue because it separates the action of the dreds of thousand of degrees of freedom, as is typical in real- boundary matrix, C, from that of the stiffness and ﬂame world thermoacoustic calculations. matrices, K and Q, as can be seen in Eq. (10). To bypass these issues, in the following section we will Together with the higher dimensions of the linearized discuss how adjoint-based methods allow us to use eigenvalue problem, the promotion of the convergence Newton’s method directly on the NLEVP equations. This towards a spurious eigenvalue is another disadvantage of will not require the evaluation of determinants nor matrix the method of in contrast to the Picard iteration (19). inverses, and provides more accurate information on the Nonetheless, the slow convergence of some modes and mapping sensitivity than ﬁnite difference approximations. the non-convergence of other modes for both ﬁxed-point iterations are inherently linked to their linear convergence rate. To overcome this limitations, we will now consider Adjoint-based Newton iteration methods with a super-linear convergence rate. By introducing an auxiliary variable λ, the NLEVP (2) can be written as L(ω) p = λY p. (24) Newton-like iteration methods A sequence generated from an iterative algorithm is said to Equation (24) can be thought of as a generalized eigenvalue have a rate of convergence m if there exists a bound G > 0 problem with eigenvalue λ depending on a parameter ω. such that From this viewpoint λ is an implicitly deﬁned function of ω, i.e. λ = λ(ω). If the parameter ω is chosen such that |ω − ω|≤ G|ω − ω| . (21) the eigenvalue problem (24) has an eigenvalue λ = 0, n+1 n then ω (and the corresponding eigenvector p) is a solution of the NLEVP (2). Since the implicitly restarted Arnoldi When the contraction condition (13) is satisﬁed, the iterative algorithm is used to solve the sparse eigenvalue problem ﬁxed-point algorithms described so far converge towards the (24), linear in λ, choosing the matrix Y to be semi-positive closest eigenvalue at a linear rate, m = 1. However, when deﬁnite has signiﬁcant advantages. Possible choices are one has |f (ω)|= 0, the rate of convergence becomes super- Y = I, the identity matrix, or Y = M, the mass matrix. In linear, with m > 1. Importantly, |f (ω)|= 0 always satisﬁes our analysis the latter will be used, since it naturally Banach’s condition (13), implying that super-linear conver- arises when discretizing the continuous thermoacoustic ging algorithms are guaranteed to converge to all eigenva- operator by means of ﬁnite elements. The following discus- lues. Super-linear convergence is a feature of Newton’s sion on the convergence of the method is however inde- method, which is therefore more robust than ﬁxed-point pendent from the speciﬁc form chosen for Y. methods (0th order), since it uses gradient information (1st To ﬁnd values of ω for which (24) has a zero eigenvalue order). Gradient information may be included also in a ﬁxed- in λ, we seek the roots of the implicit relation λ = λ(ω). point iteration by introducing a relaxation parameter Newton’s iteration on this relation reads Appenidx C.4. This procedure is however based on ﬁnite difference approximations and requires solving two eigen- ω = ω − λ(ω )/λ (ω ) = f (ω ), (25) n+1 n n n n value problems at each iteration. A direct application of Newton’s method is preferable. which requires the sensitivity of the eigenvalue with respect Since the eigenvalues of (3) are found when the deter- to the parameter, λ = dλ/dω. This sensitivity can be minant of L vanishes, a direct application of Newton’s obtained by means of adjoint methods. Starting from Eq. Mensah et al. 37 (24) it can be shown that the sensitivity of a linear eigen- m > 0 the iterative Householder step reads 22,23,27 value problem is {m−1} 1/λ (ω ) ω = ω + m . (27) † ′ n+1 n {m} p L (ω ) p 1/λ (ω ) λ (ω ) = . (26) n p Y p At ﬁrst order (m = 1), Eq. (27) corresponds to the Newton By using this expression, one can iteratively update the par- iteration (25). At second order (m = 2) one obtains ameter ω in (25) until an eigenvalue λ = 0 is found. 2λ(ω )λ (ω ) n n ω = ω − , (28) n+1 n Algorithm 3 Newton adjoint-based method ′ ′′ 2 λ (ω ) −λ(ω )λ (ω ) n n n 1: function ITERATE(ω , p , p , L, tol , maxiter) 1 ω 1 1 which is also known as Halley’s method. In order to apply 2: ω ← ∞ higher-order Householder’s methods, the evaluation of 3: n ← 1 {m} 4: while |ω − ω | > tol and n < maxiter do n n−1 ω high-order eigenvalue sensitivities λ is required. Once 5: λ , p ← EIGS L(ω ), M, 0, p n n n+1 n again, adjoint-based theory can be exploited in this ∗ H † 6: λ , p ← EIGS L (ω ), M, 0, p regard. By using perturbation theory, explicit formulas for n n n+1 n p L (ω ) p the calculation of arbitrarily high-order sensitivity have n+1 n+1 7: λ ← − n H been derived in, to which we refer the interested reader. p M p n+1 n+1 For the purpose of this study, we shall assume that a func- 8: ω ← ω − n+1 n ′ tion PERT that computes arbitrarily high-order sensitivities is 9: n ← n + 1 available. With this at hand, the pseudocode for the 10: end while Householder iterative method is given in Alg. 4, in which 11: λ ← EIGS L(ω ), M, 0, p ⊳ This line basically computes n n the function HOUSE computes the Householder update to the residual for possible later use ω at the desired order, according to Eq. (27). † n 12: return ω , p , p , λ n n n n n 13: end function Algorithm 4 Householder’s method 1: function ITERATE(ω , L, p , p , tol , maxiter, order) 1 ω 1 1 2: ω ← ∞ The algorithm for the adjoint-based Newton iteration 3: n ← 1 4: while |ω − ω | > tol and n < maxiter do is given in Alg. 3. Note that this method requires n n−1 ω 5: λ , p ← EIGS L(ω ), M, 0, p solving both the direct and the adjoint problem, which n n n+1 n ∗ H requires a numerical implementation of the adjoint 6: λ , p ← EIGS L (ω ), M, 0, p n n+1 n 7: λ ← PERT() L(ω )− λ M, order matrix L . This may cause difﬁcultiesincodeswhich derivs,n n n 8: ω ← ω + HOUSE λ , order n+1 n derivs,n are implemented in a matrix-free fashion, or in which the 9: n ← n + 1 direct and adjoint discrete equations are derived independently 10: end while from the continuous equations, and have different discretiza- 33 d 11: λ ← EIGS L(ω ), M, 0, p n n tions. In our implementation , the elements of the sparse 12: return ω , p , p, λ , n n n matrix L are explicitly stored. Furthermore, since we 13: end function employ a Bubnov-Galerkin ﬁnite elements discretization, the discretization of the continuous adjoint equation is equivalent to the adjoint of the discretized direct equation, and one can Convergence properties of Householder’s methods. By con- † H ′ show that L = L . As always for mappings obtained struction, all Houselder’smethods have f = 0at the ﬁxed from Newton’s method, the mapping deﬁned by (25) has points, implying that all eigenvalues are attractors and have super-linear convergence, i.e. f = 0. This guarantees that a non-empty basin of attraction. When considering simple all thermoacosutic eigenvalues are attractors for this iter- roots, the mth order Householder’s method has a conver- ation method, and, provided sufﬁciently good guesses are gence rate of m + 1, meaning that a lower number of itera- provided, they can always be identiﬁed. This is formally tions in generally required to identify a solution. Yet, these shown in the Appendix. methods are generally not applied to scalar equations because the increase in function evaluations per iteration outweighs the improved convergence rate when the order is increased. This limitation however does not straightfor- Householder’s methods wardly apply to large eigenvalue problems. This is because Newton’s method can be generalized by considering one needs to weight the computational effort needed for the higher-order expansions of the relation λ = λ(ω). The solution of a linear eigenvalue problem at each step – ruled resulting class of iterative root-ﬁnding methods are by the dimension of the problem only – against the one known as Householder’s methods. At an arbitrary order needed to perform all the matrix-vector products needed – 38 International Journal of Spray and Combustion Dynamics 14(1-2) Figure 3. Convergence and basins of attractions for the Householder’s methods from ﬁrst order (top, also known as Newton’s method) to ﬁfth order (bottom). The areas in orange, purple, blue and green indicate, respectively, convergence to the thermoacoustic modes number 1, 2, 3 and 4, as listed in Table 2. Modes 1 and 3 are of acoustic origin, 2 and 4 of ITA origin. The lighter the color, the less steps are needed for convergence. Gray indicates convergence to an eigenvalue outside of the considered domain. Black indicates that the algorithm does not converge with these initial conditions. Mensah et al. 39 which depends on the problem size, but also signiﬁcantly steps are generally needed to converge to the desired increases at each perturbation order. Thus, for adjoint- accuracy, as can be seen from the fact that the higher is based perturbation theory applied on large matrices, compu- the perturbation order, the brighter is its ﬁgure. This tational optimality is ruled by a non-trivial trade-off does not necessarily mean a faster convergence time, but between the number of iterations needed to converge it demonstrates the increase in convergence rate, Eq. and the cost of the perturbation method at the chosen (21), with an increase in the order of the method m. order. Numerical experiments have shown (details not Another notable feature of the Householder’s methods is reported here) that the Householder’s method of order 3 that the size of the basins of attraction of the ITA modes has, on average, optimal performances (minimum time (purple and green) grows when the order of the method and memory used) for eigenvalue problems that have a is increased. This implies that, when using a (say) fourth few thousands degrees of freedom. order Householder scheme (fourth row in Figure 3), a coarse grid of initial guesses is sufﬁcient to identify all the eigenvalues of interest. On the other hand for the Degenerate eigenvalues. Special care needs to be taken if the Newton method (top row in Figure 3), although the eigenvalue searched for is a multiple root of the character- basins of attraction of modes of ITA origin are non-empty, istic function of L, i.e., if the eigenvalue is degenerate. For they have a small size. A ﬁner mesh of initial guesses is degenerate semi-simple eigenvalues perturbation theory 32,38 therefore required if the eigenvalues are randomly can still be used to compute the necessary derivatives. searched. Higher-order Householder’s scheme suffer, To retain the high convergence rate of Householder’s however, from the existence of larger portions of the methods, the scheme must be applied to the ath root of λ, complex plane within which the method does not con- where that a is the algebraic multiplicity of the eigenvalue. verge to any eigenvalues (in black). The mth order expansion reads All these considerations suggest that a third order {m−1} a Householder scheme is on average the optimal choice, as 1/λ (ω ) ω = ω + m . (29) it provides both faster computational times and possess n+1 n {m} 1/λ (ω ) good convergence properties. To conclude, we remark that the convergence maps This however requires apriori knowledge on the multiplicity show fractal patterns at the boundaries between different of an eigenvalue. In thermoacoustic applications, degenera- basins of attraction. The fractal nature of the boundaries cies with multiplicity 2 are typically induced by symmetries of attraction (sometimes referred to as Julia sets) is a of the system and can be often predicted with intuition known feature of Newton-like iteration maps already and/or the use of symmetry breaking criteria. In most when applied to scalar equations. However, we note cases, multiplicities can be removed from the problem by a how some of the boundaries in Figure 3 are non-fractal. proper symmetry reduction scheme, such as reduction to This can be attributed to the fact that we apply Newton/ Bloch-periodic unit cells. Lastly, in case of defective eigen- Householder’s methods to eigenvalue problems. Our values, the adjoint-based schemes presented here cannot be adjoint-based algorithms require the identiﬁcation of the applied, because defective eigenvalues are not analytic in smallest eigenvalue of a linear eigenvalues problem to their parameters. However, defective eigenvalues can only the initial guess (see Alg. 3 and 4). This choice becomes appear as isolated points in the parametrized spectrum of ambiguous when the closest eigenvalue of the linear eigen- L and are, thus, not considered generic. value problem (24) becomes degenerate. This is what happens at non-fractal boundaries, and the clear-cut between convergence to two different eigenvalues when Convergence examples using very close guesses is explained by the fact that our Figure 3 shows the eigenvalue convergence maps when solution method suddenly tracks a different eigenvalue using Householder’s methods from ﬁrst order (top) to branch of the linear eigenvalue problem. ﬁfth order (bottom). The tolerance for convergence is set −10 to tol = 10 for all cases. The convergence properties have signiﬁcantly improved compared to those of the Conclusions ﬁxed-point methods shown in Figure 2. Indeed, for all considered orders of the Houselder’s method, all the In this study, we have investigated the convergence proper- four thermoacoustic eigenvalues in the considered ties of the most common methods used to identify eigenva- portion of the complex plane can be found. In fact, as pre- lues in thermoacoustic systems by means of iterative dicted by Banach’s theorem for this mapping each mode methods. By exploiting adjoint-based sensitivity, we have possesses a non-empty basin of attraction in its vicinity derived explicit equations that can be used to assess the con- – recall that f = 0. Furthermore, the higher the order of traction properties of various algorithms in the vicinity of the considered Householder iteration, the fewer iteration eigenvalues. Thanks to Banach’s ﬁxed-point theorem, this 40 International Journal of Spray and Combustion Dynamics 14(1-2) interest, since these are often observed in experiments and for information is sufﬁcient to know if a given algorithm can or these the thermoacoustic Helmholtz equation (1) is valid. cannot converge to an eigenvalue. c. There might also be ω dependent boundary conditions, as dis- For ﬁxed-point iterations, which are commonly employed cussed in. 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International Journal of Spray and Combustion Dynamics – SAGE

Published: Mar 1, 2022

Keywords: Thermoacoustics; nonlinear eigenvalue problem; ITA; basin of attraction

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Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

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Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

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