Structural topology optimization for repeated eigenvalues with the adjoint
sensitivity analysis
Gil Ho Yoon(1) , Alberto Donoso(2) , José Carlos Bellido(3) ,David Ruiz(4)(1) Hanyang University, Korea, gilho.yoon@gmail.com(2) Universidad de Castilla-La Mancha, Spain(3) Universidad de Castilla-La Mancha, Spain(4) Universidad de Castilla-La Mancha, Spain

Abstract
The sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a challenging problem.
The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can only be computed fora specific eigenvector basis, the so-called adjacent eigenvector basis. These adjacent eigenvectors depend onindividual variables, which makes the eigenvector derivative calculation very elaborate and expensive from acomputational perspective for problems. We present a new method avoiding passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. Some examples, including
one in the topology optimization, are provided to validate the present approach. The present paper is based on
our research [1]Keywords: eigenvector sensitivity repeated eigenvalue topology optimization
1 INTRODUCTION
Eigenproblems, i.e., problems regarding eigenvalues and/or eigenvectors are widely present today in differingresearch fields [1, 2, 3]. Whenever they constitute or form a part of the physics of optimization problems withnumerous design variables, then a computationally efficient sensitivity analysis must be performed to enhance
the performance. Topology optimization problems [4], in which a prescribed amount of a material is distributed
in a design space to optimize an objective function while fulfilling some constraints, need to be particularlyhighlighted. Their discretized forms are finite-dimensional mathematical programming problems in which thenumber of design variables (called densities) is large because it typically coincides with the number of finite
elements of the mesh. In such cases, the cost associated with the computation of the derivatives may be criticalsometimes. In general, computing eigenvalue derivatives with respect to either a single or vector design variable
is inexpensive, whereas computing eigenvector sensitivities is an issue that is relatively much more complex andexpensive. In the case of single eigenvalues, the eigenvectors are differentiable, and there are well establishedmethods for computing the eigenvector derivatives. Comparatively, the case of eigenvectors corresponding tomultiple eigenvalues is more difficult because in this scenario, there exist infinitely many eigenvectors basis (orthonormal with respect to the mass matrix) associated with the multiple eigenvalues. This adversely affects thedifferentiability, and even the continuity, of the eigenvectors with respect to the design variables. In this case,when a single design variable is perturbed, the multiple eigenvalues split into several single eigenvalues. Moreover, it can be shown that the eigenvector basis corresponding to the multiple eigenvalues (before perturbing thedesign variable) closer to the eigenvectors of the single eigenvalues (after perturbing the design variable) is differentiable. This particular basis is called the adjacent basis, and in general, and it is a very important feature,
which depends on each design variable. Any other eigenvector corresponding to the multiple eigenvalues can be
expressed as a linear combination of the eigenvectors of the adjacent basis, for which the partial derivatives may
be computed. Moreover, the partial derivatives of the target eigenvector can be obtained as a linear combinationof those of the adjacent eigenvectors. In practical cases, e.g., in optimization, we are interested in computing
the sensitivities of a functional depending of certain eigenvectors, which probably follows a prescribed referencemode shape, and with the existing methods in the literature, this can be achieved using the adjacent eigenvec-6239

tors basis. From the implementation perspective, when we have to compute the partial derivatives with respect
to numerous design variables, obtaining these results becomes computationally expensive and elaborate because
as highlighted above, the adjacent eigenvectors depend on each design variable. In this paper, we propose amethod that overcomes this difficulty.
2 AN EFFICIENT GENERAL METHOD FOR SENSITIVITY ANALYSIS OF EIGENVECTORSLet λ be an eigenvalue of multiplicity m and Φ a matrix whose columns φ 1 , φ 2 , · · · , φ m are a basis of Morthonormalized eigenvectors (ΦT MΦ = Im ) associated with λ . In such a case, the eigenproblem may be expressed as follows:KΦ = MΦΛ, Φ ∈ Rn×m , Λ = λ Im ∈ Rm×m
(1)
We start by computing the derivatives of the adjacent eigenvectors, Z, verifying
KZ = MZΛ,Z T MZ = Im

(2)

For this purpose, let us consider the following objective function:G̃ = Z,(3)

which indeed is differentiable, and is next modified to conveniently consider the augmented function,1G̃ = Z + PT (KZ − MZΛ) − ΞT (Z T MZ − Im ),2(4)with the help of the adjoint method, where P and Ξ are the vector Lagrange multipliers. Differentiating andrearranging the terms, we arrive at
1G̃0 = PT (K 0 − λ M 0 )Z − ΞT Z T M 0 Z,
2being the pair (P, Ξ), the unique solution of the so-called adjoint system, 
K − λ M −MZ
P−In=Ξ
0m×n−Z T M
0(5)

(6)

which is not singular, as already proved using different techniques in [3].
Two-element cantilever beam with multiple eigenvaluesA two-element cantilever beam with multiple eigenvalues is considered, as shown in Fig. 1. This is a reference
example in the computation of sensitivities of eigenvectors with repeated eigenvalues [2, 3]. The design variable
is the z-axis area moment of the second element with the material properties shown in Fig. 1.
Without loss of generality, both the global stiffness and mass matrices with clamped boundary conditions aregiven by 24I
6Iz
zz000 − 12I
00LL2L2
24Iy
12I
6I
000− L2y − Ly0 L26Iy
8Iy
002Iy0 LE
8Iz − 6ILz
002Iz K= (7)
12Iz
L00− 6ILz 
2L
6Iy12Iy
sym0 L
L2

4Iy0 4Iz

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Figure 1. Two-element cantilever beam with multiple eigenvalues.





ρAL 
M=420 




312

0312

008L2

0008L2540013L
156

sym

054−13L00156

013L−3L20022L
4L2

−13L0 0 
−3L2 
−22L 0 0 
4L2(8)The displacement vector for the second and third elements is
q = [y2 , z2 , θy2 , θz2 , y3 , z3 , θy3 , θz3 ]T(9)Taking the z-axis area moment of the second element as the design variable, b = Iz , and using the material
properties shown in Fig.1, the differentiation of the above matrices is as follows: 1266 − L122 0 000LLL2

00000 0 0 
0000 0 0 
4 − L6 0 0 2 dKEdM
=(10)126 ,dbL
db = 000−
L L2

sym0 0 0 


0 0 
4
In our computation environment, the eigenvectors and eigenvalues obtained are the following−0.0026
0.0233
 0.0233
0.0026  −0.0399 −0.0044   −0.0044
0.0399 Φ̃ = φ˜1 φ˜2 =  −0.0075
0.0687  0.0687
0.0075  −0.0473 −0.0052 −0.00520.0473(11)6241

and λ1 = λ2 = 1.8414. Notice that the eigenvectors in (11) are different from the adjacent vectors reported[2, 3], but the rotation of 83.7 degrees after switching the columns of the eigenvectors converts them into theadjacent vectors found in [2, 3], i.e.,Z = Φ̃ ×cos(θ ) sin(θ )− sin(θ ) cos(θ )(12)where θ = 83.7 ◦ , with Z the adjacent basis,−0.0234
0 

0 

−0.0401 
−0.0690 0 
0 −0.0475

0 −0.0234
 0.0401

0Z =0
 −0.0690
 0.04750(13)The adjacent eigenvector sensitivities are


0
 0
  0 0∂ Z1 ∂ Z2
Z0==
 0∂b ∂b

 0
 00
and the sensitivities of the eigenvalues are given by00∗ =000.0918

−0.00120 
0 
−0.0019 
0.0020 0 
0 0.0051

(14)

(15)

Note that the off-diagonal values of Λ0 are zero. The above sensitivity analysis implies that a change in thedesign variable changes one of the eigenvalues associated with Z. To show the accuracy of the computed
sensitivity of the adjacent vectors with respect to the design variable, b, it is common to compute the ratio,
R, of the values predicted by the design sensitivity analysis and those by the central finite difference method(approximated sensitivity values). In this work, we use the following formula:
R=(Z (b + ∆b) − Z (b − ∆b))/(2∆b)× 100(%), ∆b = 10−4
dZ /db(16)

It is considered that adjacent vectors should be employed to compute the sensitivity of the eigenvectors. However, as we stated in the previous section, this is actually not correct. To prove this, which is one of themajor contributions of the present study, let us consider the following reference vectors which are the original
eigenvectors, i.e., Φre f = Φ̃.
To determine the rotational coefficients aligning the computed eigenvectors in (11) to the target vectors, theoptimization problem is solved. To compute the sensitivity analysis, i.e., d φ˜1 /db, d φ˜2 /db, the adjoint sensitivity
analysis is performed, and the values are summarized in Table 2. For the central difference approach, it isnecessary to compute the eigenvectors Φ̃(b + ∆b) and Φ̃(b − ∆b). To perform this, again two optimization
problems have to be solved for Φ̃(b + ∆b) and Φ̃(b − ∆b) with the perturbation ∆b = 10−4 .

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Table 1. Design sensitivity coefficients of the two-beam problem and their comparison with the central finite
difference method
dZ2 /db-0.0012-0.0000
0.0000-0.00200.0021-0.0000
0.00000.0051R (%)
100.00100.00100.00100.00100.00100.00100.00100.00Note that the double type digits are used for accurate representation:
−0.002557245554096
 0.023311882479460
 −0.039928315387980
 −0.004380019189142
Φ(b + ∆b) =  −0.007531949884064
 0.068661904719745
 −0.047257847757893−0.005183965914722
−0.002557219522960
 0.023311882463730
 −0.039928315361036
 −0.004379975840005
Φ(b − ∆b) =  −0.007531995054570
 0.068661904673411
 −0.047257847726001−0.005184077616588


0.023312001772246
0.002557232468092 −0.004379997393876 
0.039928514074914 
0.068661700775857 
0.007531972255999 −0.005184021614949 
0.047257339991335

0.023311763147863
0.002557232611496 −0.004379997639496 0.039928116635597 
0.068662108656116 
0.007531972678375 −0.005184021905656 
0.047258355590119

(17)

(18)

Table 2 compares the adjoint sensitivity values with the central difference method using the ratio of the sensitivity values. As illustrated, the ratios are almost 100%, showing that the sensitivities are accurately computed.
Interestingly, the derivatives of the eigenvalues, Λ0aux , become as follows
Λ0aux =

0.0011−0.0100−0.0100
0.0908(19)

and it has the non-zero off-diagonal terms because the eigenvectors are not the adjacent ones. However, thereexists a matrix, which we call here Γaux , being a solution of a small eigenvalue problem, 2 × 2, which relates
Λ0 (which is diagonal) with Λ0 aux in the following manner
Λ0aux Γaux = Γaux Λ0and, indeed, Z 0 = Φ̃0 Γaux with

−0.9940
Γaux =−0.1090

0.1090−0.9940=cos(θ ) sin(θ )− sin(θ ) cos(θ )

(20)

,

θ = 83.7◦

(21)

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Table 2. Design sensitivity analysis of the two-beam problem with an arbitrary reference vectord φ˜1 /db d φ˜2 /db∆φ˜1 /∆b ∆φ˜2 /∆bR1 (%)
R2 (%)-0.0001
0.0012 -0.0001
0.0012 100.0000 100.0066-0.0000
0.00000.0000 -0.0000 100.0000 100.0000
0.0000 -0.0000 -0.0000
0.0000 100.0000 100.0000-0.00020.0020 -0.0002
0.0020 100.0000 100.0068
0.0002 -0.0020
0.0002 -0.0020 100.9435 100.0000-0.0000
0.00000.0000 -0.0000 100.0000 100.0000
0.0000 -0.0000 -0.0000
0.0000 100.0000 100.0000
0.0006 -0.0051
0.0006 -0.0051 100.2605 100.0000

Currently, the results of our present approach match with those of Dailey’s method, but our method neither
requires the second-order derivative terms of both the stiffness and mass matrices nor needs computing theadjacent basis.
3 CONCLUSIONSIn this paper, a highly efficient method for computing the derivatives of general functionals, which could eventually be vectorial magnitudes depending on eigenvectors, is developed. Our method includes the cases of botheigenvectors with and without repeated eigenvalues, but the emphasis is on the remarkably more difficult caseof multiple eigenvalues. Our method overcomes the main difficulties of the sensitivity analysis of eigenvectors with repeated eigenvalues, providing a simple, direct, and highly efficient procedure from a computationalperspective.
Acknowledgement
This work was supported by the National Research Foundation of Korea (NRF)grant funded by the Korea
government (MSIT)(No.2018R1A5A7025522).
REFERENCES[1] Yoon, G,H., Donoso,A., Bellido, J.C., Ruiz D., An efficient , In review, 2019.
[2] Dailey, R. L. Eigenvector Derivatives with Repeated Eigenvalues, AIAA, Vol 27 (4), 1989, pp 486-491.
[3] Lee, T.H. Adjoint Method for Design Sensitivity Analysis of Multiple Eigenvalues and Associated Eigenvectors, AIAA, Vol 45 (8), 2007, pp 1998-2004.
[4] Bendsøe, M. P. and Sigmund, O. Topology optimization, Theory, methods and application, Springer-Verlag,
Berlin, 2003.

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