Efficient buckling constrained topology optimization using reduced order modeling

Let us start by considering the approximate \(\tilde{{\varvec{{\textsf {a}}}}}_o\) solution to (1) and let \(\overline{{\varvec{{\textsf {K}}}}}_L\) be the stiffness matrix from a past design iteration. In the CA approach, we utilize the Cholesky factorization of \(\overline{{\varvec{{\textsf {K}}}}}_L\), to compute \({\varvec{{\textsf {a}}}}_o\) i.e.where \(\Delta {\varvec{{\textsf {K}}}}_L \equiv {\varvec{{\textsf {K}}}}_L - \overline{{\varvec{{\textsf {K}}}}}_L\) is the stiffness change due to a design update. Applying \(\overline{{\varvec{{\textsf {K}}}}}_L^{-1}\) to both sides of (5) yieldsNow, we expand the inverse of \(({\varvec{{\textsf {I}}}} + \overline{{\varvec{{\textsf {K}}}}}_L^{-1}\Delta {\varvec{{\textsf {K}}}}_L )\) in a power series²Hence, the solution to (6) iswhich we truncate at \(k=s\ll n\), such that \(\tilde{{\varvec{{\textsf {a}}}}}_o = {\varvec{{\textsf {R}}}}{\varvec{{\textsf {y}}}}\), cf. (4). The basis vectors in \({\varvec{{\textsf {R}}}}\) are obtained recursively from (8) asAfter the basis vectors are generated, we solve the reduced \(s\times s\) problemfor \({\varvec{{\textsf {y}}}}\), and insert the result into (4). In practice, the magnitudes of \({\varvec{{\textsf {u}}}}_i\) rapidly decrease due to repeated application of \(\overline{{\varvec{{\textsf {K}}}}}_L^{-1}\Delta {\varvec{{\textsf {K}}}}\), resulting in a poorly conditioned system (10). To mitigate this issue we introduce normalization and \({\varvec{{\textsf {K}}}}_L\)-orthogonalization of the basis vectors wherefore the final basis vectors \({\varvec{{\textsf {R}}}} = \left[ {\varvec{{\textsf {v}}}}_1, {\varvec{{\textsf {v}}}}_2, \ldots {\varvec{{\textsf {v}}}}_s\right]\) are obtained recursively asDue to the orthogonalization, \({\varvec{{\textsf {R}}}}^T{\varvec{{\textsf {K}}}}_L{\varvec{{\textsf {R}}}} = {\varvec{{\textsf {I}}}}\), the reduced system in (10) boils down to \({\varvec{{\textsf {I}}}}{\varvec{{\textsf {y}}}} = {\varvec{{\textsf {y}}}} = {\varvec{{\textsf {R}}}}^T{\varvec{{\textsf {F}}}}_o\), such that the approximate solution becomesAn alternative, numerically equivalent, approach to (12) is to use the factorization of \(\overline{{\varvec{{\textsf {K}}}}}_L\) as a preconditioner in a Preconditioned Conjugate Gradients (PCG) procedure, cf. e.g. Kirsch et al. (2002) and Amir et al. (2012). However, in this work the eigenproblem (3) is solved with ROM so for consistency we use the ROM approach for the static problem (1) as well.

Next, we seek an approximation of each eigenpair to (3). We do this by approximating the eigenmodes, after which the eigenvalues are given by the Rayleigh quotient. The eigenmodes are approximated aswhere \({\varvec{{\textsf {R}}}}_j\) and \({\varvec{{\textsf {y}}}}_j\) are analogous to the quantities defined in (4) and \(j\in {\mathbb {N}}_n\). To generate the basis vectors we utilize the additive decomposition of \({\varvec{{\textsf {K}}}}_L\) in (3), i.e.where \({\varvec{{\textsf {K}}}}_G = {\varvec{{\textsf {K}}}}_G(\tilde{{\varvec{{\textsf {a}}}}}_o)\). Premultiplication of (14) by \(\overline{{\varvec{{\textsf {K}}}}}_L^{-1}\) yieldsand, using the result (7), we find from (15)The basis vectors \({\varvec{{\textsf {R}}}}_j\) for mode j are obtained by truncating the infinite sum at term \(k = s \ll n\), replacing \(\varvec{{\upphi }}_j\) on the right-hand side with \(\overline{\varvec{{\upphi }}}_j\) and ignoring the arbitrary scaling factor of \(-(\mu _k)^{-1}\), i.e.We scale the basis vectors to ensure that their relative magnitudes do not differ significantly, analogously to the static problem. Additionally we employ orthogonalization of the basis vectors with respect to lower order modes \(\tilde{\varvec{{\upphi }}}_l\) by introducing \({\varvec{{\textsf {v}}}}_{ij} = {\varvec{{\textsf {t}}}}_{ij} + \sum _{l = 1}^{j-1}\alpha _l\tilde{\varvec{{\upphi }}}_l\) and requiring \(\tilde{\varvec{{\upphi }}}_l^T{\varvec{{\textsf {K}}}}_L{\varvec{{\textsf {v}}}}_{ij}=0\) for all \(i\le s, l<j\). Thereby ensuring that the orthogonality condition of the eigenmodes is retained, that is \(\tilde{\varvec{{\upphi }}}_l^T{\varvec{{\textsf {K}}}}_L\tilde{\varvec{{\upphi }}}_{j}=\delta _{lj}\). Due to the orthogonalization, we also avoid that the \({\tilde{\mu }}_j\) approximates the lower order \(\mu _l, \; l < j\) [compare Kirsch (2008) and Bogomolny (2010)]. The old eigenmodes, \(\overline{\varvec{{\upphi }}}_l\), may be chosen instead of the current approximation, allowing for independent solution of the eigenproblems. However, in this work the latter approach is taken. The basis generation can be defined recursively aswhere \(\overline{\varvec{{\upphi }}}_j\) is the jth eigenvector corresponding to the matrix \(\overline{{\varvec{{\textsf {K}}}}}_L\). Now, inserting the (13) approximation into (3) and premultiplying by \({\varvec{{\textsf {R}}}}_j = \left[ {\varvec{{\textsf {v}}}}_{1j},{\varvec{{\textsf {v}}}}_{2j},\ldots {\varvec{{\textsf {v}}}}_{sj}\right]\) gives a reduced generalized eigenvalue problemwhere \({\varvec{{\textsf {K}}}}_G^{R_j} = {\varvec{{\textsf {R}}}}_j^T{\varvec{{\textsf {K}}}}_G(\tilde{{\varvec{{\textsf {a}}}}}_o){\varvec{{\textsf {R}}}}_j\) and \({\varvec{{\textsf {K}}}}_L^{R_j} = {\varvec{{\textsf {R}}}}_j^T{\varvec{{\textsf {K}}}}_L{\varvec{{\textsf {R}}}}_j\). The system (19) has s solutions, and we seek the one which most accurately approximates the jth eigenpair \((\mu _j, \varvec{{\upphi }}_j)\). Since the eigenpairs are solved in order of descending magnitude and \({\varvec{{\textsf {R}}}}_j\) are \({\varvec{{\textsf {K}}}}_L\)-orthogonal to \(\tilde{\varvec{{\upphi }}}_l \; l < j\), we choose eigenpair with largest magnitude eigenvalue. To summarize, for each sought eigenpair, we first generate a set of basis vectors using (18), and then solve (19) for the largest magnitude \(({\tilde{\mu }}_j, \tilde{\varvec{{\upphi }}}_j)\).

In this work a lower bound constraint is enforced on the lowest magnitude BLF, that is the largest magnitude (or critical) \(\mu _i\), which we denote \(\mu _c\). To address the non-smoothness due to coinciding eigenvalues, we approximate \(\mu _c\) using aggregation in the form of a p-norm, cf. Torii and De Faria (2017). Therefore, the constraint on the critical BLF is stated aswhere \(\mu ^\star \in {\mathbb {R}}\) is the upper bound, \(p\in {\mathbb {R}}\) and we consider \(n_{\mu }\in {\mathbb {N}}_n\) BLFs.

When computing the sensitivity of a non self-adjoint displacement dependent function, the implicit sensitivity of the displacements with respect to the design variables appears. In this work, we annihilate this implicit sensitivity through the adjoint method. Without going into details, the sensitivities of the BLFs are (Ferrari and Sigmund 2019)where it is assumed that (1) and (3) are fulfilled. An important consequence of (24) is that for each BLF we must compute an adjoint \(\varvec{{\Uplambda }}_j\) which requires solving a linear system with \({\varvec{{\textsf {K}}}}_L\). The \(\varvec{{\Uplambda }}_j\) can be estimated the same way \({\varvec{{\textsf {a}}}}_o\) is estimated from the ROM by generating a new set of basis vectors using (11) and solving (12), where in both cases \({\varvec{{\textsf {F}}}}_o\) is replaced by \({\varvec{{\textsf {b}}}}_j\). In this work, since we approximate \({\varvec{{\textsf {a}}}}_o\) and \(\varvec{{\upphi }}_j\), we must instead usewhen estimating the adjoints.

Since the sensitivity (24) assumes that (1) and (3) are solved exactly, it is not consistent when they are solved approximately. Amir et al. (2009) and Bogomolny (2010) account for the inconsistencies by replacing \(\frac{\partial \mu _j}{\partial \varvec{{\upnu }}}\) with \(\frac{\partial {\tilde{\mu }}_j}{\partial \varvec{{\upnu }}}\) and introducing additional adjoint vectors corresponding to the steps in (18). This consistent sensitivity approach is computationally expensive and has shown to not offer any benefit over estimating (24) with \(\tilde{{\varvec{{\textsf {a}}}}}_o\) and \(({\tilde{\mu }}_j, \tilde{\varvec{{\upphi }}}_j)\). Indeed, if the approximations are accurate the sensitivities are sufficiently accurate nonetheless. In this work the latter approach will be taken, and \(\varvec{{\Uplambda }}_j\) in (24) will be estimated from the ROM (12).

The first geometry we study is shown in Fig. 1a. In Fig. 1b the stiffness maximized design is depicted. For this problem we use 8, 4 and 8 basis vectors for the linear equilibrium, eigenproblem and the adjoints, respectively. The designs obtained when constraining the critical BLF using the reference and approximate approach, shown in Fig. 1c, d, are very similar. This is also seen in Table 1 showing the value of the objective and constraints computed in a postprocessing step using the reference approach (see Algorithm 1). The constraints are satisfied for both designs, and the differences in the objective is minute. Table 1 also shows the number of matrix factorizations and triangular solves without and with ROM. We see that the number of factorizations is significantly reduced using ROM, while the difference in triangular solves is very small. Figure 2 shows the evolution of \(\mu _i/\mu ^\star\) without and with ROM. We note that the evolution is similar for both approaches.

To verify that our ROM performs well for cases where a good initial guess is not available, the spire problem is solved starting from a homogeneous initial guess. The same number of basis vectors as in the previous example is used. The resulting designs are very similar to each other, but are slightly different from those in Fig. 1. The post-process analysis shows that they have the same structural qualities, and the reduction in computational effort is similar to the previous example: the number of factorizations was reduced from 352 to 62, and the number of triangular solves from 31344 to 28623.

The second geometry we study is shown in Fig. 3a. For this problem we use 8, 4 and 8 basis vectors. Again, the designs obtained without and with ROM, seen in Fig. 3c, d, are similar. The objective and constraints at the end designs are provided in Table 2. The constraints are satisfied for both approaches, and the difference in objective is, again, very small. Table 2 also shows the number of matrix factorizations and triangular solves. We note that again the number of factorizations is significantly smaller and that the number of triangular solves is slightly smaller for the approximate approach. Figure 4 shows the evolution of \(\mu _i/\mu ^\star\), which is similar for the two approaches.

The last geometry we study is shown in Fig. 5a. For this problem, the number of basis vectors for the static problem, eigenproblem, and adjoint was increased from 8, 4 and 8 to 10, 6 and 10, respectively. When constraining the smallest BLF, the designs shown in Fig. 5c, d are obtained without and with ROM, respectively. For Fig. 5c design, the buckling constraint is satisfied and the p-norm approximating the smallest BLF renders undershooting. However, for Fig. 5d design, the constraint on the p-norm of the BLFs is slightly violated. Even so, due to the undershooting, the critical BLF still satisfies the upper bound constraint. The evolution of \(\mu _i/\mu ^\star\) for Fig. 5c, d designs are shown in Fig. 6, and we note that, like in previous examples, the graphs are very similar.

Additionally, Table 3 shows the number of matrix factorizations and triangular solves without and with ROM. When using ROM the number of factorizations is reduced significantly, by over \(75\%\), whereas the total number of triangular solves increased by about \(37\%\). While this increase is noticeably larger compared to the previous examples, it still represents a small portion of the total computational effort which is dominated by the matrix factorizations.

As a final example, we demonstrate that our ROM can accurately approximate the BLFs when they switch order and coincide. For this reason, we solve the spire problem with a higher safety factor of 6, starting from the compliance initial guess. We also increase the number of buckling factors to 10 to properly take into account the grouping of BLFs near the critical BLF. The number of basis vectors is increased to 10, 6 and 10, respectively.

The resulting designs, which can be found in Fig. 7, are again very similar. Table 4 shows the value of the objective and constraints, which are, in terms of stiffness and buckling resistance, almost identical. This example shows again that the ROM approach provides a significant speedup, in this case by \(70\%\), and the number of triangular solves is increased only slightly. Figure 8 shows the evolution of the \(\mu _i/\mu ^\star\). We can clearly see that the ROM is able to resolve both coinciding BLFs and mode switching, see for example modes 6 and 7, and modes 8, 9, 10.

Figure 9 shows the relative residuals of the eigenpairs, which we define by the quotientOf course, when using the reference approach the residuals are determined by the tolerance of the eigenvalue solver, so residuals are only shown at design iterations where the approximate approach is used.

The relative residuals for the column problem are, towards the end of the optimization, in the range \(10^{-4}-10^{-2}\) and the residuals directly after a factorization update are higher compared to the other problems. Thus, to reduce the residuals and improve the quality of the approximation, the number of basis vectors could be increased further. For the spire problem the residuals are lower in general, but also have a larger spread which can be attributed to the spread in the BLFs themselves. For the V-structure problem, the residuals are comparatively lower, \(10^{-6}-10^{-3}\), and are less scattered. Figure 9 shows that the geometry has an influence on the ROM’s efficacy. While the method works well for all considered problems, some cases require more basis vectors and/or factorizations to reach the same level of accuracy as others.

The examples show that designs produced without and with ROM are very similar. In our implementation, we did not compute the consistent sensitivities that account for the inaccuracies of the ROM, nor did we compute the adjoints with full accuracy. Nevertheless, we still found designs very similar to those generated by the reference approach. We demonstrate that our ROM can, without modifications, be applied to problems where a good initial guess is not available. We also show that the ROM can accurately resolve both mode switching and coinciding eigenmodes.

Compared to the reference approach, the ROM approach reduced the number of factorizations significantly, and with exception to the column example, the number of triangular solves did not change significantly. Since the triangular solves only constitute a small portion of the total effort compared to the matrix factorizations, the computational effort is reduced significantly by the proposed ROM approach. When the required number of BLFs grows, the savings might decrease due to the increasing cost of approximating the adjoints.

The number of basis vectors and when to update the factorization was chosen heuristically. Even so, we were able to reduce the computational effort without fine-tuning the parameters. In some cases, the ROM was inaccurate, but adding more basis vectors improved the accuracy without significantly increasing the costs. Based on our numerical experiments, the optimal number of basis vectors depends on the problem and on the design. Ideally, the number of basis vectors and the frequency of factorization should be adaptive, such that the approximation is both efficient and sufficiently accurate.

The ROM-solver for the eigenvalue problem has several advantages over traditional eigenvalue solvers that iterate over a subspace: the number of basis vectors for each eigenpair can be chosen independently. Thus, the effort can be balanced based on how each eigenpair affects the problem—higher accuracy can be enforced for more important eigenpairs. Furthermore, the approximate solution of eigenpairs is embarrassingly parallelizable as the eigenpairs can be solved independently of each other, if the old eigenmodes are chosen in the orthogonalization step. This too should be considered in future work.

The proposed ROM requires a direct solve using the system matrix, limiting its use to problems which can be efficiently solved using direct solvers. Solving problems beyond this size requires other approaches, such as highly tailored multigrid preconditioned subspace iteration methods. For this we refer to the work by Peetz and Elbanna (2021), or the multigrid approach by Ferrari and Sigmund (2020).