Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

Abstract

The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple-load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global ε-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Appendices

Appendix : 1: Relation to truss topology optimization

The problem formulation (21a) has already been known in the context of truss topology optimization (Vandenberghe and Boyd 1996), for which the constraints (21b) reduce to linear matrix inequalities (LMI). Consequently, the feasible set is convex, allowing for an efficient solution of (21a) by interior point methods, for example.

A natural question then arises: What happens when the rotational degrees of freedom are neglected, solving truss topology optimization problem instead of the frame one? To this goal, however, we must first satisfy the rather restrictive assumption that the truss ground structure is capable of carrying the loads f_j(a), i.e.,

$$ \mathbf{f}_{j} (\mathbf{a}) \in \text{Im}\left( \mathbf{K}_{\mathrm{t},j}(\mathbf{a}) \right), \forall j \in \{1{\dots} n_{\text{lc}}\}, $$

(43)

where $\mathbf {K}_{\mathrm {t},j} (\mathbf {a}) = \mathbf {K}_{j,0}+ {\sum }_{i=1}^{n_{\mathrm {e}}} \mathbf {K}_{j,i}^{(1)} a_{i}$, and that with all empty rows and columns removed K_t,j(a) is positive definite for all positive a. From the mechanical point of view, we require that no moment loads are imposed, a straight bar does not have to carry transverse loads, and the ground structure is well supported.

Suppose now that $\mathbf {a}^{*}_{\mathrm {t}}$ are optimal cross-sections obtained from a solution to (21a) with the terms $\mathbf {K}_{j,i}^{(2)}$ and $\mathbf {K}_{j,i}^{(3)}$ neglected, and $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}^{*}_{\mathrm {t}}$ is the associated optimal objective function value, which can be computed from $\mathbf {a}_{\mathrm {t}}^{*}$ as

$$ \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}^{*}_{\mathrm{t}} = \sum\limits_{j=1}^{n_{\text{lc}}}\left( \omega_{j} \left[\mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*})\right]^{\mathrm{T}} \left[\mathbf{K}_{\mathrm{t},j}(\mathbf{a}_{\mathrm{t}}^{*}) \right]^{\dagger} \mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*}) \right). $$

(44)

When the optimal cross-sections of a truss structure, $\mathbf {a}^{*}_{\mathrm {t}}$, are reused in a frame structure, the resulting objective function value changes to

$$ \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}_{\mathrm{f}} = \sum\limits_{j=1}^{n_{\text{lc}}}\left( \omega_{j} \left[\mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*})\right]^{\mathrm{T}} \left[\mathbf{K}_{\mathrm{t},j}(\mathbf{a}_{\mathrm{t}}^{*}) + \mathbf{K}_{\mathrm{b},j}(\mathbf{a}_{\mathrm{t}}^{*}) \right]^{\dagger} \mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*}) \right) $$

(45)

with $\mathbf {K}_{\mathrm {b},j}(\mathbf {a}) = {\sum }_{i=1}^{n_{\mathrm {e}}}\left (c_{\text {II}} \mathbf {K}_{j,i}^{(2)} a_{\mathrm {t},i}^{2} + c_{\text {III}}\mathbf {K}_{j,i}^{(3)} a_{\mathrm {t},i}^{3}\right )$.

To state a relation between $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}^{*}_{\mathrm {t}}$ and $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {f}}$ we recall a useful lemma:

Lemma 4

(Kovanic 1979) Let $\mathbf {A} \in \mathbb {S}^{n}$ and $\mathbf {B} \in \mathbb {R}^{n\times q}$. Then,

$$ \begin{array}{@{}rcl@{}} &&\left( \mathbf{A} + \mathbf{B}\mathbf{B}^{\mathrm{T}} \right)^{\dagger} = \mathbf{A}^{\dagger} \\ &&- \mathbf{A}^{\dagger} \mathbf{B} \left( \mathbf{I} + \mathbf{B}^{\mathrm{T}}\mathbf{A}^{\dagger} \mathbf{B}\right)^{-1} \mathbf{B}^{\mathrm{T}} \mathbf{A}^{\dagger} + \left( \mathbf{B}^{\dagger}_{\perp}\right)^{\mathrm{T}} \mathbf{B}^{\dagger}_{\perp} \end{array} $$

(46)

with $\mathbf {B}_{\perp } = \left (\mathbf {I} - \mathbf {A}\mathbf {A}^{\dagger } \right ) \mathbf {B}$.

Using this lemma, we prove that $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {t}}$ provides an upper bound for $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {f}}$.

Lemma 5

Suppose that ω^Tc^∗ is the optimal objective function value of the frame structure design problem (21a) and (43) holds. Then, $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}^{*} \le \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {f}} \le \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}^{*}_{\mathrm {t}}$.

Proof

Because of $\mathbf {f}_{j}(\mathbf {a}_{\mathrm {t}}^{*}) \in \text {Im}\left (\mathbf {K}_{\mathrm {t},j}(\mathbf {a}_{\mathrm {t}}^{*})\right )$, we clearly have $\mathbf {f}_{j} (\mathbf {a}_{\mathrm {t}}^{*}) \in \text {Im}\left (\mathbf {K}_{\mathrm {t},j}(\mathbf {a}_{\mathrm {t}}^{*}) + \mathbf {K}_{\mathrm {b},j}(\mathbf {a}_{\mathrm {t}}^{*}) \right )$. Therefore, $\mathbf {a}_{\mathrm {t}}^{*}$ is a feasible solution to the frame structure design problem (21a) and the associated objective function is bounded from below by the global optimum ω^Tc^∗. Hence, $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}^{*} \le \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {f}}$.

For the other inequality, we express (45) using Lemma 4. To this goal, let $\mathbf {K}_{\mathrm {b},j}(\mathbf {a}^{*}_{\mathrm {t}}) = \mathbf {B}_{j} \mathbf {B}_{j}^{\mathrm {T}}$, where B_j is a real matrix because $\mathbf {K}_{\mathrm {b},j}(\mathbf {a}^{*}_{\mathrm {t}}) \succeq 0$ by definition. Then, (45) can be written as

$$ \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}_{\mathrm{f}} = \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}_{\mathrm{t}}^{*} - \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}_{\mathrm{a}} + \boldsymbol{\omega}^{\mathrm{T}} \mathbf{c}_{\mathrm{b}}, $$

(47)

where

$$ \begin{array}{@{}rcl@{}} c_{\mathrm{a},j} &{=}& \left[\mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*})\right]^{\mathrm{T}} \mathbf{A}_{j}^{\dagger} \mathbf{B}_{j} \left( \mathbf{I} {+} \mathbf{B}_{j}^{\mathrm{T}}\mathbf{A}_{j}^{\dagger} \mathbf{B}_{j}\right)^{-1} \mathbf{B}_{j}^{\mathrm{T}} \mathbf{A}_{j}^{\dagger} \mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*}), \end{array} $$

(48a)

$$ \begin{array}{@{}rcl@{}} c_{\mathrm{b},j} &{=}& \left[\mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*})\right]^{\mathrm{T}} \left( \mathbf{B}^{\dagger}_{\perp,j}\right)^{\mathrm{T}} \mathbf{B}^{\dagger}_{\perp,j} \mathbf{f}_{j}(\mathbf{a}_{\mathrm{t}}^{*}), \end{array} $$

(48b)

with $\mathbf {A}_{j} = \mathbf {K}_{\mathrm {t},j}(\mathbf {a}_{\mathrm {t}}^{*})$. Clearly, (48a) is non-negative. For (48b), $\left (\mathbf {B}^{\dagger }_{\perp }\right )^{\mathrm {T}} \mathbf {B}^{\dagger }_{\perp } \in \text {Ker}(\mathbf {A}_{j})$, so that $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {b}}$ vanishes. Hence, $\boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {f}} = \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {t}}^{*} - \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {a}} \le \boldsymbol {\omega }^{\mathrm {T}} \mathbf {c}_{\mathrm {t}}^{*}$. □

Thus, when (43) holds true, the truss topology optimization produces an upper bound to the optimal objective of the frame structure topology optimization problem.

Appendix 2: Numerical performance of bound constraints

This section illustrates the effect of using different types of bound constraints, (24a) or (24b), on performance of the moment-sum-of-squares hierarchy.

We start by re-evaluating the problem in Section 4.3. Our numerical experiments in Table 6 reveal that for this specific problem the quadratic constraints (24b) are substantially tighter in terms of generated lower bounds, they require smaller relaxation degree to converge, and are computationally more efficient. Also notice that fewer constraints are needed when using (24b).

Table 6 Performance of the hierarchy on the problem with self-weight using different bound constraints. LB denotes lower bound, and PO stands for polynomial optimization. The bold lower bounds are certified global optima. In addition, n_c stands for the number of semidefinite constraints of the size m and n is the number of variables. The entries a_i denote the cross-section areas constructed from the first-order moments

For the problem in Section 4.4 discretized by Euler-Bernoulli frame elements, we observe that the difference in performance becomes less noticeable (see Table 7): the quadratic constraints lead to a better lower bound in the third relaxation only, but are still computationally more efficient because fewer constraints are needed.

Table 7 Performance of the hierarchy on the cantilever problem using different bound constraints. LB denotes lower bound, and PO stands for polynomial optimization. The bold lower bounds are certified global optima. In addition, n_c stands for the number of semidefinite constraints of the size m and n is the number of variables. The entries a_i denote the cross-section areas constructed from the first-order moments