Truss topology design under harmonic loads: Peak power minimization with semidefinite programming

TL;DR

The paper addresses peak-power minimization in truss topology optimization under harmonic loads by formulating the problem within a semidefinite representable ($SDr$) framework. It extends prior single-frequency results to multifrequency and out-of-phase loads using convex relaxations and penalty-based Lagrange-type approaches, with exactness for simple cases and suboptimal but feasible solutions otherwise. The authors validate the method on multiple benchmarks, showing substantial peak-power reductions and providing open-source code for replication. This approach offers a tractable path to robust, lightweight vibrational designs in engineering applications where loads are periodic and multi-harmonic. The work highlights the practical impact of convexification techniques in structural optimization under complex dynamic loading scenarios, while outlining computational and sparsity-driven avenues for scaling.

Abstract

Designing lightweight yet stiff structures that can withstand vibrations is a crucial task in structural optimization. Here, we present a novel framework for truss topology optimization under undamped harmonic oscillations. Our approach minimizes the peak power of the structure under harmonic loads, overcoming the limitations of single-frequency and in-phase assumptions found in previous methods. For this, we leverage the concept of semidefinite representable (SDr) functions, demonstrating that while compliance readily conforms to an SDr representation, peak power requires a derivation based on the non-negativity of trigonometric functions. Finally, we introduce convex relaxations for the minimization problem and provide promising computational results.

Figures (13)

• Figure 1: Boundary conditions and ground structure of the $21$-element truss structure introduced in heidari_optimization_2009.
• Figure 2: (a) Time-varying loads $f_R(t)$ and $f_I(t)$ and the powers associated with the (b) optimal designs under in-phase loads, and (c) the same designs under out-of-phase loads. Figure (d) shows the power of a design optimized for out-of-phase loads.
• Figure 3: Optimized topology for the (a) in-phase loads $f_R$, (b) in-phase loads $f_I$, and (c) out-of-phase loads $f_R$ and $f_I$ acting according to Fig. \ref{['fig:heidari_bc']}.
• Figure 4: Optimal solutions of convex relaxation with penalty parameter $\eta=10$.
• Figure 5: SDP relaxation with $80$ different values of $\eta$ for the Heidari truss with not-in-phase single harmonic load. The first figure shows the evolution of the trace difference $\mathop{\mathrm{Tr}}\nolimits\{\mathbf{X}-\mathbf{F}^*\mathbf{L}_{1,\omega}(\mathbf{a})^\dagger \mathbf{F}\}$ as $\eta$ ranged from $10^{-9}$ to $10$, the trace difference is indeed small as $\eta$ is large enough. The second figure shows the activation of mass constraint, the optimal structure uses all the available mass as $\eta$ is large enough. In the third figure, the blue (orange) curve shows respectively the evolution of $\theta$ (resp. the actual peak power computed at optimal design) as $\eta$ is increased. They agree only when $\eta$ is large enough. The last figure shows the evolution of the KKT residual defined in \ref{['eq:lp']}.

Theorems & Definitions (23)

• Lemma 4.1: Schur's complement Lemma with $\mathbf{C}$ invertible wolkowicz_handbook_2000
• Lemma 4.2: Generalized Schur's complement Lemma boyd_convex_2004
• Definition 4.1: SDr function, ben-tal_lectures_2001
• Definition 4.2: SDr function, second representation
• Example 4.1: Compliance minimization
• Example 4.2: Peak power minimization under single harmonic load heidari_optimization_2009
• Example 4.3: Peak power minimization
• Example 4.4: Compliance minimization, continued
• Lemma 4.3
• proof