Computing multiple solutions of topology optimization problems

Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions; however, these methods can fail even in the simplest cases. In this paper, we present an algorithm to perform a systematic exploratory search for the solutions of the optimization problem via second-order methods without a good initial guess. The algorithm combines the techniques of deflation, barrier methods and primal-dual active set solvers in a novel way. We demonstrate this approach on several numerical examples, observe mesh-independence in certain cases and show that multiple distinct local minima can be recovered.

💡 Research Summary

**
The paper addresses a fundamental difficulty in topology optimization: the lack of convexity often leads to many local minima, and traditional gradient‑based continuation methods—where a continuation parameter (e.g., the SIMP penalization factor) is increased stepwise and the previous solution is used as the initial guess—can become trapped in a suboptimal basin. To overcome this limitation, the authors propose a novel algorithm called the deflated barrier method that systematically explores the solution space without requiring a good initial guess.

The method integrates three well‑established techniques in a new way:

1. Barrier formulation – The box constraints on the material density ρ (0 ≤ ρ ≤ 1) and the volume constraint ∫Ω ρ dx = γ|Ω| are incorporated via logarithmic barrier terms. A barrier parameter μ is introduced; starting from a relatively large value, μ is decreased gradually, tracing a central path that remains interior to the feasible set.

2. Primal‑dual active‑set solver – The Karush‑Kuhn‑Tucker (KKT) conditions for the PDE‑constrained optimization problem (including pressure, Lagrange multipliers for the volume and pressure‑integral constraints) are solved with a Newton‑type active‑set method. This solver handles inequality constraints efficiently and provides fast quadratic convergence when the active set is correctly identified.

3. Deflation – After a stationary point has been obtained for a given μ, a rank‑one modification of the Jacobian (the deflation operator) is applied. This modification makes the previously found solution a singular point of the modified system, preventing the Newton iteration from converging back to it. The resulting perturbed system yields a new feasible initial guess that is “orthogonal” to the already discovered solution.

The algorithm proceeds iteratively: (i) choose an initial μ and a random feasible ρ, (ii) solve the KKT system with the primal‑dual active‑set Newton method, (iii) once convergence is achieved, apply deflation to generate a new starting point, (iv) reduce μ and repeat steps (ii)–(iii). The process is continued until a desired number of distinct stationary points have been collected.

The authors demonstrate the approach on three representative PDE‑constrained topology‑optimization problems:

• Navier‑Stokes power‑dissipation minimization in a 2‑D rectangular domain containing five decagonal holes. Using the deflated barrier method they recover 42 distinct stationary material distributions, each corresponding to a different pipe configuration. The solutions exhibit clear visual differences and a range of objective values, illustrating the richness of the solution landscape.

• Stokes flow optimization with the same geometric setting, again yielding multiple symmetric and asymmetric channel layouts.

• Compliance minimization of elastic structures (the classic MBB beam and cantilever examples) using the SIMP model with penalization exponent p = 3. The method finds several qualitatively different material layouts, and a mesh‑refinement study shows that the discovered solutions are essentially mesh‑independent.

Compared with standard continuation, the proposed method (a) does not rely on a good initial guess, (b) avoids the “poisoned” continuation path that can lock the algorithm into a suboptimal basin, and (c) remains stable as the barrier parameter approaches zero, eliminating the typical ill‑conditioning of pure barrier methods. The paper also includes a theoretical result proving the equivalence of the primal‑dual active‑set strategy of Hintermüller et al. and the reduced‑space active‑set method of Benson and Munson, and an appendix describing the new feasible‑tangent predictor‑corrector scheme.

In summary, the deflated barrier method provides a robust, scalable framework for systematically locating multiple local minima in non‑convex, inequality‑constrained, infinite‑dimensional optimization problems arising from topology optimization. By delivering a portfolio of viable designs, it enables engineers to make post‑processing decisions based on additional criteria such as manufacturability, cost, or aesthetic preferences, thereby expanding the practical utility of topology optimization in engineering design. Future work may extend the approach to three‑dimensional, multi‑physics problems and integrate data‑driven selection mechanisms.