A Penalty Method for Non-Self-Adjoint Topology Optimization

Abstract

We propose a novel penalty method framework for the non-self-adjoint topology optimization problems, taking compliant mechanism problems as an example, by incorporating a convex nonlocal perimeter approximation scheme. We rigorously analyze the existence of solutions to the optimization problem derived from the penalty method. Furthermore, we establish that the discrete problem \(Γ\)-converges to the continuous problem, ensuring consistency across scales. To solve the discrete problem, we develop a projected gradient method that guarantees strict monotonic descent of the objective function. We also extend the framework to the heat dissipation problem and propose a generalized material interpolation function (GMIF), which allows for a targeted control of the topological connectivity in the resulting optimal design. Numerical experiments on the compliant mechanism and heat dissipation problems validate the effectiveness of the proposed method. This framework provides a robust approach to addressing complex optimization challenges in computational mathematics with potential applications in engineering design.

Figures (14)

• Figure 1: Behavior of the generalized material interpolation function $Y(k_1, k_2, p, \chi)$ for different values of the exponent $p$.
• Figure 2: (adapted from Lops2016) Dashed lines denote homogeneous tangential Neumann boundaries, thick solid lines indicate homogeneous Dirichlet boundaries, $q_{\text{in}}$ and $q_{\text{out}}$ represent inhomogeneous Neumann boundaries, remaining parts are homogeneous Neumann boundaries.
• Figure 3: The final optimal shape.
• Figure 4: Convergence history of the objective functional.
• Figure 5: The final optimal shape.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof