2025 Theses Doctoral

Topology Optimization for Problems with Design-Following Neumann Boundary Conditions

Kostopoulos, Christos

Topology optimization has reshaped modern structural design by enabling unprecedented freedom in the exploration of complex geometries. Despite its broad applicability, many physical problems, such as aerodynamic pressure loading, heat transfer, and torsion, are governed by Neumann boundary conditions that evolve with the changing design. In finite element analysis, this coupling renders the forcing vector boundary dependent, requiring the optimization procedure to track the boundary and update the applied loads at every iteration. This work introduces a computationally efficient methodology for addressing such design-dependent Neumann boundary conditions. The proposed framework is demonstrated through applications to torsional mechanics, a representative class of problems in which boundary dependent Neumann conditions play a central role, and expanded in heat transfer problems.

Initially, a novel efficient topology optimization methodology for beams’ torsion using the warping function formulation is proposed. The finite element method is used to discretize the cross-section and an efficient gradient-based optimization problem is formulated to optimize the relevant parameters corresponding to the torsion and warping constants of the beam. As a result, for the first time, one can optimize a beam for problems where the warping behavior is dominant. Density-based optimization is defined where the SIMP approach is utilized to penalize intermediate element densities. A key challenge of the optimization that arises in the warping function framework is the boundary dependent nature of the problem. To this end, a differentiable boundary recognition algorithm is proposed. The methodology is applied to design beam cross-sections in which both torsion and warping constants are of interest.

In the next chapter, a new approach is formulated to tackle topology optimization problems where both the external load and stiffness matrix are boundary-dependent. The proposed algorithm is based on the concept of density gradient methods, a class of density-based approaches that do not require additional fields or parametric curves for the boundary recognition, unlike other methods in the literature. The core idea behind density gradient methods is that boundaries can be identified as regions in the design domain in which the solution exhibits significant gradients, transitioning from material-filled areas to voids. Building on this concept, the new algorithm offers significant advantages over existing density gradient methods. First, it achieves higher accuracy by solving boundary integrals directly, avoiding the transition to volume integrals commonly employed in other methods. Second, it is substantially more efficient due to two key factors: much of the computation occurs during preprocessing, and it avoids the need for fine meshes or frequent remeshing required for accurate volume integral solutions. These benefits are enabled by the distinct algorithmic approach and design choices of this method.

Finally, a concurrent multidimensional topology optimization approach for beams under torsional loading is proposed. This formulation considers the coupling between the one-dimensional beam torsion equation in the longitudinal direction, with the two-dimensional warping function equation in the cross-section. The proposed formulation enables the 3D optimization of non prismatic beams with variable cross-sections, with significantly reduced computational burden compared to full 3D approaches. The cross-sectional torsional and warping constants are accurately computed utilizing the boundary recognition algorithm, previously developed. An iterative multidimensional optimization framework is developed in which first, the cross-sectional properties are determined by solving the two-dimensional differential equation. Then, having these properties, the one dimensional beam equation is solved for the rotations along the beam.