Computationeel efficiënte topologie-optimalisatie op basis van Proper Generalized Decomposition

Topology optimization is a powerful numerical method for determining the optimal material distribution within a given design domain while accounting for relevant constraints. Despite its proven effectiveness, its widespread adoption in the industry remains limited due to the high computational cost of solving the large-scale mathematical models involved in real-world applications. This thesis investigates the use of Model Order Reduction (MOR) to improve the computational efficiency of topology optimization. MOR techniques reduce the complexity of large-scale simulations by projecting the problem onto a lower-dimensional basis, significantly decreasing computation time and storage while preserving essential model characteristics. In particular, we use Proper Generalized Decomposition (PGD), a MOR approach that constructs a reduced-order representation on-the-fly during the solution process. Unlike projection-based techniques such as Singular Value Decomposition, which require full-order solutions to generate a basis, PGD incrementally builds basis functions as the problem is solved. While PGD has been applied successfully to solve various types of partial differential equations (PDEs), its potential in topology optimization remains largely unexplored. PGD alleviates the computational burden of solving high-resolution PDEs by approximating the solution as a sum of products of lower-dimensional basis functions, reducing problem complexity. This is particularly beneficial in structural dynamics problems, where fine discretization is required for both spatial and frequency domains. Unlike conventional methods that solve the governing PDEs separately for each frequency, PGD solves the problem for all frequencies simultaneously. This leads to a higher-dimensional problem, which is then reduced by expressing the solution as a sum of products of basis functions in the spatial and frequency domains. These basis functions are computed on-the-fly, following a successive enrichment scheme. They are obtained by solving a system of nonlinear equations, which is efficiently solved by means of an alternating direction method. To facilitate gradient based design, design sensitivities are computed using the adjoint method. Although integrating the adjoint method with PGD formulations introduces additional complexity due to the use of an enrichment scheme, the resulting adjoint problem simplifies to a linear system that is computationally less expensive than the forward problem. To ensure consistency, the design sensitivities are fully derived and validated using the discretize-then-differentiate approach. The methodology is tested with two topology optimization case studies: the minimum-weight design of a 2D clamped beam under dynamic loading and a 3D ribbed floor slab subjected to excitation at the base. Beyond space-frequency separation, PGD also enables space-space separation, which is particularly useful for problems with separable geometries. This allows the Young's modulus of a problem to be expressed as a product of lower-dimensional functions, leading to further computational savings. To demonstrate the effectiveness of this approach, PGD is applied to the topology optimization of a 3D ribbed floor for minimum elastic compliance and a heat sink for minimum thermal compliance. Results show that PGD can reduce computational costs by factors of 100 to 500 compared to full 3D direct solvers and by a factor of 5 compared to state-of-the-art iterative solvers such as the multigrid conjugate gradient method.