Topology optimization of geometrically nonlinear structures

Abstract

Incorporating geometric nonlinearity into topology optimization arises two main challenges: (1) high computational costs of solving the nonlinear governing equations, and (2) convergence difficulties in the analysis due to “low-density areas" compressed by neighboring stiffer regions. These challenges are addressed in this thesis, which then, focuses on applying topology optimization of geometrically nonlinear structures to design compliant mechanisms tracing user-definedmotion paths.

First of all, Chapter 2 addresses the challenge of high computational cost by introducing reduced-order models (ROMs). The proposed method targets ROM bases consisting of a small set of base vectors, while maintaining accuracy. To this end, several fully automated techniques are developed and integrated for updating and maintaining the ROM basis, with path derivatives incorporated to better capture the behavior of highly flexible structures. In parallel, approximate sensitivity analysis methods are introduced to simplify computations and improve the efficiency of the optimization process. The effectiveness of the ROMs is demonstrated by numerical examples, which showsubstantial reductions in computational effort.

However, the efficiency of the proposed ROMs can deteriorate when faced with the second challenge, i.e., convergence difficulties arise fromthe compression of low-density regions. Such compression typically leads to “inside-out" elements in 2D structures or spurious local buckling in shells and plates. The former, i.e., “inside-out" elements, often causes computational divergence, while the latter, i.e., spurious local buckling, though not always divergent, can significantly increase the number of iterations required for convergence. These spurious instability modes are inevitably incorporated into the proposed ROM basis, which can render ROM analyses even less efficient than full-order ones. To mitigate this problem, two strategies are investigated in Chapter 3: (1) removing spurious instability modes from the ROM basis, and (2) eliminating them directly from the underlying physics. The latter approach is also applicable to standard FOM analyses. Their effectiveness is demonstrated through shell model examples, which provide detailed insights into the benefits and limitations of each approach.

To move beyond algorithmic developments, Chapters 4 and 5 apply geometrically nonlinear topology optimization to the practical design of path-generation compliant mechanisms. A main challenge in this context is ensuring material connectivity among the input, output, and support fixtures within the design domain. This challenge is addressed in Chapter 4 using a simple yet effective formulation that combines compliance and volume constraints. Here, compliance upper bounds are specified according to engineering requirements, while volume constraints are enforced through a proposed three-phase scheme. Building on this foundation, Chapter 5 applies the formulation to design path-generation mechanisms capable of tracing long-distance motion paths. In particular, shells and plates are explored because their compactness and flexible nature make them especially effective for achieving such motions. Finally, experiments on 3D printed prototypes validate the effectiveness of the proposed formulations in producing functional designs.