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.

We propose a topology optimization (TO) formulation and related optimization scheme for designing compliant mechanisms following a user-defined trajectory. To ensure the broad applicability and achieve precisely control of the outputs, geometric nonlinearity with incremental solutions are considered. A challenge in the design optimization of these structures is the development of formulations with satisfactory balance between (i) precise trajectory control and (ii) proper connectivity between the input/output ports and the support. Previously proposed density-based topology optimization formulations typically lack the promotion of the desired load-transferring connections, or usually complicate the design using mixed shape, size, and topology variables to enforce a minimum connectivity. To simplify design progress using exclusive topology variables, i.e., purely density-based TO methods, we propose a relatively straightforward formulation involving commonly used response functions, such as compliance and volume as constraints. For the constraints, the paper provides a scheme for defining corresponding upper limits. Numerical examples of challenging shell and plate design optimization problems demonstrate the effectiveness of the proposed formulation and scheme in the generation of load-transferring connections while limiting the impact on the performance of the path generation functionality.

High computational costs are encountered in topology optimization problems of geometrically nonlinear structures since intensive use has to be made of incremental-iterative finite element simulations. To alleviate this computational intensity, reduced-order models (ROMs) are explored in this paper. The proposed method targets ROM bases consisting of a relatively small set of base vectors while accuracy is still guaranteed. For this, several fully automated update and maintenance techniques for the ROM basis are investigated and combined. In order to remain effective for flexible structures, path derivatives are added to the ROM basis. The corresponding sensitivity analysis (SA) strategies are presented and the accuracy and efficiency are examined. Various geometrically nonlinear examples involving both solid as well as shell elements are studied to test the proposed ROM techniques. Test cases demonstrates that the set of degrees of freedom appearing in the nonlinear equilibrium equation typically reduces to several tenth. Test cases show a reduction of up to 6 times fewer full system updates.

Stiffened shells and plates are widely used in engineering. Their performance is highly influenced by the arrangement, or layout, of stiffeners on the base shell or plate and the geometric features, or topology, of these stiffeners. Moreover, modular design is beneficial, since it allows for increased quality control and mass production. In this work, a method is developed that simultaneously optimizes the topology of stiffeners and their layout on a base shell or plate. This is accomplished by introducing a fixed number of modular stiffeners, which are subject to density-based topology optimization and a mapping of these modules to a ground structure. To illustrate potential applications, several stiffened plates and shell examples are presented. All examples demonstrated that the proposed method is able to generate clear topologies for any number of modules and a distinct layout of the stiffeners on the base shell or plate.

A large number of deployable space structures involve multibody system dynamics, and in order to effectively analyze and optimize dynamic performance, the sensitivity information of multibody systems is often required. At present, the sensitivity analysis methods of multibody systemdynamics, which have been widely used, are mainly finite difference method, direct differentiation method, and adjoint variable method. Among them, the finite difference method is an approximate method; the direct differentiation method and the adjoint variable method are analytical methods. Based on the dynamic problems of the multibody system in the form of differential–algebraic equations, the semi-analytical sensitivity analysis method for multibody system dynamics is proposed in this paper, which combines the simplicity of the finite difference method with the accuracy of the analytical methods. It includes the local semianalytical method based on the element level and the global semi-analytical method based on the system level, of which the latter has higher computational efficiency. Through two numerical examples, the effectiveness and numerical stability of the method are verified. This method not only retains the accuracy and efficiency of the analytical methods, but also simplifies the derivation and coding of analytical formulas by combining with the existing programs. It has stronger versatility and is beneficial to the sensitivity calculation of large-scale complex multibody systems.