Topology optimization of compliant mechanisms with multiple degrees of freedom

Research output: ThesisDissertation (TU Delft)

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High-tech equipment critically relies on the precise and reliable fine alignment of components such as mirrors and lenses for calibration and adaptation of instrumentation. To meet the ever-increasing requirements on precision, engineers typically resort to monolithic compliant mechanisms. These mechanisms gain mobility by deformation of the material, eliminating any friction and backlash. The design of compliant mechanisms with multiple degrees of freedom, so-called multi-DOF compliant mechanisms, is complex, and the resulting designs are sensitive to exhibit crosstalk between the actuation modes. The manual manipulation of coupled mechanisms is unintuitive and time-consuming, and automated actuation requires complex control scenarios. Computational approaches can greatly improve designing multi-DOF compliant mechanisms without such undesired characteristics. Topology optimisation methods take a mathematical approach to designing a structure. Such methods optimize the material layout in a design domain for a given performance measure, considering a provided set of boundary conditions, loads and design constraints. Topology optimization methods have demonstrated capable as synthesis tools for designing single-DOF compliant mechanisms. The development of topology optimisation approaches for solving multi-DOF compliant mechanism design problems is relatively undeveloped and comes with severe challenges. These design problems typically involve many different loading conditions and stringent design requirements, increasing the complexity of the optimisation problem and required computational effort. Available formulations only partly address these issues and tend to be complex to understand, implement, and use or have limited applicability. This dissertation focuses on developing topology optimisation approaches for synthesising multi-DOF compliant mechanisms with relatively short strokes, which justifies the use of linear elasticity theory. The objective is the development of a topology optimisation problem formulation that is simple to understand, implement and use, applicable to a wide range of problems and relatively computationally efficient. When parts of the structure are forced into a prescribed motion, the energy contained in a compliant system is an indirect measure of the resistance to this motion. One can thus capture the characteristic stiffness of arbitrarily complex kinematics using a single energy measure. The main discovery of this study is that topology optimisation problem formulations based on specific combinations of such energy measures provide a unique combination of simplicity, versatility and computationally efficiency. While similar to the classic compliance minimisation problem, the proposed generalisation for compliant mechanism problems holds similar advantageous optimisation properties. It minimises the number and strictness of design constraints simplifying the optimisation problem. Despite the advantages, such integrated measures come with the loss of exact control over individual displacements and stiffnesses. This dissertation demonstrates the broad applicability of this formulation to the design of high-resolution decoupled multi-DOF compliant mechanisms, as well as flexures and shape-morphing structures. Furthermore, this dissertation studies the impact of design for additive manufacturing constraints on the optimization of compliant mechanisms. A critical observation to designing practically relevant compliant mechanisms is that design for additive manufacturing considerations predominantly impacts thin flexural elements. One may exploit the observation of local impact to reduce the typically negative impact of design for additive manufacturing constraints on the performance of the optimised compliant system. This dissertation introduces a computationally efficient approach to redesign the most critical regions of compliant mechanisms considering design for additive manufacturing constraints while minimizing the negatively influence on the mechanism performance. This redesign approach allows for high-resolution design and accurate modelling of sensitive flexures, providing solutions that are superior to imposing the same restrictions on the entire design domain without substantial additional computational cost. This dissertation also addresses the aspect of computation effort. The relationship between input and output ports defines the working principle of a compliant mechanism. As a result, the response functions standard in multi-DOF design problems are typically a function of the motion at those ports, and the loads often apply to the same ports. This property provides the possibility to reduce computational costs. Such optimisation problems are typically characterised by multiple combinations of boundary and loading conditions and many constraint functions, substantially increasing the computational cost of calculating the response functions and accompanying sensitivity analysis. By exploiting the characteristics of the multi-DOF compliant mechanism design problem and using static condensation, we demonstrate increased computational efficiency in solving problems with different boundary conditions. Although this is a well-known technique, the use of static condensation and corresponding advantages have not been studied in-depth in this context. The sensitivities of the procedure can be calculated without solving other systems of equations of high dimensionality, making this approach very suitable for use in gradient-based optimisation methods. In addition to problems with varying boundary conditions, there is a significant potential for reducing the computational cost for problems involving similar boundary conditions, common in multi-DOF compliant mechanism design problems. Although not commonly detected, such problems contain linear dependencies between the encountered applied loads and adjoint loads. Manually keeping track of such dependencies becomes tedious for real-world design problems that become increasingly involved. This dissertation introduces a linear-dependency-aware-solver that can efficiently detect such linear dependencies between all loads to automatically avoid solving unnecessary equations. In summary, insights and tools are provided to efficiently and effectively (re)design practically relevant high-resolution three-dimensional multi-DOF compliant mechanisms. Energy-based measures under prescribed motion scenarios offer a versatile and straightforward basis for optimising problem formulations, allowing quantitative control over mechanism stiffness and motion transmission. We envision that such problem formulations will find widespread use in industry to design complex compliant systems such as implants, optical mounts and manipulation stages.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Delft University of Technology
  • van Keulen, A., Supervisor
  • Langelaar, Matthijs, Supervisor
Thesis sponsors
Award date28 Nov 2022
Electronic ISBNs978-94-6366-627-5
Publication statusPublished - 2022




  • Topology optimization
  • compliant mechanisms
  • Computational mechanics


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