A promising new treatment for large and complex bone defects is to implant specifically designed and additively manufactured synthetic bone scaffolds. Optimizing the scaffold design can potentially improve bone in-growth and prevent under- and over-loading of the adjacent tissue. This study aims to optimize synthetic bone scaffolds over multiple-length scales using the full-scale topology optimization approach, and to assess the effectiveness of this approach as an alternative to the currently used mono- and multi-scale optimization approaches for orthopaedic applications. We present a topology optimization formulation, which is matching the scaffold's mechanical properties to the surrounding tissue in compression. The scaffold's porous structure is tuneable to achieve the desired morphological properties to enhance bone in-growth. The proposed approach is demonstrated in-silico, using PEEK, cortical bone and titanium material properties in a 2D parameter study and on 3D designs. Full-scale topology optimization indicates a design improvement of 81% compared to the multi-scale approach. Furthermore, 3D designs for PEEK and titanium are additively manufactured to test the applicability of the method. With further development, the full-scale topology optimization approach is anticipated to offer a more effective alternative for optimizing orthopaedic structures compared to the currently used multi-scale methods.

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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. @en

Real-world structural optimisation problems involve multiple loading conditions and design constraints, with responses typically depending on states of discretised governing equations. Generally, one uses gradient-based nested analysis and design approaches to solve these problems. Herein, solving both physical and adjoint problems dominates the overall computational effort. Although not commonly detected, real-world problems can contain linear dependencies between encountered physical and adjoint loads. Manually keeping track of such dependencies becomes tedious as design problems become increasingly involved. This work proposes using a Linear Dependency Aware Solver (LDAS) to detect and exploit such dependencies. The proposed algorithm can efficiently detect linear dependencies between all loads and obtain the exact solution while avoiding unnecessary solves entirely and automatically. Illustrative examples demonstrate the need and benefits of using an LDAS, including a run-time experiment.

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In topology optimization, the state of structures is typically obtained by numerically evaluating a discretized PDE-based model. The degrees of freedom of such a model can be partitioned in free and prescribed sets to define the boundary conditions. A multi-partition problem involves multiple partitions of the same discretization, typically corresponding to different loading scenarios. As a result, solving multi-partition problems involves multiple factorization/preconditionings of the system matrix, requiring a high computational effort. In this paper, a novel method is proposed to efficiently calculate the responses and accompanying design sensitivities in such multi-partition problems using static condensation for use in gradient-based topology optimization. A main problem class that benefits from the proposed method is the topology optimization of small-displacement multi-input–multi-output compliant mechanisms. However, the method is applicable to any linear problem. We present its formulation and an algorithmic complexity analysis to estimate computational advantages for both direct and iterative solution methods to solve the system of equations, verified by numerical experiments. It is demonstrated that substantial gains are achievable for large-scale multi-partition problems. This is especially true for problems with both a small set of number of degrees of freedom that fully describes the performance of the structure and with large similarities between the different partitions. A major contribution to the gain is the lack of large adjoint analyses required to obtain the sensitivities of the performance measure.

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High-tech equipment critically relies on flexures for precise manipulation and measurement. Through elastic deformation, flexures offer extreme position repeatability within a limited range of motion in their degrees of freedom, while constraining motion in the degrees of constraint. Topology optimization proves a prospective tool for the design of short-stroke flexures, providing maximum design freedom and allowing for application-specific requirements. State-of-the-art topology optimization formulations for flexure synthesis are subject to challenges like ease of use, versatility, implementation complexity, and computational cost, leaving a generally accepted formulation absent. This study proposes a novel topology optimization formulation for the synthesis of short-stroke flexures uniquely based on strain energy measures under prescribed displacement scenarios. The resulting self-adjoint optimization problem resembles great similarity to ‘classic’ compliance minimization and inherits similar implementation simplicity, computational efficiency, and convergence properties. Numerical examples demonstrate the versatility in flexure types and the extendability of additional design requirements. The provided source code encourages the formulation to be explored and applied in academia and industry.

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Compliant mechanisms are crucial components in current and future high-precision applications. Topology optimization and additive manufacturing offer freedom to design complex compliant mechanisms that were impossible to realize using conventional manufacturing. Design for additive manufacturing constraints, such as the maximum overhang angle and minimum feature size, tend to drastically decrease the performance of topology optimized compliant mechanisms. It is observed that, among others, design for additive manufacturing constraints are only dominant in the flexure regions. Flexures are most sensitive to manufacturing errors, experience the highest stress levels and removal of support material carries the highest risk of failure. It is crucial to impose these constraints on the flexure regions, while in others part of the compliant mechanism design, these constraints can be relaxed. We propose to first design the global compliant mechanism layout in the full domain without imposing any design for additive manufacturing constraints. Subsequently we redesign selected refined local redesign domains with design for additive manufacturing constraints, whilst simultaneously considering the mechanism performance. The method is applied to a single-input-multi-output compliant mechanism case study, limiting the maximum overhang angle, introducing manufacturing robustness and limiting the maximum stress levels of a selected refined redesign domain. The high resolution local redesigns are detailed and accurate, without a large additional computational effort or decrease in mechanism performance. Thereto, the method proves widely applicable, computationally efficient and effective in its purpose.

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This paper focuses on density-based topology optimization and proposes a combined method to simultaneously impose Minimum length scale in the Solid phase (MinSolid), Minimum length scale in the Void phase (MinVoid) and Maximum length scale in the Solid phase (MaxSolid). MinSolid and MinVoid mean that the size of solid parts and cavities must be greater than the size of a prescribed circle or sphere. This is ensured through the robust design approach based on eroded, intermediate and dilated designs. MaxSolid seeks to restrict the formation of solid parts larger than a prescribed size, which is imposed through local volume restrictions. In the first part of this article, we show that by proportionally restricting the maximum size of the eroded, intermediate and dilated designs, it is possible to obtain optimized designs satisfying, simultaneously, MinSolid, MinVoid and MaxSolid. However, in spite of obtaining designs with crisp boundaries, some results can be difficult to manufacture due to the presence of multiple rounded cavities, which are introduced by the maximum size restriction with the sole purpose of avoiding thick solid members in the structure. To address this issue, in the second part of this article we propose a new geometric constraint that seeks to control the minimum separation distance between two solid members, also called the Minimum Gap (MinGap). Differently from MinVoid, MinGap introduces large void areas that do not necessarily have to be round. 2D and 3D test cases show that simultaneous control of MinSolid, MinVoid, MaxSolid and MinGap can be useful to improve the manufacturability of maximum size constrained designs.

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The stringent and conflicting requirements imposed on optomechanical instruments and the ever-increasing need for higher resolution and quality imagery demands a tightly integrated system design approach. Our aim is to drive the thermomechanical design of multiple components through the optical performance of the complete system. To this end, we propose a new method combining structural-thermal-optical performance analysis and topology optimization while taking into account both component and system level constraints. A 2D two-mirror example demonstrates that the proposed approach is able to improve the system’s spot size error by 95% compared to uncoupled system optimization while satisfying equivalent constraints.

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