Structural topology optimisation for additive manufacturing using body-fitted mesh

Topology optimisation (TO) has the potential to be widely applied in additive manufacturing (AM) to produce innovative and efficient structures, allowing engineers to optimise the aesthetics and performance in the conceptualisation stage. However, challenges arise in generating smooth boundaries to improve the finite element analysis (FEA) accuracy and achieve structural aesthetics. This demand has demonstrated a vital practicability concern that needs to be addressed before TO results can be used in AM. The primary aim of this thesis is to introduce a body-fitted triangular/tetrahedral mesh into TO, significantly increasing boundary smoothness and optimisation accuracy. The void elements are excluded from planar/spatial optimisation to save computation time, even though a versatile meshing algorithm based on solving a force equilibrium generates them. Based on the zero-level contour, both the level set (LS) method and the bi-directional evolutionary structural optimisation (BESO) method can be combined with the proposed method, which avoids the grey-scale problem and closely matches elegant solid-void interfaces. Numerical examples in 2D and 3D converge within dozens of iterations and illustrate ultra-smooth boundaries, verifying the effectiveness and robustness of the proposed method. This project was the first to report the TO methods incorporating reaction-diffusion equation and nonlinear diffusion regularisation using the body-fitted mesh to address the issue of zig-zagging shapes obtained using structural rectangular mesh. The key finding was that the zig-zag interfaces between the void and solid phases in the optimised pattern greatly affect the structural aesthetics and analysis accuracy, especially in fluid, optic, and electromagnetic optimisation problems. The proposed method is computationally complex due to the mesh generation procedure before FEA in each iteration. However, adopting the bi-section method and large element removal ratio (ERR) boosts convergence and reduces the computational cost. The significance of this research is to improve the TO accuracy and efficiency, ensuring elegant configurations that can be directly produced by AM without post-processing. The brief of the four topics included in this thesis are listed as follows: The first research topic integrates the reaction diffusion-based level set (RDLS) method and the body-fitted mesh for TO. The LS optimisation method can express smooth boundaries with the zero-level contour of the LS function. However, most applications still use rectangular/hexahedral mesh in FEA, resulting in unsmooth solid-void interfaces. Besides genuinely expressing smooth boundaries, such a body-fitted mesh can increase FEA accuracy. Unlike the traditional upwind algorithm, the proposed method breaks through the constraint of Courant–Friedrichs–Lewy stability condition with an updating scheme based on FEA. The presented benchmark compliance minimisation and compliant mechanism results are aesthetically very pleasing. This developed method can be employed in AM to generate elegant, smooth, and high-quality structures with better performance in the future. The second research topic illustrates good novelty in terms of combining BESO with body-fitted meshes for the first time. The BESO method effectively uses basic strategies of removing and adding material based on element sensitivity. However, challenges remain in generating smooth boundaries to improve the FEA accuracy and achieve structural aesthetics. This work develops a body-fitted triangular/tetrahedral mesh generation algorithm to yield smooth boundaries in the BESO method. The optimisation problem is regularised by adding a diffusion term in the objective function. We found that the first has the best regularisation effect of Lorentzian, Tikhonov, Perona–Malik, Huber, and Tukey functions. The ultrasmooth boundaries of the optimised structures in 2D/3D scenarios are naturally obtained from the proposed method, not from smoothing post-processing. Compared with the optimisation toolbox in Abaqus, the example of the automotive control arm demonstrates smoother boundaries and lower average mean compliance. The third research topic implemented TO in Matlab using the body-fitted mesh. This chapter adopted a 172-line Matlab code TriTOP172 to implement the BESO method in the unstructured triangular mesh. Its most significant feature is the elimination of zig-zag boundaries essentially existing in the commonly-used rectangular mesh. The code uses 40 lines for preliminary setup and optimisation iterations and a 78-line function to obtain the body-fitted mesh by solving the balance of a truss network. It also has a 20-line function of nonlinear diffusion to further smooth boundaries and control structure complexity and a 34-line function of finite element analysis. Numerical examples of compliance minimisation are provided to assist readers in understanding the algorithm and its implementation. This code can be employed with further extensions to solve complicated conceptual design problems efficiently in several engineering fields. The educational Matlab program is accessible on the website and displayed in Appendix D. The fourth research topic is the narrow band level set (NBLS) method, directly moving the narrow band points around the solid-void interfaces. Currently, most numerical examples are designed inside regular design domains (e.g., rectangle in 2D, hexahedron in 3D). The design flexibility and adaptivity are required to be improved in topology optimisation before combining AM. Thus, we proposed a new optimisation method named narrow band-based level set (NBLS) topology optimisation. The basic concept of the proposed method is to move the boundary points using a simulated velocity field to generate new topologies in each iteration. Compared with other optimisation methods, the proposed NBLS method produces structures with better objective function values and improves efficiency while ensuring boundary smoothness. In summary, the research outcomes demonstrate that the proposed innovative topology optimisation methods can be combined with the body-fitted mesh to improve optimisation efficiency and effectiveness. The optimisation results can be directly exported to the computer-aided design software and then fabricated using AM.