Structural Topology Optimisation using Reaction Diffusion based B-spline Level Set Method

With the rapid progress in additive manufacturing, structural topology optimisation has shown significant promise for crafting structures that are simultaneously lightweight and sturdy. However, traditional methods of structural representation may result in zig-zag boundaries, posing a risk of inaccuracies in structures produced through additive manufacturing. Thus, this project seeks to introduce an innovative structural topology optimisation approach and apply it to tackle diverse engineering challenges with unique objective functions. Inspired by isogeometric analysis, the primary purpose of this thesis is to avoid the drawbacks of geometric dependence on control points and develop a novel parametric level set method based on the framework of B-splines. Therefore, the B-spline coefficients are chosen to be the design variables, and the B-spline based level set function is applied to generate the zero-level contour. In order to converge faster, a reaction diffusion updating scheme is utilised. All the sub-topics of this thesis are built based on the reaction-diffusion based B-spline level set method. The brief introductions of the five sub-topics are listed as follows: The first sub-topic is to propose the novel structural topology optimisation method. The B-spline level set representation framework is presented from B-spline curves to B-spline surfaces. Then, the updating scheme for B-spline coefficients is introduced and the predetermined matrices are created. Next, the algorithm is extended to 3D situations. 2D and 3D numerical examples show the robustness and efficiency of the proposed algorithm. The second sub-topic is to explore the potential of combining RDBLS with body-fitted mesh. The level set-based optimisation method generally uses the rectangular/hexahedral mesh in finite element analysis despite its zero-level contour crossing these elements. Hence, adaptive triangular mesh is employed to improve the similarity of the finite element model with the smooth structure presented by the level set method. Also, the parametric level set function is still a linear combination of cubic B-spline basis functions, which can increase optimisation flexibility and structural smoothness. The third sub-topic is to propose a simple and compact MATLAB code for beginners interested in exploring the parametric level set method. Combining the advantages of the B-spline-based level set function and the reaction-diffusion updating scheme, the proposed algorithm achieves a smooth structural profile and shows fast convergence. Additionally, an extension code by use of repeated knots is also provided to deal with the boundary connection of neighbouring design domains. The fourth sub-topic is to apply the abovementioned MATLAB code in designing porous materials with prescribed strength. The inverse homogenization method can tailor some mechanical and physical effective properties by laying out materials in a periodic representative volume element. However, studies on strength design are yet to be developed because of the difficulties in numerically retrieving its value. Unlike traditional asymptotic homogenization, the fast Fourier transform-based homogenization method based on the augmented Lagrangian approach uses a Green operator in the frequency domain to replace time-consuming finite element analysis and inherently meet the periodic boundary conditions. Thus, it is developed in this work to retrieve material strength in terms of the von Mises yield criterion. The fifth sub-topic is to extend the abovementioned strength design in bone scaffold design. Large bone defect is a major challenge in orthopaedic surgery, where bone scaffold shows great potential to tackle osseous defects. However, the commonly used scaffold materials, such as metals and alloys, have much higher stiffness and mismatched strength than natural bones. The equivalent Young's modulus and yield stress must be adjusted when using these bulk materials to avoid stress shielding. Therefore, a multi-objective problem is generated to satisfy the requirements of stiffness and strength simultaneously. The inverse homogenisation is achieved by reaction diffusion-based B-spline level set method, where FFT-based homogenisation is applied in both stiffness design and strength design. The 2D & 3D numerical examples are displayed to show the effect of mechanical properties mimicry. Overall, the research outcomes demonstrate that the proposed reaction diffusion-based B-spline level set method can be applied in structural topology optimisation and export smooth boundaries and display highly efficiency. Applications in 3D complex structures, strength design, bone scaffold design prove that it has strong compatibility to solve different practical problems.