The concept of topology optimization has gained considerable popularity after three decades of development due to its innovative design abilities and economical material consumption. Particularly, the topology optimization has become a viable and effective tool for design and optimization of cellular materials and structures, which are characterized with downscale porous microstructures. By virtue of the topology optimization for the downscale structure design, cellular materials can be tailored to exhibit extraordinary or extreme properties and cellular structures can be optimized to achieve high performance and excellent multi-functional responses.

In the topology optimization of cellular materials and structures, the density-based method has been mostly discussed and widely...[ Read more ]

The concept of topology optimization has gained considerable popularity after three decades of development due to its innovative design abilities and economical material consumption. Particularly, the topology optimization has become a viable and effective tool for design and optimization of cellular materials and structures, which are characterized with downscale porous microstructures. By virtue of the topology optimization for the downscale structure design, cellular materials can be tailored to exhibit extraordinary or extreme properties and cellular structures can be optimized to achieve high performance and excellent multi-functional responses.

In the topology optimization of cellular materials and structures, the density-based method has been mostly discussed and widely applied for representing the structures, due to its conceptual simplicity and numerical efficiency. However, the discretized designs may forbidden a further modification requirement common in continuous models. Moreover, this method may yield designs with ambiguous models and geometric defects, which are typically inapplicable for realistic applications. In the cellular material design, the optimization will eventually lead to results with stepwise boundaries, even for ideal black-and-white solutions. In the cellular structure design, the disconnection or mismatch between adjacent microstructures becomes a common issue in reported studies. From our point of view, all these problems can be attributed to the elimination of the geometric boundaries. By simplifying the optimization problem to a material distribution problem, the geometric boundaries disappear and thus they cannot be well treated by the optimization. Alternatively, the level set method has shown great potential in dealing with modeling and geometric problems, due to its boundary-based representation in essence. Therefore, this thesis aims to facilitate the cellular structure modeling and fix the geometric defects that may encountered in the topology optimization of cellular materials and structures by leveraging the level set representation and developing new level set methods.

Firstly, to prevent property deviations when interpreting the numerical solutions of cellular materials involving stepwise boundaries, a two-step design method is developed. Through a fast iteration using the density-based method and then a refinement using the level set method, the optimized microstructures will possess explicit and smooth boundaries, which assures that they can be 3D printed without tedious post processing. Consequently, the method can effectively tailor cellular materials with desired mechanical properties in an efficient and highly consistent way for Additive Manufacturing. Without loss of generality, three microstructures with different Poisson’s ratios are designed. A high property consistency between the design results, the numerical estimations and the experimental tests is observed, which demonstrates the method for creating reliable microstructures for 3D printing and practical use.

Next, to design cellular structures with functionally graded distribution and guarantee perfect connection between adjacent microstructures, a variable cutting level set method is developed. Two basic functions are adopted by this method. One is the basic level set function for representing the periodic cellular microstructures, which can be either built directly or obtained by for instance the above two-step method. The other is the cutting function for determining the gradient pattern of the microstructures and attaching a solid covering skin to the cellular structure. Due to the periodicity of the microstructures and the continuity of the cutting function, the resulted cellular structures are guaranteed to possess perfect connections between adjacent microstructures, without any extra constraints. The meaningfully and functionally graded patterns with connected microstructures are obtained in all the numerical examples with various optimization setups.

Finally, to facilitate the modeling of the cellular structures and overcoming the major difficulties in the classical level set method, a new cellular level set method in B-splines is developed. To support the cellular structure design in a natural way, the design domain is divided into a set of subdomains. In each subdomain, a level set function in B-splines is defined independently and separately. Benefit from the intrinsic properties of the B-splines, several powerful functionalities, e.g. fast reconstruction and discretization adjustment, are provided for modeling of cellular structures. During the optimization, this B-spline parameterization also simplifies the level set evolution and offers a set of simple equality constraints for guaranteeing the connection and controlling the smoothness between neighboring microstructures. All these modeling convenience and optimization advantages are demonstrated by numerical examples.

To sum up, this thesis exploits the level set method for solving the typical geometric issues occurred in the topology optimization of cellular materials and structures. By virtue of the level set representation, the cellular material can present explicit and smooth boundaries and thus maintain a high property consistency. By introducing a variable cutting function, the cellular structures are allowed to achieve grade distribution with perfect connection. Moreover, a new B-spline based parameterization scheme is developed, which provides unprecedented advantages and convenience for general modeling, manipulating and optimization of cellular structures. Through the work of this thesis, the powerful ability of the level set method in handling geometric issues is demonstrated, which paves a way for the future study on the topology optimization of cellular materials and structures.