Research Of Topology Optimization For Continuum And Multiple Material Structures Based On Meshless And Nodal Variable Method

It is of great importance for the development of new methods to create the best possible topology or layout for given design objectives and constraints at a very early stage of the design process(the conceptual and project definition phase). Thus, over the past two decades, substantial efforts of fundamental research have been devoted to the field of development of efficient and reliable procedures for solution of such problems. It is worth noting that element-wise design variables are conventionally used to formulate topology approaches. It means that the FEM method is used to interpolate the displacement fields as well as material distribution inside the design domain simultaneously. The material properties are assumed constant within each element. Although using element-based design variables seems natural in the formulation of optimal material distribution problems, it still presents some difficulties. For the aforementioned numerical difficulties to be pressed, many attempts have emerged several alternative methods for topology optimization of structures based on meshless method. Currently, multiphase materials have been widely used in many advanced engineering applications such as aerospace, aircraft, biomedical, electronic equipment, superconducting coil and so forth. The question whether freedom to use multiple materials available to a designer would lead to a different and more optimal topology has prompted the present investigation. With multiple candidate materials, the multi-material topology optimization problem is posed as one of seeking not only the distribution of various materials to be used in the different regions of the optimized form but also of simultaneously finding the optimum form of the structure within a given envelope or a packaging space. Using a novel Young’s modulus interpolation scheme, the corresponding modified topological derivative is derived and used to deal with multiple materials topology optimization.The author’s major contributions are outlined as follows:1. A meshless Galerkin pareto optimal method is proposed for topology optimization of continuum structures in this paper. The compactly supported radial basis function(CSRBF) is used to create shape functions. Considering the pareto optimality theory, the initial single objective topology optimization problem is transformed into multi objective problem. The optimum solution is traced via the pareto optimal frontier in a computationally effective manner. Finally, several examples are used to prove the validity and effectiveness of the proposed approach.2. A nodal variable ESO(BESO) method is proposed for topology optimization of continuum structures in this paper. The initial discrete-valued topology optimization problem is established as an optimization problem based on continuous design variables by employing a material density field into the design domain. The density field, with the Shepard family of interpolation, is mapped on the design space defined by a finite number of nodal density variables.3. An improved meshless density variable approximation is incorporated into BESO method for topology optimization of continuum structures in this paper. The essential boundary condition is enforced by using the compactly supported radial basis function(CSRBF). The Shepard function is used to create a physically meaningful dual-level density approximation. Numerical examples show that the proposed method is feasible and fidelity for the topology optimization of continuum structures. The common numerical instabilities of BESO method do not exist in the final results.4. The aforementioned method is successful extended to multiple materials topology optimization. By applying a material density field into the design domain, the initial discrete-valued topology optimization problem is established as a continuous-variables optimization problem. It is flexible to handle complex topologies. Numerical examples demonstrate the validity and effectiveness of the improved method. The characteristics of multiple material topology optimizations, comparing to single-material topology optimization, are shown simultaneously.5. A Young’s modulus interpolation scheme is employed as an alternative approach to the popular Young’s modulus methods of rule of mixtures for multi-material modeling. Therefore, a modified topological derivative method is adapted for material and topology representation based on the Young’s modulus interpolation scheme.6. Considering the new and modified topological derivative, a new multiple materials optimization model is yielded based on Pareto optimality theory. With a penalty material interpolation scheme, the above-mentioned optimization model is used to deal with various types of topologies involving possibly several materials.