Isogeometric topology optimization for auxetic metamaterials and structures

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It is known that topology optimization is located at the conceptual design phase, which can effectively determine the numbers, connectivity and existence of holes in the structural design domain and evolve design elements to improve the concerned performance. General speaking, topology optimization works as an important tool to seek for the optimal material distribution, which has been identified as one of the most promising sub-field of structural optimization due to its superior features occurring in the conceptual design stage without prior knowledge of the design domain. In the current work, the main intention is to propose a novel numerical method for the topology optimization with more effectiveness and efficiency for the single-material structures and structures with multiple materials. Meanwhile, the proposed topology optimization method is also applied to implement the rational design of auxetic metamaterials and auxetic composites. In Chapter 1, we provide a brief description for the main intention of the current work. In Chapter 2, the comprehensive review about the developments of topology optimization, isogeometric topology optimization and the rational design of auxetic materials is provided. In Chapter 3, a more effective and efficient topology optimization method using isogeometric analysis is proposed for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF mainly involves two steps: (1) the smoothness of nodal densities is improved by the Shepard function; (2) the higher-order NURBS basis functions are combined with the smoothed nodal densities to construct the DDF with the continuity. A topology optimization formulation to minimize the structural mean compliance is developed using the DDF and isogeometric analysis (IGA) is applied to solve structural responses. An integration of the geometry parametrization and numerical analysis offer several benefits for the optimization. The Chapter 4 intends to develop a Multi-material Isogeometric Topology Optimization (M-ITO) method. Firstly, a new Multi-material Interpolation model is established with the use of NURBS (Non-uniform Rational B-splines), termed by the “N-MMI” model, which mainly involves three components: (1) Multiple Fields of Design Variables (DVFs); (2) Multiple Fields of Topology Variables (TVFs); (3) Multi-material interpolation. Two different M-ITO formulations are developed using the N-MMI model to address the problems with multiple volume constraints and the total mass constraint, respectively. The decoupled expression and serial evolving of the DVFs and TVFs can effectively eliminate numerical difficulties in the multi-material problems. In Chapter 5, the proposed ITO method is applied for the systematic design of both 2D and 3D auxetic metamaterials. An energy-based homogenization method (EBHM) to evaluate the macroscopic effective properties is numerically implemented by IGA, with the imposing of periodic boundary formulation on material microstructure. An ITO formulation for 2D and 3D auxetic metamaterials is developed using the DDF, where the objective function is defined as a combination of the homogenized elastic tensor. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. In Chapter 6, the proposed M-ITO method is applied for the systematic design of both 2D and 3D auxetic composites. The homogenization, that evaluates macroscopic effective properties of auxetic composites, is numerically implemented by IGA, with the imposing of the periodic boundary formulation on composite microstructures. The developed N-MMI model is applied to describe the multi-material topology and evaluate the multi-material properties. A topology optimization formulation for the design of both two-dimensional (2D) and three-dimensional (3D) auxetic composites is developed. Finite element simulations of auxetic composites are discussed using the ANSYS to show different deformation mechanisms. Finally, conclusions and prospects are given in Chapter 7.
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