Explicit Topology Optimization Approaches Based On Isogeometric Analysis And Applications

Structural optimization design,an effective structural/material design tool,has been widely used in industrial equipments including aerospace,automobile,ship and so on,and achieved some important research results.Structural optimization design can be divided into size optimization,shape optimization and topology optimization.Topology optimization is the core of structural optimization design and can help designers and engineers to propose novel and efficient designs in the conceptual design stage,determining the final topologies,shapes and properties of equipment structures.Considering the problems that the number of design variables is too much,the geometric constraint can NOT be imposed on structures directly,the geometric information can NOT be extracted from the optimal structures and the FEA based on low-order shape functions may result in numerical instabilities in traditional topology optimization methods,we developed two explicit isogeometric topology optimization methods,TOP-IGA-MMC and TOP-IGA-MMV.Based on the TOP-IGA-MMC,we obtained negative Poisson's ratio multi-cell energy absorption structures and negative Poisson's ratio automobile crash boxes.The main contents of this theis are as follows:Firstly,we presented a set of concise and efficient MATLAB codes.The codes can calculate the B-spline basis function values at multiple parameters simultaneously and generate element stiffness matrices rapidly,which lays the foundation for the implementation of the explicit isogeometric topology optimization methods in the following sections.Then,we also proposed a numerical algorithm for searching trimmed elements for the problem that a rectangular structured grid is trimmed by some trimming curves.In this algorithm,boundaries of every element in the grid were segmented into 12 parts,and the grid were trimmed by appropriate discrete trimming curves instead of the original trimming curves.All trimmed elements can be divided into 156 different types according to the intersection points between discrete trimming curves and element boundaries.Numerical examples showed that the algorithm can search trimmed elements of any rectangular structured grid effectively and efficiently.Secondly,in order to address the problems that the number of design variables is too much,the geometric constraint can NOT be imposed on structures directly and the FEA based on low-order shape functions may result in numerical instabilities,we developed an explicit isogeometric topology optimization approach based on MMC,TOP-IGA-MMC.The structural topology is represented by several moving morphable components which are determined by sevral explicit geometric parameters(central coordinates,lengths,widths and inclined angles).TOP-IGA-MMC employed NURBS to model the given design domain and the NURBS-based IGA for structural analysis.Benchmark examples demonstrated its effectiveness and robustness.Comparing with the original MMC-based topology optimization method,the numerical stability of TOP-IGA-MMC is much higher.In addition,considering the problem that the geometric information can NOT be extracted from the optimal structures,we also developed an explicit isogeometric topology optimization approach based on MMV,TOP-IGA-MMV.The void material regions were represented by MMVs with closed B-spline boundary curves,and the control points of closed B-spline curves are determined by several explicit geometric parameters(center coordinates and distances from the central point to independent control points).We also employed NURBS to model the design domain,so we can select different identification points(control points,nodal points,Gaussian points and Greville points)to form the Young's modulus and volume fraction of NURBS elements on a sparse mesh and take more dense identification points for plotting.As a result,we can obtain explicit clear structural boundaries and arbitrary high-resolution topologies.Numerical benchmark examples demonstrated the effectiveness of TOP-IGA-MMV.By comparing with the traditional SIMP method and the TOP-IGA-MMC method,the efficiency of TOP-IGA-MMV was also demonstrated.Finally,we derived and implemented the energy-based isogeometric homogenization.numerical examples showed that the homogenized elasticity matrix obtained by the energy-based isogeometric homogenization method is equal to that obtained by the finite element homogenization method within a tolerated error.On this basis,we applied the isogeometric SIMP method and the TOP-IGA-MMC method to the optimization design of negative Poisson's ration base cells.Considering that the optimization and design of traditional negative Poisson's ratio automobile crash boxes is only on the level of size optimization,we applied one of the base cells obtained by the TOP-IGA-MMC method to the design of automobile crash boxes and obtained two kinds of negative Poisson's ratio energy absorption structures.After that,we simulated the RCAR low-speed crash(front impact)in Abaqus for the obtained energy absorption structures.The results showed that the obtained energy absorption structures can meet the standard of RCAR low-speed crash and have smaller maximum collision reaction force and higher average collision reaction force,which not only demonstrates the correctness of the proposed algorithms and extended their applications,but also present a novel base cell in the topology optimization stage and provide a reference for the design and manufacturing of automobile crash boxes.