Meshless Analysis-based Moving Morphable Component Approach For Structural Topology Optimization

In the field of structural engineering,the efficient use of material can bring great economic value.Therefore,topology optimization,which aims at placing available material within a prescribed design domain in order to achieve optimized structural performances,has attracted more and more attention.With the rapid development of the theories,many topology optimization algorithms emerge gradually and some significant research results have been obtained.However,most of the existing numerical analysis models of topology optimization are based on fixed meshes.In order to avoid the singular stiffness matrix in numerical analysis,weak material is usually introduced to mimic voids.Although this processing greatly facilitates the structural analysis under the constantly changing topology,it also inevitably increases additional calculation costs and even causes localized modes in eigenvalue analysis problems,which leads to the failure of optimization iteration.On the other hand,in order to capture the structural response more accurately and efficiently,different types of elements are usually adopted to discrete structures according to the geometric characteristics and stress states of the structure.However,the traditional topology optimization method based on implicit geometric description is difficult to directly extract the geometric information of the structure,and the solid element is always employed to discrete structure.Therefore,a fine mesh is often required to capture the structural response,which increases the computational cost of structural analysis.Based on the above analysis,in this thesis,an explicit topology optimization framework without weak material is constructed by employing the Moving Morphable Component method(MMC)and Meshless method.Under this framework,the optimization problems of minimization of the structural compliance,maximization of the fundamental frequency,and the minimization of the combined model-based stiffened plate compliance are studied.The specific research content is as follows:Firstly,the Meshless-based Moving Morphable Component method(ML-MMC)is proposed.In this method,the topology variation of the structure is achieved by optimizing the layout and deformation of a series of explicitly described components.On this basis,the meshless analysis model is only established on the solid region occupied by components,and there is no need to introduce weak material during the numerical optimization process.By adjusting the influence domain of meshless shape function adaptively,the singularity problem of the stiffness matrix caused by the disconnection of components is avoided effectively.The problem of minimizing the structural compliance for solids is studied.Compared with the analysis model of the whole design domain,the proposed method effectively reduces the degree of freedoms of the equilibrium equation and significantly improves the efficiency of numerical analysis.Secondly,under the framework of ML-MMC topology optimization,the problem of maximizing the fundamental frequency for solids is studied.The optimization formulation of the fundamental frequency maximization problem under the proposed framework is given,and the localized model problem in the fundamental frequency topology optimization is discussed.Because the proposed method does not need to introduce weak material during the optimization process,the localized vibration modes caused by weak material can be avoided in essence,and the optimization iteration can be carried out smoothly.Numerical examples demonstrate the effectiveness of the proposed method.Compared with traditional topology optimization methods,the proposed method can obtain an explicit optimal structural design with fewer design variables and degree of freedoms.This advantage is more noticeable when the volume of available solid material is less.Finally,the ML-MMC method is used to optimize the plate and beam combined modelbased stiffened plate structure.The stiffeners,which is the building blocks of topology optimization,is expressed by the topology description function of components in the MMC methodization.On this basis,the beam model is used to simulate the stiffeners,and the geometric characteristics of the beam can be satisfied by controlling the geometric parameters of the components.According to the compatibility of displacements,the multi-point constraint equations between the base plate and the beam are established by employing the meshless numerical analysis technology.Compared with the solid model-based stiffened plate structure,the combined model can not only capture the structural response more accurately,but also easily consider the influence of the cross-section shape of the stiffeners.