形状记忆合金结构拓扑优化设计方法研究

A topology optimization method for shape memory alloy structures

形状记忆合金由于其优良的力学特性得到了广泛关注,并形成了一系列具有变革性的创新应用。为了充分提升形状记忆合金结构的力学性能,提出了一种基于实体各向同性材料惩罚模型SIMP(Solid Isotropic Material with Penalization)的形状记忆合金结构拓扑优化方法。基于ZM宏观唯象本构模型,考虑形状记忆合金材料特性,对拓扑优化过程中引入的中间密度材料的奥氏体和马氏体弹性模量以及相变转变应力进行插值。同时,考虑形状记忆合金本身的材料非线性和结构在大变形下的几何非线性效应,以获得准确的力学响应。采用三密度场法来避免最终设计结果出现的棋盘格现象、网格依赖性和大量中间密度单元。利用超单元法来改善由于低密度单元引起的非线性有限元分析过程的数值不稳定问题。利用伴随法对优化模型中的响应函数进行灵敏度分析。最后,通过二维和三维的数值算例验证了本文的优化设计方法,结果表明本文提出的拓扑优化框架能够对预期性能的形状记忆合金结构方案进行求解。

Shape memory alloy(SMA)has drawn wide attention due to its excellent mechanical properties,and has been innovatively applied in many fields.In order to improve mechanical properties of SMA structures,a topology optimization method based on solid isotropic materials with penalization(SIMP)was proposed.The ZM model was adopted,and the elastic modulus of austenite and martensite as well as the phase change yield stresses of intermediate density materials introduced during the topology optimization process are interpolated.In order to obtain accurate mechanical responses,the material nonlinearity effect and geometric nonlinearity effect are considered.The three-field scheme are used to avoid checkerboarding patterns,mesh-dependency issues and intermediate density elements in the final results.The super element method is used to avoid the numerical instability caused by low density elements in nonlinear finite element analysis.Sensitivities are calculated via adjoint method.Finally,2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed optimization method.The results show that the SMA structures with expected performance can be designed by the topology optimization framework proposed in this paper.