面向应力约束的独立连续映射方法

大多数已有的拓扑优化研究为系统刚度最大化设计,尤其以体积比约束下的静态柔顺度最小化问题为典型.从工程角度出发,结构强度设计至关重要.以往的应力研究表明,应力约束拓扑优化存在着奇异性、约束数目庞大、高度非线性特性等诸多数值困难.为了实现应力约束下的拓扑优化设计,采用归一化p范数应力指标以减少单元应力约束数目.遵循独立连续映射建模方式,引入密度变量的倒变量函数作为设计变量.推导了应力约束函数和体积目标函数对设计变量的敏度,并基于一阶和二阶泰勒近似得到各自的显式表达式.通过构造的系列二次规划子问题,原拓扑优化问题采用序列二次规划算法高效求解.二维数值算例考察了结构刚度和强度设计结果的异同,以及不同应力约束上限值对应力约束拓扑优化结果的影响.通过提出方法与传统变密度法结果的比较,说明提出的独立连续映射方法在应力约束下具有可行性和有效性.优化结果也表明了考虑应力约束的连续体拓扑优化具有必要性. 

INDEPENDENT CONTINUOUS MAPPING METHOD FOR STRESS CONSTRAINT

Most existing study on topology optimization have concentrated on maximizing the system stiffness. Especially, the minimization of static compliance subject to the volume fraction is widespread in the formulation. From the engineering point of view, structural strength design is of vital importance. Past study on stress constraint have shown that an amount of numerical difficulties with the stress-constrain topology optimization exist including the so-called singularity, vast of stress constraints, highly nonlinear behavior and so on. To achieve the topological design under stress constraint requirement, the normalized stress measure using p-norm function is adopted for the reduction of stress constraints. Following the modeling manner of independent continuous mapping method, the reciprocal function of relative density is regarded as the design variables. The sensitivities of stress constraint and volume objective with respect to the design variable are derived, and their explicit expressions are formulated based on the first-order and second-order Taylor approximation respectively. By setting up the sub-problem in the form of a quadratic program, the original topology optimization problem is efficiently solved using the sequential quadratic programming approach. The difference between stiffness and strength design, as well as the effect of various upper bounds of stress value on the optimized results for stress constraint are investigated in 2D numerical examples. Through the comparison of the proposed method and traditional variable density method, the feasibility and effectiveness of the proposed optimization approach in stress constrained problems are verified. The results also demonstrate that the consideration of stress constraint in continuum structure is indispensable.