考虑载荷不确定性和刚度不确定性的鲁棒优化问题
发布时间：2018-10-08 09:20
【摘要】：一般的优化问题都假定材料参数、载荷及边界条件等是确定的,然而由于测量误差、加工误差、工艺水平等原因,导致材料或载荷与设计值的偏差在工程中普遍存在,而这往往是结构或材料失效的重要原因,因此在结构和材料设计中考虑不确定性的优化问题十分必要。尤其是随着纳米技术和复合材料的发展,桁架结构材料、周期材料、多空泡沫材料等超轻材料在日常生活以及工业生产中都表现出了其非凡的优势,比如具有吸能抗冲击的能力、较高的比强度等。在各类超轻材料中,多尺度超轻材料的性能尤为突出。因此本文针对考虑载荷不确定性的多尺度优化问题以及桁架结构刚度不确定性问题进行了相关研究。考虑载荷不确定性多尺度优化问题和刚度不确定性的优化问题均是双层优化问题。在双层优化问题中,通过下层优化问题得到最不利情况,利用上层优化问题进行设计变量的更新。对于一个双层优化问题,最主要的是关于下层优化问题的求解,并且由于存在计算效率低、稳定性差、易陷入局部最优解等缺点,使得一般算法很难保证双层优化问题的可置信性。在本文中,对于考虑载荷不确定性的多尺度优化问题,通过SDP放松技巧将下层优化问题转化为一个半定规划问题,这样不仅可以利用已有算法直接求解,同时还可以保证解的可置信性。数值算例表明微观结构中各向同性材料和Kagome类型单胞具有较好的抵抗不确定性载荷的能力,同时也表明了本文提出算法具有较快的收敛速度。而对于考虑刚度不确定性的双层优化问题,则将响应函数在不确定变量的名义值处进行泰勒展开,并利用施瓦兹不等式近似表示出下层最不利情况,将原来双层优化问题转化成一个类似确定性优化的单层优化问题。数值算例表明该算法有较好的精度,尤其是对于大规模结构大大提高了计算效率。
[Abstract]:General optimization problems assume that material parameters, loads and boundary conditions are determined. However, because of measurement error, processing error and technological level, the deviation between material or load and design value is widely existed in engineering. This is often an important reason for the failure of structures or materials, so it is necessary to consider the optimization problem of uncertainty in the design of structures and materials. In particular, with the development of nanotechnology and composite materials, ultra-light materials such as truss structure materials, periodic materials, and porous foam materials have shown their extraordinary advantages in daily life and industrial production. Such as the ability to absorb energy and resist impact, high specific strength and so on. Among all kinds of ultra-light materials, the performance of multi-scale ultra-light materials is particularly outstanding. Therefore, the multi-scale optimization problem with load uncertainty and stiffness uncertainty of truss structures are studied in this paper. The multi-scale optimization problem with load uncertainty and the optimization problem with stiffness uncertainty are bilevel optimization problems. In the bilevel optimization problem, the most disadvantageous case is obtained through the lower level optimization problem, and the design variables are updated by the upper optimization problem. For a bilevel optimization problem, the most important problem is the solution of the lower level optimization problem, and because of the shortcomings of low computational efficiency, poor stability and easy to fall into the local optimal solution, etc. It is difficult for the general algorithm to guarantee the confidence of the bilevel optimization problem. In this paper, the lower level optimization problem is transformed into a semi-definite programming problem by SDP relaxation technique, which can not only be solved directly by using existing algorithms. At the same time, the confidence of the solution can be guaranteed. Numerical examples show that isotropic materials and Kagome type cells in microstructure have better resistance to uncertain loads, and that the proposed algorithm has a faster convergence rate. For the bilevel optimization problem considering stiffness uncertainty, the response function is expanded at the nominal value of the uncertain variable, and the lowest disadvantage is expressed approximately by using Schwartz inequality. The original bilevel optimization problem is transformed into a single-layer optimization problem similar to deterministic optimization. Numerical examples show that the algorithm has good accuracy, especially for large scale structures.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O232
本文编号：2256243
[Abstract]:General optimization problems assume that material parameters, loads and boundary conditions are determined. However, because of measurement error, processing error and technological level, the deviation between material or load and design value is widely existed in engineering. This is often an important reason for the failure of structures or materials, so it is necessary to consider the optimization problem of uncertainty in the design of structures and materials. In particular, with the development of nanotechnology and composite materials, ultra-light materials such as truss structure materials, periodic materials, and porous foam materials have shown their extraordinary advantages in daily life and industrial production. Such as the ability to absorb energy and resist impact, high specific strength and so on. Among all kinds of ultra-light materials, the performance of multi-scale ultra-light materials is particularly outstanding. Therefore, the multi-scale optimization problem with load uncertainty and stiffness uncertainty of truss structures are studied in this paper. The multi-scale optimization problem with load uncertainty and the optimization problem with stiffness uncertainty are bilevel optimization problems. In the bilevel optimization problem, the most disadvantageous case is obtained through the lower level optimization problem, and the design variables are updated by the upper optimization problem. For a bilevel optimization problem, the most important problem is the solution of the lower level optimization problem, and because of the shortcomings of low computational efficiency, poor stability and easy to fall into the local optimal solution, etc. It is difficult for the general algorithm to guarantee the confidence of the bilevel optimization problem. In this paper, the lower level optimization problem is transformed into a semi-definite programming problem by SDP relaxation technique, which can not only be solved directly by using existing algorithms. At the same time, the confidence of the solution can be guaranteed. Numerical examples show that isotropic materials and Kagome type cells in microstructure have better resistance to uncertain loads, and that the proposed algorithm has a faster convergence rate. For the bilevel optimization problem considering stiffness uncertainty, the response function is expanded at the nominal value of the uncertain variable, and the lowest disadvantage is expressed approximately by using Schwartz inequality. The original bilevel optimization problem is transformed into a single-layer optimization problem similar to deterministic optimization. Numerical examples show that the algorithm has good accuracy, especially for large scale structures.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O232
【参考文献】
相关期刊论文 前1条
1 Ole Sigmund;;Manufacturing tolerant topology optimization[J];Acta Mechanica Sinica;2009年02期
相关博士学位论文 前1条
1 白巍;可置信性结构鲁棒优化设计若干问题的研究[D];大连理工大学;2009年
，本文编号：2256243
本文链接：https://www.wllwen.com/kejilunwen/yysx/2256243.html
