【摘要】：一般的優(yōu)化問題都假定材料參數(shù)、載荷及邊界條件等是確定的,然而由于測(cè)量誤差、加工誤差、工藝水平等原因,導(dǎo)致材料或載荷與設(shè)計(jì)值的偏差在工程中普遍存在,而這往往是結(jié)構(gòu)或材料失效的重要原因,因此在結(jié)構(gòu)和材料設(shè)計(jì)中考慮不確定性的優(yōu)化問題十分必要。尤其是隨著納米技術(shù)和復(fù)合材料的發(fā)展,桁架結(jié)構(gòu)材料、周期材料、多空泡沫材料等超輕材料在日常生活以及工業(yè)生產(chǎn)中都表現(xiàn)出了其非凡的優(yōu)勢(shì),比如具有吸能抗沖擊的能力、較高的比強(qiáng)度等。在各類超輕材料中,多尺度超輕材料的性能尤為突出。因此本文針對(duì)考慮載荷不確定性的多尺度優(yōu)化問題以及桁架結(jié)構(gòu)剛度不確定性問題進(jìn)行了相關(guān)研究�？紤]載荷不確定性多尺度優(yōu)化問題和剛度不確定性的優(yōu)化問題均是雙層優(yōu)化問題。在雙層優(yōu)化問題中,通過下層優(yōu)化問題得到最不利情況,利用上層優(yōu)化問題進(jìn)行設(shè)計(jì)變量的更新。對(duì)于一個(gè)雙層優(yōu)化問題,最主要的是關(guān)于下層優(yōu)化問題的求解,并且由于存在計(jì)算效率低、穩(wěn)定性差、易陷入局部最優(yōu)解等缺點(diǎn),使得一般算法很難保證雙層優(yōu)化問題的可置信性。在本文中,對(duì)于考慮載荷不確定性的多尺度優(yōu)化問題,通過SDP放松技巧將下層優(yōu)化問題轉(zhuǎn)化為一個(gè)半定規(guī)劃問題,這樣不僅可以利用已有算法直接求解,同時(shí)還可以保證解的可置信性。數(shù)值算例表明微觀結(jié)構(gòu)中各向同性材料和Kagome類型單胞具有較好的抵抗不確定性載荷的能力,同時(shí)也表明了本文提出算法具有較快的收斂速度。而對(duì)于考慮剛度不確定性的雙層優(yōu)化問題,則將響應(yīng)函數(shù)在不確定變量的名義值處進(jìn)行泰勒展開,并利用施瓦茲不等式近似表示出下層最不利情況,將原來雙層優(yōu)化問題轉(zhuǎn)化成一個(gè)類似確定性優(yōu)化的單層優(yōu)化問題。數(shù)值算例表明該算法有較好的精度,尤其是對(duì)于大規(guī)模結(jié)構(gòu)大大提高了計(jì)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.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O232

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 Ole Sigmund;;Manufacturing tolerant topology optimization[J];Acta Mechanica Sinica;2009年02期

相關(guān)博士學(xué)位論文 前1條

1 白巍;可置信性結(jié)構(gòu)魯棒優(yōu)化設(shè)計(jì)若干問題的研究[D];大連理工大學(xué);2009年

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本文編號(hào)：2256243