結構拓樸最佳化設計變數懲罰和網格產生的新方法

Abstract

Topology optimization has been developed more than 20 years. This method has beenwidely used in traditional structural design as well as design of compliant mechanisms usedfor MEMS structures. The apparent advantage of using topology optimization is to avoid abad initial design made by inexperienced designers. Topology optimization can generate astructure for final use or as an initial design for further improvement. Usually, the resultobtained by topology optimization can not be used directly because the current method usedto generate the topology can not produce smooth boundaries that can be manufactured easily.Besides this, some uncertain elements and patterns may appear in the topology, and thereforesome post-processing treatments must be done to make it usable. Although this method hasbeen used for many years, the aforementioned drawbacks still need to be solved. The mainissues are: the existence of uncertain elements(grey elements), the zigzag pattern and thecheckerboard pattern structures. Some researchers have studied these problems and providedsome solutions. These solutions more or less solve the problems. However, they are veryhard to be implemented with commercialized finite element codes. Therefore, this projectwill propose some new ideas to reduce the number of grey elements as well as the zigzagpatterns. These new approaches can be easily incorporated with commercialized finiteelement codes so that they can be used to solve practical design problems. The main aims ofthis proposal are to reduce or eliminate the grey elements as well as the zigzag patternstructures. In the meantime, the checkerboard patterns may hopefully be reduced by thesetwo efforts simultaneously.This research project will be conducted in two years. The first year will concentrate ondeveloping some new penalty function which can penalize and also encourage the designvariables toward 0 or 1. It is different from the widely used SIMP method. In SIMP, alldesign variables are penalized. However, our approach only penalizes those design variableswith smaller values and force them to approach 0 but encourages those design variables withlarger values toward 1. Through this effort, most design variable will be expected to be 0 or 1.In this year, an interface program will also be developed to incorporated msc.nastran with thetopology optimizer.The aim of the 2nd year is to eliminate the zigzag pattern structures. The basic idea is tocreate specific finite element meshes for different design needs. In order fulfill this aim andalso reduce the computational cost, Taguchi's method for experiments will be employed sothat only a minimum number of analyses are needed to determine the idea orientation ofmeshes and the type of elements used in the given design space. The zigzag pattern isexpected to be reduced through using proper meshes and elements. A simple post-processingprocedure will also be developed to fix or eliminate those still existing zigzag patterns.Through these efforts, the result of topology optimization may be used directly.

結構拓樸最佳化(Topology optimization)方法發展至今已有二十年歷史，這個方法已被廣泛使用在傳統的結構設計上，也被應用在微機電的撓性機構(Compliant mechanism)設計上。拓樸最佳化的最大好處是排除了設計者的經驗不足可能造成的不良影響，它直接使用最佳化方法去得到一個最終設計，或是一個理想的初始設計以待進一步的改進。通常拓樸最佳化的結果並不能直接被使用，原因是目前所使用的方法除了無法產生出一個具有平滑外型且易於製造的結構外，同時結構內部也經常存在一些不確定的構件以及不合理的拓樸形狀，必須使用一定的後處理程序才可將結果實用化。縱然拓樸最佳化方法已被成功地應用在許多結構設計上，但是它仍然有上述的問題有待改善；綜合這些問題大致可分成三類，一是灰階元素(Grey element)的存在，二是鋸齒狀拓樸形狀的產生，三是棋盤式結構(Checkerboard pattern)的產生。雖然已有許多文獻探討和嘗試改善這些問題，並且也得到一些成果。但是要將這些方法和商品化的有限元素分析軟體結合，以便應用到實際結構設計上會有很大的困難，因此本計畫將提出一些新的方法去減少灰階元素和鋸齒狀結構的產生，這些方法都很容易和商品化的有限元素軟體結合，所以可直接應用在真實結構的拓樸最佳化設計上。本計畫所提出的方法主要是以消除灰階元素和減少鋸齒狀結構的形成為主要目的，但是在達成這兩個目的的同時，預期棋盤式的結構也可能同時被消滅或減少。本計畫將分成兩年執行，第一年將發展出新的設計變數懲罰和獎勵並用的方法，這個方法將有別於目前被廣泛使用的SIMP 法。SIMP 法是對所有設計變數都進行懲罰，而本計畫的方法是迫使小的設計變數趨近於0，獎勵大的設計變數趨近於1，而達成消滅或減少灰階元素的目的。第一年中也將同時完成結合msc.nastran 有限元素分析軟體的介面程式之撰寫。第二年的主要目標是消除鋸齒狀的拓樸結構，同時亦將使用第一年的研究成果來進行拓樸最佳化，研究的基本構想是針對不同的需求建構最適當的有限元素網格。為了達成此一目的，將使用田口實驗設計方法在最少的分析次數下，找出針對各個不同的設計需求，在指定的設計空間中、特定的受力和邊界條件下較理想的有限元素網格的建立方式和使用元素的種類，以期能經由適當的網格佈建和搭配適當的元素，來降低鋸齒狀拓樸結構的產生；同時亦將發展簡單的後處理技術去修補或去除仍然可能存在的少量鋸齒狀結構，以便能達成直接使用拓樸最佳化結果之理想境界。