熱傳導拓樸最佳化之材料性質模型的完整研究

A Complete Research on Material Property Models on Heat Conduction Topology Optimization

魏馬克

拓樸最佳化、熱傳導問題、材料屬性模型 ； topology optimization ； access problem ； material property models

中原大學機械工程研究所學位論文

2016年

博士

林柏廷

英文

基於密度的拓樸最佳化透過材料內插法的應用，可以解決從離散最 佳化到連續梯度的問題，但是材料插值法對狀態空間中的搜尋與最終設計 有著不利的影響，目前也尚未被探討過。在許多文獻中提到材料插值法更 準確地被稱為材料屬性模型，但是大部分的測試都不是在熱傳導領域的拓 樸最佳化中。本文分別對三個利用材料插值法的拓樸最佳化，與三個可調 性可預測有效熱傳導率的模型進行研究。拓樸最佳化的材料插值法使用： 1) 基於SIMP (Solid Isotropic Material with Penalization) 理論懲罰方法中的冪 次現象 (Power law)；2) Rational Approximation of Material Property (RAMP)； 3) 雙曲正弦函數 (Hyperbolic Sine function)。另外三種可調性可預測有效熱 傳導率的模型使用：1) Krischer’s model；2) Maxwell-Hamilton model；3) Kirkpatrick’s EMT (Effective Medium Theory) model。這六個模型在材料性 質與體積分數(或是拓樸最佳化中的密度變數)間有一個可調參數，為了進 行公平的比較，六個實驗分別定義不同的材料屬性模型，並且透過RMSE 公式，適當的可調參數在本研究模型中可以被定義，最後再將不同模型之 間的關係進行總結。 本文在熱傳的”接入問題 (access problem)”中利用Adrian Bejan的理論 ，在不同的模型中透過簡化的關係，從拓樸最佳化程序得到四種不同的網 格大小 (mesh size)共100組的最佳搜索空間。經過後處理後，共有60組的 設計結果效能提升，分別為考慮最大溫度與幾何結構相比Adrian Bejan 理 論低於2.84 倍的設計。進一步探討不同材料性質的模型，定義與比較不同 的性能指標，這些指標包含：1)經過前處理與後處理後的目標函數；2)最 高溫度；3)平均溫度；4)一個定義材料在設計設計空間中的分布的擴散值； 5)一個確定性值定義每個設計變數的質量；6)計算與迭代時間。基於這些 性能指標的觀察，不同的模型被製作出來。在研究中發現，SIMP 與雙曲 II 正弦函數在非常陡峭的模型中相當不穩定；Krischer’s 模型在中間值具有豐 富的結果，但是不適合用在後處理；並且在Maxwell-Hamilton’s 模型中同 時觀察在RAMP下的穩定性與可預測性。SIMP與Kirkpatrick’s EMT模型 產生了高性能結構，並且與經過後處理的Landauer’s EMT模型有著極大的 不同，這六個固定的模型可以代表其他所有可調性模型，同時本研究也觀 察其他幾個改變原來趨勢的模型。 為了進一步驗證這些理論，比較了在不同體積分數下的模型，發現 到較高的體積分數可以產生更好的結構；目標函數在前處理與後處理呈現 反比的趨勢，並且交界處即為Landauer EMT 狀態，本文針對此問題提供 最好的狀態去尋找高性能的離散結構。最後也發現到接入問題的解類似一 種分散的設計，並且傳遞參數不會變化太大；確定性隨著模型的陡度增加 而增加；模型陡度的增加對於計算的平均時間也略有增加。最後，本次模 型中有部分迭代次數較高，並且使用尚未經過拓樸最佳化程序的模型，在 很陡度很陡的情況下計算也能收斂。

Density-based topology optimization relies on material interpolation schemes in relaxing a discrete optimization problem to a continuous problem where gradient-based techniques in optimization can be applied. The material interpolation schemes adversely affects the state-space search process as well as the final design but has never been thoroughly investigated. Material interpolation schemes, more appropriately called material property models, are abundant in literature but most have been tested before in the field of heat conduction topology optimization. In this work, three (3) material interpolation schemes from topology optimization and three (3) tunable material property models for predicting the effective thermal conductivity of materials are investigated. The material interpolation schemes in topology optimization are: 1) the power-law interpolation from Solid Isotropic Material with Penalization (SIMP), 2) Rational Approximation of Material Property (RAMP), and 3) a Hyperbolic Sine function. The three material property models for predicting the effective thermal conductivity are: 1) Krischer’s model, 2) Maxwell-Hamilton’s model, and 3) Kirkpatrick’s Effective Medium Theory (EMT) model. These six models have one tunable parameter which determine relationship between the material property and the volume fraction (or the density variable in topology optimization). To provide a fair comparison, six (6) different fixed material property models are defined. By using a RMSE formulation, the appropriate tunable parameter values in the investigated models are determined. IV The affinity between the different models are then summarized. The ‘access problem’ in heat transfer, as defined by Adrian Bejan in his work for constructal theory, has been revisited. By simplifying the affinity between the different models and using four (4) different mesh sizes, a total of 100 search space optimal solutions were obtained from the topology optimization process. After post-processing, sixty (60) of these designs performed better with maximum temperature considered and the best design is reported to be 2.84 times lower when compared to the constructal geometry. Further investigating the different material property models, different performance metrics were defined and compared. These metrics are: 1) the compliance objective value before and after the post-processing operation, 2) the maximum temperature, 3) the average temperature, 4) a ‘spread’ measure that defines how the material is distributed in the design domain, 5) a ‘definitiveness’ measure which defines quality the resulting design based on the value of each design variable, 6) computational time and iteration. Based on the observations of these performance metrics, generalizations of the different model attributes are made. The SIMP and the Hyperbolic Sine function has encountered instabilities at very steep model conditions. Krischer’s model had results rich in intermediate values which were not suitable for post-processing operations. Stability and predictability of results under RAMP was also observed in Maxwell-Hamilton’s model. SIMP and Kirkpatrick’s EMT model produced high performing structures that are very different even after post-processing in the Landauer EMT condition. V The 6 fixed models can also be used as a good representative of the other tunable models. Several other trends are observed based on increasing fixed model steepness. To further validate these observations, comparisons were made under different volume fractions. Higher volume fractions produce better performing structures. The trend for the compliance before and after the post-processing operation was directly inverse and crosses at Landauer’s EMT condition. This gives the best condition to find best performing discrete structures for this problem. The solution to the access problem is a spread design and does not change very much. Definitiveness increases as the steepness of the model is increased. The average computational time also increases slightly as the model steepness is increased. Lastly, the required number of iterations is higher in intermediate steepness and convergence can be achieved in very steep models using alternative material property models which have not been tried before in topology optimization.

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