由於現今電子裝置的製作重心在於更輕薄，一種無核芯(coreless)的基版技術被研發及發展、稱為嵌入式基板(Embedded Trace Substrate, ETS)，並廣泛應用於各式電子產品，如:通訊用具、智慧型手錶以及各式各樣消費型產品。然而，此種設計因為不同材料間的相異的物理性質會有嚴重的缺陷，舉例而言，銅線與與其他非金屬材料的熱膨脹係數(Coefficient of thermal expansion, CTE)不匹配，會造成產品發生嚴重的翹曲(warpage)，進而影響後續封裝製程(package processing)。近年來，有限元素分析法(Finite Element Analysis, FEA)是一種受歡迎且有效的方法幫助研究人員預測板彎翹曲和機械性質的研究。生產者可以應用有限元素方法去模擬基板的改良設計，以及能提供一個特定的翹曲數值滿足後續開發所需。儘管如此，對於高精度的模擬需耗費龐大的計算成本，因此，如果必須多次執行模擬研究（例如，靈敏度分析和翹曲優化），則模擬研究將成為漫長而艱鉅的任務。

在此篇論文中，可分為二部分。一是以代表性體積元素(Representative Volume Element, RVE)研究板材的材料性質，在材料的微觀組織結構下進行數值模擬分析，在經由逆向工程或是有限元素法得到此研究所需的材料物性，另外以貝茲曲線(Bézier curve)擬合玻璃轉移溫度(Tg)在溫度變化下對物質物性的影響。其二是提出了一個新的方法針對有限元素法分析機械性質的板材模擬及流程，並設計一種優化的策略對於板彎翹曲的控制。方法如下:首先將每層基板的Gerber檔轉換成高解析度的圖像，並且透過預先劃分好的區塊掃描每個圖像的銅區域進行二值化分析。接下來使用體積平均微觀力學方法計算每個子區塊的有效材料屬性，然後堆疊所有區塊構建出用於有限元素分析的模型。有別於傳統的軌跡劃分模擬，該方法大幅地降低計算資源的需求。此外，我們還能獲得與實際實驗相符、精準的翹曲預測結果。最後，文章的末端提出了一種優化策略，該策略能在封裝製程的預處理步驟，藉由參數的調整進行板彎翹曲的優化。本文的結果顯示出此一新的模擬方法是有效、實際且能降低計算成本的。

As the electronic devices getting lighter and smaller, a coreless substrate technology, called Embedded Trace Substrate (ETS) is developed to meet the market requirement. However, this design causes severe warpage due to the large difference in CTEs (coefficient of thermal expansion) of build-up material and Cu plate. Recently, finite element analysis (FEA) is a popular and effective method used for substrate warpage prediction and mechanical studies. Manufacturers apply FEA simulation for substrate design improvements, and provide substrate warpage that satisfying the customer’s specification. Nevertheless, the computational resources needed for high-fidelity simulation are extremely expensive and time consuming. Hence the simulation study becomes a long and arduous task if it has to be performed many times, e.g., sensitivity analysis and warpage optimization.

In this paper, on the one hand, we research the material properties of composite material with RVE and carry out numerical simulation analysis under the microstructure of the material. In addition, the glass transition temperature (Tg) is fitted by Bézier curve in thermos Mechanical Analysis. On the other hand, we propose a new method for FEA modeling of mechanical behaviors of substrate, and present an optimization strategy for substrate warpage control. In the first step, the Gerber files of each layer of the substrate are converted into high resolution bitmap images, and the copper area of each image is divided and scanned by a pre-sized block window. After that, the effective material properties for each block are calculated with volume average micromechanics approach, and then all blocks are stacked-up to build block-based analysis model for FEA simulation. As comparing with conventional trace mapping simulation, the proposed method significantly decreases the demands of computing resource. Besides, we gained accurate warpage prediction results as validated by a real substrate experiment. Finally, we presented an optimization strategy that manipulates the thickness of each layer for substrate warpage optimization in pre-processing steps of packaging. In conclusion, the results show that the methodology for substrate simulation in this paper is practical, effective, and costless.

第一章 緒論...1
1.1前言...1
1.1.1 嵌入式基板...3
1.1.2 翹曲與殘留應力...4
1.2 文獻回顧...9
1.3研究動機與目的...12
1.4文章架構...13
第二章 研究方法...14
2.1 Trace Mapping Method...18
2.2 等效均質理論...19
2.1.1 體積微觀力學方法(Volume average micromechanics approach)...20
2.3 Block-based Analysis Model...21
2.4 材料性質修正...27
2.4.1 貝茲曲線(Bézier Curve)...29
2.4.2 代表性體積單元(Representative Volume Element, RVE)...32
2.5 最佳化研究...37
第三章 模擬分析...41
3.1 穩態分析...50
3.1.1 案例一：A基板...50
3.1.2 案例二：B基板...62
3.2 動態分析...73
3.2.1 案例一：A基板...74
3.2.2 案例二：B基板...77
3.3 最佳化策略...80
第四章 結論與未來展望...88
4.1 結論...88
4.2 未來展望...90
第五章 參考文獻...91

[1] G. Kelly, C. Lyden, W. Lawton, J. Barrett, A Saboni, J. Exposito and F. Lamourelle, "Prediction of PQFP warpage," in 44th Electronic Components & Technology Conference - 1994 Proceedings(Proceedings - Electronic Components and Technology Conference, New York： IEEE, 1994, pp. 102-106.
[2] G. Kelly, C. Lyden, W Lawton, J. Barrett, A. Saboui, H. Pape and H.J.B. Peters, "Importance of molding compound chemical shrinkage in the stress and warpage analysis of PQFPs," in 45th Electronic Components & Technology Conference - 1995 Proceedings(Proceedings - Electronic Components and Technology Conference, New York , 1995, pp. 977-981.
[3] Package warpage measurement of surface-mount integrated circuits at elevated temperature, JESD22-B112A, 2009.
[4] Measurement Methods of Package Warpage at Elevated Temperature and Maximum Permissive Warpage, JEITA ED-7306, 2007.
[5] K. Oota and K. Shigeno, "Development of molding compounds for BGA," presented at the 45th Electronic Components & Technology Conference - 1995 Proceedings, New York, 1995.
[6] B. Z. Zhao, V. Pai, C. Brahateeswaran, G. J. Hu, S. Chew, and N. Chin, "FEA simulation and in-situ warpage monitoring of laminated package molded with green EMC using Shadow Morie system," 2006 7th International Conference on Electronics Packaging Technology, Shanghai China, 2006, pp. 176-181.
[7] N. Srikanth, C. Lim, and B. Kumar, "A Viscoelastic Warpage Analysis of Molded Packages," in Technical Symposium, Semicon Singapore, 2002, pp. 89-99.
[8] G. J. Hu, G. K. Yong, J. E. Luan, L. W. Chin, and X. Baraton, "Thermoelastic Properties of Printed Circuit Boards： Effect of Copper Trace," in 2009 European Microelectronics and Packaging Conference, New York, 2009, pp. 321-326.
[9] L. O. McCaslin, S. Yoon, H. Kim, and S. K. Sitaraman, "Methodology for Modeling Substrate Warpage Using Copper Trace Pattern Implementation," IEEE Transactions on Advanced Packaging, vol. 32, no. 4, pp. 740-745, Nov 2009.
[10] C. S. Chen, N. Kao, and D. S. Jiang, "Different Conservation Laws Utilized for Warpage Prediction of MUF FCCSP with 4L ETS," presented at the 2016 IEEE 18th Electronics Packaging Technology Conference, New York, 2016, pp. 313-319.
[11] K. Biswas, S. G. Liu, X. W. Zhang, and T. C. Chai, "Development of Numerical Modeling Approach on Substrate Warpage Prediction," presented at the Proceedings of the 2012 IEEE 14th Electronics Packaging Technology Conference, New York, 2012.
[12] C. C. Meng, S. Stoeckl, H. Pape, F. M. Yee, and T. A. Min, "Effect of Substrate Warpage on Flip Chip BGA Thermal Stress Simulation," presented at the 2010 12th Electronics Packaging Technology Conference, New York, 2010.
[13] W. Lin, B. Baloglu, and K. Stratton, "Coreless Substrate with Asymmetric Design to Improve Package Warpage," presented at the 2014 IEEE 64th Electronic Components and Technology Conference, New York, 2014.
[14] C. C. Chen, M. Z. Lin, G. C. Liao, Y. C. Ding, and W. C. Cheng, "Balanced Embedded Trace Substrate Design for Warpage Control," presented at the 2015 IEEE 65th Electronic Components and Technology Conference, New York, 2015.
[15] S. Timoshenko, "Analysis of bi-metal thermostats," (in English), Journal of the Optical Society of America and Review of Scientific Instruments, Article vol. 11, no. 3, pp. 233-255, Sep 1925.
[16] Y. H. Laio and M. C. Shih, "The Study of Warpage of a Flip Chip Embedded Trace Substrate," presented at the 2016 11th International Microsystems, Packaging, Assembly and Circuits Technology Conference, New York, 2016.
[17] M. J. Wang and B. Wells, "Substrate Trace Modeling for Package Warpage Simulation," presented at the 2016 IEEE 66th Electronic Components and Technology Conference, Los Alamitos, 2016.
[18] Q. S. Yang and W. Becker, "Numerical investigation for stress, strain and energy homogenization of orthotropic composite with periodic microstructure and non-symmetric inclusions," Computational Materials Science, vol. 31, no. 1-2, pp. 169-180, Sep 2004.
[19] Z. Hashin, "Analysis of composite materials—A Survey," Journal of Applied Mechanics, vol. 50, no. 3, pp. 481-505, Sep 1983.
[20] Y. Bachmat and J. Bear, "On the concept and size of a representative elementary volume (REV)," in Advances in Transport Phenomena in Porous Media： Springer, 1987, pp. 3-20.
[21] I. M. Gitman, H. Askes, and L. J. Sluys, "Representative volume： Existence and size determination," Engineering Fracture Mechanics, vol. 74, no. 16, pp. 2518-2534, Nov 2007.
[22] G. Andrei A, "Representative volume element size for elastic composites： A numerical study," Journal of the Mechanics and Physics of Solids, vol. 45, no. 9, pp. 1449-1459, Sep 1997.
[23] W. J. Drugan and J. R. Willis, "A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites," Journal of the Mechanics and Physics of Solids, vol. 44, no. 4, pp. 497-524, Apr 1996.
[24] S. Graham and N. Yang, "Representative volumes of materials based on microstructural statistics," Scripta Materialia, vol. 48, no. 3, pp. 269-274, Feb 3 2003.
[25] S. M. Mirkhalaf, F. M. A. Pires, and R. Simoes, "Determination of the size of the Representative Volume Element (RVE) for the simulation of heterogeneous polymers at finite strains," Finite Elements in Analysis and Design, vol. 119, pp. 30-44, Oct 15 2016.
[26] EpoxyTechnology, "Tg-Glass Transition Temperature for Epoxies," 2012.
[27] C. T. Sun and R. S. Vaidya, "Prediction of composite properties from a representative volume element," Composites Science and Technology, vol. 56, no. 2, pp. 171-179, 1996.
[28] 张研 and 张子明, 材料细观力学. 北京： 科学出版社, 2008.
[29] N. Hansen, The CMA evolution strategy： A tutorial. 2016.
[30] H. S. Jo and G. W. Lee, "Thermal expansion coefficient and Young’s modulus of silica-reinforced epoxy composite," Mechanical and Mechatronics Engineering, vol. 8, no. 11, pp. 1188-1191, 2014.
[31] C. Heinrich, M. Aldridge, A. S. Wineman, J. Kieffer, A. M. Waas, and K. Shahwan, "The influence of the representative volume element (RVE) size on the homogenized response of cured fiber composites," Modelling and Simulation in Materials Science and Engineering, vol. 20, no. 7, p. 075007, Oct 2012.
[32] N. Hansen, S. D. Muller, and P. Koumoutsakos, "Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES)," Evolutionary Computation, vol. 11, no. 1, pp. 1-18, Spring 2003.
[33] N. Hansen and A. Ostermeier, "Completely derandomized self-adaptation in evolution strategies," Evolutionary Computation, vol. 9, no. 2, pp. 159-95, Summer 2001.
[34] P. J. Ross, Taguchi techniques for quality engineering： loss function, orthogonal experiments, parameter and tolerance design, (no. TS156 R12). McGraw-Hill New York, 1988.
[35] M. Kevin and C. Bruce, Basic of Thermomechnical Analysis with TMA 4000, [Online]. Available： https：//www.perkinelmer.com/lab-solutions/resources/docs/TCH_TMA_4000.pdf?fbclid=IwAR2g-TWeH1RCZUrgTgzCWiwwyMuMvhFNJWEASbMKzTJIsABlSEPtg00EDz0