本論文之目的在於應用拉凡格式法(Levenberg-Marquardt Method)、程式語言Fortran與套裝軟體CFD-ACE+，探討具有確定已知溫度分布的加熱底板或是加熱物體時，找出符合邊界等溫設計下之最佳幾何形狀預測。
在第二章中，吾人以套裝軟體CFD-ACE+為基礎，利用拉凡格式法配合套裝軟體CFD-ACE+中CFD-GEOM裡的INTERPOLATION CURVE函數來設計表面之邊界幾何形狀(X,Y)，以預測二維模型之幾何形狀。吾人先以不限制其二維模型面積來檢驗Mayeli等人[1]中所測試的案例，以檢視本文的有效性和優秀性。並證明在延伸出有限制模型面積的情況下，亦能分別預測出所符合期望的邊界溫度設計之最佳二維幾何形狀。
結果顯示在不對模型面積作限制的情況下，與Mayeli等人[1]所研究的幾何形狀設計比較，其優點歸納如下：(i)可以避免[1]計算過程中須使用的係數C (The overall under-relaxation factor) 選擇上的困難(ii)在疊代次數上能以較少次數來達到最佳形狀設計。
在第三章中，吾人亦以套裝軟體CFD-ACE+與二維模型之研究為基礎，建立起三維模型來探討其未知的表面幾何形狀設計。同樣地，吾人亦對三維模型分別對體積不限制與有限制之下的情況，分別來預測符合邊界等溫設計下之最佳三維形狀設計。結果顯示在三維模型的研究時，仍能預測出具有邊界均勻溫度條件下之最佳幾何形狀。

A shape design problem in determining the optimal boundary shape of a conductive body with heating object or heating substrate, to yield a uniform boundary temperature, is examined in this study. The Levenberg-Marquardt Method (LMM) and a commercial package CFD-ACE+ are utilized in this shape design algorithm. The thesis is divided into two parts: first part, to predict the geometry of a two-dimensional model. When without considering the constraint of domain area, different test cases examined previously by Mayeli [1] are re-considered in this thesis to justify the validity and superiority of the present algorithm. Next, when the domain area is considered, the numerical experiments reveal that the optimal boundary shape with uniform temperature requirement can always be obtained. In the second part, it is extended to predict the geometry of a three-dimensional model. Similarly, the optimal three-dimensional shape design problems under the boundary isothermal design conditions, with and without considering the constraint of the volume, are considered. The results show that the best geometry with uniform temperature on the boundary can be estimated.

[1]. P. Mayeli, M. Nili-Ahmadabadi and H. Besharati-Foumani, Inverse shape design for heat conduction problems via the ball spine algorithm, Numerical Heat Transfer, Part B: Fundamentals, Vol. 69, pp. 249-269, 2016.
[2]. H. M Park, H. J. Shin, Shape Identification for Natural Convection Problems using the Adjoint Variable Method, Journal of Computational Physics, Vol. 186, pp. 198-211, 2003.
[3]. E. Divo, A. J. Kassab, F. Rodriquez, An efficient singular superposition technique for cavity detection and shape optimization, Numerical Heat Transfer Part B, Vol. 46, pp. 1-30, 2004.
[4]. CFD-ACE+ user’s manual, ESI-CFD Inc., 2005.
[5]. C. H. Huang, B. H. Chao, An Inverse Geometry Problem in Identifying Irregular Boundary Configurations, Int. J. Heat and Mass Transfer, Vol. 40, pp. 2045-2053, 1997.
[6]. C. H. Huang, C. C. Tsai, A Transient Inverse Two-Dimensional Geometry Problem in Estimating Time-Dependent Irregular Boundary Configurations, Int. J. Heat and Mass Transfer, Vol. 41, pp. 1707-1718, 1998.
[7]. C. H. Huang and M. T. Chaing, A Transient Three-Dimensional Inverse Geometry Problem in Estimating the Space and Time-Dependent Irregular Boundary Shapes, International Journal of Heat and Mass Transfer, Vol. 51() 5238–5246, 2008.
[8]. C. H. Huang and C. Y. Liu, A Three-Dimensional Inverse Geometry Problem in Estimating Simultaneously Two Interfacial Configurations in a Composite Domain, International Journal of Heat and Mass Transfer, Vol. 53, pp. 48-57, 2010.
[9]. C. H. Cheng, H. H. Lin, G. J. Lai, Design for geometric parameters of PEM fuel cell by integrating computational fluid dynamics code with optimization method, J. Power Sources Vol. 165, pp. 803-813, 2007.
[10]. P. F. Chen and C. H. Huang, An Inverse Hull Design Problem in Optimizing the Desired Wake of Ships, Journal of Ship Research, Vol. 46, pp. 138-147, 2002.
[11]. C. H. Huang and Y. L. Chung, An Optimal Fin Design Problem in Estimating the Shapes of Longitudinal and Spine Fully Wet Fins, CMES-Computer Modeling in Engineering & Sciences, Vol. 44, pp. 249-279, 2009.
[12]. C. H. Huang and H. M. Hsu, An Inverse Problem in Determining the Optimal Filler Shape of Composite Materials for Maximum Effective Thermal Conductivity, International Journal of Heat and Mass Transfer, Vol. 80, pp. 98–106, 2015.
[13]. S. Bigham, H. Shokouhmand, R. N. Isfahani, and S. Yazdan, Fluid Flow and Heat Transfer Simulation in a Constricted Microchannel: Effects of Rarefaction, Geometry, and Viscous Dissipation, Numerical Heat Transfer, Part A, Vol. 59, pp. 209-230, 2011.
[14]. H. Shokouhmand and S. Bigham, Slip-Flow and Heat Transfer of Gaseous Flows in the Entrance of a Wavy Microchannel, International Communications in Heat and Mass Transfer, Vol. 37, pp. 695–702, 2010.
[15]. C. H. Huang and Y. L. Chung, A Nonlinear Fin Design Problem in Estimating the Optimal Shapes of Longitudinal and Spine Fully Wet Fins, Numerical Heat Transfer, Part A, Vol. 57, pp. 749-776, 2010.
[16]. C. H. Huang and Y. L. Chung, An Inverse Design Problem of Estimating the Optimal Shape of Annular Fins Adhered to a Bare Tube of an Evaporator, Numerical Heat Transfer, Part A, Vol. 66, pp. 1195-1217, 2014.
[17]. C. H. Huang and Y. L. Chung, An Inverse Design Problem in Determining the Optimum Shape of Spine and Longitudinal Fins, Numerical Heat Transfer, Part A, Vol. 43, pp. 155-177, 2003.
[18]. P. Li and K. Y. Kim, Multiobjective Optimization of Staggered Elliptical Pin-Fin Arrays, Numerical Heat Transfer, Part A, Vol. 53, pp. 418-431, 2008.
[19]. Y. P. Cheng, Z. G. Qu, W. Q. Tao, and Y. L. He, Numerical Design of Efficient Slotted Fin Surface Based on the Field Synergy Principle, Numerical Heat Transfer, Part A, Vol. 45, pp. 517-538, 2004.
[20]. C. H. Cheng and C. Y. Wu, An Approach Combining Body-Fitted Grid Generation and Conjugate Gradient Methods for Shape Design in Heat Conduction Problems, Numerical Heat Transfer, Part B, Vol. 37, pp. 69- 83, 2000.
[21]. C. H. Lan, C. H. Cheng and C. Y. Wu, Shape Design for Heat Conduction Problems using Curvilinear Grid Generation, Conjugate Gradient, and Redistribution Methods, Numerical Heat Transfer, Part A, Vol. 39, pp. 487-510, 2001.
[22]. D. M. Marquardt, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, J. Soc. Indust. Appl. Math., Vol. 11, pp. 431-441, 1963.