本論文探討邊界元素法分析無窮域至局部無窮域的熱傳問題時的基本解。由於邊界元素法無需在內部作網格分割，因此其在無窮域分析上就有其優勢。對此有許多學者(如Wrobel【1】)有提出無窮域之基本解會有額外項，雖然提出的方法是用在等向性材料，不過利用映向點Shiah and Tan【9】可以將在無窮域平面的異向性材料扭曲成等向性材料藉此方便運算，在【1】提出半無窮等向性平面之基本解可以增加映向源點來求解，這個源點則是運用鏡射來找出。在本文中利用【9】的方法就能將異向材料轉成等效的等向性材料。再利用相同方法找四分之一無窮域，不過只能用在正交材料上，這是因為異向性材料的映射點扭曲後的四分之一域會變成非正交的楔形域，迫使為滿足邊界條件造成了額外的困難。
使用針對無窮域以及局部無窮域而修改的基本解，就不需要在無窮遠處或者局部無窮域的表面上畫網格，利用這一優勢，BEM在模擬無窮與局部無窮域熱傳問題上比起其他數值工具要來的更有效率。公式都在邊界元素法的程式碼中實現，為了驗證方法的準確性，最後給了數個範例。
關鍵字：邊界元素法、二維異向熱傳問題、無窮域、局部無窮域

This paper discusses the fundamental solutions of the boundary element method to analyze the heat conduction in an infinite/ partially infinite medium. Due to the distinctive feature of boundary discretization, the boundary element method is especially advantageous in modeling infinite domain. Many scholars (e.g Wrobel [1]) have proposed the fundamental solution of the infinite domain with an extra term. Although the proposed treatments are simply for isotropic media, the similar process can be applied to anisotropic cases when the anisotropic domain can be mapped to the equivalent “isotropic one” in infinite plane as proposed by Shiah and Tan [9]. It was proposed in [1] that the fundamental solutions to the half-infinite isotropic plane could be obtained by posing additional mapping point of the source point using the relation of mirror-mapping. In the thesis, this methodology is applied to the equivalent “isotropic” domain when the anisotropic medium is transformed by the formulations proposed in [9]. Also, the same treatment is applied to a quarter of the infinite domain; however, only orthotropic properties are allowed. This is because the mapping of anisotropy shall distort the quadrant domain into a non-orthogonal wedge-like domain, presenting extra difficulties to satisfy the boundary conditions.
By using the fundamental solutions modified for the infinite/partially infinite domain, no meshes are required to model the far-field and the partially infinite surfaces. Taking this advantage, the BEM is more efficient in modeling the heat conduction in the infinite/partially infinite plane as compared with other numerical tools. All formulations are implemented in an existing BEM code. For verifications of the veracity, several examples are studied in the end.
Keywords: Boundary Element Method, 2D anisotropic heat conduction, Infinite plane, partially infinite plane

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