本文探討三維邊界元素法中，將四邊形參數域之積分轉換流程等效套用於六節點三角形元素之可行性與應用。實際工程幾何常需使用三角形元素與四邊形元素混合建模，以兼顧複雜邊界之幾何適應性與網格品質。然而，在邊界元素法分析中，奇異積分與近似奇異積分之處理對計算結果具有重要影響，而部分特殊積分轉換方法原本多建立於四邊形參數域中，不易直接應用於三角形元素。因此，若能建立一套可將四邊形參數域積分轉換流程導入六節點三角形元素之等效處理方法，將有助於提升混合元素模型下邊界元素法程式之整合性與應用彈性。
本研究以六節點三角形元素為對象，建立四邊形參數域至三角形自然座標域之座標映射關係，並將映射所產生之 Jacobian 修正項納入積分權重中，使三角形元素可沿用原本建立於四邊形參數域之特殊積分轉換流程。本文以冪次轉換法作為主要驗證案例，將其應用於三維異彈問題中之奇異積分與近似奇異積分處理，以評估所提出等效流程在三角形元素上的實作可行性與數值表現。此外，本文亦將此等效處理概念應用於三角形與四邊形混合建模之分析中，以探討其於複雜幾何模型中的應用性。

This study investigates the feasibility and application of equivalently applying an integrationtransformation procedure originally developed in the quadrilateral parametric domain to six-node triangular elements in three-dimensional boundary element analysis. In practical engineering geometries, triangular and quadrilateral elements are often used together in mixed-element modeling to balance geometric adaptability for complex boundaries and mesh quality. However, in boundary element analysis, the treatment of singular and nearly singular integrals has a significant influence on numerical accuracy. Some special integration transformation methods are originally formulated in the quadrilateral parametric domain and cannot be directly applied to triangular elements. Therefore, establishing an equivalent treatment that introduces the quadrilateral parametric-domain integration transformation procedure into six-node triangular elements can improve the integration consistency and application flexibility of mixed-element boundary element programs.
In this study, a coordinate mapping relationship from the quadrilateral parametric domain to the triangular natural coordinate domain is established for six-node triangular elements. The Jacobian correction term generated by this mapping is incorporated into the integration weights, allowing triangular elements to adopt the special integration transformation procedure originally developed for the quadrilateral parametric domain. The power-law transformation method is used as the main verification case and is applied to the treatment of singular and nearly singular integrals in three-dimensional anisotropic elasticity problems. This is done to evaluate the implementation feasibility and numerical performance of the proposed equivalent procedure for triangular elements. In addition, the proposed concept is applied to analyses involving mixed triangular and quadrilateral element modeling in order to examine its applicability to complex geometric models.

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