本文建立一套可用於含孔洞或直線裂縫之二維異向性彈性板的史磋－特雷夫有限元素法（Stroh--Trefftz Finite Element Method，ST-FEM）。混合特雷夫有限元素法（Hybrid-Trefftz Finite Element Method，HT-FEM）以特雷夫函數（Trefftz functions）描述元素內部位移場；由於這些函數已先滿足控制微分方程，元素積分可透過修正變分原理化為邊界積分，使元素可採用較大的多邊形子區域。本文以史磋公式（Stroh formalism）所提供的矩陣形式解析解作為完備特雷夫函數（T-complete functions），建立史磋－特雷夫有限元素法的系統方程式，並將此方法延伸至多邊形孔洞與直線裂縫問題。

本文進一步建立特殊目的函數（SPF），使孔洞或裂縫的自由邊界條件可在特雷夫基底（Trefftz basis）的構造階段即被納入，從而免去在孔洞或裂縫邊界上大量配置節點的需求。推導中以映射變數ζα描述孔洞或裂縫邊界幾何；對多邊形孔洞問題，本文採用直接給定羅朗展開係數（Laurent expansion coefficients）並搭配非保角映射（nonconformal mapping）的方式取得所需的ζα，使建模流程更為簡化。

數值算例包含橢圓孔洞、多邊形孔洞與直線裂縫之異向性板，各板件均承受均勻拉伸荷載，並與邊界元素法（Boundary Element Method，BEM）及商用有限元素軟體 ANSYS 之結果進行比較。在本文測試案例中，史磋－特雷夫有限元素法於所選應力與應力強度因子量上可得到與 BEM 或 ANSYS 接近的結果，且不需於孔洞或裂縫邊界佈點。對直線裂縫算例，本文亦可直接由特殊目的函數係數回算左右尖端之廣義應力強度因子。這些結果顯示，在本文設定的幾何、材料與計時條件下，本文方法可用於含孔洞或裂縫之異向性結構基準問題。

This study develops a Stroh--Trefftz Finite Element Method (ST-FEM) for two-dimensional anisotropic elastic plates containing holes or straight cracks. The Hybrid-Trefftz Finite Element Method (HT-FEM) uses Trefftz functions to describe the intra-element displacement field. Because these functions satisfy the governing differential equations a priori, the element integrations can be transformed into boundary integrals through a modified variational principle, allowing the elements to adopt larger polygonal subregions. In this study, the matrix-form analytical solutions provided by the Stroh formalism are adopted as the T-complete functions to establish the system equations of ST-FEM, and the method is extended to polygonal-hole and straight-crack problems.

Special purpose functions (SPFs) are further constructed so that the traction-free boundary conditions of holes or cracks can be incorporated during the construction of the Trefftz basis, thereby eliminating the need to place a large number of nodes along the hole or crack boundaries. In the derivation, the mapping variables ζα are used to describe the geometry of the hole or crack boundaries. For polygonal-hole problems, the required ζα are obtained by directly prescribing the Laurent expansion coefficients and combining them with a nonconformal mapping, thereby simplifying the modeling procedure.

The numerical examples include anisotropic plates with elliptical holes, polygonal holes, and straight cracks, all subjected to uniform tensile loading, and the results are compared with those obtained using the Boundary Element Method (BEM) and the commercial finite element software ANSYS. In the cases examined in this study, ST-FEM produces results close to those of BEM or ANSYS for the selected stresses and stress intensity factors, without requiring nodes along the hole or crack boundaries. For the straight-crack examples, the generalized stress intensity factors at the left and right crack tips can also be recovered directly from the SPF coefficients. These results indicate that, under the geometry, material, and timing conditions considered in this study, the proposed method can be applied to benchmark problems involving anisotropic structures with holes or cracks.

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