Abstract This article applies the finite element over-deterministic (FEOD) method to determine the coefficients of the crack tip asymptotic fields in an anisotropic plane. This method relies on finding the best fit of the crack tip displacement equations to the displacements taken from a large number of nodes around the crack tip. We first revisit the asymptotic solution of crack tip fields in an anisotropic plane, followed by giving the exact solution of the crack tip parameters for an infinite anisotropic plane with a central crack. Using the derived displacement field, we then formulate the FEOD method to determine the crack tip parameters from a set of finite element nodal displacements. Numerical examples of cracks in finite and infinite plates are then investigated, where the FEOD results manifest a good agreement with exact or alternative numerical solutions. A convergence study is performed to examine the robustness of the methodology and the reliability of the results. By implementing a least squares error function, we also calculate the corresponding error in approximating the displacement field. The low values of errors demonstrate the high precision of the FEOD method.

中文翻译：

---

计算各向异性平面裂纹尖端渐近场系数的有限元超确定性方法

摘要 本文应用有限元过确定性(FEOD)方法确定各向异性平面内裂纹尖端渐近场的系数。这种方法依赖于寻找裂纹尖端位移方程与裂纹尖端周围大量节点的位移的最佳拟合。我们首先重新研究各向异性平面中裂纹尖端场的渐近解，然后给出具有中心裂纹的无限各向异性平面的裂纹尖端参数的精确解。使用导出的位移场，我们然后制定 FEOD 方法来确定一组有限元节点位移的裂纹尖端参数。然后研究有限和无限板中裂纹的数值例子，其中 FEOD 结果与精确或替代的数值解表现出良好的一致性。执行收敛性研究以检查方法的稳健性和结果的可靠性。通过实现最小二乘误差函数，我们还计算了近似位移场的相应误差。低误差值证明了 FEOD 方法的高精度。