Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.

断裂的相场模型是基于能量的，并采用连续场变量（相场）来指示裂纹。该字段变量在损坏区域和完整区域之间的过渡区域的宽度由正则化参数控制。对于裂缝能量的良好近似，需要狭窄的过渡区域，这涉及相场的陡峭梯度。如果使用具有标准双线性形状函数的4节点单元，则在有限元模拟中需要较高的网格密度。为了提高使用粗网格的结果的质量，引入了从一维模型的解析解导出的指数形状函数，以对相场变量进行离散化。与双线性形状函数相比，这些特殊的形状函数可以更好地逼近断裂场。不幸的是，对于双线性形状函数的积分足够精确的低阶高斯-勒根德勒正交方案不足以对指数形状函数进行准确的积分。因此，在这项工作中，分析了高阶Gauss-Legendre公式和用于数值积分的双指数公式的数值精度。