A novel method for determining the optimized parameter of the four-node incompatible generalized mixed element is presented based on the equilibrium between strain energy and complementary energy. The presented energy formulations are derived from the generalized mixed variational principle, which contains an arbitrary additional parameter. The initial solutions expressed by the displacement field are firstly assumed for the description of the energy of each generalized mixed element. Then, the identical relationship between strain energy and complementary energy is subsequently expressed at element level, which includes the arbitrary parameter. At the same time, a formulation for determining the optimized parameter at element level is proposed. Several representative examples with varying geometrical parameters, boundary and loading conditions are used to validate this method. By contrasting with the results of generalized mixed elements with different parameter values and other traditional finite elements. The effectiveness of the presented method has been demonstrated. On one hand, by ensuring the strain energy and complementary energy remain consistent under both coarse and fine meshes, the optimized parameter can adjust the stiffness of the generalized mixed element, thereby enhancing its resemblance to the real elastic body. On the other hand, the generalized mixed element has the additional advantage of conveniently introducing stress boundary conditions, thereby satisfying the requirement for zero-conditions of shear stresses on the exterior surfaces of beams. The numerical results obtained by the proposed method are accurate and stable.

中文翻译：

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确定不相容广义混合单元优化参数的能量平衡方法

提出了一种基于应变能和补能平衡的四节点不相容广义混合单元优化参数确定方法。所提出的能量公式源自广义混合变分原理，其中包含任意附加参数。首先假设位移场表示的初始解用于描述每个广义混合单元的能量。然后，应变能和互补能之间的相同关系随后在单元级别上表达，其中包括任意参数。同时，提出了一种确定单元级优化参数的公式。使用具有不同几何参数、边界和载荷条件的几个代表性示例来验证该方法。通过与不同参数值的广义混合单元和其他传统有限元的结果进行对比。所提出方法的有效性已被证明。一方面，通过保证粗网格和细网格下应变能和补能保持一致，优化参数可以调整广义混合单元的刚度，从而增强其与真实弹性体的相似性。另一方面，广义混合单元的另一个优点是可以方便地引入应力边界条件，从而满足梁外表面剪应力为零的要求。该方法得到的数值结果准确、稳定。