升阶谱有限元方法是一种典型的高阶有限元方法，在前处理、计算精度、单元矩阵的特性等方面明显优越于常规位移有限元，但在计算不规则域高阶单元时存在数值稳定性问题，高阶三角形单元目前还在用符号计算。微分求积有限元方法同微分求积方法一样，在计算精度方面有明显优势，单元的阶次也较高，但在构造三角形单元方面也存在困难，而且不具备升阶谱有限元阶次可变等优点。本项目把微分求积有限元方法的思想和技巧引入升阶谱有限元中，并利用形函数的分离变量特点，解决了升阶谱有限元方法的数值稳定性问题以及计算高阶三角形单元矩阵方面的困难。这一新的方法被命名为微分求积升阶谱有限元方法。本项目不仅研究C0单元还研究C1单元的构造。升阶谱有限元可能成为未来开发有限元软件的主流，本项目解决了求解方法自身存在的困难，从而为之奠定了理论基础。为了体现微分求积升阶谱有限元方法的优势和具体介绍其求解过程，本项目将之应用于结构振动问题。

The hierarchical finite element method (HFEM) is a typical p-version finite element method, which is superior to the h-version finite element method in pre-processing, accuracy, properties of element matrices, etc. However, the HFEM has numerical stability problems in irregular domains when the number of polynomial hierarchical functions is high. High-order triangular hierarchical elements even still use symbolic computations. The differential quadrature finite element method (DQFEM), just like the differential quadrature method, is highly accurate and does not have numerical stability problem for high-order elements. However, the DQFEM also has computational difficulty in constructing triangular elements, and it does not have the advantages of the HFEM, whose degrees of freedom of neighboring elements can be different, etc. The key ideas and skills of the DQFEM were introduced into the HFEM in this project to overcome the numerical stability problems and the difficulty in computing high-order triangular hierarchical elements in the HFEM. The separation variable characteristic of the shape functions was also used in this process. The new method was named as a differential quadrature hierarchical finite element method (DQHFEM), which is an improved version of the HFEM in essence. This project studies not only C0 but also C1 DQHFEM elements. The HFEM may become the mainstream in developing finite element software in the future. This project will solve the main problems existing in the HFEM, which will lay a theoretical foundation for HFEM software developing. The DQHFEM will be applied to structural vibration problems in this project to introduce the solving procedure and to show the advantages of the DQHFEM.