中文摘要：

作为工程和科学计算的主要工具,有限元方法已经得到了广泛的应用,但是仍然受到网格畸变敏感等固有难题的困扰,并且一直没有能够彻底根治。该文系统介绍了新型有限元方法——形状自由的高性能有限元方法研究的最新进展,包括平面问题和二维断裂问题的杂交应力函数有限元方法,中厚板问题的杂交位移函数有限元法,平面和三维问题的新型非对称有限元方法。这些方法在已有的杂交应力元法和非对称有限元法基础上,综合利用了解析试函数法、新型自然坐标方法、广义协调方法等先进技术,获得重要进展：所发展的单元模型精度高且稳定,在网格极端畸变的情况下仍可保持原有精度,具有形状自由的优异特性;同时破解了Mac Neal局限定理,解决了中厚板边缘效应计算等难题。论文的最后对上述方法的特点以及后续的研究工作进行了讨论。

英文摘要：

As the most important tool for simulation and computation, the finite element method has been widely applied in engineering and scientific problems. However, this powerful method still suffers from some inherent deficiencies, such as the sensitivity problem to mesh distortion and so on, which have not been completely solved yet. This paper systematacially introduces some research developments on new finite element methods, i.e., the shape-free finite element methods, including the hybrid stress-function elements for plane and 2D fracture problems, the hybrid displacement-function elements for Reissner-Mindlin plate bending problem, and new unsymmetric finite element for plane and 3D elasticity. Based on the existing hybrid stress and unsymmetric finite element methods, some advanced techniques, including the analytical trial function method, new natural coordinate method, generalized conforming technique, etc., were adopted in the aforementioned new approaches. The resulting new finite element models possess high precision and good robustness. In particular, they can keep their level of performance even in extremely distorted meshes. Furthermore, some historical challenges in the finite element method, such as the limitation defined by the MacNeal＇s theorem, the edge effect problem of the Resissner-Mindlin plate, were also successfully solved. At the end of this paper, some features of the new methods and further development are discussed.