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In this paper, a new finite element method, termed the generalized modal element method (GMEM), is proposed. In GMEM, the element stiffness is derived by decomposing element deformation patterns into individual element generalized modes, where different methods are used to construct the generalized modes. Specifically, three different modal construction methods, including analytical method, assumed displacement method and traditional finite element technique, are proposed for developing the element generalized modes. The concept of modal local coordinate systems is also proposed to ensure the element frame invariance when using polynomial displacement functions, which successfully enables one to use the analytical solutions derived from governing differential equations to develop high accuracy element formulations. An asymmetric hexahedral solid element and a symmetric hexahedral solid element are subsequently derived by using GMEM. The displacement functions of the elemental 24 generalized modes are expressed in terms of Cartesian coordinates so that the element behavior is independent of mesh distortions. Furthermore, the first 21 generalized modes are derived from analytical method making the element capable of avoiding common locking phenomena. Several benchmark problems are performed to demonstrate the accuracy and performance of the new element formulations in linear static and frequency analysis. [ABSTRACT FROM AUTHOR]