Performances of Variable-Node Elements With the Base Force Element Method Under the Complementary Energy Principle

To address the problems of uncoordinated nodal displacements at the intersection of the sparse and dense elements, complicated construction of solving equations, and poor spatial scalability, an element model with variable node numbers and positions was proposed with the base force element method (BFEM) under the complementary energy principle, and an explicit solution method in a unified form was established for arbitrary element types. First, a 2D variable mid-edge node element model was established, and the explicit expressions of the contribution of nodes compliance matrix and nodal displacements were introduced. Subsequently, the element model was extended to the 3D form, a variable mid-face node element model was proposed, and the above expressions were also extended to the 3D forms. Hereafter, a sparse and dense mesh hanging element model was established, and the numerical accuracy and applicability of the planar and spatial variable-node element model were demonstrated with the cantilever beam subjected to a bending moment load, a concentrated load, and a tensile load at the end. The numerical results show that, the variable-node element model and the hanging element model in the 2D and 3D forms based on the BFEM have high numerical accuracy. In addition, the nodal displacement coordination at the intersection interface between sparse and dense elements can be ensured only by the shared mid-edge node (2D) and the mid-face node (3D) at the interface without any processing treatment or construction of interpolation functions and constraint functions. Meanwhile, the element models and methods are independent of the element type, element dimensions, nodes' number, and nodes' distribution, etc., and have excellent spatial scalability and programmability.