Development of a four-node quadrilateral element-based high order numerical manifold method without linear dependency

Abstract

A high-order numerical manifold method (HONMM) is developed using four-node quadrilateral (QUAD4) element to enhance accuracy and computational efficiency in solid mechanical problems. In the current study, the high-order global approximations are built by increasing the order of local approximations without facing the linear dependence (LD) problem, which is one of the main issues in the partition of unity (PU)-based methods with high-order approximations. To remove the LD problem, a new and simple scheme is proposed. Four problems are utilized to evaluate the efficiency of the QUAD4-based HONMM (QHONMM). A cantilever beam example is analyzed to compare different orders of QHONMM. Also, a block example with different loads and boundary conditions is used to investigate the sensitivity of the QHONMM results. Moreover, to demonstrate the capabilities of the QHONMM in dynamic analysis, free and forced vibrations of a simple beam under distributed and moving loads are analyzed. The results showed that using QUAD4 elements, the proposed QHONMM performs better than the conventional NMM in terms of accuracy and computational costs.