Weak-Form Quadrature Element Method: A Comparative Review of Different Formulations and Its Comprehensive Assessment

As a relatively new computational method, the weak-form quadrature element method (QEM) has attracted more and more worldwide attention recently. However, there exist different formulations, even different names, for the QEM in the literature. To remove possible confusion caused by them, a comparative review is carried out in this paper. The step-by-step procedure of the formulations is examined in detail by a simple one-dimensional bar element, and the nature of each step is revealed carefully. Similarities and differences between the different formulations are summarized after the scrutiny. Based on the full understanding of the QEM, a comprehensive assessment of the method is conducted by comparing several relevant aspects. Superiorities and limitations of the QEM are discussed and its future research directions are pointed out. Finally, a chronological list of research papers related to the QEM since 2017 is presented. State-of-the-art developments and applications of the method are highlighted.