Numerical algorithm of concave polygon elements in solid mechanics: Theoretical analysis and performance evaluation of S-FEM and VEM

• The equivalence between the smoothing stiffness matrix of S-FEM and the consistent stiffness matrix of VEM is demonstrated within the framework of concave polygonal elements for the first time. • Two methods were validated in terms of accuracy, efficiency, and implementation difficulty. • Mathematically prove the equivalence of stiffness matrices for different types of polygonal elements, and physically (from the perspective of strain energy) demonstrate their differing properties. This research focuses on the theoretical and computational aspects of numerical methods for handling concave polygonal elements in solid mechanics, namely smoothed finite element method (S-FEM) and virtual element method (VEM). Commencing with the principle of virtual work, the study uniformly formulates and discretizes the governing equations using the same mathematical symbols for both methodologies, meticulously deriving and implementing their respective stiffness matrices rooted in the elemental strain energy formulations. The equivalence between the smoothing stiffness matrix of S-FEM with smoothing domains and the consistent stiffness matrix of VEM has been demonstrated through theoretical analysis and numerical examples using various types of polygonal elements. The equivalence is directly verified by using numerical values of stiffness matrices with different concave convex grids, but the different characteristics of the two methods are also verified from a physical perspective (strain energy). Subsequently, a series of benchmark problems and real-world engineering instances are leveraged to scrutinize and contrast the efficacies and constraints of S-FEM and VEM about concave polygon. Results attest to the substantial applicability and robustness of these methods on irregularly-shaped concave polygons, advancing numerical techniques for non-standard geometric shapes in solid mechanics. This scholarly work offers insights for researchers and practitioners in selecting optimal computational strategies when analyzing the behavior of concave polygon elements.