Simple Discretisation Strategies for Better Convergence of Finite Element Model Without Increasing the Number of Degrees of Freedom

The interpolation functions used in the finite element (FE) derivations are the backbone of the FE derivation, and consequently, they impact the whole FE analysis. Therefore, their derivation gains a lot of research interest. When these functions equal the solutions of differential equations, they lead to exact results, otherwise to approximate solutions. The approximate solutions’ quality is typically improved by adding degrees of freedom (DOF) to the model. This can be done either with a finer mesh of finite elements (h-method) or by increasing the displacement field accuracy in each element by implementing better interpolation functions (p-method). However, it is often overlooked for material nonlinearity problems that the results’ quality can also be improved without any of these two approaches. Namely, by replacing the frequently used mesh of uniform finite elements with finite elements of adequately modified lengths (and without any change in the number of degrees of freedom), beneficial effects on the quality of the results are achieved (with an almost negligible increase in overall computational effort due to the additional implementation of the meshing algorithm). To demonstrate this, the paper thus focuses on analysing the axial displacements of a beam made of material with a nonlinear constitutive law, analysed with a fundamental two-node linear finite element with a total of two degrees of freedom. Different dedicated algorithms for generating the mesh of finite elements are studied, considering the local quality indicators (displacement of the free end, reaction at support) as well as the global parameter (total internal strain energy). The studied examples show that an evidently better quality of results is achieved with (almost) all considered algorithms. Furthermore, with some algorithms, it is possible to achieve better results even with a smaller computational model, i.e. with fewer degrees of freedom. The research was conducted in two major steps. All considered meshing algorithms were initially tested with the same number of finite elements to see which one produced the better agreement with the exact solution. This stage was repeated for several meshes with various numbers of finite elements. After finding the best meshing algorithm, this one was applied (with various values of strain hardening parameter n) to demonstrate its efficiency against the original meshing algorithm.