Computational efficiency and accuracy of the Neighbored Element Method

Gradient-enhanced regularization is a frequently utilized method for addressing numerical issues in material modeling. As a consequence of the regularization scheme, Laplacian terms will emerge in the strong form of evolution equations for additional field variables, also called internal variables. In a series of previous works, the Neighbored Element Method (NEM) was presented as a combination of the Finite Element Method and a generalized finite difference scheme with a weighted least-square method to approximate the Laplacian. The objective is the efficient solution of the total system of equations containing Laplacian and gradient terms. The systems of equations used with the NEM have a similar structure to, e.g., the heat and diffusion equation. In this study, the NEM is evaluated even further, in comparison to a well-established FEM routine with respect to accuracy and computational efficiency through investigating a chemo-thermo-mechanical system. It is demonstrated that the accuracy improves with a reduction in element size and, for an appropriate mesh, the relative average error is below 1%. These highly accurate solutions can be achieved with a notable reduction in computational time and memory cost of up two orders of magnitude, for approximately 620000 nodes with five degrees of freedom per node. This new technique can be applied to arbitrary solid finite element types and/or irregular meshes.