Enriching the finite element method with mesh-free particles in structural mechanics

Finite element method is a widely adopted approach in structural mechanics, mainly because of its robustness, ease of application, and high efficiency. Unfortunately, the method suffers from the shortcomings related to the use of a mesh made from geometrically adjacent elements. Even with advanced software, the automatic generation of a quality grid, consistent with some regularity limitations, appears to be an issue in practical applications. In addition, issues with moving boundaries, for example those emanating from shape optimization, generally demand remeshing operations.

In a bid to fix the difficulties related to the topological connections among elements, researchers have developed a number of meshless methods, such as Element-free Galerkin; diffuse element method, and local boundary integration equations. Grid-free methods based on Galerkin or Petrov-Galerkin formulations are the most established. These methods find a numerical solution by using clouds of particles, instead some elemental partition of the domain, therefore avoiding the need for a grid where the unknown variables are estimated.

The meshless local Petrov-Galerkin method appears to be one of the most promising mesh free techniques. In this method, integration is done without background cells that were needed in the previous developments, this making it a truly mesh-free method. In addition, the method has proven to accurately compute an array of scientific problems such as unbounded domains, elastodynamics, magnetic diffusion, and unsteady flows.

Unfortunately, irrespective of these improvements over the conventional formulations, its high cost of computation, which results from the use of the moving least square-based shape functions as well as related integrals, has continued to limit its implementation in various engineering applications. It is therefore important to use meshless local Petrov-Galerkin method in a sub region of the computational domain only, wherein its precision as well as flexibility can be exploited at a limited cost.

Massimiliano Ferronato and Carlo Janna at University of Padova in collaboration with Andrea Zanette at Stanford University advanced a numerical method blending the finite element method with the mesh-free local Petrov-Galerkin method in structural mechanics. Their aim was to exploit the most attractive attributes of each method. Their research work is published in International Journal for Numerical Methods in Engineering.

The authors obtained an enriched solution by superimposing the meshless local Petrov-Galerkin solution to the finite element estimation space. They focused on 2-dimensional applications in structural mechanics, but the proposed method could be extended to other problems. The main aim of the study was for the application of the original idea to structural mechanics, where the issue of improving the stress-strain solution can be of interest in a number of applications.

Whatever the field of application, the formulation was endowed with an inherent flexibility for enhancing the accuracy of the solution at any instance of the domain without costly remeshing elements. The algorithm introduced for prescribing the boundary conditions on the hybrid Finite Element meshless local Petrov-Galerkin estimation allowed for full flexibility in adding the particles to any region of the computational domain.

The enrichment method takes advantage from an efficient numerical integration rule implemented to form the entries of the coupling blocks. The formulation avoided the complicated surface integrations related to the discontinuous integrands that emanate from the intersections of the moving least square supports with the finite elements. Instead, only elementary 1-dimensional integrals were analyzed over the edges of the finite elements.

The conceptual algorithm adopted in their study possesses unique features for enhancing finite element solution, avoiding remeshing efforts. This feature is attractive in problems where local adaptive refinements are necessary in varying positions of the domain in the course of simulation of a transient process.