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

Significance Statement

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.

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

About The Author

Andrea Zanette is a PhD candidate in the Institute for Computational and Mathematical Engineering at Stanford University. He works in the field of theoretical Artificial Intelligence and in particular in Reinforcement Learning with prof. Emma Brunskill and prof. Mykel Kochenderfer.

His research is centered around sample efficient and computationally efficient methods for learning how to make good decision under uncertainty. In particular, he is currently examining what structural properties of the environment make learning easy or difficult. He leverages such insights to devise intelligent agents that achieve tighter theoretical guarantees for specific classes of environments.

He works with algorithms that are either probably approximately correct or minimizing the cumulative regret. Before joining Stanford as a graduate student he worked in the construction sector as a consultant and also briefly in fluid dynamics at the von Karman Institute for Fluid Dynamics. He graduated with a BS in Mechanical Engineering with a thesis on meshless methods advised by prof. Massimiliano Ferronato.

About The Author

Massimiliano Ferronato got the Degree in Civil Engineering at the University of Padova (Italy) in 1998 and the PhD in Numerical Geomechanics at the Technology University of Delft (The Netherlands) in 2003. Since 2015, he is Associate Professor at the Department of Civil, Environmental and Architectural Engineering of the University of Padova with teaching duties in the Numerical Methods class. He has authored and co-authored more than 150 scientific papers published in international journals, books and proceedings of international conferences, and delivered invited talks and lectures in several renowned international symposia.

The main scientific interests concern the numerical solution of the partial differential equations governing the mechanics of saturated and partially saturated porous media, with engineering applications in the field of subsurface hydrology and petroleum engineering. He has been involved in a number of projects related to the numerical simulation of the geomechanical behavior of deep producing reservoirs, geological formations used for storage purposes, e.g., CO2 sequestration, and shallow multi-aquifer systems, including the analysis of fault activation, fissure generation, failure risk and land subsidence.

The main scientific contributions concern the development and implementation of efficient and robust numerical models, based on Finite Element, Mixed Finite Element and Finite Volume methods, for the simulation of geomechanical and fluid-dynamical processes in the subsurface. Particular care is paid to the implementation of accurate iterative solvers for the linear systems arising from such applications. He is the co-author of a number of original algorithms for both sequential and parallel computational architectures, with the aim of accelerating the convergence and improving the robustness of iterative linear solvers.

About The Author

Carlo Janna graduated in Civil Engineering at the University of Padova (Italy) in 2003 with 109 points over 110 and in the same University he got his PhD defending a thesis entitled “Numerical modeling of the mechanical behavior of regional faults in the geological sequestration of anthropogenic CO2 sequestration”. Since December 2011 he is assistant professor at the Department ICEA. The main scientific interests concern on one hand the mathematical and numerical modeling of the mechanics of porous media in both saturated and unsaturated conditions with specific applications in subsurface hydrology and petroleum industry, on the other the numerical linear algebra.

His main activity is the development and implementation of numerical models based on the Finite Element method for the simulation of subsurface coupled and uncoupled geomechanical and fluid dynamical processes in the exploitation of deep aquifer or reservoir resources. As to the linear algebra, Carlo Janna studies and develops numerical techniques for the solution of large sparse linear systems and eigenproblems and more specifically iterative methods and preconditioners. For sequential computers, he studied and developed several ad hoc preconditioners for the solution to specific problems arising in subsurface simulations.

From 2010 to 2012, Carlo Janna joined the HPC research projects PARPSEA (PARallel Preconditioners for large Size Engineering Applications), SCALPREC (SCALable PREConditioners), OPTIDAS (OPTImization and Data ASSimilation) e SPREAD (Scalable PREconditioners for Advanced Discretizations) studying and developing new preconditioners for massively parallel computers. Carlo Janna is author or co-author of more than 70 scientific papers in international refereed journals, books and conference proceedings.


A. Zanette, M. Ferronato and C. Janna. Enriching the finite element method with mesh-free particles in structural mechanics. International Journal for Numerical Methods in Engineering 2017; 110: pages 675–700.


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