On the convergence of overlapping elements and overlapping meshes

Significance 

The finite element method which has been pioneered by Professor Klaus-Jürgen Bathe, and others, is a commonly used numerical method to solve differential equations in sciences and engineering. However, despite its popularity, its full potential is yet to be realized. This can be partly attributed to the significant human knowledge and effort needed to establish the finite element mesh required to solve various problems. Therefore, reducing the meshing efforts is key in improving the efficiency of finite element methods and expanding their practical applications.

Recently, the use of overlapping finite elements has been identified as a promising approach to reduce meshing efforts. The elements are constructed as overlapping polygonal regions characterized by local interpolations established using concepts of the finite spheres method and a combination of the local fields and traditional finite element functions. This is critical in reducing the effects of mesh distortions and enhancing the accuracy of the numerical solutions. Additionally, the approach of overlapping meshes has also been proposed to complement the use of the overlapping elements. Whereas overlapping elements are less susceptible to mesh distortions and capable of producing high-order convergence, overlapping meshes are used to aid the interpolations of independent meshes naturally.

Alternative methods including fictitious domain methods, the generalized finite element method, Nitsche’s method and domain decomposition methods have also been used to alleviate various meshing issues such as compatibility. However, engineering applications like those involving finite element modeling of complex solids and structures require simple procedures with no Lagrange multipliers, numerical stabilization, and penalty coefficients. Additionally, retaining some features of traditional finite element methods is important to make the solution process more effective and user-friendly. Based on these considerations, Professor Bathe and his students proposed the use of overlapping elements, and the use of these elements in the AMORE scheme of meshing, see references.

AMORE is an acronym for Automatic Meshing with Overlapping and Regular Elements. But “amore” means in Italian “love”, and the word “amore” is similar to amor ( Spanish), amour ( French), also for “love”.

More recently Dr. Junbin Huang and Professor Bathe from the Massachusetts Institute of Technology proposed overlapping meshes and additional overlapping elements to analyze solids and structures. Their original research article is published in the journal, Computers and Structures.

The authors’ earlier papers gave brief discussions and illustrative examples on the convergence of the overlapping elements and the use of the AMORE framework. Here, Dr. Huang and Professor Bathe extended the previous studies to provide deeper insights into the convergence properties of the overlapping elements and meshes based on theoretical analyses and novel illustrative solutions. They commenced by discussing the formulations of the overlapping elements and meshes. The effectiveness of these schemes in reducing meshing efforts was demonstrated which illustrates their use. Finally, the convergence rate of these methods was analyzed and discussed.

The authors showed that through composite interpolations, the accuracy of the proposed schemes improved because they combined the advantages of both the traditional finite element methods and meshless methods. Consequently, the elements are less sensitive to mesh distortions and exhibit improved predictive capability. The improved performance could also be attributed to free overlapping of the individual meshes and convenient local enrichments. Furthermore, using standard tools of convergence analysis, overlapping elements produced convergence rates equal to the polynomial orders used as degrees of freedom. However, the global convergence rate of the overlapping meshes is governed based on the mesh with the lowest order elements. The error bound is independent of the mesh overlapping size, but the results suggested using overlapping meshes with relatively thick overlapped regions is best to improve solution accuracy.

In summary, Professor Bathe and Dr. Huang provided a comprehensive expert view of two methods for reducing meshing efforts in finite element methods and demonstrated their use in the AMORE framework to solve engineering problems. Their promising results indicate the feasibility and practicability of the presented approaches in reducing the meshing efforts. Having proved effectiveness for solving two-dimensional problems in this research, the theoretical approach is surely valuable for the solution of 3D problems. It is important to mention the new research paper authored by Professor Bathe and Dr. K.T. Kim entitled “Accurate solution of wave propagation problems in elasticity”, recently published in Computers &Structures. Here the authors give solutions to wave propagation problems using overlapping elements that should be considered benchmark solutions for use in the evaluations of other computational schemes.

On the convergence of overlapping elements and overlapping meshes - Advances in Engineering
A semi-infinite medium consisting of two elastic layers with a crack in plane strain conditions; a concentrated Ricker wavelet line force is applied on the free surface.
On the convergence of overlapping elements and overlapping meshes - Advances in Engineering
Horizontal and vertical displacements at x = (640,0) are shown. The overlapping elements and the Bathe time integration method are used.
On the convergence of overlapping elements and overlapping meshes - Advances in Engineering
Figure 13 Snapshots of von Mises stress distributions of the two-layered elastic medium with a crack at various observation times; the Ricker wavelet is applied on the free surface. The OFE-TRI2 with a non-uniform mesh of 5,972 elements and the standard Bathe method (γ=0.5) with CFL=0.125 are used.
On the convergence of overlapping elements and overlapping meshes - Advances in Engineering
Figure 14 Horizontal and vertical displacements of the two-layered elastic medium with a crack subjected to the step loading on the free surface. The time histories of the displacements at x = (640,0) are shown. The OFE-TRI2 with non-uniform meshes of 9,186 elements, 36,272 elements, and 144,336 elements and the β_1 \/β_2 -Bathe method (γ=0.5 and β_1=0.35 with β_2=2β_1) with CFL=0.125 are used.
On the convergence of overlapping elements and overlapping meshes - Advances in Engineering
Figure 15 Snapshots of von Mises stress distributions of the two-layered elastic medium with a crack at various observation times; the step loading is applied on the free surface. The OFE-TRI2 with a non-uniform mesh of 36,272 elements and the β_1 \/β_2 -Bathe method (γ=0.5 and β_1=0.35 with β_2=2β_1) with CFL=0.125 are used.

About the author

Klaus-Jürgen Bathe is Professor of Mechanical Engineering at the Massachusetts Institute of Technology. Professor Bathe is also the Founder of the company ADINA R & D, Inc. He has published numerous articles, six textbooks, and is an editor-in-chief of the international journal Computers and Structures. His research results, his book “Finite Element Procedures” and the computer programs he developed have been pioneering contributions in computational engineering.

He has been honored by the ASME, ASCE, U.S. National Academy of Engineering, M.I.T., and many honorary doctorates for his teaching, his pioneering and widely used contributions in engineering and the sciences, and for bridging the world between Academia and Industry. He is also the Founder of the Klaus-Jürgen Bathe Leadership Program at the University of Cape Town, South Africa. See KJ Bathe Wikipedia, see KJ Bathe Google Scholar 

References

J. Huang & K.J. Bathe, (2021). On the Convergence of Overlapping Elements and Overlapping MeshesComputers & Structures, 244, 106429.

Go To Computers & Structures

K.J. Bathe, (2016). The Finite Element Method with ‘Overlapping Finite Elements’. Proceedings Sixth International Conference on Structural Engineering, Mechanics and Computation — SEMC 2016, Cape Town, South Africa (A. Zingoni, ed.).

K.J. Bathe & L. Zhang, (2017). The Finite Element Method with Overlapping Elements. A New Paradigm for CAD Driven Simulations, Computers & Structures. 182, 526-539.

L. Zhang, K.T. Kim & K.J. Bathe,( 2018). The New Paradigm of Finite Element Solutions with Overlapping Elements in CAD — Computational Efficiency of the Procedure. Computers & Structures, 199, 1-17, 2018.

K.J. Bathe, ( 2019) The AMORE Paradigm for Finite Element Analysis. Advances in Engineering Software, 130, 1-13.

K.T. Kim, K.J.Bathe, (2021). Accurate Solution of Wave Propagation Problems in Elasticity. Computers & Structures, Volume 249, 106502.

Go To Computers & Structures

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