Application of Nonconforming Quadtree Grids in the Finite Element Method

Document Type : Research Article

Author

Abstract

In the standard finite element method, the edges of the adjacent elements are aligned to each other, and the corner of an element does not locate on the edges of another one. If this constraint violates, the mesh is called non-conforming and the use of such meshes in the finite element method requires specific techniques. In the present paper, a new method is suggested for treating non-conforming meshes. Non-conforming meshes appear generally in adaptive mesh refinement processes especially in the quadtree mesh refinement algorithm. The quadtree is a data structure with an extremely fast recursive algorithm and is used to divide a two-dimensional domain into sub-regions or elements. In the present paper, a new approach is proposed to construct the shape functions of such elements. In this method, the shape functions are considered harmonic functions and the Laplace boundary value problem is defined and its solution is used as the shape functions of the non-conforming elements. To evaluate the applicability and accuracy of the proposed method, two numerical examples are solved and the results are presented. The results show that the proposed method can be used to effectively apply the non-conforming meshes in the finite element method.

Keywords

Main Subjects

References

[1] M.J. Kazemzadeh-Parsi and F. Daneshmand, Solution of geometric inverse heat conduction problems by smoothed fixed grid finite element method, Finite Elements in Analysis and Design, 45 (2009) 599-611.
[2] M.J. Kazemzadeh-Parsi and F. Daneshmand, Three dimensional smoothed fixed grid finite element method for the solution of unconfined seepage problems, Finite Elements in Analysis and Design, 64 (2013) 24-35.
[3] M.J. Kazemzadeh-Parsi, Optimal shape design for heat conduction using smoothed fixed grid finite element method and modified firefly algorithm, Iranian Journal of Science and Technology, 39(M2) (2015) 367-387.
[4] H. Samet, Application of Spatial Data Structure, Addison-Wesley, NewYork, 1990.
[5] N. Provatas, N. Goldenfeld, J. Dantzig, Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, Journal of Computational Physics 148(1) (1999) 265-290.
[6] P.D. Stolfo, A. Schroder, N. Zander, S. Kollmannsberger, An easytreatmentofhangingnodesin hp-finiteelements, Finite Elements in Analysis and Design, 121 (2016) 101-117.
[7] A. Tabarraei, N. Sukumar, Adaptive computations on conforming quadtree meshes, Finite Elements in Analysis and Design, 41 (2005) 686-702.
[8] S. Natarajan, E.T. Ooi, C. Song, Finite element computations over quadtree meshes: strain smoothing and semi-analytical formulation, International Journal of Advances in Engineering Sciences and Applied Mathematics, 7(3) (2015) 124-133.
[9] T.P. Fries, A. Byfut, A. Alizada, K.W. Cheng, A. Schroder, Hanging nodes and XFEM, International Journal for Numerical Methods in Engineering, 86 (2011) 404-430.
[10] M.J. Kazemzadeh-Parsi and F. Daneshmand, Finite element method: A practical course, Islamic Azad University Press, Shiraz Branch, Shiraz, 2011, (In Persian).
[11] N.H. Asmar, Partial differential equations with Fourier series and boundary value problems, Pearson Prentice Hall, New Jersey, 2005.
Amirkabir Journal of Mechanical Engineering
Volume 53, Issue 2
May 2021
Pages 673-686

Files

Share

How to cite

Statistics