# Development of simple effective cloud of nodes and triangular mesh generators for meshless and element-based analyses – implementation in Matlab

### Abstract

This paper is devoted to the development of the Matlab software dedicated to the generation of 2D arbitrarily irregular clouds of nodes and triangular meshes. They may be applied in numerical analyses of boundary value problems, based on both meshless and finite element discretization techniques, especially in the case of numerical homogenization in which the domain partitioning into disjoint subdomains may be required. Several Matlab functions are extended on the basis of the simple computational geometry-based ideas and concepts of engineering nature. A set of Matlab functions, attached to this paper, is discussed in detail, and examined on selected boundary value problems.

### Keywords

cloud of nodes, mesh generation, Delaunay triangularization, meshless methods, finite element method, implementation in Matlab,### References

[1] M. Shimrat. Algorithm 112: position of point relative to polygon. Communications of the ACM, 5(8), 434 pages, 1962.[2] T. Liszka, J. Orkisz. The finite difference method at arbitrary irregular grids and its applications in applied mechanics. Computers & Structures, 11: 83–95, 1980.

[3] P. Lancaster, K. Salkauskas. Surfaces generated by moving least squares method. Mathematics of Computation, 155(37): 141–158, 1981.

[4] T. Liszka. An interpolation method for an irregular net of nodes. Int. J. Num. Meth. Eng., 20: 1599–1612, 1984.

[5] Matlab C Math Library. User’s Guide. The MathWorks Inc., 1984-2015.

[6] F.P. Preparata, M.I. Shamos. Computational Geometry: An Introduction. Springer-Verlag Berlin- Heidelberg, 1985.

[7] K.E. Atkinson. An Introduction to Numerical Analysis. Wiley Ed., NewYork, 1988.

[8] B.A. Barsky, T.D. DeRose. Geometric continuity of parametric curves: three equivalent characterizations. IEEE Comput. Graph. and Appl., 9: 60–68, 1989.

[9] L. Piegl. Modifying the shape of rational B-splines. Part 1: curves. Computer-Aided Design, 21(8): 509–518, 1989.

[10] J. Krok, J. Orkisz. A unified approach to the FE and generalized variational FD methods in nonlinear mechanics, concepts and numerical approach. Discretization Methods in Structural Mechanics, 1: 353–362, 1990.

[11] P. Lancaster, K. Salkauskas. Curve and Surface Fitting. Academic Press Inc., 1990.

[12] K. Weiler. An incremental angle point in polygon test. [In:] Paul S.Heckbert [Ed.], Graphics Gems IV, pp. 16–23. Academic Press Professional Inc., San Diego, CA, USA, 1994.

[13] T. Belytchko, Meshless methods: an overview and recent developments. Comp. Meth. Appl. Mech. Engng., 139: 3–47, 1996.

[14] D. Hegen. Element-free Galerkin methods in combination with finite element approaches. Comp. Meth. Appl. Mech. Engng., 135: 143–166, 1996.

[15] J.R. Shewchuk. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. Applied Computational Geometry Towards Geometric Engineering, Springer, Berlin-Heidelberg, pp. 203–222, 1996.

[16] M. Ainsworth. J.T. Oden. A-posteriori error estimation in finite element analysis. Comp. Meth Appl. Mech Engng., 142: 1–88, 1997.

[17] F. Hecht. BAMG: bidimensional anisotropic mesh generator. INRIA Report, 1998.

[18] J. Orkisz. Finite Difference Method (Part III). [In:] M. Kleiber [Ed.], Handbook of Computational Solid Mechanics, pp. 336–431, Springer-Verlag, Berlin, 1998.

[19] J. Alberty, C. Carstensen, S.A. Funken. Remarks around 50 lines of Matlab: short finite element implementation. Numerical Algorithms, 20: 117–137, 1999.

[20] N. Gershenfeld. The Nature of Mathematical Modeling. Cambridge University Press, ISBN 0-521-57095-6, 1999.

[21] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery. Numerical Recipes in Fortran 90. The Art of Parallel Scientific Computing. Cambridge Univ. Press, 1996.

[22] A. Huerta, S. Hernandez-Menez. Enrichment and coupling of the finite element and meshless methods. Int. J. Numer. Methods Eng., 48: 1615–1630, 2000.

[23] X.Y. Li, S.H. Teng, A. ¨Ung¨or. Point placement for meshless methods using sphere packing and advancing front methods. [In:] Proceedings of ICCES’00 – International Conference on Computational Engineering Science, Los Angeles, 2000.

[24] S. Teng, X.Y. Li, A. ¨Ung¨or. Generating a good quality point set for the meshless methods. Comput. Model. Eng. Sci., 1(1): 1017, 2000.

[25] T.P. Fries, H.G. Matthies. Classification and Overview of Meshfree Methods. Technische Universit¨at Braunschweig, 2004.

[26] R. L¨ohner, E. O˜nate. A general advancing front technique for filling space with arbitrary objects. International Journal for Numerical Methods in Engineering, 61(12): 1977–1991, 2004.

[27] O.C. Zienkiewicz, R.L. Taylor. Finite Element Method: Its Basis and Fundamentals, Elsevier, 6th edition, 2005.

[28] C. Carstensen, S. Bartels, A. Hecht. P2Q2Iso2D= 2D Isoparametric FEM in Matlab. J. Comput. Appl. Math., 192: 219–250, 2006.

[29] C. Drumm, S. Tiwari, J. Kuhnert, H.-J. Bart. Finite pointset method for simulation of the liquid-liquid flow field in an extractor. Computers and Chemical Engineering, 32(12): 2946–2957, 2008.

[30] J.A. Cottrel, T.J.R. Hughes, Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley and Sons, ISBN 978-0-470-74873-2, 2009.

[31] Y. Nie, W. Zhang, Y. Liu, L. Wang. A node placement method with high quality for mesh generation. IOP Conference Series: Materials Science and Engineering, Vol. 10, IOP Publishing, 2010.

[32] C. Pearce, L. Kaczmarczyk, J. Novak. Multiscale modelling strategies for heterogeneous materials. Comuputational Technology Reviews, 2: 23–49, 2010.

[33] S. Funken, D. Praetorius, P. Wissgott. Efficient implementation of adaptive P1-FEM in MATLAB. Computational Methods in Applied Mathematics, 11(4): 460–490, 2011.

[34] S. Milewski. Meshless finite difference method with higher order approximation-applications in mechanics. Archives of Computational Methods in Engineering, 19(1): 1–49, 2012.

[35] I. Jaworska. On the ill-conditioning in the new higher order multipoint method. Computers and Mathematics with Applications, 66(3): 238–249, 2013.

[36] S. Milewski. Selected computational aspects of the meshless finite difference method. Numerical Algorithms, 63: 107–126, 2013.

[37] Z. Ullah, W. Coombs, C. Augarde. An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 267: 111–132, 2013.

[38] S. Kumar, I. Singh, B. Mishra. A coupled finite element and element-free Galerkin approach for the simulation of stable crack growth in ductile materials. Theoretical and Applied Fracture Mechanics, 70: 49–58, 2014.

[39] Y. Nie, W. Zhang, N. Qi, Y. Li. Parallel node placement method by bubble simulation. Computer Physics Communications, 185(3): 798–808, 2014.

[40] P.O. Persson, G. Strang. A simple mesh generator in MATLAB. SIAM Review, 46(2): 329–345, 2014.

[41] J. Jaśkowiec, S. Milewski. The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions. Computers and Mathematics with Applications, 70(5): 962–979, 2015.

Published

Mar 25, 2018

How to Cite

MILEWSKI, Sławomir.
Development of simple effective cloud of nodes and triangular mesh generators for meshless and element-based analyses – implementation in Matlab.

**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 24, n. 3, p. 157–180, mar. 2018. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/192>. Date accessed: 26 jan. 2022. doi: http://dx.doi.org/10.24423/cames.192.
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