×

Integration-free Coons macroelements for the solution of 2D Poisson problems. (English) Zbl 1159.65362

Summary: Large isoparametric macroelements with closed-form cardinal global shape functions under the label ‘Coons-patch macroelements’ (CPM) have been previously proposed and used in conjunction with the finite element method and the boundary element method. This paper continues the research on the performance of CPM in conjunction with the collocation method. In contrast to the previous CPM that was based on a Galerkin/Ritz formulation, no domain integration is now required, a fact that justifies the name ‘integration-free Coons macroelements’. Therefore, in addition to avoiding mesh generation, and saving human effort, the proposed technique has the additional advantage of further reducing the computer effort. The theory is supported by five test cases concerning Poisson and Laplace problems within 2D smooth quadrilateral domains.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Software:

Mfree2D
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ritz, Über eine neue Methode zur Lösung gewisser Variations probleme der mathematischen Physik, Zeitschrift für Reine und Angewandte Mathematik 135 (1) pp 1– (1909)
[2] Zienkiewicz, The Finite Element Method (1977) · Zbl 0435.73072
[3] Wachspress, A Rational Finite Element Basis (1975) · Zbl 0322.65001
[4] Malsch, Interpolations for temperature distributions: a method for all non-concave polygons, International Journal of Solid and Structures 41 pp 2165– (2004) · Zbl 1050.80005
[5] Brebbia, Boundary Elements: An Introductory Course (1992)
[6] Kita, Trefftz method: an overview, Advances in Engineering Software 24 (1-3) pp 3– (1995) · Zbl 0984.65502
[7] Li, Trefftz, collocation, and other boundary methods-a comparison, Numerical Methods for Partial Differential Equations 23 pp 93– (2007) · Zbl 1223.65093
[8] Atluri, The Meshless Method (MLPG) for Domain and BIE Discretizations (2004) · Zbl 1105.65107
[9] Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (2003) · Zbl 1031.74001
[10] Schramm, The coupling of geometric descriptions and finite element using NURBS-a study in shape optimization, Finite Elements in Analysis and Design 15 pp 11– (1993) · Zbl 0801.73074
[11] Hughes, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 pp 4135– (2005) · Zbl 1151.74419
[12] Inoue, A NURBS finite element method for product shape design, Journal of Engineering Design 16 (2) pp 157– (2005)
[13] Provatidis, Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems, International Journal for Numerical Methods in Engineering 65 pp 111– (2006) · Zbl 1122.74518
[14] Provatidis C. Stress analysis of 3D solid structures using large boundary elements derived from 2D-Coons interpolation. In Proceedings of ASME-Greek Section Conference, Patras, Greece, Drakatos PA (ed.), 17-20 September 2001 (Paper ANG1/P129).
[15] Provatidis C. Analysis of three-dimensional sound radiation problems using trimmed patch boundary elements. In Proceedings 4th GRACM Congress on Computational Mechanics, vol. I, Patras, Greece, Tsahalis DT (ed.), 27-29 June 2002; 402-409.
[16] Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide (1990) · Zbl 0702.68004
[17] Provatidis, Three-dimensional Coons macroelements in Laplace and acoustic problems, Computers and Structures 83 (4-5) pp 1572– (2005)
[18] Provatidis, Analysis of box-like structures using 3-D Coons’ interpolation, Communications in Numerical Methods in Engineering 21 (8) pp 443– (2005) · Zbl 1151.74422
[19] Provatidis, Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems, Computers and Structures 79 pp 1769– (2001)
[20] Provatidis, Coons-patch macroelements in potential Robin problems, Forschung im Ingenieurwesen 67 pp 19– (2002) · Zbl 1099.65090
[21] Provatidis, Analysis of axisymmetric structures using Coons’ interpolation, Finite Elements in Analysis and Design 39 pp 535– (2003)
[22] Provatidis, Coons-patch macroelements in two-dimensional eigenvalue and scalar wave propagation problems, Computers and Structures 82 (4-5) pp 383– (2004)
[23] Provatidis, Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolation, Communications in Numerical Methods in Engineering 20 (7) pp 521– (2004) · Zbl 1048.65115
[24] Provatidis, Coons-patch macroelements in two-dimensional parabolic problems, Applied Mathematical Modelling 30 (4) pp 319– (2006) · Zbl 1099.65090
[25] Provatidis, Free vibration analysis of two-dimensional structures using Coons-patch macroelements, Finite Elements in Analysis and Design 42 (6) pp 518– (2006) · Zbl 1120.74516
[26] Provatidis, Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements, International Journal of Solids and Structures 43 (22-23) pp 6688– (2006) · Zbl 1120.74516
[27] Fairweather, Mathematics for Large Scale Computing pp 297– (1989)
[28] De Boor, A Practical Guide to Splines (1978) · doi:10.1007/978-1-4612-6333-3
[29] Ascher, Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations (1995) · Zbl 0843.65054 · doi:10.1137/1.9781611971231
[30] Bialecki, Orthogonal spline collocation methods for partial differential equations, Journal of Computational and Applied Mathematics 128 (1-2) pp 55– (2001) · Zbl 0971.65105
[31] Johnson, Higher order B-spline collocation at the Greville Absissae, Applied Numerical Mathematics 52 (1) pp 63– (2005) · Zbl 1063.65072
[32] Liu, Radial point interpolation collocation method (RPICM) for the solution of nonlinear Poisson problems, Computational Mechanics 36 pp 298– (2005) · Zbl 1099.65119
[33] Jator, A high order B-spline collocation method for linear boundary value problems, Applied Mathematics and Computation 191 pp 100– (2007) · Zbl 1193.65126
[34] Van Blerk, Numerical solution of partial differential equations on curved domains by collocation, Numerical Methods for Partial Differential Equations 9 pp 357– (1993) · Zbl 0780.65069
[35] Gordon, Transfinite element methods: blending-function interpolation over arbitrary curved element domains, Numerische Mathematik 21 pp 109– (1973) · Zbl 0254.65072
[36] Gordon, Blending functions methods of bivariate multivariate interpolation and approximation, SIAM Journal on Numerical Analysis 8 pp 158– (1971) · Zbl 0237.41008
[37] Elansari, Boundary solution of Poisson’s equation using radial basis function collocated on Gaussian quadrature nodes, Communications in Numerical Methods in Engineering 17 pp 455– (2001) · Zbl 0987.65129
[38] Liu, A point interpolation meshless method based on radial basis functions, International Journal for Numerical Methods in Engineering 54 pp 1623– (2002) · Zbl 1098.74741
[39] Liszka, Hp-meshless cloud method, Computer Methods in Applied Mechanics and Engineering 139 pp 263– (1996) · Zbl 0893.73077
[40] Zhang, Meshless methods based on collocation with radial basis function, Computational Mechanics 30 pp 396– (2003)
[41] Chen, The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, Journal of Sound and Vibration 257 (4) pp 667– (2002)
[42] Chen, Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Computational Mechanics 29 pp 392– (2002) · Zbl 1146.76622
[43] Zienkiewicz, The coupling of the finite element method and boundary solution procedures, International Journal for Numerical Methods in Engineering 11 (2) pp 355– (1977) · Zbl 0347.65048
[44] Zielinski, Generalized finite element analysis with T-complete boundary solution functions, International Journal for Numerical Methods in Engineering 21 pp 509– (1985) · Zbl 0594.65081
[45] Frind, A collocation finite element method for potential problems in irregular domains, International Journal for Numerical Methods in Engineering 14 pp 681– (1979) · Zbl 0397.65075
[46] Carslaw, Conduction of Heat in Solids (1959)
[47] Karamcheti, Principles of Ideal-fluid Aerodynamics pp 383– (1980)
[48] Finlayson, The Method of Weighted Residuals and Variational Principles (1972) · Zbl 0319.49020
[49] Provatidis, Free vibration analysis of elastic rods using global collocation, Archive of Applied Mechanics 78 (4) pp 241– (2008) · Zbl 1161.74398
[50] Provatidis, Time- and frequency-domain analysis using lumped mass global collocation, Archive of Applied Mechanics · Zbl 1161.74399 · doi:10.1007/s00419-008-0203-z
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.