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A new cloud-based \(hp\) finite element method. (English) Zbl 0956.74062
Conclusions: We explore the use of lower-order finite element approximations to generate partitions of unity on which hierarchical \(hp\)-cloud approximations can be constructed. The resulting methodology has a number of useful features. Among these are that non-uniform \(hp\)-meshing with variable and hierarchical order \(p\) over clouds can be easily generated. The spectral convergence of \(p\)- and \(hp\)-methods is retained and the method is very robust, the accuracy being quite insensitive to mesh distortion. Also, by building cloud approximations on finite element meshes, Dirichlet boundary conditions are easily handled. The \(hp\)-convergence properties seem to differ from traditional \(p\)-version elements, but exponential convergence is attained. Applications to problems with singularities are easily handled using cloud schemes. In all, this hybrid finite-element/cloud methodology appears to have a number of useful and attractive features that could prove to be important in broad engineering applications.

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
[1] Babuska, I.; Melenk, J.M., The partition of unity finite element method, () · Zbl 0949.65117
[2] Duarte, C.A.M.; Oden, J.T., hp clouds—a meshless method to solve boundary-value problems, () · Zbl 0956.74062
[3] Duarte, C.A.M.; Oden, J.T., An hp adative method using clouds, (), Comput. Methods Appl. Mech. Engrg., to appear · Zbl 0956.74062
[4] Duarte, C.A.M.; Oden, J.T., hp clouds—an hp meshless method, Numer. methods partial diff. eqns., 12, 673-705, (1996) · Zbl 0869.65069
[5] Lee, Nam-Sua; Bathe, Klaus-Jurgen, Effects of element distortions on the performance of isoparametric elements, Int. J. numer. methods engrg., 36, 3553-3576, (1993) · Zbl 0800.73465
[6] J.M. Melenk and Ivo Babuska, The partition of unity finite element method: Basic theory and applications, Comput. Methods Appl. Mech. Engrg., to appear. · Zbl 0881.65099
[7] Oden, J.T.; Duarte, C.A.M., Solution of singular problems using h-p clouds, ()
[8] Peano, A.G., Hierarchies of conforming finite elements for elasticity and plate bending, Comput. math. applic., 2, (1976) · Zbl 0369.73071
[9] Szabo, B.A.; Babuska, I., Computation of the amplitude of stress singular terms for cracks and reentrant corners, (), 101-124
[10] Zienkiewicz, O.C.; de S.R. Gago, J.P.; Kelly, D.W., The hierarchical concept in finite element analysis, Comput. struct., 16, 53-65, (1983) · Zbl 0498.73072
[11] Zienkiewicz, O.C.; Irons, B.M.; Campbell, J.; Scott, F.C., Three dimensional stress analysis, (), 413-432
[12] Zienkiewicz, O.C.; Taylor, R.L., ()
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