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Use of higher-order shape functions in the scaled boundary finite element method. (English) Zbl 1115.74053
The scaled boundary finite element method is a novel semi-analytical technique, whose versatility, accuracy and efficiency are not only equal to, but potentially better than the accuracy of finite element method (FEM) and boundary element method (BEM) for certain problems. The method is suitable for solving linear elliptic, parabolic and hyperbolic partial differential equations. Smooth solutions are obtained analytically in the radial direction, while solutions in the circumferential direction are obtained numerically in the normal finite element sense by introducing boundary discretization and polynomial shape functions. The method combines advantageous features of FEM and BEM. Like in FEM, no fundamental solution is required. Like in BEM, the data preparation and computational time are reduced since only the boundary need to be discretized. The author investigate the possibility of using higher-order polynomial functions for the shape functions. The spectral element approach is used with Lagrange interpolation functions, and the hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the $$p$$-version of FEM. To check the accuracy of the proposed procedures, the authors employ a plane strain problem for which an exact solution is available. The rates of convergence of these examples under $$p$$-refinement are compared with the corresponding rates of convergence achieved when a uniform $$h$$-refinement is used. The results show that it is advantageous to use higher-order elements, and that higher rates of convergence can be obtained using $$p$$-refinement instead of $$h$$-refinement.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74S15 Boundary element methods applied to problems in solid mechanics
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##### References:
  The Scaled Boundary Finite Element Method. Wiley: Chichester, 2003.  Wolf, International Journal for Numerical Methods in Engineering 39 pp 2189– (1996)  . Finite-Element Modelling of Unbounded Media. Wiley: Chichester, 1996. · Zbl 0879.73002  Deeks, Computational Mechanics 28 pp 489– (2002)  Deeks, International Journal for Numerical Methods in Fluids 41 pp 721– (2003)  Song Ch, International Journal for Numerical Methods in Engineering 45 pp 1403– (1999)  Deeks, International Journal for Numerical Methods in Engineering 54 pp 585– (2002)  Deeks, International Journal for Numerical Methods in Engineering 54 pp 557– (2002)  Song Ch, Computer Methods in Applied Mechanics and Engineering 147 pp 329– (1997)  Chebyshev and Fourier Spectral Methods. Springer: New York, 1989. · doi:10.1007/978-3-642-83876-7  . Finite Element Analysis. Wiley: New York, 1991.  Demkowicz, Computer Methods in Applied Mechanics and Engineering 77 pp 79– (1989)  , . Higher-Order Finite Element Methods. Chapman & Hall/CRC: Boca Raton, FL, 2004.  Deeks, Structural Engineering and Mechanics 14 pp 99– (2002) · doi:10.12989/sem.2002.14.1.099  Lagrange interpolating polynomial. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html  Maitre, Numerische Mathematik 74 pp 69– (1996)  Legendre polynomial. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/LegendrePolynomial.html  Melenk, Journal of Computational and Applied Mathematics 139 pp 21– (2002)  . Theory of Elasticity (2nd edn). McGraw Hill: New York, 1951.
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