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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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