Use of higher‐order shape functions in the scaled boundary finite element method

Abstract

SUMMARYThe 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 finite element method and the boundary element method for certain problems. This paper investigates the possibility of using higher-order polynomial functions for the shape functions. Two techniques for generating the higher-order shape functions are investigated. In the first, the spectral element approach is used with Lagrange interpolation functions. In the second, hierarchical polynomial shape functions are employed to add new degrees of freedom into the domain without changing the existing ones, as in the pversion of the finite element method. To check the accuracy of the proposed procedures, a plane strain problem for which an exact solution is available is employed. A more complex example involving three scaled boundary subdomains is also addressed. The rates of convergence of these examples under p-refinement are compared with the corresponding rates of convergence achieved when uniform h-refinement is used, allowing direct comparison of the computational cost of the two approaches. 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.

Development of a fundamental‐solution‐less boundary element method for exterior wave problems

“…An additional benefit of the nodal quadrature lumping procedure is that a high-order element with Gauss-Lobatto-Legendre shape functions is formulated. This type of high-order element has been successfully applied to the scaled boundary finite-element method for statics by Vu and Deeks [35]. They demonstrated that higher rates of convergence can be obtained using p-refinement than using h-refinement.…”

A spectral element approach for the stability of delay systems

“…These sets of nodes are more clustered at the end of the elements with an asymptotic density proportional to (1−u 2 ) −1/2 as n →∞ [52]. More specifically, the n +1 LGL points are the roots of the polynomial (1−u 2 )L n (u) where u ranges from −1 to 1 and L n (u) is the Legendre polynomial of order n [54]. These points can be shifted to an arbitrary interval [a, b] through the relatioñ…”

“…The Lagrange polynomial i corresponding to the ith node has the property Figure 4 shows the resulting seventh-order trial functions using eight nodes that are equally spaced in graph (a) while the LGL distribution is used in graph (b). These plots are generated over the range [−1, 1], and they match those in Reference [54]. Equation (16) comes from the typical form of Lagrange interpolation which is only recommended for a small number of nodes.…”

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

“…Consequently, it helps to save time and computational effort in data preparation and storage. Different types of hierarchical shape functions were investigated [5], including the higher-derivative-based shape functions [11] and Lobatto polynomials [7,10]. The study recommended the use of the Lobatto polynomials due to their ability to produce accurate solutions for high polynomial order [5].…”