ORIGINAL PAPER A. J. Deeks  C. E. Augarde A meshless local Petrov-Galerkin scaled boundary method Received: 10 March 2004 / Accepted: 16 August 2004 / Published online: 28 February 2005 Ó Springer-Verlag 2005 Abstract The scaled boundary ﬁnite-element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. The method weakens the governing diﬀerential equations by intro- ducing shape functions along the circumferential coordi- nate direction(s). The weakened set of ordinary diﬀerential equations is then solved analytically in the radial direction. The resulting approximation satisﬁes the governing diﬀerential equations very closely in the radial direction, and in a ﬁnite-element sense in the circumfer- ential direction. This paper develops a meshless method for determining the shape functions in the circumferential direction based on the local Petrov-Galerkin approach. Increased smoothness and continuity of the shape func- tions is obtained, and the solution is shown to converge signiﬁcantly faster than conventional scaled boundary ﬁnite elements when a comparable number of degrees of freedom are used. No stress recovery process is necessary, as suﬃciently accurate stresses are obtained directly from the derivatives of the displacement ﬁeld. Keywords Meshless methods  Scaled boundary ﬁnite- element method  Computational mechanics  Stress singularities  Unbounded domains 1 Introduction The scaled boundary ﬁnite-element method is a semi- analytical method developed relatively recently by Wolf and Song [15–18]. Initially derived to compute the dy- namic stiﬀness matrix of unbounded domains, the method proved far more versatile than initially envis- aged, and was extended to static problems and bounded domains. The rather complicated mathematics of the method in comparison with the ubiquitous ﬁnite element method hindered uptake of the method by other researchers. A virtual work derivation for elastostatics [4] increased the transparency of the method consider- ably, leading to the development of stress recovery and error estimation procedures [5], which in turn allowed adaptive procedures to be implemented [6]. Using these procedures, direct comparison of computational cost for achievement of a prescribed level of accuracy was pos- sible, and the method was shown to be more eﬃcient than the ﬁnite element method for problems involving unbounded domains and for problems involving stress singularities or discontinuities. For unbounded static problems an extension was made allowing problems of plane strain with non-self-equilibrating loads to be addressed [7]. (Displacements for such problems are inﬁnite, but the stress ﬁeld is well-behaved and displacements can be computed relative to any particu- lar point.) The method itself and many of the recent developments are described by Wolf [19]. Extension to unbounded domains in which the material properties vary with depth [8] and to axisymmetric geometries with general loading [9] allowed fruitful application of the method to many foundation problems [10]. In two-dimensional and axisymmetric cases the scaled boundary ﬁnite-element method works by weakening the governing diﬀerential equations in one (circumfer- ential) coordinate direction s through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) coordinate direction n. These coordinate directions are deﬁned by the geometry of the domain and a scaling centre. Unbounded domains not containing the scaling centre (Fig. 1a) and bounded domains containing the scaling centre (Fig. 1b) may be treated, along with domains which are bounded on two sides by lines radiating from the scaling centre (Figs. 1c Comput Mech (2005) 36: 159–170 DOI 10.1007/s00466-004-0649-y A. J. Deeks (&) School of Civil & Resource Engineering, The University of Western Australia, Crawley, Western Australia 6009 E-mail: deeks@civil.uwa.edu.au Tel.: +61 8 6488 3033 Fax: +61 8 6488 1044. C. E. Augarde School of Engineering, University of Durham, Durham, UK