H-Adaptive Local Radial Basis Function Collocation Meshless Method

• Published in: Computers, materials & continua Vol. 26; no. 3; pp. 227 - 253
• Main Authors: Kosec, G, Šarler, B
• Format: Journal Article
• Language: English
• Published: Henderson Tech Science Press 01.01.2011
• Subjects: Accuracy; Burgers equation; Collocation; Collocation methods; Derivatives; Domains; Finite element method; Mathematical analysis; Mathematical models; Meshless methods; Nodes; Radial basis function; Topology; Transport equations; Transport phenomena
• ISSN: 1546-2218; 1546-2226
• DOI: 10.3970/cmc.2011.026.227

This paper introduces an effective H-adaptive upgrade to solution of the transport phenomena by the novel Local Radial Basis Function Collocation Method (LRBFCM). The transport variable is represented on overlapping 5-noded influence-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the variable are calculated from the respective derivatives of the RBFs. The transport equation is solved through explicit time stepping. The H-adaptive upgrade includes refinement/derefinement of one to four nodes to/from the vicinity of the reference node. The number of the nodes added or removed depends on the topology of the reference node vicinity. The refinement/derefinement is triggered by an error indicator, which very simply depends on the ratio between the norm of the collocation coefficients and collocation matrix. The refinement/derefinement is proportional with the growth/decay of this indicator. Such adaptivity much increases the accuracy/performance ratio of the method. The performance of the method is numerically tested on two-dimensional Burger's equation. The results are compared with different numerical solutions, published in literature. Outstanding CPU efficiency and accuracy are clearly demonstrated from the results. The paper probably for the first time shows such a simple and effective H-adaptive meshless method, designed on five noded influence domain. The advantages of the represented meshless approach are its simplicity, accuracy, similar coding in 2D and 3D, straightforward applicability in non-uniform node arrangements, and native parallel implementation.