Coupled formulation of non-uniform rational B-spline and radial point interpolation method for 2D problems

Abstract

Simulation-based design in engineering is becoming very important nowadays due to the advancement of computing technology. In this arena, computer-aided design (CAD) for modelling and computer-aided engineering (CAE) for analysis are the two major components. They evolve independently despite dealing with the same object-of-interest. The non-collaborative nature of CAD and CAE has resulted in more manpower and less computer time being used in the steps involved during data transfer for the modelling-analysis process, which can lead to many errors. Ideally, this process should be performed entirely by a computer without human intervention. In bridging the gap between the two, isogeometric analysis (IGA) was proposed to perform both modelling and analysis using the same basis functions, i.e., non-uniform rational B-spline (NURBS). However, NURBS is formulated through the operation of tensor products, thus the refinement in the analysis process is found to be expensive due to excessive overhead of control points. This study presents the idea to develop more efficient methods by considering the NURBS only for modelling while the analysis is developed based on the Meshfree radial point interpolation method (RPIM). The main objective of this study is to construct and formulate a complete procedure for coupling NURBS and RPIM formulations, and written as N-RPIM. Computer code implementing N-RPIM is developed with MATLAB programming language. The N-RPIM is constructed based on Galerkin weak form formulation and possess the Kronecker delta property, hence enabling easy imposition of essential boundary conditions. Furthermore, parametric studies of two-dimensional planar analysis are conducted to determine the optimum range and value of parameters in ensuring the best performance of the N-RPIM method. The method is validated by employing heat transfer and plane stress problems, and is then extended to model a cellular beam with complex geometry due to the existence of web-holes along its span. Two types of performance are assessed; the convergence rate for displacements and stresses predictions. The presented result shows that, the N-RPIM works well and provides a favourable comparison against established numerical method, i.e., finite element method (FEM). The converged solution is achieved faster and provides an exact solution of less than 90% of the number of nodes compared to FEM. The convergence of displacement is achieved when the total number of nodes reaches approximately 5,000 nodes with an error of 0.005%, while more than 20,000 nodes required for the FEM to converge. In the prediction of stresses throughout the beam, the N-RPIM stress functions are readily continuous over the domain, whereas some post-processing would be required in FEM for smoothing the stresses value over the domain. This shows the potential of N-RPIM as an alternative numerical method in bridging the substantial bottleneck between CAD and CAE. In addition, by taking advantage of the exact geometry presented by NURBS and, with the flexibility and adaptivity of RPIM to determine field variables, this new method promises highly effective solutions when dealing with irregular domain problems.