Radial Basis Function Networks for Viscoelastic Flow Analysis

This project is concerned with the application of radial basis function networks (RBFNs) for the analysis of viscoelastic flows. Many well-known discretization techniques for viscoelastic flow analysis, such as the finite difference method (FDM), finite element method (FEM) and finite volume method (FVM), are usually based on low order approximation schemes, e.g. linear and quadratic ones. In these low-order methods, any continuous quantity is approximated by a set of piecewise continuous functions defined over a finite number of subdomains identified as elements or control volumes (mesh), which is widely seen as a difficult task in the modelling process. The idea of developing numerical methods for the solution of PDEs governing engineering problems without using a mesh has attracted considerable attention in recent years. Consequently, the number of meshless/element-free numerical methods has been proposed recently. In this project, global and local meshless methods, which are based on RBFNs, are developed to predict how viscoelastic fluids behave when they flow in complex geometries. Some polymer processing applications will also be simulated. Although the proposed method can achieve high convergence rates, it is still demanding great computer resources due to the nature of polymer problems.

Principal Investigator

Nam Mai-Duy
School of Aerospace, Mechanical and Mechatronic Engineering
University of Sydney

Project

f88

RFCD Codes

290501

Significant Achievements, Anticipated Outcomes and Future Work

We have recently introduced prior conversions of the multiple spaces of network weights into the single space of function values in the IRBFN approach and therefore have kept the system matrix size small and comparable to that associated with the DRBFN approach [5,6].
We have derived an integrated-RBFN (IRBFN) formulation for numerical solving viscoelastic flow problems. The IRBFN method has been verified through the simulation of the corrugated tube flow of a Newtonian fluid, Power-law fluid and Oldroyd-B fluid. Flow resistance predictions obtained have been in good agreement with the benchmark solutions [4].
We have proposed a new approach of implementing the multiple boundary conditions. The normal derivative boundary conditions have been imposed by means of integration constants. This approach has been verified successfully through the solution of high-order ODEs [1] and high-order PDEs [2].

 

Computational Techniques Used

Nonlinear problems lead to nonlinear systems of equations which must be solved iteratively. Three iterative techniques, namely the Picard iteration scheme, the Newton iteration scheme and the trust region scheme, are often employed here to handle the nonlinearity of the system. Which scheme to be applied depends on the problem to be solved. For example, the Picard algorithm is simple but its convergence is often slow. The Newton algorithm converges quadratically but it requires that the starting point should be close to the solution. The trust region algorithm is relatively complicated, but provides globally q-quadratically convergent. The last two procedures are available in, e.g. MATLAB and NAG library.

 

Publications, Awards and External Funding

External Funding and Awards

None

Publications

Journal papers:
[1] N. Mai-Duy, Solving high order ordinary differential equations with radial basis function networks, International Journal for Numerical Methods in Engineering, 62, 2005, 824-852.
[2] N. Mai-Duy, R.I. Tanner, Solving high order partial differential equations with radial basis function networks, International Journal for Numerical Methods in Engineering, 63, 2005, 1636-1654.
[3] N. Mai-Duy, R.I. Tanner, An effective high order interpolation scheme in BIEM for biharmonic boundary value problems, Engineering Analysis with Boundary Elements, 29, 2005, 210-223.
[4] N. Mai-Duy, R.I. Tanner, Computing non-Newtonian fluid flow with radial basis function networks, International Journal for Numerical Methods in Fluids, 48, 2005, 1309-1336.
[5] N. Mai-Duy, T. Tran-Cong, An efficient indirect RBFN-based method for numerical solution of PDEs, Numerical Methods for Partial Differential Equations, 21, 2005, 770-790.
Conference papers:
[6] N. Mai-Duy, R.I. Tanner, A meshless indirect RBF collocation method for steady viscous flows at high Reynolds numbers, The VIIIth US National Congress on Computational Mechanics (Meshfree and particle methods mini-symposium), US Association for Computational Mechanics, 2005.