Integrated radial basis function methods for structural, fluid flow and fluid-structure interaction analyses

Integrated radial basis function methods for structural, fluid flow and fluid-structure interaction analyses

The present research is concerned with the development of new numerical methods based on integrated radial basis function network (IRBFN) and collocation techniques for solving structural, fluid-flow and fluid-structure-interaction problems. Simply and multiply-connected domains with rectangular or nonrectangular shapes are discretised by means of Cartesian grids.

An effective one-dimensional integrated radial basis function network collocation technique, namely 1D-IRBFN, is developed for the free vibration analysis of laminated composite plates using the first order shear deformation theory (FSDT). Instead of using conventional differentiated RBF networks, 1D-IRBF networks are employed on grid lines to approximate the field variables. A number of examples concerning various thickness-to-span ratios, material properties and boundary conditions of the composite plates are investigated.

A novel local moving least square - one-dimensional integrated radial basis function network method, namely LMLS-1D-IRBFN, is proposed for solving incompressible viscous flow problems. The method is demonstrated with the
analyses of lid-driven cavity flow and flow past a circular cylinder using streamfunction - vorticity formulation. In this approach, the partition of unity method
is employed as a framework to incorporate the moving least square (MLS) and 1D-IRBFN techniques. The major advantages of the proposed method include:
(i) a banded sparse system matrix which helps reduce the computational cost;
(ii) the Kronecker-δ property of the constructed shape functions, which helps impose the essential boundary conditions in an exact manner; and
(iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local RBF approximations.

The LMLS-1D-IRBFN method is then developed to study natural convection flows in multiply-connected domains in terms of stream function, vorticity and temperature. The unknown stream function value on the inner boundary is
determined by using the single-valued pressure condition (Lewis, 1979). The LMLS-1D-IRBFN method is further extended and applied to solve time dependent problems such as Burgers’ equation, unsteady flow past a square cylinder
in a horizontal channel and unsteady flow past a circular cylinder. For fluid flow problems, the diffusion terms are discretised by using LMLS-1DIRBFN method, while the convection terms are explicitly calculated by using
1D-IRBFN method. The present numerical procedure is combined with a domain decomposition technique to handle large-scale problems. Flow parameters such as drag coefficient, length of recirculation zone, Strouhal number and the effect of blockage ratio on the behaviour of the flow field behind the cylinder are investigated.

A numerical procedure based on 1D-IRBFN and local MLS-1D-IRBFN methods is proposed for solutions of fluid-structure interaction problems. A combination
of Chorin’s method and pseudo-time subiterative technique is presented for a transient solution of 2-D Navier-Stokes equations for incompressible viscous flow in terms of primitive variables. The fluid solver is first verified through a solution of mixed convection in a lid-driven cavity with a hot-temperature lid and a cold-temperature bottom wall. The FSI numerical procedure is then
applied to simulate flows in a lid-driven open-cavity with a flexible bottom wall. The Newmark’s method is employed for structural analysis of the flexible bottom wall based on the Euler-Bernoulli theory.

Numerical results obtained in the present research are compared with corresponding analytical solutions, where possible, and numerical results by other techniques in the literature.