A Radial Basis Function Partition of Unity Method for Steady Flow Simulations

TL;DR

This paper introduces a novel radial basis function partition of unity method for solving steady flow problems in unbounded domains, validated on classical fluid mechanics benchmarks and tested on complex geometries.

Contribution

The paper presents a new numerical approach combining domain transformation, RBF partition of unity discretization, and trust-region nonlinear solvers for steady flow simulations.

Findings

Accurate flow solutions for flow past a circular cylinder.

Effective handling of less smooth obstacles with some oscillation issues.

Method's validation against established benchmarks.

Abstract

A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles - rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities.