The use of quasi-Newton methods is studied for the solution of the nonlinear finite element equations that arise in the analysis of incompressible fluid flow. An effective procedure for the use of Broyden’s method in finite element analysis is presented. The quasi-Newton method is compared with the commonly employed successive substitution and Newton-Raphson procedures, and it is concluded that the use of Broyden‘s method can constitute an effective solution strategy.

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The application of quasi‐Newton methods in fluid mechanics

Natural convection in steady solidification - Finite element analysis of a two-phase Rayleigh-Benard problem

SUMMARY Stream function-vorticity finite element solution of two-dimensional incompressible viscous flow and natural convection is considered. Steady state solutions of the natural convection problem have been obtained for a wide range of the two independent parameters. Use of boundary vorticity formulae or iterative satisfaction of the no-slip boundary condition is avoided by application of the finite element discretization and a displacement of the appropriate discrete equations. Solution is obtained by Newton-Raphson iteration of all equations simultaneously. The method then appears to give a steady solution whenever the flow is physically steady, but it does not give a steady solution when the flow is physically unsteady. In particular, no form of asymmetric differencing is required. The method offers a degree of economy over primitive variable formulations. Physical results are given for the square cavity convection problem. The paper also reports on earlier work in which the most commonly used boundary vorticity formula was found not to satisfy the no-slip condition, and in which segregated solution procedures were attempted with very minimal success. Most finite difference studies of incompressible viscous flow and convection in two dimensions have used the stream function-vorticity formulation. This formulation has the advantage of satisfying mass conservation exactly and reducing the number of simultaneous partial differential equations by one (from three to two for uncoupled viscous flow alone). With the finite element method however the majority of studies have used the primitive variable approach. In one of the first finite element works Taylor and Hood',* examined both forms of approach, but dropped stream function-vorticity in favour of primitive variables. Primitive variables were also chosen by Oden and Wellf~rd,~ and then by Kawahara et aL4 and Gartling and Beckers amongst others. Stream function-vorticity was chosen in the early works of Cheng6 and Baker.7 The difficulty with the stream function-vorticity approach is that, of two second order partial differential equations, one, a Poisson equation for stream function, has two boundary conditions instead of the one required, and the other, the vorticity transport equation, has no boundary condition. Finite difference workers have for long met this difficulty by defining boundary vorticity from the variation along the boundary normal of the stream function field determined in a previous iteration. We shall call this the boundary vorticity formula method.

Finite element, stream function–vorticity solution of steady laminar natural convection

Semidirect solution techniques can be an effective alternative to the more conventional iterative approaches used in many finite difference methods. This paper summarises several semidirect techniques which generally have not been applied to the Navier-Stokes and energy equations in finite difference form. The methods presented use both successive substitution and Jacobian-based updates as well as two variations of Broyden's full matrix update. A hybrid method is also presented, as is a norm-reducing search technique that can be used to enhance the convergence characteristic of any semidirect approach. See also the following Abstract.

Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Laminar and turbulent free convection in vertical cylinders with internal heat generation

A finite element numerical solution of natural convection in enclosed cavities

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A finite element numerical solution of natural convection in enclosed cavities