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We present a numerical technique based on the coupling of boundary and finite element methods for the steady Oseen equations in an unbounded plane domain. The present paper deals with the implementation of the coupled program in the two-dimensional case. Computational results are given for a particular problem which can be seen as a good test case for the accuracy of the method.

The coupling of boundary and finite element methods has recently been shown to be a very effective tool for solving a certain class of physical problems with infinite (or even large) domains, for which the traditional numerical analysis techniques are unsuitable (cf. [

On this subject we have studied in detail the numerical solutions of the two-dimensional exterior Oseen equations for a steady-state incompressible viscous flow. Essentially, the coupling method involves the choice of an artificial smooth boundary separating an interior inhomogeneous region from an exterior homogeneous region. An integral equation over this interface, representing the solution in the exterior region in terms of a single-layer potential, is incorporated into a variational formulation in the primitive variables velocity-pressure for the interior region. This allows a discretization along the artificial boundary together with a typical discretization by finite elements to be employed. This paper is concerned with the implementation of these coupled boundary and finite element methods for the steady Oseen problem in a completely general form and without using the standard finite element software.

One of the difficulties encountered in assessing the performance of the algorithm for the approximate solution of the exterior Oseen equations is that of finding a suitable analytical solution with which a comparison may be made. The flow, due to an infinitely long circular cylinder, rotating uniformly about its axis in an infinite mass of viscous incompressible fluid, is one such solution in the two-dimensional case.

The major aim of the present work is the development of a computational program for our coupled boundary element-finite element methods. In this sense the study of the previous example may serve as a test case of the applicability of this technique to more complicated models. A series of numerical results demonstrates the accuracy of the method.

Let

One of the main difficulties in solving (

Then an appropriate coupling technique between boundary element and finite element may be used to solve (

Since the numerical analysis of this coupling procedure has already been developed, for the sake of brevity, only its essential features will be presented here. For additional mathematical details, the reader is referred to work [

Let us introduce the Hilbert spaces

with the standard norms.

We recall the mixed variational formulation of our problem described in [

appearing as the density of the single-layer potential, which represents the solution in the exterior domain, is identified with the local stress force of the flow; here

We have

where

where

where

is essentially the zero-order Hankel function of the first kind. Here

It is worth noting that the pressure

We must point out, in contrast to other methods, when the Oseen problem is formulated in this manner, the stress force distribution, which is normally a quantity of interest in such calculations, is determined directly, and the accurate results are shown in Section

For the numerical approximation of our problem, we construct and study a finite element method based on the mixed variational formulation developed in Section

For simplicity we restrict here the discussion to the case where

From now on,

We define

In addition, we introduce the subspace

With these spaces, problem (

We now construct finite dimensional spaces

where

We choose the following finite element spaces (refer to [

Recalling [

Here some statements are borrowed from Sequeira [

and where the arising matrix

Here

To summarize, the computational structure of the coupled system is very different and this leads to some difficulties on solving it. As can be expected, the finite element system is typically large but sparse and the boundary element system is small but dense. It is therefore of interest to design solution methods that exploit these attributes to maximum advantage.

Before proceeding, let us give a short look at the numerical effort needed to derive the boundary interface nodal coefficients to be assembled. As with finite elements, a global numbering system is used for these nodes.

We start with the boundary element terms in order to obtain submatrix

and use the relations

To write the above coefficient

which become, in terms of parameters

where

Computing these integrals and using the relationship

it is a simple matter to derive the full matrix

Let us now examine the coupling terms

As it has been noted previously, once the elemental matrix calculations have taken into consideration all the internal and boundary interfacial nodes, ensuring compatibly between the finite and boundary element meshes, the coupled analysis is carried out as in the standard finite element process. Of course, the global assembly and solution procedure we have used do not ignore the large zero blocks that arise, in order to increase the computational efficiency of this method.

We recall that problem (

and, more explicitly, in terms of the block structure of matrix

The performance of the numerical model described above has been tested on the traditional example of the Oseen flow past a rotating infinitely long circular cylinder of the radius

the velocity distribution is then

since

From the expressions (

Computations were carried out with

Representative parameters of four basic meshes.

Meshes | ||||||
---|---|---|---|---|---|---|

3 | 6 | 36 | 84 | 192 | 202 | |

4 | 8 | 64 | 144 | 328 | 342 | |

5 | 10 | 100 | 220 | 500 | 518 | |

6 | 12 | 144 | 312 | 708 | 730 |

Here,

In Tables

Relative error of numerical solution on four basic meshes (

Meshes | size | ||||
---|---|---|---|---|---|

0.0524273 | 0.0664172 | 0.1427025 | 0.3090692 | ||

0.0342385 | 0.0547278 | 0.1177296 | 0.2275821 | ||

0.0243689 | 0.0474219 | 0.0998918 | 0.1813625 | ||

0.0184596 | 0.0427631 | 0.0876485 | 0.1488314 |

Relative error of numerical solution on four basic meshes (

Meshes | size | ||||
---|---|---|---|---|---|

0.0535782 | 0.0662608 | 0.1495650 | 0.3102600 | ||

0.0358974 | 0.0580217 | 0.1277890 | 0.2304795 | ||

0.0262548 | 0.0516052 | 0.1121745 | 0.1904752 | ||

0.0241313 | 0.0457448 | 0.0876485 | 0.1675414 |

From Tables

This work is supported by the Natural Science Foundation of China (no. 10971166) and the National Basic Research Program (no. 2005CB321703) and Sichuan Science and Technology project (no. 05GG006-006-2).