Conservative interpolation for the boundary integral solution of the Navier-Stokes equations

Abstract

The Dual Reciprocity Method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach the non-linear terms are usually approximated by mathematical interpolation applied to the convective terms of the form of the Navier±Stokes equations. In this paper we introduce a conservative interpolation scheme that satis®es the continuity equation and performs better than pure mathematical interpolation. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the non-linear terms in the Navier±Stokes equations for moderate Reynolds number. IntroductionThe application of the boundary element method (BEM) to non-linear partial differential equations such as the Navier± Stokes set, leads to domain integrals that can be evaluated by using the cell integration approach (Cell-BEM) [1]. Although this approach is effective and general it makes the BEM lose its boundary only nature, resulting in a numerical scheme that is several orders of magnitude more time consuming than other domain techniques as the ®nite differences (FDM) and ®nite element methods (FEM).A handful of methods have been developed to take the domain integrals to the boundary. One of the most popular is the dual reciprocity method (DRM) [9] that has been particularly attractive in recent years because of the advances in multi-dimensional interpolation with radial basis functions (RBF) [6], global functions such as those presented by Cheng et al. [2] and augmented thin plate splines or hybrid functions applied by Partridge and Sensale [10] to the solution of some elasticity problems.The DRM approximates the nonlinear and nonhomogeneous terms of a partial differential equation as a series of vector valued interpolation functions, and ®nds a particular solution for the problem that can be used together with the Green's identities to convert the domain integrals into boundary integrals. Additionally, when the nonlinear terms include derivatives of the velocities another interpolation must be used on the velocity ®eld. Thus, the resulting interpolation functions are differentiated to obtain the convective terms [9]. This commonly used interpolation does not satisfy the continuity equation, and this is more notorious in coarse meshes. The only solution to overcome this de®ciency is to increase the mesh density especially in the zones where high velocity gradients are expected. On the other hand, a mass conservative interpolation method would allow to obtain better results even in coarse meshes and it would also improve the convergence properties of the numerical solution method.The BEM solution of large problems can be dramatically improved by using domain decomposition techniques. The original domain is divided into sub-regions, and in each one the full integral representation formula is applied. At the common interface between the adjacent sub-regions, the corresponding matching conditions are enforced to ensure the conti...

Numerical solution of thermal convection problems using the multidomain boundary element method

“…As was explained in previous articles [22,23], the nonhomogeneous terms that appear within the domain integrals in Eqs. (16) and (23) can be approximated using a series of particular solutions and interpolation functions.…”

“…In the mass conservative interpolation scheme the velocity derivatives are approximated in terms of the components of the velocity field, i.e., u 1 , u 2 in such way that the mass balance (1) is satisfied in the domain. The main results can be summarized in the following equations (for more details see [23]):…”

“…(17) requires not only the values of the temperatures but also the calculation of the velocity derivatives at the collocation nodes. In this work we have used a velocity interpolation scheme that satisfies the mass conservation principle and that was presented in [23]. In the mass conservative interpolation scheme the velocity derivatives are approximated in terms of the components of the velocity field, i.e., u 1 , u 2 in such way that the mass balance (1) is satisfied in the domain.…”

“…we can apply Green's identities to both the auxiliary velocity field and the auxiliary thermal field to obtain integral representation formulas only in terms of boundary integrals [8,22,23,25]; thus,…”

“…This is probably due either to the nonlinearity of the convective term in the momentum equations that prevents the method from reaching an acceptable numerical solution or to the numerically ill-conditioned system of discretized integral equations that arises when the boundary elements are very short (Refs. [22,23,30]). …”