Fast BEM based methods for heat transfer simulation

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Learn more. In this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented.

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The algorithm solves the velocity—vorticity formulation of the incompressible Navier—Stokes equations in 3D. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells.

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Despite the great amount of research work developed in the last thirty years, there is still a wide unexplored field of research concerning applications of Boundary Element Method BEM to Engineering. There are important classes of problems which require more consistent approaches; in particular, one can mention the area of study of Transport Phenomena White, and specifically diffusive-advective problems Ramachandran, The most consistent and elegant formulation to deal with diffusive-advective problems in two-dimensions employs a fundamental solution of a similar diffusive-advective problem, with a concentrated source applied at a domain point Honna et al.

Such formulation is capable of solving with great accuracy problems where the velocity field is constant; however, the approach is not adequate anymore if the velocity field is dependent on spatial coordinates.

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Another drawback concerning this formulation is the difficulty to apply it to time dependent problems, as the fundamental solution for this case is quite cumbersome. The first BEM general formulation developed to diffusive advective problems was presented by Partridge et al. This formulation is based on the Dual Reciprocity procedure proposed originally by Nardine and Brebbia Utilizing a diffusive fundamental solution, the formulation presented by Partridge et al, overcame the restriction of the previous formulation which could only be applied if the velocity field was constant; however, it presented a serious limitation when accuracy is concerned, its application being recommended only for fluid flow with low Peclet numbers Bejan, ; Bennet and Myers, The structure of the QDR is similar to that of the Dual-Reciprocity, especially when the use of auxiliary interpolation radial basis functions is concerned Karur and Ramachandran, ; Partridge, ; however, a special treatment concerning the advective terms was proposed aiming at improving the accuracy of the numerical results for high Peclet numbers.

This new approach also employs a diffusive fundamental solution like the DRF and the QDR, but it employs operational procedures which include a special transformation of variables that makes the computational process quicker and accurate. This transformation is based on the governing equation equivalence between potential advective problems and inhomogeneous scalar problems, that is, Laplace's problems; this equation can be easily written in the inverse integral form if strategic operations put it in harmonic form.

Unhappily the HTT procedure is not general, being limited to one-dimensional problems and special two-dimensional cases, but it can be applied in some situations in hydraulic engineering, such as water seepage in regular soils and fluid transport in pipes and gutters. In addition, within the context of the Boundary Element Method approach, for some problems which the transient process is the focus of the computational simulation, it can be very interesting to apply the HTT together with DRF, because the HTT does not introduce meaningful numerical disturbance in the spatial model allowing a better performance of time discretization techniques.

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It is the case of inhomogeneous heat transfer problems Pasqueti and Caruso, ; Sutradhar et al. Problems with circumferential symmetry are also accessible to HTT approach. For all these situations, the HTT formulation shows significant advantages. It presents accuracy superior to the QDR and DRF, eliminates the need of using interpolation algorithms and, therefore, does not require matrix inversion. This last feature is the most important, because the reduction of CPU time is substantial.

The governing equation and boundary conditions can be presented in a more simple form; as mentioned before, the core of HTT is one-dimensional applications. On the other hand, in BEM approaches, it is very common to use two-dimensional codes to solve also more simple situations.

For this reason, a mathematical model in two dimensions is presented here.

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Then, here it is considered a homogeneous fluid medium, with mass movement which permits characterization of a continuous flow, subjected to a temperature gradient. Herein it is assumed incompressible and inviscid flow and steady state conditions Batchelor, In Eq. The boundary conditions are thus described by:. Governing Equation Compaction. To develop the formulation, it is convenient to rewrite the governing equation, Eq.

The last expression is the inhomogeneous Laplace's equation. Next it is shown the steps required to reach the contracted form of the governing equation, given by Eq. First, the derivative indicated in Eq. Comparing Eq. The transformation presented in Eq. It is a sufficient condition for one-dimensional cases, but for two-dimensional ones the incompressibility and irrotationality conditions to the fluid flow are required.

Finite Element Methods (FEM)

The Harmonic Transformation Technique. Starting from the compact form of the governing equation, Eq. The objective is to write the governing equation in a harmonic equivalent form, suitable to a BEM approach. If Eq. The classical BEM procedure Brebbia, leads to the following matrix equation:. The vector can be determined directly from Eq.

If k and k - 1 are considered to be two consecutive nodes of a discretized boundary, the following approximation is valid:. The trapezoidal rule was employed to perform the integration indicated in Eq. Since the equation 16 is the simplest approach to process the integration given by the equation 15 , some more accurate schemes could easily be imagined to improve it, especially considering some weighted effect of adjacent nodal values.

However, the most effective way to improve the accuracy of this integration without loss of simplicity is the use of boundary elements of the highest order, since the improvement of the proposed scheme would happen naturally. Considering now the boundary being discretized through n constant boundary elements Brebbia and Walker, and applying Eq.

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For the next interval, one can write:. Substituting Eq. When this procedure is repeated to all boundary nodes, one obtains the following equations:. Equation 20 can be written in matrix form as:.

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Equation 21 is rewritten in compact notation:. As is a homogeneous vector, having in mind that the sum of all terms of a row of matrix H is null Brebbia, one has:. This last equation can be easily solved by the usual methods employed to solve linear algebraic system of equations. Although the set of equations above refer to a boundary discretized into constant elements, it's easy to establish new forms to the equations above when higher-order elements are used.

In this way, a new set of equations related to a more precise integration scheme can be obtained. This topic discussion concerns three test-cases whose analytical solutions are known, thereby used to evaluate the performance of the formulation proposed. In the three applications, it is shown the behavior of the numerical solution for different mesh refinement and Peclet numbers.

Another important point to be highlighted concerning the numerical simulations carried out is related to the need for poles internal nodes, basis for global interpolation.

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  • It is well known in the literature, e. However, the approach proposed here HTT does not require poles. The first example consists of a heat transfer problem with a one-dimensional constant velocity field, subjected to the following boundary conditions: null diffusive flux on the horizontal edges, and prescribed temperatures on the vertical ones, as shown in Fig.

    The analytical solution of this problem is given by:. The results obtained from the numerical simulations are described by graphs which illustrate the behavior of the numerical solution through values of the mean error, described in percentage with respect to the analytical value, the following expression being used in the computations:. Equation 28 will be employed for all examples shown here.