Transient heat conduction by the boundary element method: D‐BEM approaches

Abstract

SUMMARYThis work is concerned with the development of different domain-BEM (D-BEM) approaches to the solution of two-dimensional diffusion problems. In the first approach, the process of time marching is accomplished with a combination of the finite difference and the Houbolt methods. The second approach starts by weighting, with respect to time, the basic D-BEM equation, under the assumption of linear and constant time variation for the temperature and for the heat flux, respectively. A constant time weighting function is adopted. The time integration reduces the order of the time derivative that appears in the domain integral; as a consequence, the initial conditions are directly taken into account. Four examples are presented to verify the applicability of the proposed approaches, and the D-BEM results are compared with the corresponding analytical solutions.

The boundary element method applied to the solution of the anomalous diffusion problem

“…In this section, the same domain and mesh as those in the previous example are considered, under a zero initial condition and subject to the following boundary conditions: This is a very interesting diffusion example, with an oscillatory solution resulting from the oscillatory boundary condition. Ochiai et al [34], and Carrer et al [35] also analysed this problem, with the latter also providing the analytical solution for the case α = 1.0. In this example, a finite difference grid size ∆x = 0.0125 was used, more refined than that used the previous example.…”

One-dimensional scalar wave propagation in multi-region domains by the boundary element method

“…The main drawback of the D-BEM formulation is the entire domain discretization and, consequently, the assemblage of a much larger system of equations, as now not only the boundary nodes but also all the internal points contain the unknowns to be determined, see, for instance, Carrer and Costa [5]. The D-BEM formulation, however, is very versatile; for example, the pure diffusion problem can easily be solved by simply substituting the acceleration by the first-order time derivative of the variable of interest and choosing an appropriate time approximation for it; see Carrer et al [8].…”

Transiente diffusion equation: an alternative finite difference approach

“…The work published by Rahaman et al (2018) deals with the numerical solution of the diffusion equation using MDF, regarding the stability of the method. A discussion involving a similar approach is initially presented by Carrer et al (2012) for the diffusion equation and later by Cunha et al (2016), but in this work the time integral approximation is used within the Boundary Elements Method -BEM for the Diffusive-Advective transport equation. More detailed studies in relation to BEM can be found in (BREBBIA et al, 1984, CARRER & MANSUR, 2010and CARRER et al, 2012.…”