Application of BEM to Identification of Distributions of Thermal Conductivities of Inhomogeneous Media

摘要

In this paper, distributions of thermal conductivities in three-dimensional inhomogeneous media such as functionally graded materials are identified by mean of the boundary element method. The distribution of thermal conductivity on both the boundary and the domain are identified from the temperature and the heat flux both of which are assumed to be known on the boundary. The governing equation has an inhomogeneous term originated from the inhomogeneity of the thermal conductivity; hence the corresponding boundary integral representation has a domain integral term for it. This domain integral is converted to boundary integrals by using the dual reciprocity method [1]. The inhomogeneous term is given by1/(k(x)) ▽k(x)·▽u(x), where k denotes the thermal conductivity, u the temperature and x a point in the medium, has derivatives of the temperature. This term is regarded as an unknown source term, then, the problem can be treated as a source identification problem. By approximating it with a linear combination of radial basis functions, the values of its coefficients become what should be evaluated instead of the values of those inhomogeneous terms.A similar approach employed in references is utilized to determine the coefficients from the boundary quantities. However, the boundary integral equation involves the unknown thermal conductivities on the boundaries. To guess these values, the thermal conductivities at the boundary nodes are also approximated with radial basis functions and the unknown coefficients are determined from the given temperatures. Using these values, the inhomogeneous source term is evaluated to calculate the temperatures at internal collocation points by means of the boundary element method with the dual reciprocity method [2]. Finally, the temperatures at all boundary and internal points and the thermal conductivities at boundary collocation points are used to evaluate the thermal conductivities at all the collocation points.The effectiveness of the present approach is demonstrated through some numerical test examples for three-dimensional media.