Optimal specification of target temperature points for inverse design of an indoor thermal environment

Inverse models based on computational fluid dynamics (CFD) are able to determine the required boundary wall convective heat fluxes in enclosed environments by performing a regularized inversion to the governing matrix. The governing matrix describes the cause-effect relationship between the boundary wall heat fluxes and the exhibited temperatures. Some target temperatures must be specified as known inputs to infer the unknown thermal boundary conditions. However, the current target temperature specification is determined by intuition or experience rather than by systematic derivation. This investigation proposes to evaluate the specification of target temperature points using the eigenvalue decomposition to the governing matrix. A connection was established between the eigenvalues and the condition number of the matrix that describes the posedness of the problem. The effects of the number and locations of the target temperature points on the inverse solution were analyzed based on a two-dimensional ventilation enclosure. The results showed that the proposed inverse model provides accurate solutions with appropriately specified target temperature points. Locating the target temperature points downstream of the flows that sweep boundary walls can help reduce the posedness of the governing matrix. A constituted matrix with a smaller condition number responds better to design targets. There is no need to adopt more target temperature points than the number of unknown thermal boundaries. In some circumstances, if more target temperature points than the number of unknown boundaries are provided, it is recommended to select the points that correspond to eigenvalues of the matrix far from zero.