Thermal systems modelling via singular value decomposition: direct and modular approach

Abstract

Singular Value Decomposition (SVD) technique consists of obtaining the decomposition of a given matrix M=U M Σ M V M ∗ where U M and V M are unitary matrices and Σ M is a matrix of the same size as M with nonnegative real diagonal entries which are called singular values or principal values. We show in this article that if M is the gramian matrix of the temperature field of a thermal system, a small number of columns of U M , associated to the greatest singular values, used as a basis to project the temperature field, allows us to reconstitute an accurate solution of the heat transfer problem. Thus the number of differential equations to resolve in order to simulate or to control the evolution of the temperature field is drastically reduced. A bound of the spectral norm of the introduced error is derived. The method is applied to reduce a model of thermal bridge, and in a modular approach to reduce a model of a multizones building.