Abstract : This work is a combined analytical and numerical study of the extreme values of the thermal and elastic fields occurring in a propellant composite material, a granular medium containing dense self-assembled spheroidal grains embedded in a matrix. First, a 3D microtomography image of a representative sample is segmented using morphological operators. Second, the local temperature and heat flux in the quasi-static thermal response is computed numerically, making use of the segmented microstructure and of experimental values of the heat conductivity in each phase. Such fields are readily derived by means of a Fast Fourier Transform method. Emphasis is put on the maximum values of the local fields: even at low contrast of properties between the grain and the matrix, the heat flux patterns is made of hot-spot zones located in-between grains that are close to each other with preferential directions. Third, the local extrema of the fields are investigated in the context of linear elasticity. Finally, analytical approximations are examined at low and high contrast of properties, on a simple hard-core model of cylinders, which is tantamount to a 2D granular microstructure.