Where does the shape function come from in tolerance modeling of parabolic heat conduction in layered composites?

Abstract Shape functions are widely used in all scientific and engineering fields that involve solving differential equations to describe the physical behavior of various materials and geometries. Thus, they are crucial in modeling physical issues in layered structures, where the choice of an appropriate shape function directly affects the accuracy and efficiency of calculations. Therefore, the aim of this paper is to demonstrate the conceptual origin of the fluctuation shape function in tolerance modeling, which is shown to derive from the cell problem formulated within the homogenization framework and is illustrated on the example of thermal problems in periodic and functionally graded composites. It is proven that, for periodic or stepwise functionally graded structures with inhomogeneity in one direction, the classical sawtooth shape function correctly reflects the nature of such two‐phase materials, even though it assumes constant values at the interfaces. However, for multi‐component composites, a modified version of the shape function is recommended—one that incorporates the thermal conductivities of the individual constituent materials.