At the studying the Mannesmann piercing process, we unify two approaches to the problem solving. Namely, the commercial FEM software procedure and the mathematical model of the process via the mathematical model of the considered FEM-simulation bound by certain unifying symmetries. Such phenomenon seemingly exists only if the FE-mesh is initiated to be physically interpreted. We shortly outline, how to come to the slightly modified Cahn-Hilliard equation as to the mathematical model of the FE-simulation possessing quasi-symmetry given by a lattice of colloidal assembly formed by the chosen FE-mesh. Separation of two cylindrical surfaces of the pierced product together with the inpainting role of the piercing plug are described with respect to the background given by the Navier-Stokes equations related to the flow between the both surfaces. Influence of the involved groups related to the considered quasi-symmetry is illustrated by the convergence/divergence of the Newton-Raphson number in the CPU-time.