Linearization of the Weak Forms – Tangent Matrices in a Covariant Form

Abstract

The full Newton iterative method will be applied to solve the global equilibrium equations on the stage of the numerical solution. This requires the full linearization of the corresponding weak forms representing the equilibrium conditions on the contact boundaries described in Chapter 5 for all contact cases. Linearization is obtained using the covariant derivatives in the corresponding local surface coordinate system, where derivatives of contact tractions are taken in covariant forms as described in Chapter 6 and derivatives of corresponding convective coordinates are described in Chapter 4. Linearized weak forms are the basis to create tangent matrices in the form independent of any approximations of the object (surfaces, curves, beams etc.). All tangent matrices are formulated then via the abstract approximation operator A. The matrices are split into several parts and possessing a clear geometrical structure for all studied geometrical contact cases – Surface-To-Surface, Point-To-Curve, Curve-To-Curve including also various constitutive relations for contact tractions.