Modifications of the Newton-Raphson method for finite element simulations in ferroelectroelasticity

The system of equations arising in finite element simulations of components made of ferroelectroelastic materials is non-linear if the loading is sufficiently high. The Newton-Raphson method represents a widely used iterative technique to solve this system of non-linear equations. However, if the scalar potential formulation is utilised, convergence difficulties may occur. This circumstance can be primarily attributed to the specific form of the non-linear response of typical ferroelectroelastic materials being subjected to electrical loading. The present paper is devoted to modifications of the Newton-Raphson method, which are capable of improving the convergence behaviour experienced in the finite element iteration. We extend an existing modification to the fully coupled, ferroelectroelastic case. Additionally, a new modification of the Newton-Raphson method is proposed. This method applies an iteration algorithm, which is virtually equivalent to the iteration algorithm of the unmodified Newton-Raphson method combined with the vector potential formulation. An important feature of both modifications is that they are applied on the integration point level. Therefore, the global non-linear finite element iteration scheme remains unchanged. Finally, the practicability of the modifications discussed in the paper is shown in a numerical example. (C) 2012 Elsevier Ltd. All rights reserved.