Abstract

This paper exploits the concept of stabilized finite element methods to formulate stable mixed stress/displacement and strain/displacement finite elements for the solution of nonlinear solid mechanics problems. The different assumptions and approximations used to derive the methods are exposed. The proposed procedure is very general, applicable to 2D and 3D problems. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation.

Abstract

This paper deals with the computational modeling and sub-grid scale stabilization of incompressibility and convection in the numerical simulation of the material flow around the probe tool in a friction stir welding (FSW) process. Within the paradigmatic framework of the multiscale stabilization methods, suitable pressure and convective derivative of the temperature sub-grid scale stabilized coupled thermomechanical formulations have been developed using an Eulerian description. Norton-Hoff and Sheppard-Wright thermo-rigid-viscoplastic constitutive material models have been considered. Constitutive equations for the sub-grid scale models have been proposed and an approximation of the sub-grid scale variables has been given. In particular, algebraic sub-grid scale (ASGS) and orthogonal sub-grid scale (OSGS) methods for mixed velocity, pressure and temperature P1/P1/P1 linear elements have been considered. Furthermore, it has been shown that well known classical stabilized formulations, such as the Galerkin least-squares (GLS) for incompressible (or quasi-incompressible) problems or the Streamline Upwind/Petrov-Galerkin (SUPG) method for convection dominant problems, can be recovered as particular cases of the multiscale stabilization framework considered. Using a product formula algorithm for the solution of the coupled thermomechanical problem, the resulting algebraic system of equations has been solved using a staggered procedure in which a mechanical problem, defined by the linear momentum balance equation, under quasi-static conditions, and the incompressibility equation, is solved first at constant temperature. Then a thermal problem, defined by the energy balance equation, is solved keeping constant the mechanical variables, i.e. velocity and pressure. The computational model has been implemented in an enhanced version of the finite element software COMET, developed by the authors at the International Center for Numerical Methods in Engineering (CIMNE). Two numerical examples have been considered. The first one deals with the numerical simulation of a coupled thermomechanical flow in a 2D rectangular domain. Steady-state and transient conditions have been considered. The goal of this numerical example has been the comparison between different sub-grid scale stabilization methods for the velocity and temperature equations. In particular, using a GLS stabilization method for the pressure equation, a comparison between SUPG and OSGS convective stabilization methods has been performed. Additionally, using a SUPG stabilization method for the temperature equation, a comparison between GLS and OSGS pressure stabilization methods has been done. The second example deals with the 3D numerical simulation of a representative FSW process. Numerical results obtained have been compared with experimental results available in the literature. A good agreement on the temperature distribution has been obtained and predicted peak temperatures compare well, both in value and position, with the experimental results available.