تجزیه و تحلیل حساسیت و بهینه سازی فرآیندهای حرارتی الاستو پلاستیک با برنامه های کاربردی برای طراحی بخاری سمت جوش

A computational scheme is presented to optimize quasi-static weakly coupled thermo-elasto-plastic processes in three dimensional Lagrangian reference frames. Sensitivity formulations are developed from the radial return algorithm based thermo-elasto-plastic finite element equations using the direct differentiation method. These formulations are exemplified in the optimization of the side heaters in the transient thermal tensioning welding process for minimum residual stress. The results of the direct differentiation sensitivity analysis are validated by comparing with finite difference sensitivity calculations. Optimization is performed using the BFGS line search method.

In material processes such as welding and laser forming, the resultant permanent transformations of the materials are dependent on design variables such as heat input and travel speed. An empirical approach to optimize material processes is time consuming and costly. Therefore, computational optimization methodologies are required. For computationally intensive problems, gradient optimization methods are computationally efficient. However, they require the evaluation of sensitivities of the solution with respect to each design variable. Sensitivity analysis has been widely used in many design optimization problems [1], [2], [3], [4], [5], [6], [7] and [8]. Sensitivity analysis can be performed by analytical or by finite difference techniques [9]. The analytical methods are more accurate and computationally more efficient than the finite difference method. The analytical sensitivities can be computed either by direct differentiation or by the adjoint method [2]. For transient problems, the direct differentiation method is algorithmically more efficient than the adjoint method. Sensitivity analysis for coupled systems is presented in Ref. [10]. Sensitivity analysis of thermo-elasto-plastic processes in two dimensional frames with the assumption of generalized plane strain has been implemented in minimizing welding residual stress and distortion in Ref. [11]. Sensitivity analysis for thermo-elasto-plastic processes in Eulerian reference frames has been developed in optimizing the laser forming process in Ref. [12]. These two approaches have critical assumptions in their formulations. The two dimensional approach is limited in accounting for three dimensional effects and the Eulerian approach is applicable only for steady-state processes. Michaleris et al. [13] have demonstrated a sensitivity analysis for thermo-elasto-plastic processes in Lagrangian reference frames. However, they use a stress update algorithm (fully implicit backward Euler in tensor form) that is computationally less efficient than the radial return algorithm [14] and, thus, the sensitivity algorithm is implemented only for a two dimensional example in their paper. Therefore, it is necessary to develop sensitivity equations for thermo-elasto-plastic processes with the radial return algorithm in three dimensional Lagrangian reference frames. In this paper, conventional finite element formulations for thermo-elasto-plastic analyses with the radial return algorithm in three dimensional Lagrangian reference frames are reviewed. Then, thermo-elasto-plastic sensitivity equations are developed using the direct differentiation method. The sensitivity formulations are verified by comparing with results obtained by the finite difference sensitivity analysis. The sensitivity equations are implemented to a welding optimization example. The positions and the heat input power of side heaters in the fillet welding process are optimized for minimum residual stress.

Direct sensitivity formulations for thermo-elasto-plastic processes in three dimensional Lagrangian reference frames have been developed. The sensitivity results from these formulations show good agreement with the results from the finite difference sensitivity analysis method. The sensitivity formulations are successfully implemented in an optimization procedure to determine the optimal side heater heat input power and positions for minimum welding residual stress in the transient thermal tensioning process. In this simulation, only three design variables (side heat source and positions of side heater in transverse and longitudinal directions) are considered. However, if necessary, more design variables such as side heat shape can be considered without much effort. The addition of constraints such as peak temperature in an objective region can also be easily incorporated in the optimization. Since welding residual displacement usually exceeds small strain range, large deformation theory needs to be implemented in this analysis procedure as future work. Adaptive meshing may also reduce the analysis time for large structural problems.