| Auto-body panel forming plays a very important role in automobile production. It restricts the improvement of product quality and the shortening of replacement cycle. The traditional trail and error procedure which depends on experience can not meet the demand of high quality, high efficiency and low cost. With the development of CAD and CAM technologies, computer simulation of sheet metal forming is in constant progress to evaluate the deformation paths and the forming defects such as fracture and wrinkling. As a rapid forming method, one-step simulation is widely used in evaluation of the manufacturability of the product at the design stage.One-step simulation simplifies the process of sheet metal forming. Only the initial flat blank and the final3D configuration of workpiece are considered in the calculation in which the intermediate states are ignored. The theorem of virtual work is expressed on the overall final part:Where Winte and Wexte are respectively the virtual work of internal forces and external forces;{u*} and{ε*} are the virtual displacements and virtual membrane strains, respectively,{σ) are the Cauchy stresses and {f} are the external forces.Due to the high degree of nonlinearity which can be categorized as geometrical non-linearities, material non-linearities due to plastic deformation, friction force reversals and contact mode changes. The Newton-Raphson (N-R) scheme is utilized to solve the non-linear equtions. For an opproximative solution {U1}, the residual force vector can be written as:For the standard N-R iteration, the following procedure will improve the solution:[kTi]{△Ui}={R(Ui)}(3){Ui+1}={Ui}+{△Ui}(4) [KTi] is the tangent stiffness matrix in the i-th iterate. Standard N-R iteration requires the initial solution have a high precision. But for some complicated problems, it is hard to obtain a initial solution {U0} which can make the standard N-R iteration converge. Generally a relaxation factor is introduced to restrict the iteration step.{Ui+1}={Ui}+ωi{△Ui}(0<ωi≤1)(5)Relaxation factor can keep the solution within the radius of convergence. It has a great influence on convergence and computational efficiency. But it is hard to choose proper relaxation factors. And up till now, there is not much research on how to select relaxation factor in one-step simulation.In this thesis, the influence of relaxation factor on the convergence and computational efficiency of the one-step simulation was studied with emphasis. Firstly, the author summarized the rules for selecting relaxation factors. Then, various current relaxation factor algorithms are studied. We compared and analyzed the advantages and disadvantages of these methods and put forward improved plans.Finally, A new optimized algorithm of the relaxation factor which takes into account of both robustness and calculation efficiency is proposed. This method is based on line search of the local minimum point. Then the computer program is implemented and tested on Siemens PLM NX/One-Step. Six numerical examples such as square box, disc, beam, and fender et al are chosen for the test. The result is compared with the fixed relaxation factor algorithm and "energy method", confirming the effectiveness of proposed algorithm. |
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