摩擦约束不完全变分不等原理及其微粒群优化解法

利用不完全变分不等原理，将摩擦约束纳入问题的能量泛函，同时放松体积不可压缩的约束条件ii=0，用罚因子处理体积应变。实际算例表明，不完全变分不等原理不但能计算速度场，而且能直接计算变形力，且所得到的变形力比滑移线场结合极值方法获得的解析解更接近试验结果。在有限变形情况下，仍能得到变形力的理想结果。同时，利用微粒群优化方法，直接求解变分问题的能量泛函，避开了变分不等式的求解困难问题。针对优化求解过程中出现解的随机性，利用求平均值方法，使求解过程稳定，有效地改善了求解结果，并由此扩展了微粒群算法的应用领域。

Friction Constraint Incomplete Variational Inequality Principle and Its Solution Using the Particle Swarm Optimization Method

With the presence of frictions and constraints of ii=0, it is difficult to simulate a plastic forming process involving frictional contacts by using a conventional method such as finite element analysis. However, it is shown that simulations can be performed by employing the incomplete variational inequality principle. Variational inequalities are established by introducing the friction constraints into the energy functional, allowing changes in material volume and dealing with the volume strain by means of penalty factor. Example calculations indicate that, by using the incomplete variational inequality principle, not only the velocity field but also the deformation force can be predicted. The predicted deformation force is found to be closer to that from measurement than the solution produced from the sliding line field combined with the optimization method. The incomplete variational inequality principle can generate accurate results even in the case of finite deformation. Variational inequalities derived from the incomplete variational inequality principle are highly mathematically complicated, therefore, the particle swarm optimization (PSO) method is explored in order to solve these inequalities. The randomness of solutions from PSO is depressed by performing an average process. Not only the solution process becomes more stable, but also the results are improved. A new application of the PSO is thus demonstrated.