Metal bulk forming is widely used because of its own advantages. In order to improve the production efficiency and reduce the cost, numerical simulation is often used to analyze the volume forming process. Because of the large deformation and non-linearity of the forming process, the finite element method (FEM) has the problem of element distortion, which will affect the accuracy and even lead to the failure of the analysis process. In this paper, an adaptive finite element method (AFEM) is proposed to solve this problem. Firstly, the finite element model is established, and grids are roughly divided. After the analysis, according to the error calculation model, the area with large error is determined by the standard deviation of nominal strain energy of nodes. Then, the grids are refined by dichotomy method, and the calculation is continued, repeating this step until the error meets the requirement. Finally, the numerical analysis of the forging process of the bulge formed joint is taken as an example to prove the accuracy of the proposed method.  

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.