Stabilized finite element method for viscoplastic flow: formulation and a simple progressive solution strategy

Abstract This paper presents a stabilized finite element formulation for steady-state viscoplastic flow and a simple strategy for solving the resulting non-linear equations with a Newton–Raphson algorithm. An Eulerian stabilized finite element formulation is presented, where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A local reconstruction method is used for computing derivatives of the stress field needed when higher order elements are used. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. In order to get convergence in the Newton–Raphson algorithm, a trial solution is needed which is within the radius of convergence. An effective strategy for progressively moving inside the radius of convergence for highly non-linear viscoplastic constitutive equations, typical of metals, is presented. Numerical experiments using the stabilization method with both linear and quadratic shape functions for the velocity and pressure fields in viscoplastic flow problems show that the stabilized method and the progressive convergence strategy are effective in non-linear steady forming problems. Finally, conclusions are inferred and extensions of this work are discussed.

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