A semi-exact primal-dual method for steady viscoplastic flows

Constitutive laws for viscoplastic materials involve a multivalued nonlinearity. Duality methods for such equations are known to converge, but the convergence is slow, requiring a large number of iterations. We introduce here a new iterative method of implicit primal-dual type for a class of variational problems, with a particular asymmetrical form in terms of the primal and dual unknowns, with an exact resolution of one relation and an approximate resolution of the other (semi-exact method). Two fast algorithms are proposed. The first includes an adaptive choice of a variable parameter, and the second includes a Newton-like linearized correction procedure. Applying the semi-exact method to a steady viscoplastic problem leads at each iteration to the resolution of a Laplace type equation, thus equivalent in terms of cost per iteration to the well-known Augmented Lagrangian or dual FISTA methods. Numerical tests show that the semi-exact method achieves a faster convergence in terms of number of iterations, compared to the augmented Lagrangian method or to the FISTA* method with acceleration, for Bingham or Herschel–Bulkley laws. This is particularly true when a large zero order term is present, as is the case when solving a time dependent problem.