ИСТИНА

Viscoplastic or yield stress fluids are materials which behave like a solid below critical yield stress and flow like a viscous fluid for stresses higher than this threshold. The numerical solution of yield stress fluid flows involves nonsmooth convex optimisation problems. Traditionally, augmented Lagrangian methods (ALM) developed in the 1980-s have been used for this purpose. The main drawback of these algorithms is their frustratingly slow convergence. Beck and Teboulle (2009) present fast iterative shrinkage-thresholding algorithms (FISTA) for solving linear inverse problems arising in signal/image processing. This method, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. The proposed acceleration is of the form first proposed by Nesterov, for gradient descent methods. Later was present accelerated variant of ALM that exhibit faster convergence than ALM. Many viscoplastic fluids slip at the wall with a yield slip. The fluid slip when the tangential stress exceeds a critical value called the yield slip, and otherwise, the fluid sticks at the wall. We exploit the analogy of structure between the slip law and the viscoplastic constitutive law and apply accelerated ALM to both the viscoplastic model and the yield slip equation. The traditional ALM converges with rate O(1/√ k), an accelerated variant converges with the higher and provably optimal bound O(1/k) convergence, where k is the iteration counter. This accelerated version is obtained at a negligible extra computational cost. The proposed method is used to simulate the axisymmetric squeeze flow of Bingham, Casson, and Herschel-Bulkley fluids with the slip yield boundary condition at the wall.