Efficient numerical computations of yield stress fluid flows using second-order cone programming

Abstract

International audienceThis work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel-Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative op-timization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel-Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal-dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress-velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing meth-ods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables to solve large-scale problems on a personal computer within seconds only. The pro-posed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid

Advances in the simulation of viscoplastic fluid flows using interior-point methods

“…It usually requires reformulating the considered problem as the minimization of a linear function under specific convex constraints: linear equality/inequality constraints for LP, second-order cone constraints for SOCP and semi-definite constraints for SDP. As shown in [12], the viscoplastic problem can be rewritten into a second-order cone program by introducing additional optimization variables and conic constraints to fit the standard formulation required by industrial IP softwares like Mosek. However, we will take advantage of the particular structure of the problem, namely an objective function which can be decomposed into a smooth (viscous) and a non-smooth (plastic) part to reformulate only the non-smooth part using second-order cone constraints and keep the non-linear but smooth part of the objective function as such.…”

“…However, we will take advantage of the particular structure of the problem, namely an objective function which can be decomposed into a smooth (viscous) and a non-smooth (plastic) part to reformulate only the non-smooth part using second-order cone constraints and keep the non-linear but smooth part of the objective function as such. This leads to an important saving on the number of auxiliary variables compared to what has been proposed in [12].…”

“…If this reformulation is quite straightforward in the case of a quadratic viscous potential for a Bingham fluid, it is less obvious in the Herschel-Bulkley case. A specific procedure has been described in [12] to tackle the Herschel-Bulkley case with a rational power-law exponent. In addition to increasing the number of optimization variables, one drawback of such an approach is that the form of the reformulation is strongly dependent on the value of the exponent and has to be performed on a case-by-case basis.…”

“…In [12], it has first been proposed to use interior-point algorithms to solve the velocity minimum principle satisfied by the solution to a viscoplastic fluid flow. It follows the general trend that has been observed in the field of computational limit analysis/yield design theory in which interior-point methods (IPM) have been largely and successfully used for estimating the collapse load of various structures (continuum, beams, plates, shells, etc.)…”

“…In the field of limit analysis, many yield criteria can be reformulated using such SOCP constraints and, in particular, the von Mises criterion. In [12], it has been shown that the quadratic viscous term for the Bingham model could also be easily reformulated so that the associated optimization problems also falls in the SOCP framework. It was therefore proposed to use the industrial interior-point software Mosek [23] as a black-box solver for computing the viscoplastic flows.…”