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Practical guidelines for fast, efficient and robust simulations of yield-stress flows without regularisation using accelerated proximal gradient or augmented Lagrangian methods Treskatis, Timm; Roustaei, Ali; Frigaard, Ian; Wachs, Anthony

Abstract

The mathematically sound resolution of yield stress fluid flows involves nonsmooth convex optimisation problems. Traditionally, augmented Lagrangian methods developed in the 1980's have been used for this purpose. The main drawback of these algorithms is their frustratingly slow O(1/√k) worst-case convergence, where k is the iteration counter. Recently, an improved 'dual FISTA' algorithm (short: FISTA*) was introduced, which achieves the higher and provably optimal rate of O(1/k). When implementing these algorithms in two finite-element packages (FreeFem++ by Frédéric Hecht, UPMC Paris and Rheolef by Pierre Saramito, UGA Grenoble), we observed that these theoretical convergence rates are not generally attained. In this article, we present four common numerical pitfalls that adversely impact the convergence of the optimisation algorithms. By means of constructive and practical guidelines we point out how a careful implementation can not only recover the full order of convergence, but also reduce the computational cost per iteration for further efficiency gains. Furthermore, we assess the performance and accuracy of FISTA* for the practical case of flow in wavy walled channel and demonstrate significant speed-up when FISTA* is employed instead of the classical augmented Lagrangian method.

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Title
Practical guidelines for fast, efficient and robust simulations of yield-stress flows without regularisation using accelerated proximal gradient or augmented Lagrangian methods
Creator
Treskatis, Timm; Roustaei, Ali; Frigaard, Ian; Wachs, Anthony
Date Issued
2017-11-27
Description
The mathematically sound resolution of yield stress fluid flows involves nonsmooth convex optimisation problems. Traditionally, augmented Lagrangian methods developed in the 1980's have been used for this purpose. The main drawback of these algorithms is their frustratingly slow O(1/√k) worst-case convergence, where k is the iteration counter. Recently, an improved 'dual FISTA' algorithm (short: FISTA*) was introduced, which achieves the higher and provably optimal rate of O(1/k). When implementing these algorithms in two finite-element packages (FreeFem++ by Frédéric Hecht, UPMC Paris and Rheolef by Pierre Saramito, UGA Grenoble), we observed that these theoretical convergence rates are not generally attained. In this article, we present four common numerical pitfalls that adversely impact the convergence of the optimisation algorithms. By means of constructive and practical guidelines we point out how a careful implementation can not only recover the full order of convergence, but also reduce the computational cost per iteration for further efficiency gains. Furthermore, we assess the performance and accuracy of FISTA* for the practical case of flow in wavy walled channel and demonstrate significant speed-up when FISTA* is employed instead of the classical augmented Lagrangian method.
Subject
yield-stress flows; viscoplastic fluids; augmented Lagrangian method; FISTA*; finite elements
Genre
Article; Preprint
Type
Text
Language
eng
Date Available
2017-11-29
Provider
Vancouver : University of British Columbia Library
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International
DOI
10.14288/1.0360784
URI
http://hdl.handle.net/2429/63760
Affiliation
Applied Science, Faculty of; Science, Faculty of; Chemical and Biological Engineering, Department of; Mathematics, Department of; Mechanical Engineering, Department of
Peer Review Status
Unreviewed
Scholarly Level
Faculty; Postdoctoral
Rights URI
http://creativecommons.org/licenses/by-nc-nd/4.0/
Aggregated Source Repository
DSpace

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Attribution-NonCommercial-NoDerivatives 4.0 International