Residual stresses couple microscopic and macroscopic scales

We show how residual stresses emerge in a visco-elastic material as a signature of its past flow history, through an interplay between flow-modified microscopic relaxation and macroscopic features of the flow. Long-lasting temporal-history dependence of the microscopic dynamics and nonlinear rheology are incorporated through the mode-coupling theory of the glass transition (MCT). The theory's integral constitutive equation (ICE) is coupled to continuum mechanics in a finite-element method (FEM) scheme that tracks the flow history through the Finger tensor. The method is suitable for a calculation of residual stresses from a "first-principles" starting point following well-understood approximations. As an example, we calculate within a schematic version of MCT the stress-induced optical birefringence pattern of an amorphous solid cast into the shape of a slab with a cylindrical obstacle and demonstrate how FEM-MCT can predict the dependence of material properties on the material's processing history.Comment: 5 pages, 3 figure

Fast proximal algorithms for applications in viscoplasticity.

Numerical flow simulations for viscoplastic fluids have posed, and continue to pose major challenges. The large scale of industrially relevant flow problems coupled with the highly nonlinear and nonsmooth nature of viscoplastic materials still poses too high an obstacle even for modern computer clusters. This research aims to provide more efficient numerical schemes for flow simulations of Bingham, Casson and Herschel-Bulkley fluids without perturbing their viscoplastic behaviour by smoothing or regularisation. Two main contributions form the focus of this thesis: firstly, a new dual formulation of such problems and secondly, their numerical solution by proximal gradient or proximal Newton-type methods. To this end, we initially study a class of generic convex optimisation problems in Hilbert spaces. We design dual-based algorithms in the appropriate function spaces and derive properties of the primal problem that guarantee their applicability and convergence. ‘Fast’ or ‘accelerated’ proximal gradient methods can be adapted to viscoplastic flow problems, to yield strong convergence of order O(1=k), as the iteration counter k ! 1. This contrasts to O(1= p k) convergence of state-of-the-art solvers in viscoplasticity. Accelerated second-order methods of Newton type are particularly advantageous for resolving the additional nonlinearity that arises in Casson and Herschel-Bulkley flow problems. We observe that these algorithms can converge several times faster than classical alternatives. Simulations of stationary and time-dependent flows through pipe cross-sections and two-dimensional cavities demonstrate the viability and efficiency of this approach. One may anticipate that these new numerical methods bring us an important step closer towards the industrial applicability of computational viscoplasticity

Fast and Exact: an accelerated dual gradient method for Bingham flow

Non UBCUnreviewedAuthor affiliation: University of CanterburyGraduat

Practical guidelines for fast, efficient and robust simulations of yield-stress flows without regularisation using accelerated proximal gradient or augmented Lagrangian methods

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.Applied Science, Faculty ofScience, Faculty ofChemical and Biological Engineering, Department ofMathematics, Department ofMechanical Engineering, Department ofUnreviewedFacultyPostdoctora