Four-field finite element solver and sensitivities for quasi-Newtonian flows

International audienceA computationally efficient finite element algorithm for power law fluid is elaborated in view of extensive direct and inverse simulations. We adopt a splitting technique to simplify the nonlinear structure of the fluids equations and derive a four-field saddle point formulation for which we prove the existence of a solution. The resolution of the corresponding variational inequalities is based on an augmented Lagrangian method and a mixed finite element discretization. The resulting iterative solver reveals to be fast and robust with low memory consumption. The time-saving provided by the algorithm compared to the standard algorithms of fixed point and Newton increases with the number of degrees of freedom and the nonlinearity of the problem. It is therefore well-suited for the solution of large problems with a great number of elements and for corresponding adjoint-based computations. Bidimensional numerical experiments are performed on two realistic situations of gravity flows: an experimental viscoplastic steady wave and a continental glacier. In the present study, results emphasize that for both cases, the modeling at bottom plays a strongly dominant role. Using surface velocitiy observations, the sensitivity analysis with respect to a spatially varying power-law exponent highlights the importance of an accurate knowledge of the rheology at high shear rate. The one on the basal sliding allows to detect the presence of a short wavelength (two times the thickness) free-slip area indetectable from surface velocities

A new approach to hyperbolic inverse problems

We present a modification of the BC-method in the inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper we prove the crucial local step. The global step of the proof will be presented in the forthcoming paper.Comment: We corrected the proof of the main Lemma 2.1 by assuming that potentials A(x),V(x) are real value

Simulating (electro)hydrodynamic effects in colloidal dispersions: smoothed profile method

Previously, we have proposed a direct simulation scheme for colloidal dispersions in a Newtonian solvent [Phys.Rev.E 71,036707 (2005)]. An improved formulation called the ``Smoothed Profile (SP) method'' is presented here in which simultaneous time-marching is used for the host fluid and colloids. The SP method is a direct numerical simulation of particulate flows and provides a coupling scheme between the continuum fluid dynamics and rigid-body dynamics through utilization of a smoothed profile for the colloidal particles. Moreover, the improved formulation includes an extension to incorporate multi-component fluids, allowing systems such as charged colloids in electrolyte solutions to be studied. The dynamics of the colloidal dispersions are solved with the same computational cost as required for solving non-particulate flows. Numerical results which assess the hydrodynamic interactions of colloidal dispersions are presented to validate the SP method. The SP method is not restricted to particular constitutive models of the host fluids and can hence be applied to colloidal dispersions in complex fluids

Pore-scale Modeling of Viscous Flow and Induced Forces in Dense Sphere Packings

We propose a method for effectively upscaling incompressible viscous flow in large random polydispersed sphere packings: the emphasis of this method is on the determination of the forces applied on the solid particles by the fluid. Pore bodies and their connections are defined locally through a regular Delaunay triangulation of the packings. Viscous flow equations are upscaled at the pore level, and approximated with a finite volume numerical scheme. We compare numerical simulations of the proposed method to detailed finite element (FEM) simulations of the Stokes equations for assemblies of 8 to 200 spheres. A good agreement is found both in terms of forces exerted on the solid particles and effective permeability coefficients

Theoretical analysis and numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows

The theoretical formulation of the smoothed particle hydrodynamics (SPH) method deserves great care because of some inconsistencies occurring when considering free-surface inviscid flows. Actually, in SPH formulations one usually assumes that (i) surface integral terms on the boundary of the interpolation kernel support are neglected, (ii) free-surface conditions are implicitly verified. These assumptions are studied in detail in the present work for free-surface Newtonian viscous flow. The consistency of classical viscous weakly compressible SPH formulations is investigated. In particular, the principle of virtual work is used to study the verification of the free-surface boundary conditions in a weak sense. The latter can be related to the global energy dissipation induced by the viscous term formulations and their consistency. Numerical verification of this theoretical analysis is provided on three free-surface test cases including a standing wave, with the three viscous term formulations investigated

A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control

A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases

Genetic and environmental risk for major depression in African-American and European-American women

It is unknown whether there are racial differences in the heritability of major depressive disorder (MDD) because most psychiatric genetic studies have been conducted in samples comprised largely of white non-Hispanics. To examine potential differences between African-American (AA) and European-American (EA) young adult women in (1) DSM-IV MDD prevalence, symptomatology and risk factors and (2) genetic and/or environmental liability to MDD, we analyzed data from a large, population representative sample of twins ascertained from birth records (n= 550 AA and n=3226 EA female twins) aged 18–28 years at the time of MDD assessment by semi-structured psychiatric interview. AA women were more likely to have MDD risk factors; however, there were no significant differences in lifetime MDD prevalence between AA and EA women after adjusting for covariates (Odds Ratio = 0.88, 95% confidence interval: 0.67–1.15 ). Most MDD risk factors identified among AAs were also associated with MDD at similar magnitudes among EAs. Although the MDD heritability point estimate was higher among AA than EA women in a model with paths estimated separately by race (56%, 95% CI: 29%–78% vs. 41%, 95% CI: 29%–52%), the best-fitting model was one in which additive genetic and nonshared environmental paths for AA and EA women were constrained to be equal (A = 43%, 33%–53% and E = 57%, 47%–67%). Despite a marked elevation in the prevalence of environmental risk exposures related to MDD among AA women, there were no significant differences in lifetime prevalence or heritability of MDD between AA and EA young women

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem