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

Constrained optimization in seismic reflection tomography: a Gauss-Newton augmented Lagrangian approach

International audienceS U M M A R Y Seismic reflection tomography is a method for determining a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint , the problem consists in minimizing a non-linear least-squares function measuring the mismatch between observed traveltimes and those calculated by ray tracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as inequality constraints in the optimization problem. This technique is based on a Gauss-Newton (GN) sequential quadratic programming approach. At each GN step, a solution to a convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian algorithm. Our choice for this optimization method is motivated and its original aspects are described. First applications on real data sets are presented to illustrate the potential of the approach in practical use of reflection tomography

Migration of a sphere in tube flow

The cross-stream migration of a single neutrally buoyant rigid sphere in tube flow is simulated by two packages, one (ALE) based on a moving and adaptive grid and another (DLM) using distributed Lagrange multipliers on a fixed grid. The two packages give results in good agreement with each other and with experiments. A lift law L=CUs (Ωs— Ωse) analogous to L=ρUΓ which was proposed and validated in two dimensions is validated in three dimensions here; C is a constant depending on material and geometric parameters, Us is the slip velocity and it is positive, Ωs is the slip angular velocity and Ωse is the slip angular velocity when the sphere is in equilibrium at the Segré–Silberberg radius. The slip angular velocity discrepancy Ωs— Ωse is the circulation for the free particle and it changes sign with the lift. A method of constrained simulation is used to generate data which is processed for correlation formulas for the lift force, slip velocity, and equilibrium position. Our formulae predict the change of sign of the lift force which is necessary in the Segré–Silberberg effect. Our correlation formula is compared with analytical lift formulae in the literature and with the results of two-dimensional simulations. Our work establishes a general procedure for obtaining correlation formulae from numerical experiments. This procedure forms a link between numerical simulation and engineering practice

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

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

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

Uniform stability estimates for the discrete Calderon problems

In this article, we focus on the analysis of discrete versions of the Calderon problem in dimension d \geq 3. In particular, our goal is to obtain stability estimates for the discrete Calderon problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we shall prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that there, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical Complex Geometric Optics (CGO) approach, we can thus derive discrete CGO solutions, but with limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderon problems.Comment: 38 pages, 2 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

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

Numerical analysis and simulation of the dynamics of mountain glaciers

In this chapter, we analyze and approximate a nonlinear stationary Stokes problem that describes the motion of glacier ice. The existence and uniqueness of solutions are proved and an a priori error estimate for the finite element approximation is found. In a second time, we combine the Stokes problem with a transport equation for the volume fraction of ice, which describes the time evolution of a glacier. The accumulation due to snow precipitation and melting are accounted for in the source term of the transport equation. A decoupling algorithm allows the diffusion and the advection problems to be solved using a two-grids method. As an illustration, we simulate the evolution of Aletsch glacier, Switzerland, over the 21st century by using realistic climatic conditions