Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems

We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations. We can compute branches of solutions with limit points, bifurcation points, etc. Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations

A Simulation Method to Resolve Hydrodynamic Interactions in Colloidal Dispersions

A new computational method is presented to resolve hydrodynamic interactions acting on solid particles immersed in incompressible host fluids. In this method, boundaries between solid particles and host fluids are replaced with a continuous interface by assuming a smoothed profile. This enabled us to calculate hydrodynamic interactions both efficiently and accurately, without neglecting many-body interactions. The validity of the method was tested by calculating the drag force acting on a single cylindrical rod moving in an incompressible Newtonian fluid. This method was then applied in order to simulate sedimentation process of colloidal dispersions.Comment: 7pages, 7 figure

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

FINITE-ELEMENT APPROXIMATION OF THE PARABOLIC P-LAPLACIAN

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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

Finite element approximation of some degenerate monotone quasilinear elliptic systems

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