Finite difference method

Abstract: The problem of concentrated brine transport arises in the study of transport of pollutants released from a repository in a rock salt formation. An important characteristic of brine, as compared to other solutions normally encountered in groundwater problems, is that it contains a high concentration of solutes. This factor requires special attention in the development of mathematical models for brine transport problems. In this work we discuss certain important physical and mathematical differences between low- and high-concentration situations. In particular, we consider three primary aspects of a model: basic equations, boundary conditions, and numerical techniques. Recognizing the fact that in high-concentration situations, the fluid motion is not independent of the solutes movement, a new formulation of Darcy's and Fick's law are proposed. The basic equations comprise a set of two nonlinear coupled partial differential equations to be solved for the pressure p and the solute mass fraction ω. These equations have to be solved by means of iterative methods. Various possibilities involving finite difference methods have been studied. In one case, after discretizing the equations in a fully implicit way, the Newton-Raphson method has been employed to solve the system of nonlinear difference equations simultaneously. In another case, after removing part of the nonlinearity by a transformation of the dependent variable ω, a procedure of sequential solution of the two equations by successive substitution is employed. It turns out that the latter method is considerably faster than the former one as a result of the quasi-linearization. Finally, considering boundary conditions, it is shown that often they are also nonlinear and coupled. Appropriate conditions for a rock salt boundary and an outflow boundary are developed and their significance in high-concentration situations are discussed. In particular, a nonlinear time-dependent boundary condition at a rock salt boundary is developed which takes into account the process of salt dissolution and cap rock formation.

Abstract: The MAC (Marker and Cell) discretization of fluid flow is analysed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard ${\bf H}^1 $ and $L^2 $ norms are given for the velocity and pressure, respectively.

Abstract: The solution of the elastohydrodynamic lubrication (EHL) problem involves the simultaneous resolution of the hydrodynamic (Reynolds equation) and elastic problems (elastic deformation of the contacting surfaces) Up to now, most of the numerical works dealing with the modeling of the isothermal EHL problem were based on a weak coupling resolution of the Reynolds and elasticity equations (semi-system approach) The latter were solved separately using iterative schemes and a finite difference discretization Very few authors attempted to solve the problem in a fully coupled way, thus solving both equations simultaneously (full-system approach) These attempts suffered from a major drawback which is the almost full Jacobian matrix of the nonlinear system of equations This work presents a new approach for solving the fully coupled isothermal elastohydrodynamic problem using a finite element discretization of the corresponding equations The use of the finite element method allows the use of variable unstructured meshing and different types of elements within the same model which leads to a reduced size of the problem The nonlinear system of equations is solved using a Newton procedure which provides faster convergence rates Suitable stabilization techniques are used to extend the solution to the case of highly loaded contacts The complexity is the same as for classical algorithms, but an improved convergence rate, a reduced size of the problem and a sparse Jacobian matrix are obtained Thus, the computational effort, time and memory usage are considerably reduced

Abstract: The nonlinear quasi-Poisson equation that describes static magnetic fields in saturable iron is solved approximately by minimizing the corresponding nonlinear energy functional. The minimization is performed by means of the method of finite elements, using firstorder elements and a quadratically convergent iterative solution method. The method is applied to a turboalternator and used to predict all the normal shop-floor test results. Excellent agreement is found between experimental and computed values. Computing times are found to be extremely fast, and it is concluded that this method is capable of producing results comparable to those obtained by finite difference methods, but at very much reduced cost.