Nonlinear Multigrid Methods for Numerical Solution of the Variably Saturated Flow Equation in Two Space Dimensions

The need of accurate and efficient numerical schemes to solve Richards’ equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards’ equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated–unsaturated soils with hydraulic properties described by van Genuchten–Mualem models. The numerical results obtained are compared with those provided by the modified Picard–preconditioned conjugated gradient (Krylov subspace) approach.     

On the Capillarity Equation in Two Dimensions

We study the capillarity equation from the global point of view of behavior of its solutions without explicit regard to boundary conditions. We show its solutions to be constrained in ways, that have till now not been characterized in literature known to us.     

A Lattice Gas Automaton Simulation of the Nonlinear Diffusion Equation: A Model for Moisture Flow in Unsaturated Porous Media

We investigate a two-dimensional lattice gas automaton (LGA) for simulating the nonlinear diffusion equation in a random heterogeneous structure. The utilility of the LGA for computation of nonlinear diffusion arises from the fact that, the diffusion coefficient in the LGA depends on the local density of fluid particles which statistically determines the collision rate and thus, the mean free path of the particles at the microscopic scale. The LGA may therefore be used as a physical analogue to simulate moisture flow in unsaturated porous media. The capability of the LGA to account for unsaturated flow is tested through a set of numerical experiments simulating one-dimensional infiltration in a simplified semi-infinite homogenous isotropic porous material. Different mechanisms of interactions are used between the fluid and the solid phase to simulate various fluid–solid interfaces. The heterogeneous medium, initially at low density is submitted to a steep density gradient by continuously injecting fluid particles at high concentration and zero velocity along one face of the model. The propagation of the infiltration front is visualized at different time steps through concentration profiles parallel to the applied concentration gradient and the infiltration rate is measured continuously until steady-state flow is reached. The numerical results show close agreement with the classical theory of flow in unsaturated porous media. The cumulative absorption exhibits the expected t ^1/2 dependence. The evolution of the effective diffusion coefficient with the particle concentration is estimated from the measured density profiles for the various porous materials. Depending on the applied fluid–solid interactions, the macroscopic effective diffusivity may vary by more than two orders of magnitude with density.     

The Quantum Hydrodynamics System in Two Space Dimensions

In this paper we study global existence of weak solutions for the quantum hydrodynamics system in two-dimensional energy space. We do not require any additional regularity and/or smallness assumptions on the initial data. Our approach replaces the WKB formalism with a polar decomposition theory which is not limited by the presence of vacuum regions. In this way we set up a self consistent theory, based only on particle density and current density, which does not need to define velocity fields in the nodal regions. The mathematical techniques we use in this paper are based on uniform (with respect to the approximating parameter) Strichartz estimates and the local smoothing property.     

Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions

A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.     

Slow Dynamics for the Cahn‐Hilliard Equation in Higher Space Dimensions: The Motion of Bubbles

It is known that the Van der Waals‐Cahn‐Hilliard (W‐C‐H) dynamics can be approximated by a Quasi‐static Stefan problem with surface tension. It turns out that the Stefan problem has a manifold of equilibria equal in dimension to that of the domain Ω: any sphere of fixed radius with interface contained in the domain is an equilibrium (indistinguishable from the point of view of the perimeter functional). We resolve this degeneracy by showing that at the W‐C‐H level this manifold is replaced by a quasi‐invariant stable manifold, on which the typical solution preserves its “bubble” like shape until it reaches the boundary. Moreover, we show that the “bubble” moves superslowly. We also obtain an equation that determines those special spheres that correspond to equilibria at the W‐C‐H level. Our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions. (Accepted February 12, 1996)     

多重网格法与Newton-Raphson法耦合求线弹流数值解

利用多重网格算法,在最粗网格上,采用Newton-Raphson迭代法模式精解线接触弹流非线性方程组,充分利用了多重网格法与Newton-Raphson法各自的优点,计算实践表明,求解弹流问题的数值解过程在收敛与稳定性方面均有较大改善,且有相当宽的载荷参数适用范围。     

Linear and Nonlinear Problems on the Elastic Deformation of Complex Shells and Methods of Their Numerical Solution

Some approaches to the solution of problems on the elastic deformation of thin-walled solids with a complex shape are analyzed on the basis of linear and geometrically nonlinear models. The general characteristic of the classical approaches to the solution of the problems is discussed. Approaches employing new classes of surfaces are considered. Solutions to some problems on the stress state of complex shell elements are presented     

The Relativistic Vlasov-Maxwell System in Two Space Dimensions: Part I

A collisionless plasma is modeled by the Vlasov‐Maxwell system. For the relativistic Vlasov‐Maxwell system in the plane, integral representations of the electromagnetic fields are derived. Using these representations it is shown that smooth solutions can break down only if particles of the plasma approach the speed of light. These results are analogous to those of Glassey & Strauss[10] in three dimensions. (Accepted July 5, 1996)     

两种线接触非稳态部分弹流模型数值解的比较

本文利用一种快速求解弹流问题完全数值方法，求解了包含挤压项的非稳态部分弹流的Ｐａｔｉｒ和Ｃｈｅｎｇ的平均流动模型以及吴承伟的郑林庆的接触因子模型，研究了微峰接触动态变化特点以及这两种部分膜型的数值解异同，为更加直观、简单的部分弹流问题的接触因子模型的工程应用提供了理论依据。     

煤层气在非饱和水流阶段的非定常渗流摄动解

煤层甲烷由煤层的割理裂隙系统流入生产井一般经历：单相水流、非饱和流和气、水两相饱和流三个阶段，在非饱和流阶段，储层压力降至临界解吸压力之后，储存在煤基质中的吸附气体少量被解吸出来形成互不连续的气泡并阻止水的流动，含气量尚未达到饱和程度。同时煤层甲烷运移包含渗流场、变形场和应力场的动态耦合过程。本文考虑渗流过程中水-气两相不溶混流体与固体耦合作用，建立了非饱和水流阶段非定常渗流问题的流固耦合数学模型，对该强非线性一维数学模型采用摄动法和积分变换法进行解析求解，并讨论了其压力动态特性，分析了压力随饱和度S及时间t变化的规律和气相及耦合作用的影响，这些研究对煤层气、石油和天然气的开采等地下工程领域具有一定的指导意义。     

求解非线性动力系统周期解大范围收敛方法

对于多自由度非线性动力系统,提出一种求解周期解的大范围收敛方法,这种算法对处理非线性动力系统有较强的功能。结合数值延拓算法,为求解具有系统参数的非线性动力系统在整个系统参数范围内的周期解提供了有效的方法。     

Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow

Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear and yet current models for it use linear filters to select the eddies that will be damped. In this report we consider for the first time nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow problems. The particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to calculating a linear filter of similar form. We then analyze nonlinear filter based stabilization for the Navier?CStokes equations. We give a precise analysis of the numerical diffusion and error in this process.     

Transient Heat and Moisture Flow Around Heat Source Buried in an Unsaturated Half Space

This article presents solutions for the transient heat and moisture transport due to both disk heat source and cylindrical heat source buried in an unsaturated half space. The solutions are presented in Hankel–Laplace transform domain and in dimensionless style. Coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low-permeability medium. Parametric study has been performed to assess the effects of five independent dimensionless parameters on flow field. The stability and accuracy of the present solutions are demonstrated from the comparison between the results obtained from these solutions and those by using a well-established finite element code CODE_BRIGHT. Despite the simplified assumptions required in order to obtain analytical solutions in Hankel–Laplace transform domain, the results incorporate the main mechanisms involved in the coupled thermo-hydraulic (T-H) problem, and they may be eventually used for validation purposes.     

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Numerical Model of Transient Unsaturated Flow Under Centrifugation Based on Mass Balance

Numerical modeling of unsaturated flow in porous media under centrifugation is studied. A precise and numerically efficient approximation is presented for the mathematical model, based on Richards’ nonlinear and degenerate equation expressed in terms of effective saturation using the Van Genuchten–Mualem ansatz. The main difference with other methods is the utilization of a nonlocal condition based on mass balance. The method is suitable for determination of soil parameters, including the saturated conductivity, via the solution of an inverse problem in an iterative way. First, the fully saturated sample is centrifugated with a free outflow boundary during some time interval. Next, the output boundary is sealed and the sample is centrifugated for a prescribed time interval, or up to the creation of an equilibrium. Finally, the centrifugation is continued with a free outflow boundary. This procedure can be repeated to increase the information to drive the inverse problem. The application of the present method requires only non-intrusive, cheap measurements: rotational momentum and/or gravitational center of the sample, and optionally, the amount of expelled water.     

Solution of the Reynolds Equation for Viscous-Fluid Flow in the Clearance between a Rotating Ellipsoid and a Surface

Slow viscous-fluid flows in the narrow clearance (i) between a moving ellipsoid and a straight tube of elliptic cross section and (ii) between a rotating ellipsoid and a toroidal tube, including the case of an ellipsoid near a plane, are considered. A solution of the boundary-value problem for the Reynolds equation describing the flow in the clearance is found. The similarity of the pressure profiles in the “ellipsoid-plane” and “ cylinder-plane” systems is indicated.     

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

We consider the problem of solving numerically the stationary incompressible Navier–Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called “artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct – by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke – a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitude. Peter Wittwer: Supported in part by the Fonds National Suisse.