MIXED TRANSFORM FINITE ELEMENT METHOD FOR SOLVING THE NON-LINEAR EQUATION FOR FLOW IN VARIABLY SATURATED POROUS MEDIA

A new computational method is developed for numerical solution of the Richards equation for flow in variably saturated porous media. The new method, referred to as the mixed transform finite element method, employs the mixed formulation of the Richards equation but expressed in terms of a partitioned transform. An iterative finite element algorithm is derived using a Newton–Galerkin weak statement. Specific advantages of the new method are demonstrated with applications to a set of one— dimensional test problems. Comparisons with the modified Picard method show that the new method produces more robust solutions for a broad range of soil– moisture regimes, including flow in desiccated soils, in heterogeneous media and in layered soils with formation of perched water zones. In addition, the mixed transform finite element method is shown to converge faster than the modified Picard method in a number of cases and to accurately represent pressure head and moisture content profiles with very steep fronts. © 1997 by John Wiley & Sons, Ltd.     

THE ‘NO BOUNDARY CONDITION’ OUTFLOW BOUNDARY CONDITION

We use a one-dimensional model problem of advection– diffusion to investigate the treatment recently advocated by Papanastasiou and colleagues to deal with boundary conditions at artificial outflow boundaries. Using finite elements of degree p, we show that their treatment is equivalent to imposing the condition that the (p+1 )st derivative of the dependent variable should vanish at a point close to the outflow. This is then shown to lead to errors of order 𝒪((h+1/Pe)1.6^p+1) in the numerical solutions (where h is the maximum element size and Pe is the global Peclet number), which is superior to the errors of order 𝒪(h^p+1+1/Pe) obtained using a standard no-flux outflow condition. These findings are verified by numerical experiments. © 1997 by John Wiley and Sons, Ltd.     

AN ADAPTIVE FINITE ELEMENT SCHEME FOR HYDRAULIC FRACTURING WITH PROPPANT TRANSPORT

A mathematical model and adaptive finite element scheme are developed for describing the distribution of proppant in a propagating hydraulic fracture. The governing equation for proppant concentration is derived by applying the conservation law of mass to the proppant and to the proppant-laden fluid. Shah's empirical equation, which relates the proppant concentration and the indices of the non-Newtonian fluid, is used to describe the proppant-laden fluid. The proppant distribution inside a hydraulic fracture can then be obtained by solving the proppant concentration equation together with the governing equations of fluid and elasticity for a hydraulic fracturing. A novel moving grid scheme is developed that combines grid point insertion with redistribution. Four examples corresponding to different in situ stress distributions are computed to demonstrate the scheme. © 1997 by John Wiley & Sons, Ltd.     

Computation of the stabilization parameter for the finite element solution of advective–diffusive problems

In a previous paper a general procedure for deriving stabilized finite element schemes for advective type problems based on invoking higher order balance laws over finite size domains was presented. This provides an expression for the element stabilization parameter in terms of the solution residual and its first derivatives in a kind of iterative or adaptative manner. Details of the application of this procedure to 1D and 2D advective–diffusive problems are given. Some examples of applications showing the potential of the new approach are presented. © 1997 John Wiley & Sons, Ltd.     

Synthesis and Solid-State Structure of 2,6-Trip2C6H3Tl (Trip=2,4,6-iPr3C6H2): A Monomeric Arylthallium(I) Compound with a Singly Coordinated Thallium Atom

A “one-legged thallium” is observed in the arylthallium(I ) compound 2,6-Trip₂C₆H₃Tl (Trip=2,4,6-iPr₃C₆H₂), which was synthesized from the corresponding organolithium compound and thallium chloride. X-ray structure analysis reveals that 2,6-Trip₂C₆H₃Tl is monomeric in the solid state and contains a singly coordinated thallium atom (see space-filling model on the right).     

Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem

Viscous flow in elastic (collapsible) tubes is a large-displacement fluid-structure interaction problem frequently encountered in biomechanics. This paper presents a robust and rapidly converging procedure for the solution of the steady three-dimensional Stokes equations, coupled to the geometrically non-linear shell equations which describe the large deformations of the tube wall. The fluid and solid equations are coupled in a segregated method whose slow convergence is accelerated by an extrapolation procedure based on the scheme's asymptotic convergence behaviour. A displacement control technique is developed to handle the system's snap-through behaviour. Finally, results for the tube's post-buckling deformation and for the flow in the strongly collapsed tube are shown. © 1998 John Wiley & Sons, Ltd.     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

Gauss point mass lumping schemes for Maxwell's equations

Finite element and finite difference methods for approximating the Maxwell system propagate numerical waves with slightly incorrect velocities, and this results in phase error in the computed solution. Indeed this error limits the type of problem that can be solved, because phase error accumulates during the computation and eventually destroys the solution. Here we propose a family of mass-lumped finite element schemes using edge elements. We emphasize in particular linear elements that are equivalent to the standard Yee FDTD scheme, and cubic elements that have superior phase accuracy. We prove theorems that allow us to perform a dispersion analysis of the two common families of edge elements on rectilinear grids. A result of this analysis is to provide some justification for the choice of the particular family we use. We also provide a limited selection of numerical results that show the efficiency of our scheme. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 63–88, 1998     

Harmonic oscillator tensors. V. The doubly degenerate harmonic oscillator

The step operators of the two-dimensional isotropic harmonic oscillator are shown to be separable into the basis elements of two disjoint Heisenberg Lie algebras. This separability leads to two sets of irreducible tensors, each of which is based upon its associated underlying Heisenberg Lie algebra. The matrix elements of these tensors are evaluated, along with those of some vibrational operators of physical interest. The possibility of other irreducible tensors are discussed and their usefulness is compared with that of those found here. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 343–357, 1998     

贵金属基纳米催化剂在乙醇电催化氧化中的应用

乙醇电催化氧化反应是直接乙醇燃料电池的核心步骤之一,制备高效稳定的电催化材料已经成为提升其电催化反应效率和选择性的关键。贵金属基纳米催化剂以其独特的物理和化学特性,在乙醇电催化氧化中表现出优异的电催化性能,在燃料电池领域具有重要的应用前景。近来贵金属基乙醇氧化催化剂受到广泛关注并取得系列重要研究进展。本文主要介绍催化剂元素组成调变形成多元素协同作用、形貌调控暴露高指数晶面和载体选择提升分散性等三个方面对贵金属基纳米催化剂性能的影响,为后续研究设计高效稳定的直接乙醇燃料电池催化剂提供参考。