An unstructured mesh finite volume method for modelling saltwater intrusion into coastal aquifers

In this paper, a two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular background interpolation mesh is developed for analysing the evolution of the saltwater intrusion into single and multiple coastal aquifer systems. The model formulation consists of a ground-water flow equation and a salt transport equation. These coupled and non-linear partial differential equations are transformed by FVUM into a system of differential/algebraic equations, which is solved using backward differentiation formulas of order one through five. Simulation results are compared with previously published solutions where good agreement is observed.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

Implementation of a finite‐volume method for the determination of effective parameters in fissured porous media

Many studies have proposed one‐equation models to represent transport processes in heterogeneous porous media. This approach is based on the assumption that dependent variables such as pressure, temperature, or concentration can be expressed in terms of a single large‐scale averaged quantity in regions having very different chemical and/or mechanical properties. However, one can also develop large‐scale averaged equations that apply to the distinct regions that make up a heterogeneous porous medium. This approach leads to region‐averaged equations that contain traditional convective and dispersive terms, in addition to exchange terms that account for the transfer between the different media. In our approach, the fissures represent one region, and the porous media blocks represent the second region. The analysis leads to upscaled equations having a domain of validity that is clearly identified in terms of time and length‐scale constraints. Closure problems are developed that lead to the prediction of the effective coefficients that appear in the region averaged equations, and the main purpose of this article is to provide solutions to those closure problems. The method of solution makes use of an unstructured grid and a joint element method in order to take care of the special characteristics of the fissured network. This new numerical method uses the theory developed by Quintard and Whitaker and is applied on considerably more complex geometries than previously published results. It has been tested for several special cases such as stratified systems and “sugarbox” media, and we have compared our calculations with other computational methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 237–263, 2000     

Simulating Three-Dimensional Free Surface Viscoelastic Flows using Moving Finite Difference Schemes

An efficient finite difference framework based on moving meshes methods is developed for the three-dimensional free surface viscoelastic flows. The basic model equations are based on the incompressible Navier-Stokes equations and the Oldroyd-B constitutive model for viscoelastic flows is adopted. A logical domain semi-Lagrangian scheme is designed for moving-mesh solution interpolation and convection. Numerical results show that harmonic map based moving mesh methods can achieve better accuracy for viscoelastic flows with free boundaries while using much less memory and computational time compared to the uniform mesh simulations.     

Nonlinear static analysis of arbitrary quadrilateral plates in very large deflections

Through a linear mapping, an arbitrary quadrilateral plate is transformed into a standard square computational domain in which the deformation and director fields are developed together with the general forms of the uncoupled nonlinear equations. By proper interpolation of displacement and rotation fields on the whole domain, such that the boundary conditions are satisfied, a mathematical model based on the elastic Cosserat theory, is developed to analyze very large deformations of thin plates in nonlinear static loading. The principle of virtual work is exploited to present the weak form of the governing differential equations. The geometric and material tangential stiffness matrices are formed through linearization, and a step by step procedure is presented to complete the method. The validity and the accuracy of the method are illustrated through certain numerical examples and comparison of the results with other researches.     

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.     

Methods for the rapid solution of the pricing PIDEs in exponential and Merton models

The pricing equations for options on assets that follow jump-diffusion processes contain integrals in addition to the usual differential terms. These integrals usually make such equations expensive to solve numerically. Although Fast Fourier Transform methods can be used to to evaluate the integrals at all mesh points simultaneously, they are costly since the computational region must be extended in order to avoid problems with wrap around. Other numerical difficulties arise when the density function for the jump size is not smooth, as in the Kou double exponential model. We present new solution methods which are based on the fact that even when the problems contain time-dependent parameters the integrals often satisfy easily solved ordinary or parabolic partial differential equations. In particular, we show that by using the operator splitting method proposed by Andersen and Andreasen it is possible to reduce the solution of the pricing equation in the Kou and similar models to a sequence of ordinary differential equations at each time step. We discuss the methods and present results of numerical experiments.     

采用三角形面积坐标的四边形17节点样条单元

利用二元4次样条插值基和三角形面积坐标构造17节点四边形单元．这个新单元具有4次完备阶,通过一些算例测试表明了该单元有较高精度并对网格畸变不敏感．     

一阶线性和拟线性双曲型方程的格点模型

一阶线性和拟线性双曲型方程的格点模型李元香（武汉大学软件工程国家重点实验室）黄樟灿（武汉工学院）ＬＡＴＴＩＣＥＭＯＤＥＬＳＦＯＲＦＩＲＳＴＯＲＤＥＲＬＩＮＥＡＲＡＮＤＱＵＡＳＩ－ＬＩＮＥＡＲＨＹＰＥＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＬｉＹｕａｎ－ｘｉａｎ...     

Solving systems of fractional differential equations using differential transform method

This paper presents approximate analytical solutions for systems of fractional differential equations using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series with easily computable components. Some examples are solved as illustrations, using symbolic computation. The numerical results show that the approach is easy to implement and accurate when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

A finite volume simulation model for saturated–unsaturated flow and application to Gooburrum,Bundaberg, Queensland,Australia

In this paper, a two-dimensional control volume finite-element computational model is developed for simulating saltwater intrusion in a heterogeneous coastal alluvial aquifer system at Gooburrum located near Bundaberg in Queensland, Australia. The model consists of a coupled system of two non-linear partial differential equations. The first equation describes the flow of a variable-density fluid, and the second equation describes the transport of dissolved salt via a form of the Fokker–Planck equation. The outcomes of the work demonstrate that transport simulation techniques provide excellent tools for hydraulic investigations even when complex transition zones are involved.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory

In the present paper, a non-classical model for functionally graded annular sector microplates under distributed transverse loading is developed based on the modified couple stress theory and the first-order shear deformation plate theory. The model contains a single material length scale parameter which can capture the size effect. The material properties are graded through the thickness of plates according to a power-law distribution of the volume fraction of the constituents. The equilibrium equations and boundary conditions are simultaneously derived from the principle of minimum total potential energy. The system of equilibrium equations is then solved using the generalized differential quadrature method. The effects of length scale parameter, power-law index and geometrical parameters on the bending response of annular sector plates subjected to distributed transverse loading are investigated.     

Stability and bifurcation analysis of a buckled beam via active control

In this paper, we focus on applying active control to nonlinear dynamical beam system to eliminate its vibration. We analyzed stability using frequency-response equations and bifurcation. The analytical solution of the nonlinear differential equations describing the above system is investigated using multiple time scale method (MTSM). All resonance cases were extracted from second order approximations. Numerical solutions of the system are included. The effects of most system parameters were investigated. The results demonstrated that proposed controller is efficient to suppress the vibrations. Increasing the quadratic stiffness coefficient term vanished the multi-valued solution. Bifurcation diagrams refiled the effects of various system parameters on its stability showing different bifurcation cases. Finally, we conclude that for low values of natural frequencies dynamical system, the controller is more effective. The results show that the analytical solutions of the system are in good agreement with the numerical solutions.     

Unconditional Superconvergent Analysis of Quasi-Wilson Element for Benjamin-Bona-Mahoney Equation

This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney (BBM) equation. For the generalized rectangular meshes including rectangular mesh, deformed rectangular mesh and piecewise deformed rectangular mesh, by use of the special character of this element, that is, the conforming part (bilinear element) has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order $O(h^2)$, one order higher than its interpolation error, the superconvergent estimates with respect to mesh size $h$ are obtained in the broken $H^1$-norm for the semi-/ fully-discrete schemes. A striking ingredient is that the restrictions between mesh size $h$ and time step $\tau$ required in the previous works are removed. Finally, some numerical results are provided to confirm the theoretical analysis.     

OSCILLATION CRITERIA FOR A FOURTH ORDER SUBLINEAR DYNAMIC EQUATION ON TIME SCALE

Some new criteria for the oscillation of a fourth order sublinear and/or linear dynamic equation on time scale are established. Our results are new for the corresponding fourth order differential equations as well as difference equations.     

A vertex-centered,dual discontinuous Galerkin method

This note introduces a new version of the discontinuous Galerkin method for discretizing first-order hyperbolic partial differential equations. The method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements. At lowest order, the method reduces to a vertex-centered finite-volume method with control volumes based on a dual mesh, and the method can be implemented using an edge-based data structure. The method provides therefore a strategy to extend existing vertex-centered finite-volume codes to higher order using the discontinuous Galerkin method. Preliminary tests on a model linear hyperbolic equation in two-dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions.     

Steady state numerical simulation of the particle collection efficiency of a new urban sustainable gravity settler using design of experiments by FVM

The aim of this work is to analyze the efficiency of a new sustainable urban gravity settler to avoid the solid particle transport, to improve the water waste quality and to prevent pollution problems due to rain water harvesting in areas with no drainage pavement. In order to get this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (sand particles) and fluid phase (water). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, a particle transport model termed as Lagrangian particle tracking model is used, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 2,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. The entire FVM model is built and the design of experiments (DOE) method was used to limit the number of simulations required, saving on the computational time significantly needed to arrive at the optimum configuration of the settler. Finally, conclusions of this work are exposed.