Reduced order borehole induction modelling

This article presents a new reduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic fields generated by an array induction tool – an instrument that transmits and receives electrical signals along different positions within a borehole. The motivation for this approach is in the generation of an efficient ‘forward model’ (which provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations (which typically require a large number of forward solutions) that are used to determine surrounding material properties. This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to an established method.     

A reduced‐order approach for optimal control of fluids using proper orthogonal decomposition

In this article, a reduced‐order modeling approach, suitable for active control of fluid dynamical systems, based on proper orthogonal decomposition (POD) is presented. The rationale behind the reduced‐order modeling is that numerical simulation of Navier–Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. The possibility of obtaining reduced‐order models that reduce the computational complexity associated with the Navier–Stokes equations is examined while capturing the essential dynamics by using the POD. The POD allows the extraction of a reduced set of basis functions, perhaps just a few, from a computational or experimental database through an eigenvalue analysis. The solution is then obtained as a linear combination of this reduced set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations (PDEs). It is used here in active control of fluid flows governed by the Navier–Stokes equations. In particular, flow over a backward‐facing step is considered. Reduced‐order models/low‐dimensional dynamical models for this system are obtained using POD basis functions (global) from the finite element discretizations of the Navier–Stokes equations. Their effectiveness in flow control applications is shown on a recirculation control problem using blowing on the channel boundary. Implementational issues are discussed and numerical experiments are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element solution of multi-dimensional two-phase flow through porous media with arbitrary heating conditions

Mechanistic models for flow regime transitions and drag forces proposed in an earlier work are employed to predict two-phase flow characteristics in multi-dimensional porous layers. The numerical scheme calls for elimination of velocities in favor of pressure and void fraction. The momentum equations for vapor and liquid then can be reduced to a system of two partial differential equations (PDEs) which must be solved simultaneously for pressure and void fraction.

Solutions are obtained both in two-dimensional cartesian and in axi-symmetric coordinate systems. The porous layers in both cases are composed of regions with different permeabilities. The finite element method is employed by casting the PDEs in their equivalent variational forms. Two classes of boundary conditions (specified pressure and specified fluid fluxes) can be incorporated in the solution. Volumetric heating can be included as a source term. The numerical procedure is thus suitable for a wide variety of geometry and heating conditions. Numerical solutions are also compared with available experimental data.     

On Dirichlet Problem for a Class of Delayed Reaction–Diffusion Equations with Spatial Non-locality

We consider a very general class of delayed reaction–diffusion equations in which the reaction term can be non-monotone as well as spatially non-local. By employing comparison technique and a dynamical system approach, we study the global asymptotic behavior of solutions to the equation subject to the homogeneous Dirichlet condition. Established are threshold results and global attractiveness of the trivial steady state, as well as the existence, uniqueness and global attractiveness of a positive steady state solution to the problem. As illustrations, we apply our main results to the local delayed diffusive Mackey–Glass equation and the nonlocal delayed diffusive Nicholson blowfly equation, leading to some very sharp results for these two particular models.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

A generalized variational formulation for convective heat transfer

The Scope of this paper is to develop the basic equations for a variational formulation which can be used to solve problems related to convection and/or diffusion dominated flows. The formulation is based on the introduction of a generalized quantity defined as the hear displacement. The governing equation is expressed in terms of this quantity and a variational formulation is developed which leads to a system of equations similar in form to Lagrange's equations of mechanics. These equations can be used for obtaining approximate solutions, though they are of particular interest for application of the finite element method. As an example of the formulation two finite element models are derived for solving convectiondiffusion boundary value problems. The performance of the two models is investigated and numerical results are given for different cases of convection and diffusion with two types of boundary conditions. The applications of the developed formulations are not limited to convection-diffusion problems but can also be applied to other types of problems such as mass transfer, hydrodynamics and wave propagation.     

Preventing Blow up by Convective Terms in Dissipative PDE’s

We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger’s type equations, convective Cahn–Hilliard equation, generalized Kuramoto–Sivashinsky equation and KdV type equations. The following common scenario is established: adding sufficiently strong (in comparison with the destabilizing nonlinearity) convective terms to equation prevents the solutions from blowing up in a finite time and makes the considered system globally well-posed and dissipative and for weak enough convective terms the finite time blow up may occur similar to the case, when the equation does not involve convective term. This kind of result has been previously known for the case of Burger’s type equations and has been strongly based on maximum principle. In contrast to this, our results are based on the weighted energy estimates which do not require the maximum principle for the considered problem.     

Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation

The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial differential equations (e.g.,fluid flows). It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference (FD) scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.     

TWISTING STATICS AND DYNAMICS FOR CIRCULAR ELASTIC NANOSOLIDS BY NONLOCAL ELASTICITY THEORY

The torsional static and dynamic behaviors of circular nanosolids such as nanoshafts, nanorods and nanotubes are established based on a new nonlocal elastic stress field theory. Based on a new expression for strain energy with a nonlocal nanoscale parameter, new higher-order governing equations and the corresponding boundary conditions are first derived here via the variational principle because the classical equilibrium conditions and/or equations of motion can- not be directly applied to nonlocal nanostructures even if the stress and moment quantities are replaced by the corresponding nonlocal quantities. The static twist and torsional vibration of circular, nonlocal nanosolids are solved and discussed in detail. A comparison of the conventional and new nonlocal models is also presented for a fully fixed nanosolid, where a lower-order governing equation and reduced stiffness are found in the conventional model while the new model reports opposite solutions. Analytical solutions and numerical examples based on the new nonlocal stress theory demonstrate that nonlocal stress enhances stiffness of nanosolids, i.e. the angular displacement decreases with the increasing nonlocal nanoscale while the natural frequency increases with the increasing nonlocal nanoscale.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).     

Galerkin Projections and Finite Elements for Fractional Order Derivatives

Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t ²) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.     

Asymptotic Linear Stability of Solitary Water Waves

In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities.     

Nonlinear Porous Medium Flow with Fractional Potential Pressure

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator:

$\partial_t u-\nabla\cdot(u\nabla p)=0, \quad p=(-\Delta)^{-s}u,\quad 0 < s < 1.$

We pose the problem for \({x\in {\mathbb{R}^n}}\) and t > 0 with bounded and compactly supported initial data, and prove the existence of weak and bounded solutions that propagate with finite speed, a property that is not shared by other fractional diffusion models.

     

Co-located equal-order control-volume finite element method for two-dimensional axisymmetric incompressible fluid flow

The formulation of a control-volume-based finite element method (CVFEM) for axisymmetric, two-dimensional, incompressible fluid flow and heat transfer in irregular-shaped domains is presented. The calculation domain is discretized into torus-shaped elements and control volumes. In a longitudinal cross-sectional plane, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the mid-points of the sides. Two different interpolation schemes are proposed for the scalar-dependent variables in the advection terms: a flow-oriented upwind function, and a mass-weighted upwind function that guarantees that the discretized advection terms contribute positively to the coefficients in the discretized equations. In the discretization of diffusion transport terms, the dependent variables are interpolated linearly. An iterative sequential variable adjustment algorithm is used to solve the discretized equations for the velocity components, pressure and other scalar-dependent variables of interest. The capabilities of the proposed CVFEM are demonstrated by its application to four different example problems. The numerical solutions are compared with the results of independent numerical and experimental investigations. These comparisons are quite encouraging.     

Finite element solution of an enclosed turbulent diffusion flame

A finite element formulation of enclosed turbulent diffusion flames is presented. A primitive variables approach is preferred in the analysis. A mixed interpolation is employed for the velocity and pressure. In the solution of the Navier-Stokes equations, a segregated formulation is adopted, where the pressure discretization equation is obtained directly from the discretized continuity equation, considering the velocity-pressure relationships in the discretized momentum equations. The state of turbulence is defined by a κ–? model. Near solid boundaries, a wall function approach is employed. The combustion rates are estimated using the eddy dissipation concept. The expensive direct treatment of the integrodifferential equations of radiation is avoided by employing the moment method, which allows the derivation of an approximate local field equation for the radiation intensity. The proposed finite element model is verified by investigating a technical turbulent diffusion flame of semi-industrial size, and comparing the results with experiments and finite difference predictions.     

Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

Pulses in binary wave guide arrays and long wave PDE approximations

Binary waveguide arrays are linear arrays of optical waveguides with binary alternation of parameters, and have been of recent interest. They can be modeled by systems of nonlinear ODEs with forms related to the discrete nonlinear Schrödinger equation. Such equations can also arise in semi-classical molecular models of polymers with excitable states in each monomer, and coupling between these.An important class of solutions arises from an initially highly localized signal, such as input to a single element of the array. Simulations show that for a wide array of parameter values and of such initial data, a pulse is generated that travels approximately as a traveling wave. After a suitable phase shift in the variables, this pulse quickly develops a slow spatial variation, leading to a long-wave approximation by a system of coupled third order PDEs; one each for nodes of even and odd indices.This system of PDEs is presented, and verified to quite accurately reproduce the pulse propagation seen in the ODE system; further there is often a strong tendency for the behavior of the two PDE components to converge, with a corresponding convergence of the even and odd index parts of the ODE system solution. The PDE model gives some indication of why this occurs.     

On the Loss of Continuity for Super-Critical Drift-Diffusion Equations

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts, we prove that solutions satisfy a modulus of continuity depending only on the local L ¹ norm of the drift, which is a super-critical quantity.     

非线性粘弹性Timoshenko梁动力学行为的分布

本文讨论了有限变形粘弹性Timoshenko梁的动力学行为。首先由Timoshenko梁的理论和分数导数型本构关系给出了梁的控制方程。其次为了便于求解,采用Galerkin方法对系统进行了简化,并比较了1阶和2阶截断系统的动力学性质,它们具有相同的定性性质,说明Galerkin方法的合理性。给出了求解包含分数积分的积分－微分方程的一种新方法,以便求解系统的长时间的解。综合利用非线性动力系统中的经典方法,揭示了梁在有限变形情况下丰富的动力学行为,并分别考察了载荷参数的材料参数对结构的动力学行为的影响。     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.