Adaptive Extended Isogeometric Analysis for Steady-State Heat Transfer in Heterogeneous Media

Steady-state heat transfer problems in heterogeneous solid are simulated by developing an adaptive extended isogeometric analysis(XIGA)method based on locally refined non-uniforms rational B-splines(LR NURBS).In the XIGA,the LR NURBS,which have a simple local refinement algorithm and good description ability for complex geometries,are employed to represent the geometry and discretize the field variables;and some special enrichment functions are introduced into the approximation of temperature field,thus the computational mesh is independent of the material interfaces,which are described with the level setmethod.Similar to the approximation of temperature field,a temperature gradient recovery technique for heterogeneous media is proposed,and based on the Zienkiewicz–Zhu recovery technique a posteriori error estimator is defined to automatically identify the locally refined regions.The convergence and performance properties of the developed method are verified by using three numerical examples.The numerical results show that(1)The convergence speed of the adaptive local refinement is faster than that of the uniform global refinement;(2)The convergence rate of the high-order basis functions is faster than that of the low-order basis functions;and(3)The existing inclusions change the local distributions of the temperature,and the extreme values of the temperature gradients take place around the inclusion interfaces.

Born-series approximation to volume-scattering wave for piecewise heterogeneous media

An efficient approximate scheme is presented for wave-propagation simulation in piecewise heterogeneous media by applying the Born-series approximation to volume-scattering waves. The numerical scheme is tested for dimensionless frequency responses to a heterogeneous alluvial valley where the velocity is perturbed randomly in the range of 5 %–25 %,compared with the full-waveform numerical solution. Then,the scheme is extended to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies Numerical experiments indicate that the convergence rate of this method decreases gradually with increasing velocity perturbations. The method has a fast convergence for velocity perturbations less than 15 %. However,the convergence becomes slow drastically when the velocity perturbation increases to 20 %. The method can hardly converge for the velocity perturbation up to 25 %.

Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media.The kernel K is assumed to depend on the media.First,we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues.Second,we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

In this paper,we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards’equation in porous media with highly heterogeneous conductivity fields.It is known that in such cases the contrast,ratio between the highest and lowest values of the conductivity,can adversely affect the performance of the preconditioners and,consequently,a design of robust preconditioners is important for many practical applications.The proposed iterative solvers consist of two kinds of iterations,outer and inner iterations.Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state.As a result of the linearization,a large-scale linear system needs to be solved.This linear system is solved iteratively(called inner iterations),and since it can have large variations in the coefficients,a robust preconditioner is needed.First,we show that under some assumptions the number of outer iterations is independent of the contrast.Second,based on the recently developed iterative methods,we construct a class of preconditioners that yields convergence rate that is independent of the contrast.Thus,the proposed iterative solvers are optimal with respect to the large variation in the physical parameters.Since the same preconditioner can be reused in every outer iteration,this provides an additional computational savings in the overall solution process.Numerical tests are presented to confirm the theoretical results.

A Layer-IntegratedModel of Solute Transport in HeterogeneousMedia

This study presents a numerical solution to the three-dimensional solute transport in heterogeneous media by using a layer-integrated approach.Omitting vertical spatial variation of soil and hydraulic properties within each layer,a threedimensional solute transport can be simplified as a quasi-three-dimensional solute transport which couples a horizontal two-dimensional simulation and a vertical onedimensional computation.The finite analytic numericalmethod was used to discretize the derived two-dimensional governing equation.A quadratic function was used to approximate the vertical one-dimensional concentration distribution in the layer to ensure the continuity of concentration and flux at the interface between the adjacent layers.By integration over each layer,a set of system of equations can be generated for a single column of vertical cells and solved numerically to give the vertical solute concentration profile.The solute concentration field was then obtained by solving all columns of vertical cells to achieve convergence with the iterative solution procedure.The proposed model was verified through examples from the published literatures including four verifications in terms of analytical and experimental cases.Comparison of simulation results indicates that the proposed model satisfies the solute concentration profiles obtained from experiments in time and space.

RECENT ADVANCES OF UPSCALING METHODS FOR THE SIMULATION OF FLOW TRANSPORT THROUGH HETEROGENEOUS POROUS MEDIA

We review some of our recent efforts in developing upscaling methods for simulating the flow transport through heterogeneous porous media. In particular, the steady flow transport through highly heterogeneous porous media driven by extraction wells and the flow transport through unsaturated porous media will be considered.

AN ANALYTICAL SOLUTION FOR AN EXPONENTIAL- TYPE DISPERSION PROCESS

The dispersion process in heterogeneous porous media is distance-dependent, which results from multi-scaling property of heterogeneous structure. An analytical model describing the dispersion with an exponential dispersion function is built, which is transformed into ODE problem with variable coefficients, and obtained analytical solution for two type boundary conditions using hypergeometric function and inversion technique. According to the analytical solution and computing results the difference between the exponential dispersion and constant dispersion process is analyzed

Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media.The model for three-phase flow considered in this work takes into account capillary forces,general relations for the relative permeability functions and variable porosity and permeability fields.In our numerical procedure a high resolution,nonoscillatory,second order,conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases.This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem.This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves(called transitional or undercompressive shock waves)in two-dimensional heterogeneous flows,thereby extending previous results for one-dimensional flow problems.Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.

TAILORED FINITE CELL METHOD FOR SOLVING HELMHOLTZ EQUATION IN LAYERED HETEROGENEOUS MEDIUM

In this paper, we propose a tailored finite cell method for the computation of two- dimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme based on a local approximation of the solution to Helmholtz equation. This provides a computational tool of achieving high accuracy with coarse mesh even for large wave number （high frequency）. The stability analysis and error estimates of this method are also proved. We present several numerical results to show its efficiency and accuracy.

COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

A NUMERICAL SIMULATION OF CHANNEL RESERVOIRS CONTAINING VERTICAL AND HORIZONTAL WELLS

In this article, the bounding surfaces of channels were modeled by Bayesian stochastic simulation, which is a boundary-valued problem with observed valley erosion thickness at the locations of wells （hard data）. In this study, it was assumed that the cross-section of the channel shows a parabolic shape, and the case that the vertical well and the horizontal well are located in the channel was considered. Peaceman＇s equations were modified to simultaneously solve both the vertical well problem and the horizontal well problem. In porous media, a 3D fluid equation was solved with iteration in the spatial domain, which had channels, vertical wells, and horizontal wells. As an example, the spatial distributions of pressure were calculated for channel reservoirs containing vertical and horizontal wells.

MixedMultiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media.Some of the methods developed using the framework are already known[20];others are new.New insight is gained for the known methods and extra flexibility is provided by the new methods.We give as an example a mixed MsFV on uniform mesh in 2-D.This method uses novel multiscale velocity basis functions that are suited for using global information,which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features.The method efficiently captures the small effects on a coarse grid.We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media.Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.