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.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

A fast algorithm to solve viscous two-phase flow in an axisymmetric rocket nozzle

A numerically fast algorithm has been developed to solve the viscous two-phase flow in an axisymmetric rocket nozzle. A Eulerian–Eulerian approach is employed in the computation to couple the gas–particle flow. Turbulence closure is achieved using a Baldwin–Lomax model. The numerical procedure employs a multistage time-stepping Runge–Kutta scheme in conjunction with a finite volume method and is made computationally fast for the axisymmetric nozzle. The present numerical scheme is applied to compute the flow field inside JPL and AGARD nozzles. © 1998 John Wiley & Sons, Ltd.     

On some variational acceleration techniques and related methods for local refinement

This paper shows that the well-known variational acceleration method described by Wachspress (E. Wachspress, Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equations of Reactor Physics, Prentice-Hall, Englewood Cliffs, NJ, 1966) and later generalized to multilevels (known as the additive correction multigrid method (B.R Huthchinson and G.D. Raithby, Numer. Heat Transf., 9 , 511–537 (1986))) is similar to the FAC method of McCormick and Thomas (S.F McCormick and J.W. Thomas, Math. Comput., 46 , 439–456 (1986)) and related multilevel methods. The performance of the method is demonstrated for some simple model problems using local refinement and suggestions for improving the performance of the method are given. © 1998 John Wiley & Sons, Ltd.     

A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS

An unstructured grid, finite volume method is presented for the solution of two-dimensional viscous, incompressible flow. The method is based on the pressure-correction concept implemented on a semi-staggered grid. The computational procedure can handle cells of arbitrary shape, although solutions presented herein have been obtained only with meshes of triangular and quadrilateral cells. The discretization of the momentum equations is effected on dual cells surrounding the vertices of primary cells, while the pressure-correction equation applies to the primary-cell centroids and represents the conservation of mass across the primary cells. A special interpolation scheme s used to suppress pressure and velocity oscillations in cases where the semi-staggered arrangement does not ensure a sufficiently strong coupling between pressure and velocity to avoid such oscillations. Computational results presented for several viscous flows are shown to be in good agreement with analytical and experimental data reported in the open literature.     

A PARALLEL BLOCK-STRUCTURED MULTIGRID METHOD FOR THE PREDICTION OF INCOMPRESSIBLE FLOWS

In this paper a parallel multigrid finite volume solver for the prediction of steady and unsteady flows in complex geometries is presented. For the handling of the complexity of the geometry and for the parallelization a unified approach connected with the concept of block-structured grids is employed. The parallel implementation is based on grid partitioning with automatic load balancing and follows the message-passing concept, ensuring a high degree of portability. A high numerical efficiency is obtained by a non-linear multigrid method with a pressure correction scheme as smoother. By a number of numerical experiments on various parallel computers the method is investigated with respect to its numerical and parallel efficiency. The results illustrate that the high performance of the underlying sequential multigrid algorithm can largely be retained in the parallel implementation and that the proposed method is well suited for solving complex flow problems on parallel computers with high efficiency.     

AN ADAPTIVE UNSTRUCTURED TRI-TREE ITERATIVE SOLVER FOR MIXED FINITE ELEMENT FORMULATION OF THE STOKES EQUATIONS

An iterative adaptive equation solver for solving the implicit Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structure which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element grid the Stokes equations are solved. The approximate solution at each level is projected onto the next finer grid and used as a start vector for the iterative equation solver at the finer level. When the finest grid is reached, the equation solver is iterated until a tolerated solution is reached. In order to reduce the overall work, the element matrices are integrated analytically beforehand. The efficiency and behaviour of the present adaptive method are compared with those of the previously developed iterative equation solver which is preconditioned by incomplete LU factorization with coupled node fill-in. The efficiency of the incomplete coupled node fill-in preconditioner is shown to be largely dependent on the global node numbering. The preconditioner is therefore tested for the natural node ordering of the tri-tree grid generator and for different ways of sorting the nodes.     

AN ALGEBRAIC MULTIGRID SOLVER FOR NAVIER–STOKES PROBLEMS IN DISCRETE SECOND–ORDER APPROXIMATION

An algebraic multigrid scheme is presented for solving the discrete Navier–Stokes equations to second-order accuracy using the defect correction method. Solutions for the driven cavity and asymmetric, sudden expansion test problems have been obtained for both structured and unstructured meshes, the resolution and resolution grading being controlled by global and local mesh refinements. The solver is efficient and robust to the extent that, for problems attempted so far, no underrelaxation of variables has been required to ensure convergence. Provided that the computational mesh can resolve the flow field, convergence characteristics are almost mesh-independent. Rates of convergence actually improve with refinement, asymptotically approaching mesh-independent values. For extremely coarse meshes, where dispersive truncation errors would be expected to prevent convergence (or even induce divergence), solutions can still be obtained by using explicit underrelaxation in the iterative cycle.     

利用改进的ADI方法实现单机千万节点的油水两相渗流数值模拟

针对实际油藏的非均质分布特征及其复杂的边界条件,本文通过引入迭代参数的压缩因子和放大因子,对现有的交替方向迭代法(ADI)进行改进,提出一种适用于大规模油藏数值模拟的新算法．改进的ADI 方法计算精度可靠,且与现有的算法相比计算效率有所提高．更为关键的是,ADI 算法将求解三维压力方程的七对角矩阵分解为三个方向的三对角矩阵依次迭代求解,所需的运算存储量大幅减少,最大的计算规模有了大幅的提升．使用改进的ADI 方法,在单机上成功实现了千万节点的油水两相渗流数值模拟．计算实例表明,在同等单机硬件条件下,改进ADI 算法的最大计算规模是现有算法的1.7 倍以上．     

A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties

This work proposes a domain adaptive stochastic collocation approach for uncertainty quantification, suitable for effective handling of discontinuities or sharp variations in the random domain. The basic idea of the proposed methodology is to adaptively decompose the random domain into subdomains. Within each subdomain, a sparse grid interpolant is constructed using the classical Smolyak construction [S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240–243], to approximate the stochastic solution locally. The adaptive strategy is governed by the hierarchical surpluses, which are computed as part of the interpolation procedure. These hierarchical surpluses then serve as an error indicator for each subdomain, and lead to subdivision whenever it becomes greater than a threshold value. The hierarchical surpluses also provide information about the more important dimensions, and accordingly the random elements can be split along those dimensions. The proposed adaptive approach is employed to quantify the effect of uncertainty in input parameters on the performance of micro-electromechanical systems (MEMS). Specifically, we study the effect of uncertain material properties and geometrical parameters on the pull-in behavior and actuation properties of a MEMS switch. Using the adaptive approach, we resolve the pull-in instability in MEMS switches. The results from the proposed approach are verified using Monte Carlo simulations and it is demonstrated that it computes the required statistics effectively.