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.     

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.     

Sparse p-version BEM for first kind boundary integral equations with random loading

We consider the weakly singular boundary integral equation on a deterministic smooth closed curve with random loading g(ω). Given the kth order statistical moment of g, the aim is the efficient deterministic computation of the kth order statistical moment of u. We derive a deterministic formulation for the kth statistical moment. It is posed in the tensor product Sobolev space and involves the k-fold tensor product operator . The standard full tensor product Galerkin BEM requires unknowns for the kth moment problem, where N is the number of unknowns needed to discretize Γ. Extending ideas of [V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. (1989) vi+121. A translation of Trudy Mat. Inst. Steklov 178 (1986)], we develop the p-Sparse Grid Galerkin BEM to reduce the number of unknowns from to .     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.     

非定常Stokes问题的矩形Crouzeix-Raviart型各向异性非协调元变网格方法

该文给出了关于速度-压力型非定常Stokes问题的一个 矩形 Crouzeix-Raviart 型各向异性非协调有限元的变网格逼近格式.并用一些新的技巧和方法导出了各向异性网格下的有关速度和压力的最优误差估计.     

Computational properties of vertical grids for a nonhydrostatic anelastic model

Computational dispersion properties of all vertically staggered grids, which are presently available, are analysed in terms of frequency and group velocity components using the second-order centre difference scheme for a nonhydrostatic anelastic approximation system with a general method. The inertial-gravitational waves with a horizontal scale of a hundred-, ten- and one-kilometres are considered. The comparison analysis shows that the Charny-Phillips (CP) and Lorenz grids are suitable for waves at all abovementioned horizontal scales, while the Lorenz time staggered and Charny-Phillips time staggered grids are applicable only to waves with a horizontal scale less than 10 km. The unstaggered (N) grid is not suitable for simulating waves at any horizontal scale. In an idealised flow numerical test, the result on the CP grid has much less error than that on the N grid.     

An Adaptive Grid Eulerian Method for the Computation of Free Surface Flows

A numerical scheme for the prediction of free surface flows is presented and investigated. The method is based on an adaptive grid Eulerian finite-volume method, where non-orthogonal boundary-fitted moving grids are employed to follow the free surface. The underlying flow solver consists in a pressure-correction scheme of SIMPLE type with multigrid acceleration, which is iteratively combined with the moving grid technique. Several numerical examples are considered to illustrate the capabilities of the approach.     

A PRESSURE-BASED EULER SCHEME FOR TRANSONIC INTERNAL AND EXTERNAL FLOW SIMULATION

SUMMARY

The development of a pressure-based Euler scheme, based on a collocated grid arrangement and solving for Cartesian mass flux components is described. An implicit numerical dissipation model, which includes second and fourth difference terms expressed in pressure, is demonstrated. A smoothing function, which includes a two-level filter, is used to adjust the second order-dissipation. The successful calculations for the inviscid transonic aerodynamic flow around airfoil NACA0012 are presented. The transonic quasi-one-dimensional flow calculation in a converging-diverging nozzle is chosen as an example of an internal flow simulation.