Spectral Element Methods for Transitional Flows in Complex Geometries

We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex three-dimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled with a stabilized high-order time-split formulation of the semi-discrete Navier–Stokes equations. Simulations of transition in a model of an arteriovenous graft illustrate the potential of this approach in biomechanical applications.     

Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We propose a simple modification of standard weighted essentially non-oscillatory (WENO) finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.     

Computer Simulation of Compositional Flow Using Unstructured Control Volume Finite Element Methods

In this paper, we study the computer simulation of gas cycling in a rich retrograde condensate reservoir. Two prediction cases are studied. The first case is gas cycling with constant sales gas removal, and the second case is cycling with some gas sales deferral to enhance pressure maintenance in the early life of this reservoir. In this problem the great majority of cycling takes place at pressure below the dew point pressure, indicating the need for modeling the compositional three-phase, multicomponent flow in the reservoir. This compositional model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. The control volume finite element (CVFE) method on unstructured grids is used to discretize the model governing equations for the first time. Numerical experiments are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). The PVT (pressure-volume-temperature) data are based on a real fluid analysis.     

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.     

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order \(\mathcal{O}(\tau ^{r}+h^{4})\), where \(r\ge 2\) is a positive integer and \(\tau \) and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of \(2\le r\le 6\), and the optimal error estimates in the weighted \(H^{1}\), \(L^{2}\) and \(L^{\infty }\) norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.     

Adaptive Nonlinear Finite Elements for Deformable Body Simulation Using Dynamic Progressive Meshes

Realistic behavior of deformable objects is essential for many applications such as simulation for surgical training. Existing techniques of deformable modeling for real time simulation have either used approximate methods that are not physically accurate or linear methods that do not produce reasonable global behavior. Nonlinear finite element methods (FEM) are globally accurate, but conventional FEM is not real time. In this paper, we apply nonlinear FEM using mass lumping to produce a diagonal mass matrix that allows real time computation. Adaptive meshing is necessary to provide sufficient detail where required while minimizing unnecessary computation. We propose a scheme for mesh adaptation based on an extension of the progressive mesh concept, which we call dynamic progressive meshes.     

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems     

钛合金超声振动辅助钻削有限元仿真中不同刀具几何的评价

针对钛合金(Ti6A14V)难加工材料采用普通麻花钻传统钻削过程中切削力过大及切削温度过高的问题,对传统钻削和超声振动辅助钻削钛合金进行了有限元仿真试验,分析了进给量、主轴转速等钻削参数对三种不同的点角钻头在钻削过程中产生的切削力、等效应力及温度的影响。结果表明:相比于传统钻削,超声振动辅助钻削明显降低钻削力、最高切削温度,分别降低了13-22%和7-15%,为钻头刀具几何形状、钻削参数和切削性能的优化提供依据。     

Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations

In this paper, a class of distributed-order time fractional diffusion equations (DOFDEs) on bounded domains is considered. By L1 method in temporal direction, a semi-discrete variational formulation of DOFDEs is obtained firstly. The stability and convergence of this semi-discrete scheme are discussed, and the corresponding fully discrete finite element scheme is investigated. To improve the convergence rate in time, the weighted and shifted Grünwald difference method is used. By this method, another finite element scheme for DOFDEs is obtained, and the corresponding stability and convergence are considered. And then, as a supplement, a higher order finite difference scheme of the Caputo fractional derivative is developed. By this scheme, an approach is suggested to improve the time convergence rate of our methods. Finally, some numerical examples are given for verification of our theoretical analysis.     

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method. This research was supported by the Israel Science Foundation (grant No. 1362/04).     

时域有限差分中各向异性完全匹配层的实现方法

本文讨论了如何在给定边界层数的条件下,通过调整各介质层吸收层的介质参数来实现入射波的最佳吸收,并利用自适应遗传算法对算法中的边界参数进行优化。     

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re \((s)\ge 0\), two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at \(s=0\). We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.     

Front-Tracking Finite Difference Methods for the Valuation of American Options

This paper is concerned with the numerical solution of the American option valuation problem formulated as a parabolic free boundary/initial value model. We introduce and analyze a front-tracking finite difference method and compare it with other commonly used techniques. The numerical experiments performed indicate that the front-tracking method considered is an efficient alternative for approximating simultaneously the option value and free boundary functions associated with the valuation problem.     

复杂系统仿真方法及应用

文章讨论复杂系统仿真方法及其应用,内容包括：组成系统仿真的要素及活动：传统系统仿真方法学：复杂性及复杂系统仿真方法等。     

High Order Finite Difference and Finite Volume Methods for Advection on the Sphere

Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significant potential gain in computational efficiency. We here present two classes of high order methods for discretization on the surface of a sphere, first finite difference schemes satisfying the summation-by-parts property on the cube sphere grid, secondly finite volume discretizations on unstructured grids with polygonal cells. Furthermore, we also implement the seventh order accurate weighted essentially non-oscillatory (WENO7) scheme for the cube sphere grid. For the finite difference approximation, we prove a stability estimate, derived from projection boundary conditions. For the finite volume method, we develop the implementational details by working in a local coordinate system at each cell. We apply the schemes to compute advection on a sphere, which is a well established test problem. We compare the performance of the methods with respect to accuracy, computational efficiency, and ability to capture discontinuities.     

枕寰枢复合体有限元模型的建模研究

在生物力学的研究中,枕寰复合体易损伤,针对结构的安全性和可靠性,建立合理有效的枕寰枢复合体有限元模型,提高结构的精度.通过CT断层扫描图片,应用医学影像处理技术、逆向造型等计算机辅助建模方法,建立了包括韧带及软骨的枕寰枢复合体有限元模型;对关键技术进行了研究,对模型进行了相应技术指标的评价和有效性验证.验证结果表明,可以在节约大量前处理时间的基础上,建立具有精细解剖结构的高精度八节点六面体有限元模型,为枕寰枢复合体有限元分析及其生物力学的研究工作奠定基础.