厚板动力响应分析的一种辛算法

将厚板动力分析从Lagrange体系改换为Hamilton体系.根据古典阴阳互补和现代对偶互补的基本思想,首次建立了线性阻尼情形下厚板动力学的相空间非传统Hamilton变分原理.这种变分原理不仅能反映这种动力学初值—边值问题的全部特征,而且它的欧拉方程具有辛结构的特征.基于该变分原理,提出一种称之为辛空间有限元—时间子域法的辛算法.这种新方法是由空间域采用有限元法与时间子域采用La-grange插值多项式插值的时间子域法相结合而成.文中用这种辛算法分析了四种支承条件下厚板的动力响应问题.算例的计算结果表明,这种新方法的稳定性、收敛性、计算精度和效率都明显高于国际上常用的W ilson-θ法和Newm ark-β法.     

Iterative methods for domain decomposition using spectral tau and collocation approximation

In this paper we consider the numerical approximation of elliptic problems by spectral methods in domains subdivided into substructures. We review an iterative procedure with interface relaxation, reducing the given differential problem to a sequence of Dirichlet and mixed Neumann-Dirichlet problems on each subdomain. The iterative procedure is applied to both tau and collocation spectral approximations. Two optimal strategies for the automatic selection of the relaxation parameter are indicated. We present several numerical experiments showing the convergence properties of the iterative scheme with respect to the decomposition. A multilevel technique for domain decomposition methods is proposed.     

Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces

High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.     

Polynomial collocation using a domain decomposition solution to parabolic PDE's via the penalty method and explicit/implicit time marching

A domain decomposition method is examined to solve a time-dependent parabolic equation. The method employs an orthogonal polynomial collocation technique on multiple subdomains. The subdomain interfaces are approximated with the aid of a penalty method. The time discretization is implemented in an explicit/implicit finite difference method. The subdomain interface is approximated using an explicit Dufort-Frankel method, while the interior of each subdomain is approximated using an implicit backwards Euler's method. The principal advantage to the method is the direct implementation on a distributed computing system with a minimum of interprocessor communication. Theoretical results are given for Legendre polynomials, while computational results are given for Chebyshev polynomials. Results are given for both a single processor computer and a distributed computing system.     

无穷凹角区域椭圆边值问题的重叠型区域分解算法

§1.引言许多科学和工程计算问题都可归结为无界区域上的偏微分方程边值问题,数值求解无界     

Piecewise constant level set method for topology optimization of unilateral contact problems

The paper deals with the structural optimization of the elastic body in unilateral contact with a rigid foundation using the level set approach. A piecewise constant level set method is used to represent the evolution of interfaces rather than the standard method. The piecewise constant level set function takes distinct constant values in each subdomain of a whole design domain. Using a two-phase approximation the original structural optimization problem is reformulated as an equivalent constrained optimization problem in terms of the piecewise constant level set function. Necessary optimality condition is formulated. Finite difference and finite element methods are applied as the approximation methods. Numerical examples are provided and discussed.     

A Non-Overlapping Domain Decomposition Method for the Advection-Diffusion Problem

The application of a non-overlapping domain decomposition method to the solution of a stabilized finite element method for elliptic boundary value problems is considered. We derive an a-posteriori error estimate which bounds the error on the subdomains by the interface error of the subdomain solutions. As a by-product, some foundation is given to the design of the interface transmission condition. Numerical results support the theoretical results. Furthermore, we adapt a recent result on a-posteriori estimates for singular perturbation problems in order to obtain an a-posteriori estimate for the discrete subdomain solutions.     

Exploring or reducing noise?

We consider the problem of the global minimization of a function observed with noise. This problem occurs for example when the objective function is estimated through stochastic simulations. We propose an original method for iteratively partitioning the search domain when this area is a finite union of simplexes. On each subdomain of the partition, we compute an indicator measuring if the subdomain is likely or not to contain a global minimizer. Next areas to be explored are chosen in accordance with this indicator. Confidence sets for minimizers are given. Numerical applications show empirical convergence results, and illustrate the compromise to be made between the global exploration of the search domain and the focalization around potential minimizers of the problem.     

A 3D Crouzeix-Raviart mortar finite element

A mortar finite element discretization of the second order elliptic problem in three dimensions, on non-matching grids, using the 3D Crouzeix-Raviart (CR) finite element in each subdomain, is proposed in this paper. The overall discretization is based on using only the nodal values on the mortar side of a subdomain interface for the calculation of the mortar projection, as opposed to applying the conventional approach where some nodal values in the interior of a subdomain are also required. Since the interior degrees of freedom disappear completely from the computation of the mortar projection, the proposed algorithm becomes less intricate and more flexible as compared to the conventional approach. An error estimate is given, and some numerical experiments are presented.     

A Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary

In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.     

Boundary-Conforming Discontinuous Galerkin Methods via Extensions from Subdomains

A new way of devising numerical methods is introduced whose distinctive feature is the computation of a finite element approximation only in a polyhedral subdomain D{\mathsf{D}} of the original, possibly curved-boundary domain. The technique is applied to a discontinuous Galerkin method for the one-dimensional diffusion-reaction problem. Sharp a priori error estimates are obtained which identify conditions, on the subdomain D{\mathsf{D}} and the discretization parameters of the discontinuous Galerkin method, under which the method maintains its original optimal convergence properties. The error analysis is new even in the case in which D=W{\mathsf{D}}=\Omega . It allows to see that the uniform error at any given interval is bounded by an interpolation error associated to the interval plus a significantly smaller error of a global nature. Numerical results confirming the sharpness of the theoretical results are displayed. Also, preliminary numerical results illustrating the application of the method to two-dimensional second-order elliptic problems are shown.     

A novel method for modeling Neumann and Robin boundary conditions in smoothed particle hydrodynamics

We present a novel smoothed particle hydrodynamics (SPH) method for diffusion equations subject to Neumann and Robin boundary conditions. The Neumann and Robin boundary conditions are common to many physical problems (such as heat/mass transfer), and can prove challenging to implement in numerical methods when the boundary geometry is complex. The new method presented here is based on the approximation of the sharp boundary with a diffuse interface and allows an efficient implementation of the Neumann and Robin boundary conditions in the SPH method. The paper discusses the details of the method and the criteria for the width of the diffuse interface. The method is used to simulate diffusion and reactions in a domain bounded by two concentric circles and reactive flow between two parallel plates and its accuracy is demonstrated through comparison with analytical and finite difference solutions. To further illustrate the capabilities of the model, a reactive flow in a porous medium was simulated and good convergence properties of the model are demonstrated.     

Iterative methods for model reduction by domain decomposition

It is proposed a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including Laplace equation with non-linear boundary conditions and compressible Euler equations.     

A meshless multilayer model for a coastal system by radial basis functions

A multilayer computational model for simulating three-dimensional tidal flows in coastal waters is presented in this paper. A truly meshless numerical scheme based on radial basis functions (RBFs) is employed to obtain an accurate approximation to the solution of the model. For computational efficiency in solving large-scale problems, a noniterative domain decomposition method is combined with the use of the RBFs scheme. To smooth the numerical simulation of the advection across each subdomain, the commonly used upstream technique is also incorporated. Finally, for numerical verification, the proposed multilayer model is successfully used to obtain a stable and convergent simulation of the water flows in the Pearl River Estuary of the South China Sea. The numerical approximations exhibit good agreement with the observed tidal and current data.     

Non-linear explicit transient finite element analysis on the Intel Delta

Many large-scale finite element problems are intractable on current generation production supercomputers. High-performance computer architectures offer effective avenues to bridge the gap between computational needs and the power of computational hardware. The biggest challenge lies in the substitution of the key algorithms in an application program with redesigned algorithms which exploit the new architectures and use better or more appropriate numerical techniques. A methodology for implementing non-linear finite element analysis on a homogeneous distributed processing network is discussed. The method can also be extended to heterogeneous networks comprised of different machine architectures provided that they have a mutual communication interface. This unique feature has greatly facilitated the port of the code to the 8-node Intel Touchstone Gamma and then the 512-node Intel Touchstone Delta. The domain is decomposed serially in a preprocessor. Separate input files are written for each subdomain. These files are read in by local copies of the program executable operating in parallel. Communication between processors is addressed utilizing asynchronous and synchronous message passing. The basic kernel of message passing is the internal force exchange which is analogous to the computed interactions between sections of physical bodies in static stress analysis. Benchmarks for the Intel Delta are presented. Performance exceeding 1 gigaflop was attained. Results for two large-scale finite element meshes are presented.     

Degenerate Two-Phase Incompressible Flow IV: Local Refinement and Domain Decomposition

This is the fourth paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we describe a finite element approximation for this system on locally refined grids. This adaptive approximation is based on a mixed finite element method for the elliptic pressure equation and a Galerkin finite element method for the degenerate parabolic saturation equation. Both discrete stability and sharp a priori error estimates are established for this approximation. Iterative techniques of domain decomposition type for solving it are discussed, and numerical results are presented.     

A monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics

A new hybrid multi-time method for multi-time scales structural dynamics simulations is described. A monolithic method in a Schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain. In the proposed method, zero numerical dissipation is ensured at the interface. This implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface. For that purpose, velocity continuity is ensured in a weak sense at the interfaces, and time integrators (Newmark, HHT, Simo, Krenk, Verlet) are introduced in a unified framework (incremental velocity formulation). Furthermore, dynamics governing equations are introduced from a weak formulation in time. In other words, equilibrium equation is no more ensured in a strong sense at a given time step, but rather on average on a time interval. Some numerical examples illustrate the efficiency and the robustness of the proposed method, for ratio of time scales close to 1000 without any numerical dissipation at the interfaces.     

Aerodynamics of high speed trains passing by each other

A three-dimensional flow field induced by two trains passing by each other inside a tunnel is studied based on the numerical simulation of the three-dimensional compressible Euler/Navier-Stokes equations formulated in the finite difference approximation. A domain decomposition method with the FSA (fortified solution algorithm) interface scheme is used to treat this moving-body problem. The computed results show the basic characteristics of the flow field created when two trains pass by each other. The history of the pressure distributions and the aerodynamic forces acting on the trains are the main areas discussed. The results indicate that the phenomenon is complicated due to the interaction of the flow induced by the two trains. Strong side forces occur between the two trains when the front portion of the opposite train passes by. The forces fluctuate rapidly and the maximum suction force occurs when two trains are aligned side by side. The results also indicate the effectiveness of the present numerical method calculating moving boundary problems.     

Porting an industrial sheet metal forming code to a distributed memory parallel computer

The parallel version of the sheet metal forming semi-implicit finite element code ITAS3D has been developed using the domain decomposition method and direct solution methods at both subdomain and interface levels. IBM Message Passing Library is used for data communication between tasks of the parallel code. Solutions of some sheet metal forming problems on IBM SP2 computer show that the adopted DDM algorithm with the direct solver provides acceptable parallel efficiency using a moderate number of processors. The speedup 6.7 is achieved for the problem with 20000 degrees-of-freedom on the 8-processor configuration.     

Accelerated domain decomposition iterative procedures for mixed methods based on Robin transmission conditions

A nonoverlapping domain decomposition iterative procedure based on Robin transmission conditions applicable to elliptic boundary problems was first introduced by P.~L.~Lions and later discussed by a number of authors under the assumption that the weighting of the flux and the trace of the solution in the Robin interface condition be independent of the iterative step number. Recently, the authors [6] studied a finite difference method for a Dirichlet problem and introduced a cycle of weights for the flux in this interface condition and proved that an acceleration in the convergence rate similar to that occurring for alternating-direction iteration using a cycle of pseudo-time steps results. The objects of this paper are to describe an analogous procedure for a mixed finite element approximation for a model Neumann problem and to consider an overlapping subdomain of the iteration, while retaining the variable parameter cycle. It will be shown that a greater acceleration of the iteration can be obtained by combining overlap and the parameter cycle than by the separate use of either. Received: November 1997 / Revised version: December 1997