Implicit-explicit schemes for nonlinear consolidation

A general analytical procedure capable of performing linear and nonlinear consolidation analysis of saturated porous media is proposed. A brief review of the coupled field equations is included and the constitutive assumptions are stated explicitly. Time integration of the resulting nonlinear semidiscrete finite element equations is performed by using an implicit/explicit predictor/multicorrector scheme developed by Hughes and co-workers. It is shown that the algorithm can be simply and concisely implemented. The technique allows for a convenient selection of implicit and explicit elements, and for a convenient implicit-explicit split of the various operators appearing in the equations. The procedure proves to be extremely effective in dealing with consolidation problems. Numerical results which demonstrate the versatility and accuracy of the proposed procedures are presented.     

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.     

A mixed time integration scheme for virtual fabrication of steel plate girders

In this paper the use of mixed time integration is proposed to increase the efficiency of welding simulation. A technique similar to the one commonly used in sheet metal forming is applied to the welding of steel plates. The welding procedure can be divided into two very distinguishable parts with significantly different characteristics. The first part, the phase of the actual welding, is a fast paced, rapidly changing process that involves highly nonlinear material behavior. The second part, the phase of cooling down, is a slow paced, slowly changing process that does not result in dramatic changes in the material behavior. In this project, explicit time integration was used for the thermo-mechanical analysis of welding and implicit time integration was proposed for the subsequent cooling phase. This paper presents examples based on the conventional fully implicit solution and the proposed explicit integration solution using the finite element codes ABAQUS/Standard and ABAQUS/Explicit, respectively.     

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. $L^2$-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.     

Parallel hybrid particle/finite volume algorithm for transported PDF methods employing sub-time stepping

A previously presented hybrid finite volume/particle method for the solution of the joint-velocity-frequency-composition probability density function (JPDF) transport equation in complex 3D geometries is extended for parallel computing. The parallelization strategy is based on domain decomposition. The finite volume method (FVM) and the particle method (PM) are parallelized separately and the algorithm is fully synchronous. For the FVM a standard method based on transferring data in ghost cells is used. Moreover, a subdomain interior decomposition algorithm to efficiently solve the implicit time integration for hyperbolic systems is described. The parallelization of the PM is more complicated due to the use of a sub-time stepping algorithm for the particle trajectory integration. Hereby, each particle obeys its local CFL criterion, and the covered distances per global time step can vary significantly. Therefore, an efficient algorithm which deals with this issue and has minimum communication effort was devised and implemented. Numerical tests to validate the parallel vs. the serial algorithm are presented, where also the effectiveness of the subdomain interior decomposition for the implicit time integration was investigated. A 3D dump-combustor configuration test case with about 2.5 × 10⁵ cells was used to demonstrate the good performance of the parallel algorithm. The hybrid algorithm scales well and the maximum speedup on 60 processors for this configuration was 50 (≈80% parallel efficiency).     

Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices

Several finite difference schemes are discussed for solving the two-dimensional Schrodinger equation with Dirichlet’s boundary conditions. We use three fully implicit finite difference schemes, two fully explicit finite difference techniques, an alternating direction implicit procedure and the Barakat and Clark type explicit formula. Theoretical and numerical comparisons between four families of methods are described. The main advantage of the alternating direction implicit finite difference technique is that the bandwidth of the sets of equations is a fixed small number that depends only on the nature of the computational molecule. This allows the use of very efficient and very fast techniques for solving the resulting tridiagonal systems of linear algebraic equations. The unique advantage of the Barakat and Clark technique is that it is unconditionally stable and is explicit in nature. Numerical results are presented followed by concluding remarks.     

Efficient removal of boundary-divergence errors in time-splitting methods

A normal mode analysis is presented and numerical tests are performed to assess the effectiveness of a new time-splitting algorithm proposed recently in Karniadakiset al. (1990) for solving the incompressible Navier-Stokes equations. This new algorithm employs high-order explicit pressure boundary conditions and mixed explicit/implicit stiffly stable time-integration schemes, which can lead to arbitrarily high-order accuracy in time. In the current article we investigate both the time accuracy of the new scheme as well as the corresponding reduction in boundary-divergence errors for two model flow problems involving solid boundaries. The main finding is that time discretization errors, induced by the nondivergent splitting mode, scale with the order of the accuracy of the integration rule employed if a proper rotational form of the pressure boundary condition is used; otherwise a first-order accuracy in time similar to the classical splitting methods is achieved. In the former case the corresponding errors in divergence can be completely eliminated, while in the latter case they scale asO(vt)_1/2.     

A fast high-order sinc-based algorithm for pricing options under jump-diffusion processes

An implicit–explicit Euler scheme in temporal direction is employed to discretize a partial integro-differential equation, which arises in pricing options under jump-diffusion process. Then the semi-discretized equation is approximated in space by the Sinc–Galerkin method with exponential accuracy. Meanwhile, the domain decomposition method is incorporated to handle the non-smoothness of the payoff function, and the improved fast Gauss transform is applied to accelerate the evaluation of the jump integral term. An effective preconditioner is proposed for solving the resulting dense Toeplitz-related systems by the preconditioned generalized minimum residual (GMRES) method. Numerical tests are performed to illustrate the efficiency of the proposed algorithm.     

A distributed memory parallel algorithm for the efficient computation of sensitivities of differential-algebraic systems

An efficient algorithm for the evaluation of the parametric sensitivities for mixed systems of differential and algebraic equations (DAEs) on computers involving multiple processors operating in parallel is presented. The algorithm derives its efficiency by decoupling the integration of the sensitivity equations from that of the original DAE system, and by allowing tasks associated with the evaluation of sensitivities at multiple time points to overlap instead of being carried out in sequence. Numerical experiments demonstrating the efficiency of the proposed algorithm with systems of more than 850 DAEs and 45 parameters are presented. In all the cases studied, the simultaneous integration of the original DAEs and their sensitivity equations is carried out in less than 10% more time than that of the original DAEs alone.     

An asynchronous parallel mixed algorithm for linear and nonlinear equations

In this paper we propose an asynchronous parallel mixed algorithm for solving linear and nonlinear equations. This algorithm can be used not only on serial and parallel computers, but also on MIMD multiprocessor systems. The convergence of the algorithm has been proved under certain conditions. This paper gives some special cases of the algorithm which are known to us as efficient iterative methods. Numerical experiments are given to illustrate the method.     

A parallel algorithm for global magnetic reconnection studies

A new algorithm is presented for the computation of two-dimensional magnetic reconnection in plasmas. Both resistive magnetohydrodynamic (MHD) and two-fluid models are considered. It has been implemented on several parallel platforms and shows good scalability up to 32 CPUs for reasonable problem sizes. A fixed, non-uniform rectangular mesh is used to resolve the different spatial scales in the reconnection problem. The resistive MHD version uses an implicit/explicit hybrid method, while the two-fluid version uses an alternating-direction implicit (ADI) method with high-order artificial dissipation. The technique has proven useful for comparing several different theories of collisional and collisionless reconnection.     

基于有效并行求解策略的显式有限元分析并行算法

针对大规模结构非线性动力问题的有限元分析非常耗时,基于消息传递接口（MPI）机群环境,提出多种基于并行求解策略的显式有限元并行算法。基于显式消息传递的区域分解技术,采取重叠、非重叠区域分解技术及动态任务分配方法,通过将计算与通信重叠,优化处理器间的通信,对非重叠通信区域分解并行算法、重叠通信区域分解并行算法、群动态任务分配算法、动态任务分配算法及动态负载平衡算法进行研究。为在机群环境下实现非线性动力有限元分析,开发了基于有效并行求解策略的显式有限元并行算法。编写了基于消息传递编程模式的并行有限元程序,在工作站机群上实现了数值算例,分析了算法的性能,并与传统的Newmark算法进行了比较。算例表明：群动态任务分配算法的性能优于动态任务分配算法,低于区域分解算法的性能,动态负载平衡算法最优。对相同规模的问题提出的算法比Newmark算法快,优于Newmark算法。对结构非线性动力问题的有限元分析,所提出的并行算法是可行有效的。     

A time-discontinuous implicit variational integrator for stress wave propagation analysis in solids

This paper proposes a time-discontinuous and space-continuous variational integration (TDSC-VI) method for accurate elastic stress wave propagation computations in one-dimensional bar. The algorithm employs a limiter, akin to classical artificial viscosity treatment, to mitigate the deleterious Gibbs jumps across the stress discontinuities, and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the Courant stability limit, which becomes necessary for elastic unloading and internal reflections in plastic deformation problems. Stability and accuracy analyses of the proposed TDSC-VI method are carried out and compared with several well-known traditional integration algorithms. Numerical experiments are carried out with the proposed implicit and explicit methods, which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method.     

Explicit nonlinear dynamic finite element analysis on homogeneous/heterogeneous parallel computing environment

This paper presents parallel computational strategies to implement explicit nonlinear finite element analysis code onto distributed memory parallel computers for solving large-scale problems in structural dynamics. Implementation details on both homogeneous and heterogeneous parallel processing environments are considered in detail in this paper. Implementation of an explicit nonlinear finite element dynamic analysis code on homogeneous systems is discussed first and this is later moved onto heterogeneous systems. Domain decomposition with explicit message passing is preferred for parallel implementation. The message passing implementation in the parallel algorithm is based on MPI (Message Passing Interface) libraries. Implementation aspects of overlapped, non-overlapped domain decomposition techniques, Dynamic Task Allocation (DTA) and clustering techniques for DTA and their relative merits are presented. The interprocessor communications are optimised by overlapping with computations to improve the performance of the domain decomposition based explicit dynamic analysis finite element code.The issues related to implementation of finite element code for nonlinear dynamic analysis on heterogeneous parallel computing environment are later presented. A new dynamic load-balancing algorithm is developed for this purpose and it is integrated with the domain decomposition based parallel explicit finite element code to test our algorithms on a coarse grain heterogeneous cluster of workstations. Numerical experiments have been carried out on PARAM-10000, an Indian parallel computer and also on cluster of Unix workstations.     

面向半物理仿真的陈述式模型求解方法研究

通过符号操作和数值计算相结合,提出了一种求解半物理仿真模型的新方法。为了满足半物理仿真对实时性的要求,在模型编译阶段将代表数值积分的隐式离散公式插入到仿真模型中,增广后的方程系统伴随着非线性方程的出现,需要在积分的每一步对这些非线性方程进行迭代求解,而求解非线性方程的时间复杂度随维度的变大成指数增加,因此引入代数环撕裂减小代数方程块耦合变量数,以满足实时求解对粒度的要求。最后通过实例对文中提出的方法进行了验证。     

Hardware Assistance for Z-Buffer Visible Surface Algorithms

The well-known z-buffer algorithm for solving the visible surface problem has a number of points in its favor, the main one being that it can be very efficiently implemented at little additional cost in many existing frame-buffer systems. The traditional software implementation of the algorithm assumes explicit initialization of both the image buffer and the z-buffer before each image is generated. We will describe a simple technique for synchronizing initialization and image generation so the two can be performed in parallel, effectively elimination the time needed for explicit initialization of the frame buffer. The technique assumes a modest investment in additional hardware within the frame buffer.     

An Implicit Runge–Kutta Method for Integration of Differential Algebraic Equations of Multibody Dynamics

When performing dynamic analysis of a constrained mechanical system, aset of index 3 Differential Algebraic Equations (DAE) describes the timeevolution of the model. This paper presents a state space DAE solutionframework that can embed an arbitrary implicit Ordinary DifferentialEquations (ODE) code for numerical integration of a reduced set of statespace ordinary differential equations. This solution framework isconstructed with the goal of leveraging with minimal effort establishedoff the shelf implicit ODE integrators for efficiently solving the DAEof multibody dynamics. This concept is demonstrated by embedding awell-known public domain singly diagonal implicit Runge–Kutta code inthe framework provided. The resulting L-stable, stiffly accurateimplicit algorithm is shown to be two orders of magnitude faster than astate of the art explicit algorithm when used to simulate a stiffvehicle model.     

无条件稳定的显式降维非线性有限元

针对传统降维非线性有限元计算速度与精确度难以兼顾的问题,提出了一种无条件稳定的显式迭代算法。基于泰勒展开式得到速度、加速度的三阶精度差分表达式从而获得新的有限元显式迭代方程,并分析其单自由度系统下的传递矩阵谱半径。改进迭代方程使谱半径始终小于1从而满足无条件稳定的要求。实验表明,改进后的显式迭代算法在等效阻尼比的精度上优于中心差分法和隐式迭代法;在降维非线性有限元模型计算中的计算耗时优于隐式迭代方法,提高了降维非线性有限元的迭代计算速度。模型在降维后维度数值较高时,仍能维持良好的计算耗时和帧率,保证了模型的精确度。     

数值积分的一种计算方法研究

科学与工程领域经常使用数值积分，为此提出了一种求解数值积分的新方法。其主要思想是通过训练神经网络权值 并用傅立叶级数 来近似未知函数 ，然后用 来近似积分 。提出并证明了神经网络算法的收敛性定理和数值积分的求解定理。数值积分算例验证了本文算法的有效性。研究结果表明，本文提出的数值积分方法有高的计算精度，在工程实际中有较大的应用价值。     

Partitioned and Implicit–Explicit General Linear Methods for Ordinary Differential Equations

Implicit–explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge–Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit–explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.