On the Numerical Solution of Diffraction Problem by Random Spheres Using Electric Field Integral Equation

The aim of this paper is to solve the two-dimensional acoustic scattering problems by random sphere using Electric field integral equation. Some approximations for the two-dimensional case are derived. These various approximations are next numerically validated in the case of high-frequency.

Linear System Solutions of the Navier-Stokes Equations with Application to Flow over a Backward-Facing Step

Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.

Efficient solver for time-dependent Schrodinger equation with interaction between atoms and strong laser field

We present a parallel numerical method of simulating the interaction of atoms with a strong laser field by solving the time-depending Schr?dinger equation(TDSE) in spherical coordinates. This method is realized by combining constructing block diagonal matrices through using the real space product formula(RSPF) with splitting out diagonal sub-matrices for short iterative Lanczos(SIL) propagator. The numerical implementation of the solver guarantees efficient parallel computing for the simulation of real physical problems such as high harmonic generation(HHG) in these interaction systems.

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.

Neural network as a function approximator and its application in solving differential equations

A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).

A Finite Element Approximation of the Stokes Equations

In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q1 – P0 velocity/pressure ?nite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O( h2) in three dimensions. The basic tool in the analysis is the method of modi?ed equations which is applied to ?nite difference representations of the underlying ?nite element equations. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

Operator Splitting Method for Coupled Problems:Transport and Maxwell Equations

In this article a new approach is considered for implementing operator splitting methods for transport problems, influenced by electric fields. Our motivation came to model PE-CVD (plasma-enhanced chemical vapor deposition) processes, means the flow of species to a gas-phase, which are influenced by an electric field. Such a field we can model by wave equations. The main contributions are to improve the standard discretization schemes of each part of the coupling equation. So we discuss an improvement with implicit Runge- Kutta methods instead of the Yee’s algorithm. Further we balance the solver method between the Maxwell and Transport equation.

A Comparison of Techniques for Solving the Poisson Equation in CFD

CFD is a ubiquitous technique central to much of computational simulation such as that required by aircraft design.Solving of the Poisson equation occurs frequently in CFD and there are a number of possible approaches one may leverage.The dynamical core of the MONC atmospheric model is one example of CFD which requires the solving of the Poisson equation to determine pressure terms.Traditionally this aspect of the model has been very time consuming and so it is important to consider how we might reduce the runtime cost.In this paper we survey the different approaches implemented in MONC to perform the pressure solve.Designed to take advantage of large scale,modern,HPC machines,we are concerned with the computation and communication behaviour of the available techniques and in this text we focus on direct FFT and indirect iterative methods.In addition to describing the implementation of these techniques we illustrate on up to 32768 processor cores of a Cray XC30 both the performance and scalability of our approaches.Raw runtime is not the only measure so we also make some comments around the stability and accuracy of solution.The result of this work are a number of techniques,optimised for large scale HPC systems,and an understanding of which is most appropriate in different situations.

基于FluxNet的热传导方程反问题的求解器

热传导偏微分方程的求解是工业应用中一种重要的计算,为了解决传统的热传导方程正演运算耗时过长的问题,提出一个基于新型的网络结构FluxNet的求解器。通过热传导方程的正演运算获得关于温度场分布图以及对应的热流密度图的数据集,基于深度学习训练该数据集,建立一个具有卷积层和反卷积层的热流密度网络结构FluxNet模型。使用该求解器预测数据以及实际数据的热流密度图进行测试和验证,实验结果得出该求解器预测的热流密度结构相似度均达到90%以上,达到了工业应用需求。

基于EES的太阳能光热发电系统性能模拟与优化

随着全球能源结构的持续转型和对可持续能源技术的不断追求,太阳能光热发电作为一种高效利用太阳能的方式受到了广泛关注。基于工程方程求解软件(Engineering Equation Solver,EES),构建高契合度的太阳能光热发电系统数学模型,对系统中的热力过程、能量转换效率以及关键影响因素进行模拟分析,进而提出一系列具有针对性的优化措施。实践证明,基于EES的太阳能光热发电系统的性能模拟与优化有助于提高系统光热发电效率,提升系统的经济性和环境友好性,降低能源损耗,具有广阔的应用前景。

Lattice Boltzmann Flux Solver:An Efficient Approach for Numerical Simulation of Fluid Flows

A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.

OPTIMAL SOLVER FOR MORLEY ELEMENT DISCRETIZATION OF BIHARMONIC EQUATION ON SHAPE-REGULAR GRIDS

This paper presents an optimal solver for the Morley element problem for the boundaryvalue problem of the biharmonic equation by decomposing it into several subproblems and solving these subproblems optimally. The optimality of the proposed method is mathematically proved for general shape-regular grids.

A direct solver with O（N） complexity for integral equations on one-dimensional domains

An algorithm for the direct inversion of the linear systems arising from NystrSm discretization of integral equations on one-dimensional domains is described. The method typically has O（N） complexity when applied to boundary integral equations （BIEs） in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes＇ equations. The scaling coefficient suppressed by the ＂big-O＂ notation depends logarithraically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank： deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H- and H2-matrix arithmetic of Hackbusch and coworkers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

ChemNODE: A neural ordinary differential equations framework for efficient chemical kinetic solvers

Solving for detailed chemical kinetics remains one of the major bottlenecks for computational fluid dynamics simulations of reacting flows using a finite-rate-chemistry approach.This has motivated the use of neural networks to predict stiff chemical source terms as functions of the thermochemical state of the combustion system.However,due to the nonlinearities and multi-scale nature of combustion,the predicted solution often diverges from the true solution when these machine learning models are coupled with a computational fluid dynamics solver.This is because these approaches minimize the error during training without guaranteeing successful integration with ordinary differential equation solvers.In the present work,a novel neural ordinary differential equations approach to modeling chemical kinetics,termed as ChemNODE,is developed.In this machine learning framework,the chemical source terms predicted by the neural networks are integrated during training,and by computing the required derivatives,the neural network weights are adjusted accordingly to minimize the difference between the predicted and ground-truth solution.A proof-of-concept study is performed with ChemNODE for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions.It is shown that ChemNODE accurately captures the chemical kinetic behavior and reproduces the results obtained using the detailed kinetic mechanism at a fraction of the computational cost.

A New Directional Algebraic Fast Multipole Method Based Iterative Solver for the Lippmann-Schwinger Equation Accelerated with HODLR Preconditioner

We present a fast iterative solver for scattering problems in 2D,where a penetrable object with compact support is considered.By representing the scattered field as a volume potential in terms of the Green’s function,we arrive at the Lippmann-Schwinger equation in integral form,which is then discretized using an appropriate quadrature technique.The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method(DAFMM).The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel[1],and the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation(NCA)[2].The advantage of the NCA described in[2]is that the search space of so-called far-field pivots is smaller than that of the existing NCAs[3,4].Another significant contribution of this work is the use of HODLR based direct solver[5]as a preconditioner to further accelerate the iterative solver.In one of our numerical experiments,the iterative solver does not converge without a preconditioner.We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not.Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver,DAFMMbased fast iterative solver,and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem.To the best of our knowledge,this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions.In the spirit of reproducible computational science,the implementation of the algorithms developed in this article is made available at https://github.com/vaishna77/Lippmann_Schwinger_Solver.

DSMC: Fast direct simulation Monte Carlo solver for the Boltzmann equation by Multi-Chain Markov Chain and multicore programming

Direct Simulation Monte Carlo(DSMC)solves the Boltzmann equation with large Knudsen number.The Boltzmann equation generally consists of three terms:the force term,the diffusion term and the collision term.While the first two terms of the Boltzmann equation can be discretized by numerical methods such as the finite volume method,the third term can be approximated by DSMC,and DSMC simulates the physical behaviors of gas molecules.However,because of the low sampling efficiency of Monte Carlo Simulation in DSMC,this part usually occupies large portion of computational costs to solve the Boltzmann equation.In this paper,by Markov Chain Monte Carlo(MCMC)and multicore programming,we develop Direct Simulation Multi-Chain Markov Chain Monte Carlo(DSMC3):a fast solver to calculate the numerical solution for the Boltzmann equation.Computational results show that DSMC3 is significantly faster than the conventional method DSMC.

含内源污染平面二维水流-水质耦合模型

本文从紊动扩散理论出发根据浅水湖泊紊流扰动特性,推导出包含底泥污染的水质控制方程,并与浅水方程耦合,建立相应的平面二维水流-水质-底泥污染的数值模型。模型在有限体积法框架下应用黎曼近似解求得耦合方程的数值解。并以玄武湖为例进行模型验证。结果表明,数值解与实测值较吻合,证明预测结果可信。

全过程动态仿真中大型线性方程组的分块求解算法

电力系统全过程动态仿真能够将机电暂态、中期和长期动态过程有机地统一起来进行数字仿真,仿真过程中需要多次求解大型稀疏线性方程组。该方程组由电力系统设备模型的微分—代数方程式差分后的代数方程和输电网络模型的代数方程形成,其快速求解算法是电力系统全过程动态仿真的难点之一。文中提出一种利用仿真中矩阵结构特点的分块快速直接求解算法,并开发实现了大型电力系统线性方程组稀疏求解器(ESS)。该算法首先将稀疏矩阵分为4个分块矩阵,然后将其中规模最大的对角块进一步细分为多个更小的对角分块矩阵,并利用部分小分块具有相同结构的特点进行矩阵LU符号分解和数值分解,最后根据分块矩阵进行前代和回代求解计算。与现有其他求解器进行的算例对比表明,ESS具有较为明显的整体求解速度优势,特别是在矩阵LU分解方面。

常微分方程特征值问题求解器解法的改进

对于工程中的常微分方程(ODE)特征值问题,已有一套完整的算法,并据此算法开发出ODE特征值求解程序COLEGN。该程序需要用户输入直接求解(非线性)和逆幂迭代(线性)两套ODE体系,输入繁琐,不便于使用。针对这一问题,研究了这两套ODE体系间的内部联系,建立了从非线性ODE体系获取线性ODE体系的具体途径,并据此改写了COLEGN程序,简化了用户的输入,使之更易于使用,另外,还对边界条件含特征值的情况作了相应处理,拓宽了算法的适用范围。

二维三温能量方程组的高效代数解法

针对二维三温能量方程九点格式离散后形成的非线性方程组,研制了高效求解的代数解法器.主要思想是在部分Newton-Krylov(PNK)方法和Jacobi矩阵自由的Newton-Krylov(JFNK)方法的框架下,结合非精确Newton类方法和预条件Krylov子空间方法进行高效求解.数值结果显示,PNK方法比非线性块Gauss-Seidel方法快6倍以上,在PNK框架下比较了3种预条件子和4种Krylov子空间方法,得出不同组合的最佳方案.还比较了JFNK方法和PNK方法.