基于牛顿-拉夫逊的拉压不同模量问题的数值求解

针对拉压模量不同引起的材料本构非线性,首先,通过引入改进的Heaviside函数将本构方程连续光滑化;然后,基于特征值与特征向量的求导策略,推导有限元求解模型中切线刚度矩阵的列式;最终,提出基于牛顿-拉夫逊迭代格式的拉压不同模量问题的数值求解算法。数值算例验证了本文算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于工程分析中大规模计算问题的求解。     

基于牛顿-拉夫逊的拉压不同模量问题的数值求解

针对拉压模量不同引起的材料本构非线性,首先,通过引入改进的Heaviside函数将本构方程连续光滑化;然后,基于特征值与特征向量的求导策略,推导有限元求解模型中切线刚度矩阵的列式;最终,提出基于牛顿-拉夫逊迭代格式的拉压不同模量问题的数值求解算法。数值算例验证了本文算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于工程分析中大规模计算问题的求解。     

求解非线性方程组的混沌分形方法

混沌分形是动力系统普遍出现的一种现象,牛顿-拉夫森NR(Newton-Raphson)方法是重要的一维及多维迭代技术,其迭代本身对初始点非常敏感,该敏感区是牛顿-拉夫森法所构成的非线性离散动力系统Julia集,在Julia集中迭代函数会呈现出混沌分形现象,提出了一种寻找牛顿-拉夫森函数的Julia点的求解方法,利用非线性离散动力系统在其Julia集出现混沌分形现象的特点,提出了一种基于牛顿-拉夫森法的非线性方程组求解的新方法,计算实例表明了该方法的有效性和正确性.     

索杆体系的机构运动及其与弹性变形的混合问题

基于有限单元平衡方程推导了索杆体系机构运动的通解、特解及其路径跟踪的迭代求解列式，以及机构运动与弹性变形混合问题的迭代求解列式。从机构运动和弹性变形问题的比较角度出发，解释了机构运动中的若干基本定义，对索杆体系的机构运动和弹性变形混合问题作了分类。数值算例表明：采用本文方法可以非常有效地跟踪求解索杆体系机构运动、弹性变形及其混合问题的全过程路径，得到机构运动的静定状态和混合问题的平衡状态。     

成形过程数值模拟的张量时空求解策略

成形过程的数值模拟和分析涉及几何、材料、接触摩擦等高度非线性的耦合作用，采用传统的增量算法会导致巨大计算量。本文建议采用张量时间函数的非增量时空算法，在整个时间和空间域上迭代求解。由于采用新的分离变量构思，以张量表达时间函数，可提高问题求解的速度和精度。但对问题的列式和数值求解方法提出了更高的要求。文中讨论其方法与数值实施。     

基于能量优化的桁架几何非线性问题力密度法

力密度法最初是求解膜结构找形问题的方法,经发展可用于计算桁架结构的几何非线性问题。本文应用力密度法建立结构变形后的非线性平衡方程及相应的雅可比矩阵,用于迭代求解;从能量原理出发,推导出杆单元应变能、外荷载势能、结构总势能在每次迭代位移方向上关于步长λ的显式列式。相对于固定步长的牛顿法,本文将最优迭代步长λ引入求解,使结构在每次迭代位移方向上均达到总势能最小。经桁架算例验证,表明该方法可加快计算收敛进程。     

非线性复合材料杂交应力有限元的有效迭代方法

推导了面内剪应力应变关系非线性的复合材料的杂交应力有限元列式,给出了位移迭代和应力迭代的策略和步骤．提出一种非线性应力场迭代格式的改进方案,不仅提高了收敛速度,而且克服了大载荷下简单迭代法循环迭代而无法收敛的关键问题,使得所提出的非线性杂交应力元方法几乎对任意大载荷都能够收敛．数值算例表明该方法是确实可行的．     

化可分离凸规划为对偶规划显式模型的普适解法研究比较

文章旨在提升对偶规划显式模型(dual programming-explicit model, DP-EM)的建模和求解的境界. DPEM模型从一类变量可分离凸规划的特点出发,突破了对偶目标二阶采用近似的定势,推导得出显式的对偶目标函数;应用于ICM方法求解连续体结构拓扑优化问题时,其求解效率比对偶序列二次规划方法 (DSQP)和可移动渐近线方法 (MMA)求解效率更高.文章进一步把常见的一类显式模型抽象为普适的可分离凸规划列式,在需要满足的一些条件下,转换为DP-EM模型,并且提出4种处理方法:(1)对偶变量迭代逼近法;(2)指数函数形式的解法;(3)幂函数形式的解法;(4)基于变换的精确解法.为了进行数值验证,做了广泛的计算,限于篇幅,文章列出了5个具有代表性的算例,除了算例1属于纯数学问题,其余4个算例皆基于ICM方法,分别对于位移、应力、疲劳等约束和破损-安全的连续体结构拓扑优化问题,基于所提出的方法进行建模和求解,都显示了所提出方法的普适性及更高的求解效率.工作的意义在于:(1)深度方面,加深了结构优化对偶解法的研究;(2)广度方面,对数学规划对偶理论的发展做出了新的贡献.     

岩体渗透结构类型的划分及其渗透特性研究

提出了一种计算二维有限变形弹塑性摩擦接触问题形状设计灵敏度的算法. 采用主动 集策略和mortar方法处理接触边线上的约束条件. 在mortar接触边线的切线和法线方向上 采用相同的名义罚函数,提出基于名义罚函数的移动摩擦锥算法来正则化接触约束条件,发 展了一种新的二维多体有限变形摩擦接触算法. 在此基础上, 通过将离散形式的摩擦接触问题 控制方程对形状设计变量微分,得到了该路径相关问题的直接微分法解析设计灵敏度 计算格式, 其节点位移灵敏度方程在每个增量步不用迭代、直接求解. 与国际上现有 的二维多体有限变形摩擦接触问题的解析设计灵敏度算法相比,本算法不需分 解为法向和切向推导,表达式较简洁,便于编程实现. 数值算例验证了算法的精度 和有效性.     

IMPOSING DISPLACEMENT BOUNDARY CONDITIONS WITH NITSCHE＇S METHOD IN ISOGEOMETRIC ANALYSIS

等几何分析使用 NURBS 基函数统一表示几何和分析模型, 消除了传统有限元的网格离散误差, 容易构造高阶连续的协调单元. 对于结构分析, 选择合适的几何参数可以得到光滑的应力解, 避免了后置处理的应力磨平. 但是由于 NURBS 基函数不具备插值性, 难以直接施加位移边界条件. 针对这一问题, 提出一种基于 Nitsche 变分原理的边界位移条件“弱”处理方法, 它具有一致稳定的弱形式, 不增加自由度, 方程组对称正定和不会产生病态矩阵等优点. 同时给出方法的稳定性条件, 并通过求解广义特征值问题计算稳定性系数. 最后, 数值算例表明 Nitsche 方法在h细化策略下能获得最优收敛率, 其结果要明显优于在控制顶点处直接施加位移约束.}     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

A problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second‐order finite volume method. The study is motivated by further applications of the finite volume‐based stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low‐order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov‐subspace‐based iteration techniques. The same replacement in the Newton steady‐state solver also yields a robust numerical process, however, it cannot be as effective as modern preconditioned Krylov‐subspace‐based iterative solvers. Copyright © 2006 John Wiley & Sons, Ltd.     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

Newton's method for solving heat transfer,fluid flow and interface shapes in a floating molten zone

Newton's method is applied to the finite volume approximation for the steady state heat transfer, fluid flow and unknown interfaces in a floating molten zone. The streamfunction/vorticity and temperature formulation of the Navier–Stokes and energy equations and their associated boundary conditions are written in generalized curvilinear co-ordinates and conservative law form with the Boussinesq approximation. During Newton iteration the ILU(0) preconditioned GMRES matrix solver is applied for solving the linear system, where the sparse Jacobian matrix is estimated by finite differences. Nearly quadratic convergence of the method is observed. Sample calculations are reported for sodium nitrate, a high-Prandtl-number material (Pr = 9.12). Both natural convection and thermocapillary flow as well as an overall mass balance constraint in the molten zone are considered. The effects of convection and heat input on the flow patterns, zone position and interface shapes are illustrated. After the lens effect due to the molten zone is considered, the calculated flow patterns and interface shapes are compared with the observed ones and are found to be in good agreement.     

A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two‐dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi‐discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo‐time step requires the solution of a sparse linear system for the flow variables. In this study, a non‐overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson‐type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd.     

Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

This work presents an approximate Riemann solver to the transient isothermal drift ‐ flux model. The set of equations constitutes a non‐linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A . The proposed solver is validated against three benchmark solutions and two experimental pipe flow data. Copyright © 2015 John Wiley & Sons, Ltd.     

Improving the convergence behaviour of a fixed‐point‐iteration solver for multiphase flow in porous media

A new method to admit large Courant numbers in the numerical simulation of multiphase flow is presented. The governing equations are discretized in time using an adaptive θ‐method. However, the use of implicit discretizations does not guarantee convergence of the nonlinear solver for large Courant numbers. In this work, a double‐fixed point iteration method with backtracking is presented, which improves both convergence and convergence rate. Moreover, acceleration techniques are presented to yield a more robust nonlinear solver with increased effective convergence rate. The new method reduces the computational effort by strengthening the coupling between saturation and velocity, obtaining an efficient backtracking parameter, using a modified version of Anderson's acceleration and adding vanishing artificial diffusion. © 2016 The Authors. International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd.     

A Jacobian‐free Newton–Krylov method for thermalhydraulics simulations

The current paper is focused on investigating a Jacobian‐free Newton–Krylov (JFNK) method to obtain a fully implicit solution for two‐phase flows. In the JFNK formulation, the Jacobian matrix is not directly evaluated, potentially leading to major computational savings compared with a simple Newton's solver. The objectives of the present paper are as follows: (i) application of the JFNK method to two‐fluid models; (ii) investigation of the advantages and disadvantages of the fully implicit JFNK method compared with commonly used explicit formulations and implicit Newton–Krylov calculations using the determination of the Jacobian matrix; and (iii) comparison of the numerical predictions with those obtained by the Canadian Algorithm for Thermaulhydraulics Network Analysis 4. Two well‐known benchmarks are considered, the water faucet and the oscillating manometer. An isentropic two‐fluid model is selected. Time discretization is performed using a backward Euler scheme. A Crank–Nicolson scheme is also implemented to check the effect of temporal discretization on the predictions. Advection Upstream Splitting Method+ is applied to the convective fluxes. The source terms are discretized using a central differencing scheme. One explicit and two implicit formulations, one with Newton's solver with the Jacobian matrix and one with JFNK, are implemented. A detailed grid and model parameter sensitivity analysis is performed. For both cases, the JFNK predictions are in good agreement with the analytical solutions and explicit profiles. Further, stable results can be achieved using high CFL numbers up to 200 with a suitable choice of JFNK parameters. The computational time is significantly reduced by JFNK compared with the calculations requiring the determination of the Jacobian matrix. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of the Jacobian evaluation on Newton's solution of the Euler equations

Newton's method is developed for solving the 2‐D Euler equations. The Euler equations are discretized using a finite‐volume method with upwind flux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical differentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the finite‐difference perturbation magnitude and computer precision are the most important parameters that affect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved significantly by using the optimal perturbation magnitude. The effects of the accuracy of numerical Jacobians on the convergence of the flow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, flux vectors with perturbed flow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. Effects of different flux splitting methods and higher‐order discretizations on the performance of the solver are analysed. Copyright © 2005 John Wiley & Sons, Ltd.