Asymptotic–Newton method for solving incompressible flows

In this paper we present a comparative study of three non-linear schemes for solving finite element systems of Navier–Stokes incompressible flows. The first scheme is the classical Newton–Raphson linearization, the second one is the modified Newton–Raphson linearization and the last one is a new scheme called the asymptotic–Newton method. The relative efficiency of these approaches is evaluated over a large number of examples. © 1997 John Wiley & Sons, Ltd.     

非常应变模式数字相关方法的初值估计

本文给出了基于高精度非常应变子区位移模式数字相关方法的Newton-Raphson迭代法求解的新通用公式,对相关迭代算法中的初值估计问题进行了研究,提出两种初值估计方法：（1）利用“实时相减”和“精密调节”相结合的方法而获得零初值;（2）快速迭代初值估计方法,从而有效地解决了Newton-Raphson迭代算法中的初值估计问题,并提高了迭代的收敛速度。     

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.     

多重网格法与Newton-Raphson法耦合求线弹流数值解

利用多重网格算法,在最粗网格上,采用Newton-Raphson迭代法模式精解线接触弹流非线性方程组,充分利用了多重网格法与Newton-Raphson法各自的优点,计算实践表明,求解弹流问题的数值解过程在收敛与稳定性方面均有较大改善,且有相当宽的载荷参数适用范围。     

A discussion and comparison of numerical techniques used to solve the Navier–Stokes and Euler equations

This paper is intended to provide some background to a number of widely used methods for solving the Navier–Stokes and Euler equations. The difference between coupled and uncoupled iterative schemes is discussed together with methods for solving the equations. Methods covered include time marching (both explicit and implicit), pressure correction and a Newton–Raphson technique. The relationship between the methods is illustrated.     

A modified basis reduction method for limited kinematic hardening shakedown analysis under complex loads

A novel extension of the basis reduction method for kinematic hardening shakedown problem is presented. Firstly, the basis reduction method is implemented based on the modified Newton–Raphson (N-R) method. Then a new technique for the construction of back stress field is introduced, where the simultaneous influence of multiple load corners in shakedown is taken into consideration. Finally, two typical numerical examples are investigated. The results compared with previous works in literatures demonstrated that the proposed method is accurate and the performance in reducing of computation time is significant.     

Axisymmetric static and dynamic buckling of hollow microspheres

Syntactic foams are particulate composites that are obtained by dispersing thin hollow inclusions in a matrix material. The wide spectrum of applications of these composites in naval and aerospace structures has fostered a multitude of theoretical, numerical, and experimental studies on the mechanical behavior of syntactic foams and their constituents. In this work, we study static and dynamic axisymmetric buckling of single hollow spherical particles modeled as non-linear thin shells. Specifically, we compare theoretical predictions obtained by using Donnell, Sanders–Koiter, and Teng–Hong non-linear shell theories. The equations of motion of the particle are obtained from Hamilton׳s principle, and the Galerkin method is used to formulate a tractable non-linear system of coupled ordinary differential equations. An iterative solution procedure based on the modified Newton–Raphson method is developed to estimate the critical static load of the microballoon, and alternative methodologies of reduced complexity are further discussed. For dynamic buckling analysis, a Newmark-type integration scheme is integrated with the modified Newton–Raphson method to evaluate the transient response of the shell. Results are specialized to glass particles, and a parametric study is conducted to investigate the effect of microballoon wall thickness on the predictions of the selected non-linear shell theories. Comparison with finite element predictions demonstrates that Sanders–Koiter theory provides accurate estimates of the static critical load for a wide set of particle wall thicknesses. On the other hand, Donnell and Teng–Hong theories should be considered valid only for very thin particles, with the latter theory generally providing better agreement with finite element findings due to its more complete kinematics. In this context, we also demonstrate that a full non-linear analysis is required when considering thicker shells, while simplified treatment can be utilized for thin particles. For dynamic buckling, we confirm the accuracy of Sanders–Koiter theory for all the considered particle thicknesses and of Teng–Hong and Donnell theories for very thin particles.     

Nonlinear thermo-elastic buckling characteristics of cross-ply laminated joined conical–cylindrical shells

Here, the nonlinear thermo-elastic buckling/post-buckling characteristics of laminated circular conical–cylindrical/conical–cylindrical–conical joined shells subjected to uniform temperature rise are studied employing semi-analytical finite element approach. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton–Raphson iteration procedure coupled with displacement control method to trace the pre-buckling/post-buckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, material properties and number of circumferential waves on the nonlinear thermo-elastic response of the different joined shell systems.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

A study of penalty elements for incompressible laminar flows

A finite element model is developed based on the penalty formulation to study incompressible laminar flows. The study includes a number of new quadrilateral and triangular elements for 2-dimensional flows and a number of new hexahedral and tetrahedral elements for 3-dimensional flows. All elements employ continuous velocity approximations and discontinuous pressure approximations respecting the LBB condition of numerical instability. An incremental Newton–Raphson method coupled with the Broyden method is used to solve the non-linear equations. Several numerical examples (colliding flow, cavity flow, etc.) are presented to assess the efficiency of elements.     

SENSITIVITY ANALYSIS AND OPTIMIZATION OF COUPLED THERMAL AND FLOW PROBLEMS WITH APPLICATIONS TO CONTRACTION DESIGN

The finite element method and the Newton–Raphson solution algorithm are combined to solve the momentum, mass and energy conservation equations for coupled flow problems. Design sensitivities for a generalised response function with respect to design parameters which describe shape, material property and load data are evaluated via the direct differentiation method. The efficiently computed sensitivities are verified by comparison with computationally intensive, finite difference sensitivity approximations. The design sensitivities are then used in a numerical optimization algorithm to minimize the pressure drop in flow through contractions. Both laminar and turbulent flows are considered. In the turbulent flow problems the time-averaged momentum and mass conservati on equations are solved using a mixing length turbulence model.     

Implicit finite element methods for two-phase flow in oil reservoirs

The equations governing immiscible, incompressible, two-phase, porous media flow are discretized by generalized streamline diffusion Petrov–Galerkin methods in space and by implicit differences in time. Systems of non-linear algebraic equations are solved by Newton–Raphson iteration employing ILU-preconditioned conjugate-gradient-like methods to the non-symmetric matrix system in each iteration. The resulting solution methods are robust, enable complex grids with irregular nodal orderings and allow capillary effects. Several numerical formulations are tested and compared for one-, two- and three-dimensional flow cases, with emphasis on problems involving saturation shocks, heterogeneous media and curved boundaries. For reservoirs consisting of multiple rock types with differing capillary pressure properties, it is shown that traditional Bubnov-Galerkin methods give poor results and the new Petrov–Galerkin formulations are required. Investigations regarding the behaviour of several preconditioned conjugate-gradient-like methods in these type of problems are also reported.     

Continuation of travelling‐wave solutions of the Navier–Stokes equations

An efficient way of obtaining travelling waves in a periodic fluid system is described and tested. We search for steady states in a reference frame travelling at the wave phase velocity using a first‐order pseudospectral semi‐implicit time scheme adapted to carry out the Newton's iterations. The method is compared to a standard Newton–Raphson solver and is shown to be highly efficient in performing this task, even when high‐resolution grids are used. This method is well suited to three‐dimensional calculations in cylindrical or spherical geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonlinear energy-based relations and numerical procedure for multiaxial local analysis

In this paper, two sets of nonlinear energy-based approximate models are formulated and used to predict the local parameters in a two-phase composite system. A numerical procedure using Newton–Raphson method is developed to solve each system of nonlinear equations in small time steps. The procedure allows the estimation of the local parameters through various loading and unloading steps and therefore, is independent of a yield criterion. The local strain results obtained using the procedure are compared with experimental results obtained at the depth of circumferentially notched particulate metal matrix composite subjected to variable amplitude loads. The numerical results are in good agreement with corresponding experimental results for the geometry and load paths considered.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.     

Aeroelastic analysis of helicopter rotor blade in hover using an efficient reduced-order aerodynamic model

This paper presents a coupled flap–lag–torsion aeroelastic stability analysis and response of a hingeless helicopter blade in the hovering flight condition. The boundary element method based on the wake eigenvalues is used for the prediction of unsteady airloads of the rotor blade. The aeroelastic equations of motion of the rotor blade are derived by Galerkin's method. To obtain the aeroelastic stability and response, the governing nonlinear equations of motion are linearized about the nonlinear steady equilibrium positions using small perturbation theory. The equilibrium deflections are calculated through the iterative Newton–Raphson method. Numerical results comprising steady equilibrium state deflections, aeroelastic eigenvalues and time history response about these states for a two-bladed rotor are presented, and some of them are compared with those obtained from a two-dimensional quasi-steady strip aerodynamic theory. Also, the effect of the number of aerodynamic eigenmodes is investigated. The results show that the three-dimensional aerodynamic formulation has considerable impact on the determination of both the equilibrium condition and lead-lag instability.     

渐开线齿轮传动非牛顿润滑介质的线弹流数值分析研究

采用适合各种流变模型的广义Reynolds方程，通过数值联立求解非牛顿介质的线弹流润滑基本方程组，获得了渐开线齿轮啮合过程的油膜压力、膜厚、表面剪应力分布，并分析了啮合过程中非牛顿效应对齿轮传动最小油膜厚度的影响。在数值计算方向引入延拓方法，使表面煎应力迭代具有大范围收敛性。     

Online determination of natural gas propertiesCaractérisation des propriétés du gaz naturel

Methane number and Lower Heating Value (LHV) of natural gas are determined by the online measurement of thermal conductivity at two different temperatures. Natural gas is first considered as a ternary mixture of the most important components. A pseudo-ternary composition can then be calculated, using the thermal conductivity formula for mixtures derived from kinetic theory. A non-linear system is solved numerically using the Newton–Raphson method. A sensor based on thermal conductivity measurement has been developed and tested successfully for many natural gas compositions. To cite this article: C. Rahmouni et al., C. R. Mecanique 331 (2003).     

Nonlocal continuum model for large deformation analysis of SLGSs using the kp-Ritz element-free method

A geometrically nonlinear large deformation analysis of SLGSs is presented using the element-free kp-Ritz method. Classical plate theory (CLP) is applied to describe the geometrically nonlinear behavior of SLGSs. Nonlocal elasticity theory is incorporated into CLP to take the small-scale effect into consideration. The system nonlinear equations are derived from the Ritz procedure based on the total Lagrangian formulation. The modified Newton–Raphson method and arc-length continuation are employed to solve the nonlinear equations. The efficiency of the element-free kp-Ritz method is verified through comparison with results reported in previous research. Numerical cases are studied to examine the influence of boundary conditions, aspect ratio, side length and nonlocal parameters on the nonlinear large deformation behavior of SLGSs. An interesting phenomenon is observed in that the nonlocal parameter effect is related to the mathematical expression of the transverse load.     

Solution to the Riemann problem for drift-flux model with modified Chaplygin two-phase flows

In this paper, we concern about the Riemann problem for compressible no-slip drift-flux model which represents a system of quasi-linear partial differential equations derived by averaging the mass and momentum conservation laws with modified Chaplygin two-phase flows. We obtain the exact solution of Riemann problem by elaborately analyzing characteristic fields and discuss the elementary waves namely, shock wave, rarefaction wave and contact discontinuity wave. By employing the equality of pressure and velocity across the middle characteristic field, two nonlinear algebraic equations with two unknowns as gas density ahead and behind the middle wave are formed. The Newton–Raphson method of two variables is applied to find the unknowns with a series of initial data from the literature. Finally, the exact solution for the physical quantities such as gas density, liquid density, velocity, and pressure are illustrated graphically.