Global convergence of inexact Newton methods for transonic flow

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.     

A matrix‐free preconditioned Newton/GMRES method for unsteady Navier–Stokes solutions

The unsteady compressible Reynolds‐averaged Navier–Stokes equations are discretized using the Osher approximate Riemann solver with fully implicit time stepping. The resulting non‐linear system at each time step is solved iteratively using a Newton/GMRES method. In the solution process, the Jacobian matrix–vector products are replaced by directional derivatives so that the evaluation and storage of the Jacobian matrix is removed from the procedure. An effective matrix‐free preconditioner is proposed to fully avoid matrix storage. Convergence rates, computational costs and computer memory requirements of the present method are compared with those of a matrix Newton/GMRES method, a four stage Runge–Kutta explicit method, and an approximate factorization sub‐iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and the efficiency of the Newton iteration for the non‐linear system at each time step are analysed for both matrix‐free and matrix methods. Differences in the performance of the matrix‐free method for laminar and turbulent flows are highlighted and analysed. Unsteady turbulent Navier–Stokes solutions of pitching and combined translation–pitching aerofoil oscillations are presented for unsteady shock‐induced separation problems associated with the rotor blade flows of forward flying helicopters. Copyright © 2000 John Wiley & Sons, Ltd.     

Comparison of derivative free Newton‐based and evolutionary methods for shape optimization of flow problems

The object of this study is to investigate two derivative free optimization techniques, i.e. Newton‐based method and an evolutionary method for shape optimization of flow geometry problems. The approaches are compared quantitatively with respect to efficiency and quality by using the minimization of the pressure drop of a pipe conjunction which can be considered as a representative test case for a practical three‐dimensional flow configuration. The comparison is performed by using CONDOR representing derivative free Newton‐based techniques and SIMPLIFIED NSGA‐II as the representative of evolutionary methods (EM). For the shape variation the computational grid employed by the flow solver is deformed. To do this, the displacement fields are scaled by design variables and added to the initial grid configuration. The displacement vectors are calculated once before the optimization procedure by means of a free form deformation (FFD) technique. The simulation tool employed is a parallel multi‐grid flow solver, which uses a fully conservative finite‐volume method for the solution of the incompressible Navier–Stokes equations on a non‐staggered, cell‐centred grid arrangement. For the coupling of pressure and velocity a pressure‐correction approach of SIMPLE type is used. The possibility of parallel computing and a multi‐grid technique allow for a high numerical efficiency. Copyright © 2006 John Wiley & Sons, Ltd.     

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions.

     

Two‐level consistent splitting methods based on three corrections for the time‐dependent Navier–Stokes equations

Three kinds of two‐level consistent splitting algorithms for the time‐dependent Navier–Stokes equations are discussed. The basic technique of two‐level type methods for solving the nonlinear problem is first to solve a nonlinear problem in a coarse‐level subspace, then to solve a linear equation in a fine‐level subspace. Hence, the two‐level methods can save a lot of work compared with the one‐level methods. The approaches to linearization are considered based on Stokes, Newton, and Oseen corrections. The stability and convergence demonstrate that the two‐level methods can acquire the optimal accuracy with the proper choice of the coarse and fine mesh scales. Numerical examples show that Stokes correction is the simplest, Newton correction has the best accuracy, while Oseen correction is preferable for the large Reynolds number problems and the long‐time simulations among the three methods. Copyright © 2015 John Wiley & Sons, Ltd.     

A SIGNIFICANT IMPROVEMENT ON NEWTON’S ITERATIVE METHOD

For solving nonlinear and transcendental equation f(x)=0, a singnificant improvement on Newton’s method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov’s methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton’s method, and remove the monotoneity condition imposed on f(x):f′(x)≠0.     

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 Newton type iterative inverse method for heat-conduction problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

Recently, a numerical method was proposed to compute a Hopf bifurcation point in fluid mechanics. This numerical method associates a bifurcation indicator and a Newton method. The former gives initial guesses to the iterative method. These initial values are the minima of the bifurcation indicator. However, sometimes, these minima do not lead to the convergence of the Newton method. Moreover, as only a single initial guess is obtained for each computation of the indicator, the computational time to obtain a Hopf bifurcation point can be quite long. The present algorithm is an enhancement of the previous one. It consists in automatically computing several initial guesses for each indicator curve. The majority of these initial values leads to the convergence of the Newton method. This method is evaluated through the problem of the lid‐driven cavity with several aspect ratios in the framework of the finite element analysis of the 2D Navier–Stokes equations. The results prove the efficiency and the robustness of the proposed algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

A Newton type iterative method for heat-conduction inverse problems

An inverse problem for identification of the coefficient in heat-conduction equation is considered. After reducing the problem to a nonlinear ill-posed operator equation, Newton type iterative methods are considered. The implicit iterative method is applied to the linearized Newton equation, and the key step in the process is that a new reasonable a posteriori stopping rule for the inner iteration is presented. Numerical experiments for the new method as well as for Tikhonov method and Bakushikskii method are given, and these results show the obvious advantages of the new method over the other ones.     

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.     

NEWTON–GMRES ALGORITHM APPLIED TO COMPRESSIBLE FLOWS

This paper addresses the resolution of non-linear problems arising from an implicit time discretization in CFD problems. We study the convergence of the Newton–GMRES algorithm with a Jacobian approximated by a finite difference scheme and with restarting in GMRES. In our numerical experiments we observe, as predicted by the theory, the impact of the matrix-free approximations. A second-order scheme clearly improves the convergence in the Newton process.     

弹塑性接触问题数值计算的二次规划算法

针对弹塑性接触问题所推得的数值求解式子,运用二次规划法具体设计了算法,该算法采用有限的基底交换运算就可得到收敛的数值解,具有较好的收敛性及较小的计算工作量.工程计算算例结果表明文章所提出的接触问题的求解方法是有效的,针对结构中接触问题所建立的数值计算模型能真实反映实际工作状况是可靠的.文章中还详细给出了接触单元相关矩阵和向量的具体推导形式.     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

A comparative study of lattice Boltzmann methods using bounce‐back schemes and immersed boundary ones for flow acoustic problems

In order to find applicable treatments of moving boundary conditions based on the lattice Boltzmann method in flow acoustic problems, three bounce‐back (BB) methods and four kinds of immersed boundary (IB) methods are compared. We focused on fluid–solid boundary conditions for flow acoustic problems especially the simulations of sound waves from moving boundaries. BB methods include link bounce‐back, interpolation bounce‐back and unified interpolation bounce‐back methods. Five IB methods are explicit and implicit direct‐forcing (Explicit‐IB and Implicit‐IB), two kinds of partially saturated computational methods and ghost fluid method. In order to reduce the spurious pressure generated by the fresh grid node changing from solid domain to fluid domain for BB methods and sharp IB methods, we proposed two new kinds of treatments and compared them with two existing ones. Simulations of the benchmark problems prove that the local evolutionary iteration (LI) is the best one in treatments of the fresh nodes. In addition, for standing boundary problems, although BB methods have a little higher accuracy, all the methods have similar accuracy. However, for moving boundary problems, IB methods are more appropriate than BB methods, because IB methods' smooth interpolation of pressure eld produces less disturbing spurious pressure waves. With improved treatments of fresh nodes, BB methods are also acceptable for moving boundary acoustic problems. In comparative tests in respective type, unified interpolation bounce‐back with LI, Implicit‐IB, and ghost fluid with LI are the best choices. Copyright © 2013 John Wiley & Sons, Ltd.     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

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.     

A semi-implicit finite volume scheme for a simplified hydrostatic model for fluid-structure interaction

Simulating fluid-structure interaction problems usually requires a considerable computational effort. In this article, a novel semi-implicit finite volume scheme is developed for the coupled solution of free surface shallow water flow and the movement of one or more floating rigid structures. The model is well-suited for geophysical flows, as it is based on the hydrostatic pressure assumption and the shallow water equations. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating structures. Furthermore, the nonlinear volume function allows for the simultaneous existence of wet, dry and pressurized cells in the computational domain. The resulting mildly nonlinear pressure system is solved using a nested Newton method. The accuracy of the volume computation is improved by using a subgrid, and time accuracy is increased via the application of the theta method. Additionally, mass is always conserved to machine precision. At each time step, the volume function is updated in each cell according to the position of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The simulated moving objects may for example represent ships, and the forces considered here are simply gravity and the hydrostatic pressure on the hull. For a set of test cases, the model has been applied and compared with available exact solutions to verify the correctness and accuracy of the proposed algorithm. The model is able to treat fluid-structure interaction in the context of hydrostatic geophysical free surface flows in an efficient and flexible way, and the employed nested Newton method rapidly converges to a solution. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.