Rate‐dependent projection operators for frictional contact constraints

We develop rate‐dependent regularization approaches for three‐dimensional frictional contact constraints based on the Kelvin and Maxwell viscoelastic constitutive models. With the present regularization schemes, we aim to provide a basis to better model friction and to stabilize the contact analysis while keeping the contact model as simple as possible. The key feature of the regularization approaches, implemented using an implicit time integrator, is that one can recover in the limit the widely used rate‐independent elastoplastic regularization framework without encountering numerical difficulties. Intermediate contact tractions are defined in terms of the relative displacement, the relative velocity, and the regularization parameters. The projection operators operate on the intermediate tractions and yield contact tractions that satisfy all the discretized contact constraints. The use of projection operators allows a systematic implementation of the present regularization schemes. Through numerical simulations, we observed that the Maxwell‐type regularization effectively avoids convergence problems, even for relatively large time step sizes, while the Kelvin‐type regularization does not. Copyright © 2003 John Wiley & Sons, Ltd.     

解带线性或非线性约束最优化问题的混合三项记忆梯度投影算法

利用Rosen投影矩阵,结合Solodov投影技巧建立求解带线性或非线性不等式约束优化问题的混合三项记忆梯度Rosen投影算法,并证明了算法的收敛性。数值例子表明该算法是有效的。     

脉冲风洞测力系统建模与载荷辨识方法研究

将载荷辨识技术应用于脉冲燃烧风洞模型测力。用子结构综合法建立了测力试验系统的动力学模型,在时域内将动力学方程进行离散,建立起天平测量信号与模型气动载荷历程之间的线性关系,作为载荷辨识的模型。采用Tikhonov正则化和子空间投影法相结合的混合正则化方法,将高维的、不适定的载荷辨识问题转化为低维的适定问题,以利于快速求解。提出了一种新方法来确定合适的投影子空间维数,然后应用L曲线准则来寻找低维正则化问题的最优正则化参数。最后通过算例验证了系统建模方法的精度和载荷辨识算法的有效性与稳定性。     

求解非线性等式约束优化问题的共轭梯度投影算法

利用投影矩阵,对求解无约束规划的共轭梯度算法中的参数βk给一限制条件确定βk的取值范围,以保证得到目标函数的共轭梯度投影下降方向,建立了求解非线性等式约束优化问题的共轭梯度投影算法,并证明了算法的收敛性。数值例子表明算法是有效的。     

Determining the initial distribution in space-fractional diffusion by a negative exponential regularization method

In this paper, we consider the backward problem for diffusion equation with space fractional Laplacian, i.e. determining the initial distribution from the final value measurement data. In order to overcome the ill-posedness of the backward problem, we present a so-called negative exponential regularization method to deal with it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate are established under a-priori bound assumption for the exact solution. Finally, several numerical examples are proposed to show that the numerical methods are effective.     

Optimally Generalized Regularization Methods for Solving Linear Inverse Problems

In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be equilibrated or diagonally dominated to derive the optimal scales in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved that the iterative sequences are monotonically convergent to the true solution. The presently proposed optimally generalized regularization methods are able to overcome the ill-posedness of linear inverse problems, and provide rather accurate numerical solution.     

Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem for the Helmholtz equation can be regularized by various methods, such as the general regularization methods presented in this paper, but more accurate results are obtained by classical methods, such as the singular value decomposition and the Tikhonov regularization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilized Reconstruction in Signal and Image Processing

Abstract

Part II is essentially devoted to the iterative reconstruction procedures that can easily be implemented in signal and image processing when the stability conditions are fulfilled. The application of the regularization principle introduced in part I for deconvolution is examined in this context. As the convergence of the method of conjugate gradients is then superlinear, this technique proves to be very well suited to solving the least-squares problem without constraint. If need be, the non-negativity constraint can be taken into account by slightly modifying the algorithm of steepest descent with fixed step, i.e. the Bialy-Jacobi iteration. Implemented in an appropriate manner, these methods may also provide the interesting part of the spectrum of the operator A? A to be inverted. This last point is particularly useful for conducting the error analysis. The exploration of the eigenspaces is also possible. As far as the implementation of the regularization principle is concerned, the theoretical resolution limit of the reconstruction process is selected through an interactive decision procedure based on a progressive estimation of the size of the object-reconstruction error. The numerical implementation, which is illustrated with the aid of simulated one-dimensional examples, reinforces and completes, in a concrete manner, the overall analysis presented in part I. The transposition to the more general situation in which the object function is defined on a low-resolution background is outlined in this context. It is also indicated how this approach should lead to a better understanding of the other deconvolution methods.     

有关非线性图像配准的正则化(英文)

近年来,非线性图像配准各种方法的数值分析已经十分成熟;但是大多数工作都没有考虑理论上的收敛性分析.本文对非线性图像配准极小问题进行了理论上的分析证明.我们得到了极小问题正则解的存在性、稳定性及收敛性,证明了在正则化算子和正则参数分别满足一定的条件下,极小问题的正则解收敛到我们所关心的问题的解.这一理论上的证明过程及结果将对非线性图像配准的正则化算子的选取提供十分重要的依据.     

一般约束最优化超线性与二次收敛的SQP拟可行方法

讨论一般约束最优化问题,利用序列二次规划(SQP)技术和强收敛方法思想建立问题的一个新的拟可行下降算法,算法每次迭代只需解一个要求较弱的二次规划或用广义投影技术产生搜索方向。分析和论证了算法的全局收敛件、强收敛性、超线性收敛性和二次收敛率。     

An implicit integration algorithm with a projection method for viscoplastic constitutive equations

Robinson's viscoplastic model, a representative of the so-called overstress models, is integrated by use of the generalized midpoint rule. The solution of the non-linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with Newton's method. Consistent tangent moduli are calculated and the quadratic convergence of the global Newton equilibrium iteration is shown. The time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration. Various numerical examples are considered to demonstrate the efficiency of the methods.     

A high‐order accurate particle‐in‐cell method

We propose the use of high‐order weighted essentially non‐oscillatory interpolation and moving‐least‐squares approximation schemes alongside high‐order time integration to enable high‐order accurate particle‐in‐cell methods. The key insight is to view the unstructured set of particles as the underlying representation of the continuous fields; the grid used to evaluate integro–differential coupling terms is purely auxiliary. We also include a novel regularization term to avoid the accumulation of noise in the particle samples without harming the convergence rate. We include numerical examples for several model problems: advection–diffusion, shallow water, and incompressible Navier–Stokes in vorticity formulation. The implementation demonstrates fourth‐order convergence, shows very low numerical dissipation, and is competitive with high‐order Eulerian schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation

In this paper, we consider a Cauchy problem for the semi-linear elliptic equation and use a modified boundary Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability of the regularization solution are proven. Under an a-priori bound assumption for the exact solution, a convergence estimate of Hölder type for this method is obtained. Finally, an iterative scheme is proposed to calculate the regularization solution, numerical results show that this method works well.     

New solution algorithms for asymmetric traffic assignment model

Abstract

The asymmetric traffic assignment model can improve the traditional traffic assignment model by adopting detailed network representation and more realistic asymmetric cost functions. The diagonalization, streamlined diagonalization, and projection methods are three widely mentioned solution algorithms for solving asymmetric traffic assignment models. The diagonalization and streamlined diagonalization methods have the advantage of requiring less computer memory but typically require greater computational time. The projection method has the advantage of converging more rapidly but requires a large computer memory. In order to balance computer memory and computational time, we propose two new algorithms; i.e., hybrid and streamlined hybrid methods. According to our case study, the proposed algorithms show their superiority over the diagonalization and streamlined diagonalization methods in terms of computational time, and over the projection method in terms of computer memory. Both new algorithms can handle small or medium networks sized asymmetric traffic assignment problems on personal computers.     

Generalized Lavrentiev-type regularization method for the Cauchy problem of a semi-linear elliptic equation

We consider the Cauchy problem of a semi-linear elliptic equation and use a generalized Lavrentiev-type regularization method to overcome its ill-posedness. The existence, uniqueness and stability for regularized solution are proven, the convergence estimate of Hölder type for this method is derived. An iterative scheme is constructed to calculate the regularization solution, and the results of numerical simulations show that this method works well.     

A new strategy for the solution of frictional contact problems

This article is devoted to the development of a new heuristic algorithm for the solution of the general variational inequality arising in frictional contact problems. The existing algorithms devised for the treatment of the variational inequality representing frictional contact rely on the decomposition of the physical problem into two sub-problems which are then solved iteratively. In addition, the penalty function method and/or the regularization techniques are typically used in the solution of these reduced sub-problems. These techniques introduce user-defined parameters which could influence the convergence and accuracy of the solution. The new method presented in this article overcomes these difficulties by providing a solution for the general variational inequality without decomposition into sub-problems. This is accomplished using a new heuristic algorithm which utilizes mathematical programming techniques, and thus avoids the use of penalty or regularization methods. The versatility and reliability of the developed algorithm were demonstrated through implementation to the case of frictional contact of an elastic hollow cylinder with a rigid foundation. © 1998 John Wiley & Sons, Ltd.     

Direct regularization for system of integral-algebraic equations of index-1

It is known that the coupled system consisting of Volterra integral equations of the first and second kind belongs to the class of moderately ill-posed problems. In the present paper, we are interested in numerical solution of Volterra integral-algebraic equations by a direct regularization method, i.e. an approach which does not make use of the adjoint operator as well as any reduction or remodelling of the original problem. A numerical algorithm based on Lavrentiev’s regularization iterated method is constructed that preserves the Volterra structure of the original problem. The convergence analysis of the proposed method is given and its validity and efficiency are also demonstrated through several numerical experiments.     

Numerically consistent regularization of force‐based frame elements

Recent advances in the literature regularize the strain‐softening response of force‐based frame elements by either modifying the constitutive parameters or scaling selected integration weights. Although the former case maintains numerical accuracy for strain‐hardening behavior, the regularization requires a tight coupling of the element constitutive properties and the numerical integration method. In the latter case, objectivity is maintained for strain‐softening problems; however, there is a lack of convergence for strain‐hardening response. To resolve the dichotomy between strain‐hardening and strain‐softening solutions, a numerically consistent regularization technique is developed for force‐based frame elements using interpolatory quadrature with two integration points of prescribed characteristic lengths at the element ends. Owing to manipulation of the integration weights at the element ends, the solution of a Vandermonde system of equations ensures numerical accuracy in the linear‐elastic range of response. Comparison of closed‐form solutions and published experimental results of reinforced concrete columns demonstrates the effect of the regularization approach on simulating the response of structural members. Copyright © 2008 John Wiley & Sons, Ltd.     

Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact

 Two major difficulties are encountered in the meshfree solution of incompressible boundary value problems. The first is due to the employment of higher-order quadrature rules that leads to an over-constrained discrete system in incompressible problems. The second is associated with the treatment of essential boundary conditions and contact conditions owing to the loss of Kronecker delta properties in the meshfree shape functions. This paper discusses some recent enhancements in meshfree methods for incompressible boundary value problems, carries out numerical convergence analysis, and compares accuracy and efficiency improvement of these methods. Presented methods are a pressure projection method to remedy the over-constrained discrete system, and a mixed transformation method and a boundary singular kernel method for imposition of essential boundary conditions and contact constraints. Several linear and nonlinear problems were analyzed to demonstrate the effectiveness of the new approaches.