Efficient model reduction of large scale systems using Krylov-subspace iterative methods

In this work, the use of Krylov-subspace iterative methods is examined to obtain low-rank approximate solutions of Lyapunov equations for use in contol-oriented model reduction of large space structures. In particular, we examine the application of these methods to the approximate component cost analysis (CCA) of large scale systems. It is shown that using the proposed methods approximate CCA reduced-order models can be obtained with a significant reduction in the computational effort and time. In addition, we derive closed-form expressions for approximate cost-equivalent, cost-decoupled and covariance-equivalent realizations using the proposed Krylov-subspace solutions.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparison of preconditioned iterative techniques using rapid operator application against direct solution methods

The p-version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation, and high accuracy with relatively few degrees of freedom. However, the solution time and storage requirements for large 3-D problems can be very high when direct methods are used to factor global matrices. Iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed, but may not converge rapidly enough. We study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level. This resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods. We illustrate the effectiveness of this scheme on some large three-dimensional problems in structural analysis.     

New iterative reconstruction methods for fan-beam tomography

In this paper, we present a novel class of iterative reconstruction methods for severely angular undersampled or/and limited-view tomographic problems with fan-beam scanning geometry. The proposed algorithms are based on a new analytical transform which generalizes Fourier-slice theorem to divergent-beam scanning geometries. Using a non-rigid coordinate transform, divergent rays can be reorganized into parallel ones. Therefore, one can employ a simpler parallel-beam projection model instead of more complicated divergent-beam geometries. Various existing iterative reconstruction techniques for divergent-beam geometries can be easily adapted to the proposed framework. The significant advantage of this formulation is the possibility of exploiting efficient Fourier-based recovery methods without rebinning of the projections. In case of highly sparse measurements (few-view data), rebinning methods are not suitable due to error-prone angular interpolation involved. In this work, three new methods based on the novel analytical framework for fan-beam geometry are presented: the Gerchberg-Papoulis algorithm, the Neumann decomposition method and its total variation regularized version. Presented numerical experiments demonstrate that the methods can be competitive in reconstructing from few-view noisy tomographic measurements.     

High‐order iterative bracketing methods

This paper presents high‐order, single‐point, iterative methods for the numerical solution of the general non‐linear equilibrium equation. The methods are designed so that the approximations provided by the iterative process alternate about the targeted root, thus providing progressively better upper, as well as lower bounds on the sought root, which is thus numerically bracketed ever tighter. The effectiveness of the proposed methods is demonstrated by applying them to obtain close bounds on the extreme eigenvalues of the finite elements global stiffness matrix of the hanging string. Copyright © 2013 John Wiley & Sons, Ltd.     

Solution of the partial eigenproblem by iterative methods

A combined iterative method is formulated for the partial solution of the eigenproblem Ax = λ Bx that arises from the application of the finite element method. The method is a combination of the conjugate gradient (CG) method and the symmetric successive co-ordinate overrelaxation (SSCOR) method. Both of these methods belong to the same family of iterative methods which retain their vectorial form and approach the solution by seeking stationary points of the Rayleigh quotient. The formulation consists in splitting a characteristic matrix into the sum of two matrices, one of which corresponds to the SSCOR matrix, and then accelerating the associated iteration using the CG. The behaviour of the method is illustrated for five test cases and comparisons are made with other iterative methods. The influence of the relaxation parameter ω on the computational efficiency of the methods is also studied.     

采用等效有限元模型的复合材料机翼结构优化

在机翼设计过程中,将等效有限元模型(EFEM)方法应用于考虑静力学和动力学要求的机翼结构优化。提出了"三步走"的结构优化策略,将一个多变量的复杂优化问题转换为一系列少变量的简单优化问题,对某支线客机的复合材料机翼进行了优化设计。首先以位移、静强度和颤振速度作为约束条件对机翼复合材料铺层比例进行优化;然后以静强度和结构稳定性作为约束,以最小化结构质量和结构效率作为优化目标,对各翼肋之间的加强壁板进行优化设计;最后再以位移和颤振速度为约束,对机翼结构总体刚度进行优化设计。结果表明:EFEM方法具有快速建模和计算量少的优点,采用"三步走"优化策略具有更高的效率,适用于初步机翼结构优化设计。     

Scheduling methods for the statistical setup adjustment problem

Feedback control methods have been proposed in recent literature to regulate the quality characteristic of parts or products in a manufacturing process. Depending on the costs involved, adjustments may not be needed at each time instant (i.e. for every part or product). This paper presents scheduling methods to determine the optimal time instants for adjusting a process. The focus is on the set-up adjustment problem, in which it is necessary to adjust the process in order to compensate for an initial offset that occurs due to an incorrect set-up operation. The performance of three scheduling methods are compared in terms of the expected manufacturing cost and computational effort of each method. The adjustment methods considered are based on estimates of the process variance and the size of the offset. The robustness of these methods with respect to biased estimates of the process variance and of the set-up error or offset are discussed. One simple method, a backward implementation of the Silver-Meal heuristic used for inventory control is recommended based on a performance analysis.     

The convergence behaviour of iterative methods on severely stretched grids

In this paper we examine the dramatic influence that a severe stretching of finite difference grids can have on the convergence behaviour of iterative methods. For the most important classes of iterative methods this phenomenon is considered for a simple model problem with various boundary conditions and an exponential grid. It is shown that grid compression near 2 Neumann boundary or in the centre can make the convergence of some methods extremely slow, whereas grid compression near a Dirichlet boundary can be very advantageous. More theoretical insight is obtained by analysing the spectrum of the Jacob: matrix for one- and two-dimensional problems. Several bounds on dominant eigenvalues of this matrix are given. The final conclusions are also applicable to problems with a variable diffusion coefficient and convection-diffusion equations solved by central difference schemes.     

On a class of preconditioned iterative methods on parallel computers

The standard implementations of iterative solvers for finite element and finite difference methods frequently use a diagonal (Jacobi) preconditioner, particularly for element-by-element schemes. However, for such methods the actual order of the condition number with respect to mesh size is not reduced by the preconditioner. In the present paper we describe an iterative method where, in addition, the condition number is reduced by an order of magnitude. Moreover, the scheme may also be implemented as an element-by-element method. The method uses a generalized SSOR preconditioner and a wave front or multi-frontal ordering of the mesh nodes. For a general irregular finite element mesh a striped irregular wave front ordering may be used. The performance of the method as well as various iterative acceleration techniques for a parallel computer are examined in the numerical studies.     

Mesh refinement and iterative solution methods for finite element computations

Global and element residuals are introduced to determine a posteriori, computable, error bounds for finite element computations on a given mesh. The element residuals provide a criterion for determining where a finite element mesh requires refinement. This indicator is implemented in an algorithm in a finite element research program. There it is utilized to automatically refine the mesh for sample two-point problems exhibiting boundary layer and interior layer solutions. Results for both linear and nonlinear problems are presented. An important aspect of this investigation concerns the use of adaptive refinement in conjunction with iterative methods for system solution. As the mesh is being enriched through the refinement process, the solution on a given mesh provides an accurate starting iterate for the next mesh, and so on. A wide range of iterative methods are examined in a feasibility study and strategies for interweaving refinement and iteration are compared.     

A statistical view of iterative methods for linear inverse problems

In this article, we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods, from a statistical point of view. The basic purpose of the paper is to develop adaptive model selection techniques for determining the regularization parameters, i.e., the iteration index. We assume observations are taken over a fixed grid and we consider solutions over a sequence of finite-dimensional subspaces. Based on concentration inequalities techniques, we derive non-asymptotic optimal upper bounds for the mean square error of the proposed estimator.     

Measuring leader perceptions of school readiness for reforms: use of an iterative model combining classical and Rasch methods

This study examines validity of data generated by the School Readiness for Reforms: Leader Questionnaire (SRR-LQ) using an iterative procedure that combines classical and Rasch rating scale analysis. Following content-validation and pilot-testing, principal axis factor extraction and promax rotation of factors yielded a five factor structure consistent with the content-validated subscales of the original instrument. Factors were identified based on inspection of pattern and structure coefficients. The rotated factor pattern, inter-factor correlations, convergent validity coefficients, and Cronbach's alpha reliability estimates supported the hypothesized construct properties. To further examine unidimensionality and efficacy of the rating scale structures, item-level data from each factor-defined subscale were subjected to analysis with the Rasch rating scale model. Data-to-model fit statistics and separation reliability for items and persons met acceptable criteria. Rating scale results suggested consistency of expected and observed step difficulties in rating categories, and correspondence of step calibrations with increases in the underlying variables. The combined approach yielded more comprehensive diagnostic information on the quality of the five SRR-LQ subscales; further research is continuing.     

Structural reliability analysis under evidence theory using the active learning kriging model

Structural reliability analysis under evidence theory is investigated. It is rigorously proved that a surrogate model providing only correct sign prediction of the performance function can meet the accuracy requirement of evidence-theory-based reliability analysis. Accordingly, a method based on the active learning kriging model which only correctly predicts the sign of the performance function is proposed. Interval Monte Carlo simulation and a modified optimization method based on Karush–Kuhn–Tucker conditions are introduced to make the method more efficient in estimating the bounds of failure probability based on the kriging model. Four examples are investigated to demonstrate the efficiency and accuracy of the proposed method.     

Improving binary phase correlation filters using iterative techniques

An iterative technique has been used to improve the design and performance of the binary phase version of a tandem-component correlation filter. The results are compared to a regular matched filter, a phase-only filter (POF), and a binary phase POF, in terms of optical efficiency, SNR, and peak correlation intensity.     

A comparison of direct and iterative methods for determining traction-separation relations

Traction-separation relations have been used to represent the adhesive interactions at bimaterial interfaces for contact and fracture analyses. There are a variety of methods for determining these relations, which are broadly sorted into iterative and direct methods. Here we compare the traction-separation relations for a silicon/epoxy interface extracted by two such methods. Interferometric measurements of the normal crack opening displacements near the crack front in a double-cantilever beam specimen were exploited along with an augmented analytical solution for J-integral as an illustration of the direct method. As an example of the iterative method, we relied on comparisons of measured crack length and normal crack opening displacements with numerical simulations obtained from two types of candidate traction-separation relations. It was found that the shape of the traction-separation relation, in addition to the interfacial toughness and strength, was needed to bring the numerical solutions into optimal registration with the measurements. On the other hand, the direct method lived up to its name in terms of ease of parameter extraction while providing a reasonable set of parameters.     

Three iterative methods for the numerical determination of stress intensity factors

Three iterative methods for the numerical determination of stress intensity factors at crack tips (by using the method of singular integral equations with Cauchy-type kernels) are proposed. These methods are based on the Neumann iterative method for the solution of Fredholm integral equations of the second kind. Two of these methods are essentially used for the solution of the system of linear algebraic equations to which the singular integral equation is reduced when the direct Lobatto-Chebyshev method is used for its approximate solution, whereas the third method is a generalization of the first two and is related directly to the singular integral equation to be solved. The proposed methods are useful for the determination of stress intensity factors at crack tips. Some numerical results obtained in a crack problem show the effectiveness of all three methods.     

模型缩聚-模型修正迭代方法的研究

模型修正中通常需要解决自由度匹配问题,模型缩聚是解决这一问题的一种方法。当有限元建模误差较大时,模型缩聚的近似会大大降低模型修正的精度。针对这一问题,提出了模型缩聚-模型修正迭代方法,消除模型缩聚带来的误差。文中应用IRS缩聚和基于频响函数的模型修正方法对提出的迭代方法进行了具体讨论。通过板梁混合结构的数值模拟实验,比较了现有修正方法和迭代修正方法的修正精度。结果表明提出的迭代方法有效提高了修正精度,使修正后的模型频率和物理参数更逼近真实值。同时该方法具有较高的迭代收敛效率,符合实际工程应用的要求。     

新的模型修正与模态扩展迭代方法

由于有限元模型与实验模型自由度的不匹配,模型修正中经常需要使用模态扩展技术。利用动态扩充法进行模态扩展进而进行模型修正时,模型修正过程中并没有考虑到使用有限元模型对实验数据进行模态扩展而造成的误差。针对这一问题,提出一种新的迭代方法,在模型修正过程中计及了上述的误差项,并将其以修正项的形式添加到模型修正当中,同时该方法没有限定所使用模型修正的方法,具有一定的通用性。通过Cross-model Cross-mode method(CM-CM)模型修正方法对新的迭代方法进行了详细讨论。算例表明,新的迭代方法较原方法可以极大地提高模型修正的迭代收敛速度。     

A comparison of eigensolvers for large‐scale 3D modal analysis using AMG‐preconditioned iterative methods

The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift‐invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.