Modal‐based goal‐oriented error assessment for timeline‐dependent quantities in transient dynamics

This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline‐dependent quantities) is proposed. These quantities are scalar time‐dependent outputs of the transient solution, which are better suited to time‐dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline‐dependent quantities of interest. The key ingredient is the modal‐based approximation of the associated adjoint problems, which allows efficiently computing and storing the adjoint solution. The approximated adjoint solution is readily post‐processed to produce an enhanced solution, requiring only one spatial post‐process for each vibration mode and using the time‐harmonic hypothesis to recover the time dependence. Thus, the proposed goal‐oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low. Copyright © 2013 John Wiley & Sons, Ltd.     

Guaranteed error bounds on pointwise quantities of interest for transient viscodynamics problems

A new method is developed to obtain guaranteed error bounds on pointwise quantities of interest for linear transient viscodynamics problems. The calculation of strict error bounds is based on the concept of “constitutive relation error” (CRE) and the solution of an adjoint problem. The central and original point of this work is the treatment of the singularity in space and time introduced by the loading of the adjoint problem. Hence, the adjoint solution is decomposed into two parts: (i) an analytical part determined from Green’s functions; (ii) a residual part approximated with classical numerical tools (finite element method, Newmark integration scheme). The capabilities and the limits of the proposed approach are analyzed on a 2D example.     

ASAC系统中少测点条件下的近场声压误差传感策略

根据有源控制策略设计简洁、有效的误差传感非常关键。针对基于声压声辐射模态的有源控制策略,具体设计与之对应的少测点条件下近场声压误差传感策略。首先,通过分析声压声辐射模态的空间滤波特性,证明基于声压声辐射模态的有源控制策略可行;然后,利用近场测量面上有限个测点的声压分布,通过求解模态展开所构成的欠定方程组的最小模最小二乘解,获取前K阶声压声辐射模态伴随系数的近似值,从而形成与基于声压声辐射模态的有源控制策略相对应的近场声压误差传感策略;最后,以置于无限大障板上的钢质简支薄板为例,进行数值仿真分析。结果表明设计的误差传感策略可行,所获得的前K阶声压声辐射模态伴随系数具有较高的精度。     

膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节,但在数学上,膜结构的极小曲面找形分析是一个高度非线性问题,一般无法求得其解析解,因此数值方法成为重要工具。近年来,基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功,基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上,该文提出一种基于EEP法的二维非线性有限元自适应方法,并成功将之应用于膜结构的找形分析。其主要思想是,通过将非线性问题用Newton法线性化,引入现有的二维线性问题的自适应求解技术,进而实现二维有限元自适应分析技术从线性到非线性的跨越,将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率,对网格自适应地进行调整,最终得到优化的网格,其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展,并给出数值算例用以表明该方法的可行性和可靠性。     

Error estimation and model adaptation for a stochastic‐deterministic coupling method based on the Arlequin framework

The paper deals with the issue of accuracy for multiscale methods applied to solve stochastic problems. It more precisely focuses on the control of a coupling, performed using the Arlequin framework, between a deterministic continuum model and a stochastic continuum one. By using residual‐type estimates and adjoint‐based techniques, a strategy for goal‐oriented error estimation is presented for this coupling and contributions of various error sources (modeling, space discretization, and Monte Carlo approximation) are assessed. Furthermore, an adaptive strategy is proposed to enhance the quality of outputs of interest obtained by the coupled stochastic‐deterministic model. Performance of the proposed approach is illustrated on 1D and 2D numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.     

Using mesh adaption for the identification of a spatial field of material properties

It is well known that the solution of an inverse problem is ill‐posed and not unique. To avoid difficulties caused by this, when solving such a problem, Tikhonov's regularization terms are usually added to the norm quantifying the discrepancy between the model's predictions and experimental data. This regularization term however is often inadequate to perform the identification of a field of material properties that varies spatially. This is all the more difficult when dealing with the numerical solution of this inverse problem, for the sought field is spatially discretized and this discretization can influence the result of the identification. We will here examine an overall strategy using classical adaptive meshing methods used to circumvent these drawbacks. The first step consists of using two distinct meshes: one associated with the discretization of the sought spatial field and the other associated with the solution of the mechanical problems (forward and adjoint states). In the second step, we will introduce local error estimators that allow an oriented refinement of the mesh associated with the sought parameters. This general strategy is applied to a practical case study: the detection of underground cavities using experimental data obtained by an interferometric device on a satellite. We will then address the question of how the regularization terms and the error estimator driving the mesh refinement were selected. Copyright © 2011 John Wiley & Sons, Ltd.     

Weight-adaptive isogeometric analysis for solving elastodynamic problems based on space-time discretization approach

In this paper, a novel adaptive isogeometric analysis (IGA) is introduced and its application in the numerical solution of two-dimensional elastodynamic problems based on the space-time discretization (STD) approach is studied. In the STD approach, the time is considered as an additional dimension and is discretized the same as the spatial domain. The weights of control points play the main role in the proposed method. In the conventional IGA, the same set of weights is used in the modeling of geometric and solution spaces. The idea is to define two groups of weights: geometric and solution weights. Geometric weights are known and can be determined based on the position of control points, but the solution weights are considered to be unknown and can be determined using a proper strategy so that the accuracy of the solution is optimized. This strategy is based on the minimization of an error function. The results obtained from the proposed method are compared with those obtained from the conventional IGA.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

A recipe to detect the error in discretization schemes

The necessity for a reliable measure of the discretization error arises in adaptive mesh refinement and in moving mesh adaptation. The present work discusses a detector of the discretization error based on the interpolation reconstruction of the operators. The technique presented here is named operator recovery error source detector (ORESD). Its main features are: First, the technique is based on the operators being discretized and does not require any user intervention or any a priori knowledge of the solution or its properties. Second, the ORESD is an a posteriori error indicator, but it is shown to be consistent with the a priori error provided by the modified equation approach. Third, the technique is based on the operators being solved and is tailored to the specific problem at hand. Four, the technique is simple and is based on a small stencil, resulting in a very inexpensive error detection. In the present work, the ORESD is derived and applied to two tutorial examples: divergence and gradient. With the aid of the two examples and using the general derivation, the ORESD is then applied to the gas dynamics equations. Two benchmarks are used to test the performance. First, a shock tube problem is solved (Sod's benchmark) in a Lagrangian and in a Eulerian frame. Second, the Colella's wedge problem is solved using CLAWPACK. Finally, the ORESD is applied to the 2D Poisson equation on a uniform and on a non‐uniform grid to test the application to elliptic problems. In all examples the operator recovery error source detector succeeds in detecting the real sources of error. Copyright © 2004 John Wiley & Sons, Ltd.     

Goal-oriented error estimation for transient parabolic problems

This work focuses on controlling the error and adapting the discretization in the context of parabolic problems. In order to obtain a sound mathematical framework, the time domain is discretized using a Discontinuous Galerkin (DG) approach. This allows to formulate the time stepping procedure in a variational format. The error is measured in the basis of an output of interest of the solution, defined by a linear functional. A dual problem, associated with this linear output is introduced. The dual problem has to be solved backward in time. An error representation is introduced, based on the weak residual of the primal error applied to the dual solution. Two different alternatives are studied to estimate the error in the dual solution: (1) recovery based error estimators and (2) implicit residual type estimators. Once the error assessment is performed implicitly in the dual problem, the obtained estimate is plugged into the primal residual to obtain the error in the quantity of interest. The implementation of the estimator is drastically simplified by using the weak version of the residual instead of the strong version used in previous works. Thus, the output error is assessed using a mixed technique, explicit for the primal problem and implicit for the dual. In the framework of adaptive computations of transient problems, this approach is very attractive because it allows using first the implicit scheme for the dual problem and then integrating the primal problem, estimating the error explicitly and eventually adapting the space-time grid. Thus, at every time step of the time marching scheme, the estimate of the dual error is injected into the primal residual (explicit estimate for the primal problem). Partially supported by MCYT, Spain. Grant Contract: DPI2001-2204.     

陀螺系统辛子空间迭代法

转子系统的有限元分析可以导出陀螺系统的本征值问题．而陀螺本征值问题可在哈密顿体系下求解。基于辛子空间迭代法的思想，提出了一种求解陀螺系统本征值问题的算法。首先引入对偶变量，将陀螺动力系统导入哈密顿体系，将问题化为了哈密顿矩阵的本征值问题。由于稳定的陀螺系统其本征值必为纯虚数，利用这个特点。提出了对应陀螺系统的辛子空问迭代法，从而可以求出系统任意阶的本征值及其振型。算例证明了这种算法的有效性。     

Calculation of strict error bounds for finite element approximations of non‐linear pointwise quantities of interest

This paper deals with the verification of simulations performed using the finite element method. More specifically, it addresses the calculation of strict bounds on the discretization errors affecting pointwise outputs of interest which may be non‐linear with respect to the displacement field. The method is based on classical tools, such as the constitutive relation error and extraction techniques associated with the solution of an adjoint problem. However, it uses two specific and innovative techniques: the enrichment of the adjoint solution using a partition of unity method, which enables one to consider truly pointwise quantities of interest, and the decomposition of the non‐linear quantities of interest by means of projection properties in order to take into account higher‐order terms in establishing the bounds. Thus, no linearization is performed and the property that the local error bounds are guaranteed is preserved. The effectiveness of the approach and the quality of the bounds are illustrated with two‐dimensional applications in the context of elastic fatigue problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

The parametric analysis of electric grids requires carrying out a large number of power flow computations. The different parameters describe loading conditions and grid properties. In this framework, the proper generalized decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to (1) iterating algebraic solver, (2) number of terms in the separable greedy expansion, and (3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal‐oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end‐user. The paper discusses how to compute the goal‐oriented error estimates. This requires linearizing the error equation and the quantity of interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermally induced vibrations of cross-ply laminated shallow shells

Summary An exact analytical solution of the dynamic response of cross-ply laminated shallow shells subject to rapid heating is presented. The classical theory (based on Love-Kirchhoff assumption), involving three coupled partial differential equations, is used. The solution is applicable to shells whose parallel edges are simply supported and the remaining ones are clamped. A generalized modal approach is used to obtain the solution. The equations of motion are converted into a single-order system of equations by using state variables. The biorthogonality conditions of principal modes of the original and adjoint eigenfunctions are used to decouple the state space equation. Histories of deflection of graphite-reinforced aluminum shell panels are presented through numerical examples.     

基于预实验分析的箱梁模型模态测试

以箱梁模型为实验对象,使用一种基于预实验分析的方法对箱梁模型进行模态识别。首先,在有限元模型模态分析结果的基础上,利用预实验分析确定合理的布点方案及最优激励点;再由预实验分析结果,采用激振器法对自由支撑的箱梁模型进行模态识别;最后,将实验模态与有限元模态进行相关性分析,以验证实验模态数据的可信性。实验结果表明,识别的四阶模态频率值与有限元分析结果在10%以内,前三阶在5%以内;与有限元模态相关性分析表明,实验测得的模态数据与有限元结果相似度高,尤其是对应前两阶模态振型相关性高达90%以上。     

Goal-oriented error estimation and mesh adaptivity in 3d elastoplasticity problems

We propose a 3D adaptive method for plasticity problems based on goal-oriented error estimation, which computes the error with respect to a prescribed quantity of interest. It is a dual-based scheme which requires an adjoint problem. The computed element-wise errors at each load/displacement increment are utilized for the mesh adaptivity purpose. Mesh adaptivity procedure is performed based on refinement and coarsening by introducing hanging nodes in quadrilateral and hexahedral elements in 2d and 3d, respectively. Several numerical simulations are investigated and the results are compared with available analytical solutions, existing experimental data and results of mesh adaptivity based on other conventional error estimation methods.     

Goal-Oriented Error Estimates Based on Different FE-Spaces for the Primal and the Dual Problem with Applications to Fracture Mechanics

The objective of this paper is to derive goal-oriented a posteriori error estimators for the error obtained while approximately evaluating the nonlinear J-integral as a fracture criterion in linear elastic fracture mechanics (LEFM) using the finite element method. Such error estimators are based on the well-established technique of solving an auxiliary dual problem. In a straightforward fashion, the solution to the discretized dual problem is sought in the same FE-space as the solution to the original (primal) problem, i.e. on the same mesh, although it merely acts as a weight of the discretization error only. In this paper, we follow the strategy recently proposed by Korotov et al. [J Numer Math 11:33–59, 2003; Comp Lett (in press)] and derive goal-oriented error estimators of the averaging type, where the discrete dual solution is computed on a different mesh than the primal solution. On doing so, the FE-solutions to the primal and the dual problems need to be transferred from one mesh to the other. The necessary algorithms are briefly explained and finally some illustrative numerical examples are presented.     

Identification of boundary displacements in plane elasticity by BEM

The purpose of this study is to present a possible application of BEM for numerical identification of the boundary conditions for Navier equations in plane elasticity with internal measurements, based on insufficient and noisy information for unique identification. The inverse problem is re-formulated as a minimization problem by the direct variational method. The minimization problem is then recast using the gradient method into successive primary and adjoint boundary value problems in the corresponding plane elasticity problem. For numerical solution of the elasticity problems, the conventional direct boundary element method is employed. From the simple numerical examples considered, it is concluded that our identification scheme is stable and the approximate solutions are convergent to the minimum.     

大跨圆拱屋盖结构的风致响应分析

大跨屋盖特征值问题的求解是结构动力响应分析中最繁琐的一个环节，而且一些对结构响应贡献较大的高阶模态容易在传统的模态叠加法中被忽略。本文以典型的大跨圆拱屋盖为例，将里兹向量直接叠加法应用于屋盖系统特征值问题计算和风致响应分析，其特点是在误差逼近的基础上自动生成一组正交的里兹向量并用于缩减系统自由度数。与传统模态叠加法算得的结果相比，里兹向量直接叠加法只用很少数目的向量就可以得到较精确的结果，而且高阶模态的贡献不会被忽略。该方法不仅大幅度地减少了机时，而且提供了动力分析的误差估计。