An improved updating parameter selection method and finite element model update using multiobjective optimisation technique

Finite element model updating is a procedure to minimise the differences between analytical and experimental results and is usually posed as an optimisation problem. In model updating process, one requires not only satisfactory correlations between analytical and experimental results, but also maintaining physical significance of updated parameters. For this purpose, setting up of an objective function and selecting updating parameters are crucial steps in model updating. These require considerable physical insight and usually trial-and-error approaches are common to use. In conventional model updating procedures, an objective function is set as the weighted sum of the differences between analytical and experimental results. But the selection of the weighting factors is not clear since the relative importance among them is not obvious but specific for each problem. In this work, multiobjective optimisation technique is introduced to extremise several objective terms simultaneously. Also the success of finite element model updating depends heavily on the selection of updating parameters. In order to avoid an ill-conditioned numerical problem, the number of updating parameters should be kept as small as possible. Such parameters should be selected with the aim of correcting modelling errors and modal properties of interest should be sensitive to them. When the selected parameters are inadequate, then the updated model becomes unsatisfactory or unrealistic. An improved method to guide the parameter selection is suggested.     

DAMAGE IDENTIFICATION USING CHANGES OF EIGENFREQUENCIES AND MODE SHAPES

In the present paper, we describe an approach to identify the location and the extent of the damage introduced into the steel frame, using a two-step procedure. In the first step, the measured dynamic response of the original undamaged structure was used to generate a reference finite element (FE) model of the structure. The selected parameters were identified by means of a mathematical optimisation algorithm (‘updating procedure'), minimising an objective function containing the test/analyses differences of eigenfrequencies and mode shapes. The uncertain model parameters had to be chosen with care in order to retain the physical significance of the updated model. In the next step, the experimental modal data of the damaged structure were used to identify the extent of the damages. This was based on comparing the changes of stiffness parameters identified from the undamaged and the damaged structure. With the identified parameters, the FE model was able to reproduce the experimental data as close as possible and allowed the identification of the extent of the damage.     

Finite element model updating of concrete-filled steel tubular arch bridge under operational condition using modal flexibility

This paper presents finite element (FE) model updating method for real bridge structure under operational condition using modal flexibility. The theoretical background of the updating procedure is presented. The case study of a simulated simply supported beam demonstrates an effectiveness of modal flexibility in objective function. This objective function is then implemented in case study of a real concrete-filled tubular arch bridge. The bridge was tested under operational condition. Followed by the three-dimensional FE modeling of the bridge, an eigenvalue sensitivity study is carried out to select the most sensitive parameters to the concerned modes. Guyan technique is used to the mass normalization of the mode shapes extracted from ambient modal test to calculate the modal flexibility. The updated FE model of the bridge is able to produce a sufficient improvement on modal parameters of the concerned modes, which is in close agreement with the experiment results and updated parameters still preserve the physical meaning in practice.     

Comparative study of damped FE model updating methods

Most of finite element (FE) model updating techniques do not employ damping matrices and hence, cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

基于频响函数相关性的灵敏度分析的有限元模型修正

有限元模型的修正对机械结构的动态特性进行准确而可靠的预测是很重要的。利用试验测试和预测的有限元模型计算得到的频响函数（FRF），引入两种频响函数相关性的判定标准，提出基于频响相关函数的灵敏度分析的修正方程。数值实例研究结果表明，该方法利用少量的测量数据，即使测试数据含附加噪声，也可在很宽的频率范围内得到接近真实结构的有限元模型修正解。本文的方法可适用于大型复杂结构的模型修正。     

Adaptive regularization parameter optimization in output-error-based finite element model updating

In finite element (FE) model updating, regularization methods are required to alter the ill-conditioned system of equations towards a well-conditioned one. The present study addresses the regularization parameter determination when implementing the Tikhonov regularization technique in output-error-based FE model updating. As the output-error-based FE model updating results in a nonlinear least-squares problem which requires iteration for solution, an adaptive strategy that allows varying value of the regularization parameter at different iteration steps is formulated, where the optimal regularization parameter at each iteration step is determined based on the computationally efficient minimum product criterion (MPC). The performance of MPC in output-error-based FE model updating is examined and compared with the commonly used L-curve method (LCM) and the generalized cross validation (GCV) through numerical studies of a truss bridge using noise-free and noise-corrupted modal data. It is shown that MPC is effective and robust in determining the regularization parameter compared with the other two methods, especially when noise-corrupted data are used. The adaptive strategy is more efficient than the fixed strategy that uses a constant value of the regularization parameter throughout the iteration process.     

Structural crack detection without updated baseline model by single and multiobjective optimization

Identification, localization and quantification of structural damage can be performed through a model-updating procedure. Model-updating methods require a baseline finite element (FE) model of the undamaged structure, which imposes a restriction on their applicability and can become very problematic especially for large and complex civil structures. Modeling errors in the baseline model whose effects exceed the modal sensitivity to damage are critical and make an accurate estimation of damage impossible. This paper presents an identification algorithm using modal data for assessing structural damage that is based on FE-updating procedures and takes modeling error into account. To overcome its influence, differences of mode shapes and frequencies before and after damage for both numerical model and experimental measurements are used instead of the mode shapes and frequencies themselves. To formulate the objective function, two different approaches have been considered taking into account how these differences are grouped: a single-objective approach and a multiobjective approach. The effectiveness of both approaches is verified against numerical and experimental results.     

A two-level updating procedure of the component-mode synthesis model

The component-mode synthesis method is effective for reducing the degrees of freedom of a finite element model for dynamic analysis of a complex structure. However, in the resulting computational model unacceptable errors can exist due to poor modeling. The model can be corrected by applying system identification methods. By localising the parameter errors it is possible to use the updating procedure of the second author. The modal transformation performed within this updating leaves the parameters unchanged. As a result, the corrected component equations and synthesised component-mode model are derived. The general identification procedure contains many parameters to be identified for a complex system. Minimising the objective function with respect to these parameters results in a highly non-linear system of equations. In order to circumvent this, the two-level identification procedure is developed and applied. If the measurement errors of the component quantities are independent of each other, the quadratic objective function can be decomposed on the basis of these components. By introducing model and goal coordination variables, the identified parameters as subsystem units are corrected at the first level and the second level includes the co-ordination units. A parallel iterative algorithm is presented in detail.     

Further experience with model updating incorporating damping matrices

Most of the model updating techniques do not employ damping matrices and hence cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, response function method (RFM) is extended to deal with the complexity of FRF by updating damping matrices along with mass and stiffness matrices. The effectiveness of the damped FE model updating procedure is demonstrated by actual laboratory experiments of an F-shaped test structure. The updated results have shown that the damped RFM model updating procedure can be used to derive accurate model of the system. This is illustrated by matching of the complex FRFs obtained from the updated model with the experimental data.     

Robust updating of uncertain damping models in structural dynamics for low- and medium-frequency ranges

This paper deals with the robust updating of uncertain computational models in the context of structural dynamics in the low- and medium-frequency ranges of composite sandwich panels for which experimental results are available. The uncertain computational model is constructed using the non-parametric probabilistic approach which takes into account model and data uncertainties. The formulation of the robust updating problem includes the effects of uncertainties and consists in minimizing a cost function with respect to an admissible set of updating parameters. Updating is performed in two steps using several cost functions and experimental results. The results of the robust updating problem show that the method proposed is efficient for updating the uncertain computational model in both low- and medium-frequency ranges.     

Finite element model updating and structural damage identification using OMAX data

The main limitations in the finite element (FE) model updating technique lie in the ability of the FE model to represent the true behavior of the structure (modelling problem), and in the ability to identify enough modal parameters with sufficient accuracy, especially for large structures that are tested in operational conditions (identification problem). In this paper, the identification problem is solved with an OMAX approach, where an artificial force is used in operational conditions and a structural model is identified that takes both the forced and the ambient excitation into account. From an extensive case study on a real three-span bridge, it is observed that, while updating the FE model using the experimental output-only data yields a good fit, discrepancies show up when the more extensive set of OMAX data is used for validation, or even for updating. It can be concluded that an OMAX approach not only increases the well-posedness of the updating problem, it also allows to detect potential inaccuracies in the FE model.     

考虑模型偏差的结构动力学模型修正

针对结构有限元模型修正后仍可能存在模型偏差的问题,提出用待修正参数的不确定性来表征模型偏差的有限元模型修正方法。首先,基于响应面方法识别得到待修正参数的最优值,并通过计算结果与试验结果比较获得模型偏差;然后,基于响应面模型并结合灵敏度分析计算得到模型偏差对待修正参数的影响,从而得到考虑模型偏差后待修正参数的区间;最后,通过一个悬臂梁工程实例的模型修正,验证了笔者所提出方法的可行性。结果表明,考虑模型偏差的修正可以提高模型可靠性。     

RESULTS OBTAINED BY MINIMISING NATURAL FREQUENCY AND MAC-VALUE ERRORS OF A BEAM MODEL

This paper reports the procedure followed by the ‘LTAS-Vibrations et Identification des Structures’ research group to generate a low-order finite element (FE) model of the GARTEUR SM-AG19 structure. The model is made of beam elements, local inertia and rigid body elements. The philosophy of the updating method is first based on the correlation of the experimental data with the results of the FE model and on the localisation of errors in the model. Updating parameters are then selected using eigenvalue sensitivity and model error localisation analyses. After updating, the quality of the FE model is assessed in terms of accuracy of the response prediction to structural modifications.     

Model selection in finite element model updating using the Bayesian evidence statistic

This paper considers the problem of finite element model (FEM) updating in the context of model selection. The FEM updating problem arises from the need to update the initial FE model that does not match the measured real system outputs. This inverse system identification-problem is made even more complex by the uncertainties in modeling some of the structural parameters. Such uncertainty often results in a number of competing forms of FE models being proposed which leads to lack of consensus in the field. A model can be formulated in a number of ways; by the number, the location and the form of the updating parameters. We propose the use of a Bayesian evidence statistic to help decide on the best model from any given set of models. This statistic uses the recently developed stochastic nested sampling algorithm whose by-product is the posterior samples of the updated model parameters. Two examples of real structures are each modeled by a number of competing finite element models. The individual model evidences are compared using the Bayes factor, which is the ratio of evidences. Jeffrey's scale is then used to determine the significance of the model differences obtained through the Bayes factor.     

Updating non-linear components

The objective of the present investigations was updating of finite element (FE) models with local non-linearities, such as Coulomb friction, gaps, local plasticity. Parameters of non-linear elements in the input file of a FE code are updated by fitting simulated time history functions and the corresponding measurement data. The problem of estimating the initial values as well as the problem of increasing error between simulated and measured time history functions have been overcome by using the method of 'modal state observers'. State observers are known in control theory but are a new approach for FE analysis.The presented methods use least square algorithms with analytically and numerically calculated sensitivity matrices for the updating process. A program for updating on principle any parameter of the input file of a standard FE code is described. The only requirement is, that the parameters should have a significant influence on the measured time history function. All of the presented methods have been validated against test results.     

基于频响函数 Neumann 级数展开的有限元模型修正

针对有限元模型修正中测试自由度不完备、求解不适定的问题,提出了一种基于频响函数Neumann级数展开的模型修正方法。首先,利用测试和有限元分析模态数据构造完整的实测频响函数。然后,根据Neumann级数展开式建立实测频响函数与有限元模型频响函数之间的关系式,以此构建模型修正目标函数。最后,采用改进鲸鱼优化算法求解目标函数获取修正结果。通过平面桁架数值算例表明,该方法具有较强的噪声鲁棒性,模型修正精度较好,15%噪声时修正后的平均相对误差不超过1%。利用4层剪力框架结构测试算例进一步验证所提方法,结果表明,修正后的模型能反映结构真实状况。     

PRINCIPAL COMPONENT DECOMPOSITION BASED FINITE ELEMENT MODEL UPDATING FOR STRAIN-RATE-DEPENDENCE NONLINEAR DYNAMIC PROBLEMS

Thin wail component is utilized to absorb impact energy of a structure. However, the dynamic behavior of such thin-walled structure is highly non-linear with material, geometry and boundary non-linearity. A model updating and validation procedure is proposed to build accurate finite element model of a frame structure with a non-linear thin-walled component for dynamic analysis. Design of experiments （DOE） and principal component decomposition （PCD） approach are applied to extract dynamic feature from nonlinear impact response for correlation of impact test result and FE model of the non-linear structure. A strain-rate-dependent non-linear model updating method is then developed to build accurate FE model of the structure. Computer simulation and a real frame structure with a highly non-linear thin-walled component are employed to demonstrate the feasibility and effectiveness of the proposed approach.     

Modeling the powder compaction process using the finite element method and inverse optimization

This paper focuses on studying and adapting modeling techniques using the finite element method to simulate the rigid die compaction of metal powders. First, it presents the implementation of the cap constitutive model into ABAQUS FE software using the closest point projection algorithm. Then, an inverse modeling procedure was proposed to alleviate the problems raised by the interpretation of the experimental tests and to more accurately determine the material parameters. The objective function is formed, based on the discrepancy in density data between the numerical model prediction and the experiment. Minimization of the objective function with respect to the material parameters was performed using an in-house optimization software shell built on a modified Levenberg?CMarquardt method. Thus, an integrated simulation module consisting of an inverse optimization method and a finite element method was developed for modeling the powder compaction process as a whole. The simulation and identification module developed was applied to simulate the compaction of some industrial parts. The results reveal that the maximum absolute error between densities is 2.3%. It corresponds to the precision of the experimental method.     

基于矩阵逼近的模型修正方法的研究

提出一种新的以试验振动和参数辨识的数据为参数,进行有限元分析模型修正的方法。该方法基于矩阵最佳逼近理论,运用Ｂａｙｅｓ估计原理来处理试验结果误差带来的试验模态可信度问题,求取分析模对试验获得的不完备模态的谱点的最佳逼近结果,最后获得质量阵的最小修正模型。     

非稳态罚函数遗传算法及其用于机械/结构系统的健壮性设计

在机械 /结构的优化设计中 ,普遍存在约束的作用 ,且最优解往往位于可行域的边界上。由于外界环境的变化或人为因素造成设计变量扰动 ,可能使设计成为不可行。本文提出了一种基于设计变量敏感性的健壮性设计方法 ,并用非稳态罚函数遗传算法来实现多目标优化设计。