有限元模型修正方法及自由度匹配迭代技术研究
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摘要
本文对有限元模型修正领域中的修正方法及自由度匹配的迭代技术进行了深入的研究。
在工程中,通常使用有限元建模的方式来掌握结构的动力学特性。现阶段,航空、航天、大型桥梁以及其它工程结构对有限元模型的精度及可靠性提出了更高的要求。然而,结构的连接部位、边界条件以及结构阻尼的建模存在着很大的难度,其中的众多参数具有一定的不确定性,使得对复杂结构建立的有限元模型往往不精确。因此需要利用有限元模型修正技术使用实验测得的数据对有限元模型进行修正,提高有限元模型的精度及可靠性。
传统的基于灵敏度分析的模型修正方法及修正参数的选取方法尽管应用范围广泛,但是其往往只注重数学意义上的修正,存在着诸如修正参数有限、计算耗时、灵敏度大的待修正参数有时并不是真正存在建模误差的参数等不足之处。因此,有必要找到一种计算效率高,不依赖灵敏度分析,能够同时修正大量待修正参数,并且希望在修正过程中自动找到真正存在误差的参数的模型修正方法。研究发现正交模型正交模态(CMCM)法满足以上的修正需求,然而CMCM法的修正结果仅为某一单元的整体修正因子,对于某一单元存在多个待修正参数的情况则无法分别计算出各个参数的修正量,并且在修正过程中存在着分母为零或者逼近零的危险,这给修正工作带来了困难。本文在保持原有CMCM法优越性的基础上,通过公式重新推导、变形,得到了改进的正交模型正交模态法即ICMCM法。ICMCM法解决了CMCM法存在的病态问题,并且其修正对象可以是任意的材料以及几何参数,增加了待修正参数的数量。并在ICMCM法的基础上进行扩展,推导得到了含有阻尼形式的ICMCM法,扩宽了ICMCM法的应用范围。利用ICMCM法的思想,将其应用到频响型模型修正方法中,获得了新的频响型模型修正方法——正交模型正交频响函数(CMCFRF)法。通过算例证明了ICMCM法以及CMCFRF法的有效性及优越性。
另外,在使用传统的模型修正与自由度匹配相结合的迭代方法时,仅仅使用有限元模型的缩聚/扩展转换矩阵代替实验模型的缩聚/扩展转换矩阵,未对上述替代的误差进行额外的处理,因此造成使用传统的模型修正与自由度匹配相结合的迭代方法时模型修正的计算效率低,迭代收敛速度慢。针对此问题,本文进行了深入的研究并在传统迭代方法的基础提出了一种新的模型修正与模型缩聚结合应用的迭代方法即误差循环迭代缩聚(ECIMR)法以及新的模型修正与模态扩展结合应用的迭代方法即误差循环迭代扩展(ECIME)法。上述两种新的方法分别在传统迭代方法的基础上添加了误差修正项,大大加快了自由度匹配后模型修正的迭代收敛速度并减少了计算时间。ECIMR法在推导过程中未限制所使用的模型修正与模型缩聚的方法,ECIME法并未限制所使用的模型修正方法,因此二者分别具有一定的通用性。在修正过程中使用新的迭代方法较传统的迭代法具有更快的迭代收敛速度以及计算效率,这对于工程中大型、复杂结构的模型修正问题具有重要的工程应用价值,使用新的方法可以节省大量的计算时间。通过ICMCM法以及CMCFRF法证明了上述两种迭代方法的通用性以及有效性。另外,使用ECIMR法进行迭代后,在相同的迭代次数下,随着模型缩聚精度的提高,模型修正的精度也将随之提高。
为了能够进一步验证本文所提出的模型修正方法对复杂结构尤其是航天对象的模型修正能力,对评价模型修正技术的基准模型即GARTEUR benchmark模型以及某型号的火箭舱段模型进行了模型修正,并获得了非常理想的模型修正结果。修正结果表明模态扩展方法更加适合ICMCM法,此时修正的方程数量远大于使用模型缩聚方法,进而可以修正数量较大的参数,模型修正结果也远好于使用模型缩聚方法。使用ICMCM法所能修正的参数数量远大于基于灵敏度分析的模型修正方法所能修正的参数数量,尽可能包含所有真正存在建模误差的参数,并利用修正方程自动的对真正存在建模误差的参数进行修正,使得修正结果更加的符合结构真实的物理意义。
In this dissertation, model updating methods and iterative techniques for degree of freedom matching in the field of finite element model updating are studied.
Accurate finite element (FE) models of engineering structures are needed in order to predict their dynamic characteristics. Nowadays, the appearances of high precision aircrafts and spacecrafts, large scale bridges and other new engineering structures impose higher demands on the reliability and accuracy of their FE models. However, a FE model may be inaccurate especially in the case of complex structures due to difficulties in the modeling of joints, boundary conditions and structural damping. So the development of model updating method which has the ability of correcting the numerical values of individual parameters in a FE model using the data obtained from an associated experimental model is necessary. The updated model can describe the dynamic properties of the subject structure more accurately.
The traditional eigensensitivity-based approaches have extensive application, however, this kind of model updating methods pay much attention to the updating results and have many disadvantages, for example, they have limited updating parameters, time consuming, especially, the updated parameters that have high sensitivity may not be the ones that have modeling errors. So it is necessary to find a new method which has high calculation efficiency, does not need sensitivity analysis, updates more parameters simultaneously and finds the parameters that have modeling errors indeed. It is found that the cross-model cross-mode (CMCM) method meet the former requirements, however, the updating results are the correction factor of one element, so in the case of a element has more than one parameter that have modeling errors, the CMCM method lose its updating ability. On the other hand,the denominator of the updating formula may equal or close to zero which leads to the emergence of ill-conditioned problems and brings difficulties to model updating process. This paper propose a new model updating method named as improved cross-model cross-mode (ICMCM) method which based on the CMCM method. The new method avoided the disadvantage of CMCM method by changing the process of formula derivation. Moreover, the updating object can be any physical or geometric parameters of one element, which increase the number of the updating parameters. The ICMCM method with damping is obtained by extending the ICMCM method, which enlarges application range of the original method. A new model updating method based on the frequency response function named as cross-model cross-frequency response function (CMCFRF) method is obtained by using the theory of ICMCM method. Updating results show the effectiveness and superiority of ICMCM method and CMCFRF method.
In addition, using the traditional iterative method that associating the model updating method with the degree of freedom matching technique, the errors resulted from replacing the reduction/expansion matrix of the experimental model with those of the FE model are not fully considered, which needs more iterations and computing time in the model updating process. In order to reduce the errors produced in the replacement, based on the traditional methods, a new iterative method named as error cyclic iteration method based on model reduction (ECIMR) that associating the model updating method with the model reduction technique and a new iterative method named as error cyclic iteration method based on modal expansion (ECIME) that associating the model updating method with the modal expansion technique are proposed, in which the correction term related to the errors is added. The convergence rate and the computing time of the new method are significantly superior to those of the traditional iterative method. Because ECIMR method does not limit the model updating method and the model reduction method and ECIME method does not limit the model updating method, both of these two new iteration methods have generalities. One of important advantages of the new iterative methods is that they make the calculation converge with less iteration and computing time than those of the traditional methods, which are very useful for model updating of large-scale engineering structures, they can save much computing time and improve work efficiency. In addition, the model updating precision becomes higher with the precision of the model reduction upgraded after using ECIMR method.
In order to further verify the updating ability of the proposed model updating method for the complex structures especially the aerospace structures, GARTEUT model which is the benchmark model for model updating and a cabin structure of a certain type rocket are used as the updating objects, and the perfect updating results are obtained. Updating results show that the modal expansion technique is more suitable for ICMCM method. The ICMCM method using modal expansion can update far more updating parameters than using model reduction, and the updating results are better by using modal expansion than by using model reduction. ICMCM method can update more parameters than the eigensensitivity-based approaches, which can include the parameters with modeling errors as much as possible. The updating equations can automatically update the parameters that have modeling errors indeed and the updating results meet the real physical meaning of the structures using ICMCM method.
在工程中,通常使用有限元建模的方式来掌握结构的动力学特性。现阶段,航空、航天、大型桥梁以及其它工程结构对有限元模型的精度及可靠性提出了更高的要求。然而,结构的连接部位、边界条件以及结构阻尼的建模存在着很大的难度,其中的众多参数具有一定的不确定性,使得对复杂结构建立的有限元模型往往不精确。因此需要利用有限元模型修正技术使用实验测得的数据对有限元模型进行修正,提高有限元模型的精度及可靠性。
传统的基于灵敏度分析的模型修正方法及修正参数的选取方法尽管应用范围广泛,但是其往往只注重数学意义上的修正,存在着诸如修正参数有限、计算耗时、灵敏度大的待修正参数有时并不是真正存在建模误差的参数等不足之处。因此,有必要找到一种计算效率高,不依赖灵敏度分析,能够同时修正大量待修正参数,并且希望在修正过程中自动找到真正存在误差的参数的模型修正方法。研究发现正交模型正交模态(CMCM)法满足以上的修正需求,然而CMCM法的修正结果仅为某一单元的整体修正因子,对于某一单元存在多个待修正参数的情况则无法分别计算出各个参数的修正量,并且在修正过程中存在着分母为零或者逼近零的危险,这给修正工作带来了困难。本文在保持原有CMCM法优越性的基础上,通过公式重新推导、变形,得到了改进的正交模型正交模态法即ICMCM法。ICMCM法解决了CMCM法存在的病态问题,并且其修正对象可以是任意的材料以及几何参数,增加了待修正参数的数量。并在ICMCM法的基础上进行扩展,推导得到了含有阻尼形式的ICMCM法,扩宽了ICMCM法的应用范围。利用ICMCM法的思想,将其应用到频响型模型修正方法中,获得了新的频响型模型修正方法——正交模型正交频响函数(CMCFRF)法。通过算例证明了ICMCM法以及CMCFRF法的有效性及优越性。
另外,在使用传统的模型修正与自由度匹配相结合的迭代方法时,仅仅使用有限元模型的缩聚/扩展转换矩阵代替实验模型的缩聚/扩展转换矩阵,未对上述替代的误差进行额外的处理,因此造成使用传统的模型修正与自由度匹配相结合的迭代方法时模型修正的计算效率低,迭代收敛速度慢。针对此问题,本文进行了深入的研究并在传统迭代方法的基础提出了一种新的模型修正与模型缩聚结合应用的迭代方法即误差循环迭代缩聚(ECIMR)法以及新的模型修正与模态扩展结合应用的迭代方法即误差循环迭代扩展(ECIME)法。上述两种新的方法分别在传统迭代方法的基础上添加了误差修正项,大大加快了自由度匹配后模型修正的迭代收敛速度并减少了计算时间。ECIMR法在推导过程中未限制所使用的模型修正与模型缩聚的方法,ECIME法并未限制所使用的模型修正方法,因此二者分别具有一定的通用性。在修正过程中使用新的迭代方法较传统的迭代法具有更快的迭代收敛速度以及计算效率,这对于工程中大型、复杂结构的模型修正问题具有重要的工程应用价值,使用新的方法可以节省大量的计算时间。通过ICMCM法以及CMCFRF法证明了上述两种迭代方法的通用性以及有效性。另外,使用ECIMR法进行迭代后,在相同的迭代次数下,随着模型缩聚精度的提高,模型修正的精度也将随之提高。
为了能够进一步验证本文所提出的模型修正方法对复杂结构尤其是航天对象的模型修正能力,对评价模型修正技术的基准模型即GARTEUR benchmark模型以及某型号的火箭舱段模型进行了模型修正,并获得了非常理想的模型修正结果。修正结果表明模态扩展方法更加适合ICMCM法,此时修正的方程数量远大于使用模型缩聚方法,进而可以修正数量较大的参数,模型修正结果也远好于使用模型缩聚方法。使用ICMCM法所能修正的参数数量远大于基于灵敏度分析的模型修正方法所能修正的参数数量,尽可能包含所有真正存在建模误差的参数,并利用修正方程自动的对真正存在建模误差的参数进行修正,使得修正结果更加的符合结构真实的物理意义。
In this dissertation, model updating methods and iterative techniques for degree of freedom matching in the field of finite element model updating are studied.
Accurate finite element (FE) models of engineering structures are needed in order to predict their dynamic characteristics. Nowadays, the appearances of high precision aircrafts and spacecrafts, large scale bridges and other new engineering structures impose higher demands on the reliability and accuracy of their FE models. However, a FE model may be inaccurate especially in the case of complex structures due to difficulties in the modeling of joints, boundary conditions and structural damping. So the development of model updating method which has the ability of correcting the numerical values of individual parameters in a FE model using the data obtained from an associated experimental model is necessary. The updated model can describe the dynamic properties of the subject structure more accurately.
The traditional eigensensitivity-based approaches have extensive application, however, this kind of model updating methods pay much attention to the updating results and have many disadvantages, for example, they have limited updating parameters, time consuming, especially, the updated parameters that have high sensitivity may not be the ones that have modeling errors. So it is necessary to find a new method which has high calculation efficiency, does not need sensitivity analysis, updates more parameters simultaneously and finds the parameters that have modeling errors indeed. It is found that the cross-model cross-mode (CMCM) method meet the former requirements, however, the updating results are the correction factor of one element, so in the case of a element has more than one parameter that have modeling errors, the CMCM method lose its updating ability. On the other hand,the denominator of the updating formula may equal or close to zero which leads to the emergence of ill-conditioned problems and brings difficulties to model updating process. This paper propose a new model updating method named as improved cross-model cross-mode (ICMCM) method which based on the CMCM method. The new method avoided the disadvantage of CMCM method by changing the process of formula derivation. Moreover, the updating object can be any physical or geometric parameters of one element, which increase the number of the updating parameters. The ICMCM method with damping is obtained by extending the ICMCM method, which enlarges application range of the original method. A new model updating method based on the frequency response function named as cross-model cross-frequency response function (CMCFRF) method is obtained by using the theory of ICMCM method. Updating results show the effectiveness and superiority of ICMCM method and CMCFRF method.
In addition, using the traditional iterative method that associating the model updating method with the degree of freedom matching technique, the errors resulted from replacing the reduction/expansion matrix of the experimental model with those of the FE model are not fully considered, which needs more iterations and computing time in the model updating process. In order to reduce the errors produced in the replacement, based on the traditional methods, a new iterative method named as error cyclic iteration method based on model reduction (ECIMR) that associating the model updating method with the model reduction technique and a new iterative method named as error cyclic iteration method based on modal expansion (ECIME) that associating the model updating method with the modal expansion technique are proposed, in which the correction term related to the errors is added. The convergence rate and the computing time of the new method are significantly superior to those of the traditional iterative method. Because ECIMR method does not limit the model updating method and the model reduction method and ECIME method does not limit the model updating method, both of these two new iteration methods have generalities. One of important advantages of the new iterative methods is that they make the calculation converge with less iteration and computing time than those of the traditional methods, which are very useful for model updating of large-scale engineering structures, they can save much computing time and improve work efficiency. In addition, the model updating precision becomes higher with the precision of the model reduction upgraded after using ECIMR method.
In order to further verify the updating ability of the proposed model updating method for the complex structures especially the aerospace structures, GARTEUT model which is the benchmark model for model updating and a cabin structure of a certain type rocket are used as the updating objects, and the perfect updating results are obtained. Updating results show that the modal expansion technique is more suitable for ICMCM method. The ICMCM method using modal expansion can update far more updating parameters than using model reduction, and the updating results are better by using modal expansion than by using model reduction. ICMCM method can update more parameters than the eigensensitivity-based approaches, which can include the parameters with modeling errors as much as possible. The updating equations can automatically update the parameters that have modeling errors indeed and the updating results meet the real physical meaning of the structures using ICMCM method.
引文
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