Error localization method of structural modeling based on equivalent element modal energy change【推荐论文】 .doc

精品论文Error localization method of structural modeling based on equivalent element modal energy change5Zang Chaoping(College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016)Abstract: In this paper a novel method for error localization of a structural design model based onequivalent element modal energy change is introduced. The concepts of Equivalent Element Modal10Strain Energy (EEMSE) and Equivalent Element Modal Kinetic Energy (EEMKE) are discussed and two indicators, i.e. the indicator of element errors in stiffness based on EEMSE and the indicator ofelement errors in mass based on EEMKE, are developed to identify element errors in the design model.A plate structure is used as an example to demonstrate the effectiveness of the method. Various types of the plate model with meshing errors or combinations of meshing and parameter errors, together with15the refined model which is treated as a supermodel to provide reference data, have been simulated and further compared with the distributions of element strain energy and kinetic energy. The EquivalentElement Modal Strain Energy (EEMSE) and Equivalent Element Modal Kinetic Energy (EEMKE), together with the indicators of EEMKE and EEMSE, were used to identify clearly various errors in theplate model. This method shows potential for the practical application in the effective and efficient20validation of a design model.Key words: dynamics; error localization; Equivalent Element Modal Strain Energy;EquivalentElement Modal Kinetic Energy ; indicator; model validation0Introduction25In modern structural design, finite element modelling is used extensively for the structural response analysis and design prediction. However, the lack of agreement between FE predictions and the references (experimental observations or simulations from supermodels) can always be found due to the inaccuracies present in FE models. Therefore, correcting inaccurate parameters in FE models is necessary in order to improve the agreement between predictions and test results.30The processes involved in making these corrections are considered as model updating1.The model updating process includes verification and correlation between the information of an FE model and the experimental data, and updating the parameters in the FE model in order to reduce the discrepancies between the experimental data and the dynamic characteristic predicted by the model. Practically, two major problems to deal with in updating process are those of35measurement incompatibility. One is the spatial incompatibility linked to the measurement of mode shapes through a limited set of physical sensors and their analytical prediction at a large number of finite element degrees of freedom (DOFs). Another is that the number of measured modes is far fewer than the number of analytical ones calculated from the FE model. Expansion of the measured mode shape data or reduction of the size of the initial FE model to the measured40degrees of freedom is normally used to match the requirement. However, both techniques will bring some erroneous information into the process. The current extensive review2,3,4 indicates that the success of model updating is likely to remain case-dependent and applicability is bounded by the skill of the experts in choosing a correct updating procedure. Another disadvantage is that itusually takes a long time and also is very costly in the current design process to manufacture a45physical structure or a prototype of the design for vibration tests. As a result, a validated FE model through updating is not involved in the first stages but only in the late stages of the design processFoundations: Research Fund for the Doctoral Program of Higher Education of China (No. 20093218110008) Brief author introduction:Zang Chaoping (1963-),male,Professor, Main research: structural dynamics and model validation. E-mail: zangchaopingnuaa.edu.cn- 15 -and the effectiveness of its guiding of the design is extremely limited.Recently, a new method of using simulations from supermodels 5, 6 is developed to replace the experimental data as a reference for model updating. A supermodel is considered to be capable50of representing all geometric features and its dynamic properties and, therefore, can be taken as representative of those of the actual structure. As a supermodel is usually created with a refined mesh and can be ideally employed in virtual experiments, it would obviously be advantageous to help to construct and update the design model. In a model updating process, based on a supermodel, more modes can be produced and more accurate mode shapes can be represented by55as many DOFs as required. Such features can overcome the drawback of measurement incompleteness and generally avoid ill-conditioning problems in updating solutions. However, the method for updating-parameter selection based on reference information from supermodels still needs to be explored. Currently, the selection of updating parameters is generally based on the engineers judgment and experience. The sensitivity matrix in iterative updating methods can help60to select the updating parameters. However, the parameters to be selected only indicate those with high sensitivity and not those with erroneous values. In this paper, a new method to localise errors in the design model for model updating via supermodels will be presented and explored.1Error localization of a model based on equivalent element modal energy change651.1Overview of modeling errorsA design model, usually with a course mesh, is considered to include three kinds of error7: (1) discretisation errors, (2) configuration errors, and (3) parameter errors. Discretisation errors are those that arise because the mass and stiffness matrices for each element of the model are formed in a way that is different from the equation of motion for a continuous system. Configuration70errors are those that arise because some physical features are omitted when model simplifications are made in representing complicated parts in a structure. Correction of discretisation and configuration errors is normally involved in the scope of design updating. Parameter errors are those that result from inaccurate estimation for some parameters while constructing the model. In the updating process, such errors can be corrected in order that the coefficients in the model are75sufficiently accurate to enable the model to provide an acceptably correct description of the subject structure's behaviour. As all three types of error are mixed in a design model, a big challenge is to recognise them properly and find the causes of the discrepancies in the model.1.2Concepts of element strain energy and element kinetic energyFor a discrete finite element system, the dynamic equation of an element can be expressed as80me u&&e + ce u& e + k e ue = fe (1)where ue is the vector of generalized nodal displacements which may include translationaldisplacements, rotational displacements or other quantities.fe is the equivalent nodal forcevector of externally-applied forces. Matrices ke ,ce ,and me are defined as the elementstiffness, damping, and mass matrices respectively. If the relationship between the element nodal85displacement ue and the system (or global) nodal displacements are established throughgeometrical relations as below:where Te ue = Te is a transformation (or connectivity) matrix for element e.(2)Then, the element strain energy (ESE) can be given byESE = 1 u T ku = 1 T T T kT e 2 e e e 2e e ee= 1 T K902(3)where K = T T kT which is the element stiffness matrix associated with the globale e e enodal displacement vector .Similarly, the element kinetic energy (EKE) can be given byEKE = 1 u&T m u& = 1 & T T T m T & e 2 e e e 2e e ee= 1 & T M & 2 (4)e e e e95where M = T T m T displacement vector .is the element mass matrix associated with the global nodal1001.3Element modal strain energy and element modal kinetic energyFor each element of the finite element model, Equations (3) and (4) show the element strain and kinetic energy under the equivalent nodal force vector of the externally applied forces. Both energies include the contribution of all vibration modes excited by the externally applied forces in dynamics.In order to investigate the elemental strain energy and kinetic energy caused by each mode, the Element Modal Strain Energy (EMSE) and the Element Modal Kinetic Energy are introduced. Considering the jth element for contribution of the ith mode, the Element Modal StrainEnergy (EMSE) and the Element Modal Kinetic Energy (EMKE) can be written asEMSE= 1 f T K f105ij 2 ie j i(5)1 & T &1 2 T MEMKEij = 2 f ie j f= wi fMi 2 ie j f i(6)jwhere Ke is the jth element stiffness matrix andM e is the jth element mass matrix.wiji2 is the eigenvalue of the ith mode of the structure and fis the associated global nodal110115displacement of the ith mode corresponding tothe jth element.1.4Equivalent element modal strain energy and equivalent element modal kinetic energyThe EMKE and EMSE of a mode in Equations (5), and (6) can tell the energy distribution over all the elements. As the energy distribution of a model is calculated using its own dynamic features (eigenvalues and eigenvectors), direct comparison of the EMKE and EMSE distributions of two models (for example, a design (coarse) model and a supermodel (refined model) may not able to straightforward describe the source of the discrepancies between them. It is because that the EMKE and EMSE of supermodel and design model are calculated independently. To build a120relationship between two models, a new method will be developed to calculate the Equivalent Element Modal Strain Energy (EEMSE) and the Equivalent Element Modal Kinetic Energy (EEMKE) using the modal data of the eigenvalues and eigenvectors from the supermodel and the element stiffness matrices and mass matrices from the design model.The equation of calculatingthe Equivalent Element Modal Strain Energy (EEMSE) is written as1 s T éd ù s EEMSEij = 2 fi ëKeû j f i(7)125And the equation of calculating the Equivalent Element Modal Kinetic Energy is expressed as1 &s T éd ù &s 1 ( s )2 s T éd ù s EEMKEij = 2 fi ëMe û j f= wii 2f i ëMe û j f i(8)whereK d ù is the jth element stiffness matrix of a design model andûe jsM d ùis the jthûejelement mass matrix of a design model. wiis the eigenvalue of the ith mode of the supermodeliand f s is the associated global nodal displacement of the ith mode of the supermodel130135corresponding to the jth element of a design model.1.5Indicators of error localizationBased on the definitions of the Equivalent Element Modal Strain Energy (EEMSE) and the Equivalent Element Modal Kinetic Energy (EEMKE), two indicators are proposed in order to localise possible discrepancies in a design model.The first indicator is called the indicator of Equivalent Element Modal Strain Energy that aims to localise the possible errors in the elemental strain matrix of a design model. The second indicator is the indicator of Equivalent Element Modal Kinetic Energy that indicates possible errors in the elemental mass matrix of a designmodel. Both indicators are written ascstiffness =EEMSE d - EMSE sij ijEMSE scmass =ijEEMKE d - EMKE sijijEMKE s(9)140ij (10)ijwhereEEMSEdis the equivalent element modal strain energy of the jth element of the designmodel on the corresponding ith mode of the supermodel.ijEEMKEdis the equivalent elementmodal kinetic energy of the jth element of the design model on the corresponding ith mode of theijsupermodel.EMSE sis the element modal strain energy of the ith mode of the corresponding145region of the supermodel matching or approximately matching the same area of an jth individualijelement of the design model.EMKEsis the element modal kinetic energy of the ith mode of150the corresponding region of the supermodel matching or approximately matching the same area of an jth individual element of the design model. Theoretically, the indicators will equal to zero if there is no discrepancies between the design model and the supermodel. When the discrepancy errors exist in the model, the values of indicators will describe the relative differences of energy distribution between two models compared with the energy distribution of the supermodel, whichindirectly indicate the mass or stiffness difference between both models2Model validation with a plate structure1551601651701751802.1ConfigurationThree FE models of a plate structure, which are Models A, B, and C, with the same dimensions are built and illustrated in Figure 1.Model A and B are created with the same mesh of 24 elements and 39 nodes and Model C with refined mesh of 384 elements and 441 nodes. In addition, material and property errors in some elements were added in Model B. The value of Youngs Modules in element 2 was increased by 50% while the value in element 18 was reduced by 30%. In the same time, the values of mass density in element 12 and 18 were reduced by 20%. The refined model C is considered as the supermodel and is used as a reference and the coarse model A and model B are treated as the design models. Model B also includes the mixture of discrepancy errors caused by the mesh and parameter errors.Fig. 1 FE models of a plate structureThe correlations of Models A and B against Model C were undertaken respectively. The natural frequency differences and the MAC values are listed in Tables 1 and 2. Notes in both tables describe the actual mode shapes of the mode. For instance, ZB1 refers to the first bending