Mesh Density Effects on Finite Element Model Updating

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1 Mesh Density Effects on Finite Element Model Updating M. Imregun and D. J. Ewins Imperial College of Science, Technology and Medicine Mechanical Engineering Department London SW1 2BX United Kingdom ABSTRACT. This paper investigates the feasibility of updating finite element models with coarser-than-usual meshes and examines the consequences of using such meshes from a model updating viewpoint. A channel-section structure with relatively simple geometry was modelled using three different mesh densities and the resulting analytical models were correlated with the experimental model obtained from measurements on an actual test structure. All three finite element models were then updated using the inverse eigensensitivity method and the success of the updated models was investigated with particular emphasis on mesh density. Although the model with the highest mesh density produced the best updated model, because of the good initial agreement between the analytical and experimental modal models, it was concluded that the coarse mesh route was worth pursuing, especially in the light of the prohibitive computational requirements associated with large finite element models. NOMENCLATURE [A] : Modal parameter variation vector [AM] : Mass correction matrix [M] : Mass matrix [K] : Stiffness correction matrix [AK] : Stiffness matrix [S] : Sensitivity matrix A e Ne P X : Analytical : Element : Number of individual finite elements : Correction factor : Experimental 1. INTRODUCTION Growing demands for quality and reliability in all types of structure and machine generate a need for the ability to predict the dynamic characteristics of engineering structures. This need can only be satisfied by the availability of suitable mathematical representations of the structures under study and state-of-the-art finite element technology is addressing the problem with varying levels of success trl. Although structural dynamics models resulting from such representations need to be validated against measured data, this is not an end in itself since an indication of poor (or even reasonable) performance of the mathematical model is not of much use to the design engineer who needs to know the dynamic response of the structure with good accuracy. Hence, the primary requirement is the correction of the finite element model in the light of measured data, a process known as model updating [al. One of the prerequisites in model updating is an initial closeness of the analytical and experimental models. In finite element modelling, the model 1372

2 quality is very often associated with the mesh density, an axiom which remains unchallenged so far. However, when the objective is to correct or to update the initial finite element model of a structure, there are enormous computational advantages in keeping this initial model small while retaining a basic correlation with the experimental model. Indeed, large finite element models are often vety difficult to update because of the relative insignificance of the individual elements which must be corrected one by one. This is especially true for both response and sensitivity-based methods which assign and compute correction factors for each elemental matrix. One is then confronted with the dilemma of (i) updating a large (and hopefully fullyconverged) model and accepting the computational consequences, or (ii) starting with a small (and probably not fully-converged) model and reducing the computational requirements by several orders of magnitude. If both approaches above lead to comparable updated models in terms of representing the dynamic behaviour of the actual structure, then there will be a very good argument in favour of choosing route (ii) because of its obvious computational advantages. In any case, if the finite element model is going to be corrected in a global sense by multiplying its individual elements by arbitrary correction factors, the initial discretization, determined by the mesh size, becomes of secondary importance since the updated model will not have all the discretization properties of the initial one. Therefore, the main purpose of this paper is to explore route (ii) and to investigate the possibility of updating finite element models with relatively coarse meshes with the eventual aim of tackling very large-size problems. element sense and the investigation will focus on whether it is possible to update them in such a way such that their performance is comparable to that of the updated model obtained from the third finite element model. To make the exercise more realistic, it is also proposed to use true measured FRF data, rather than simulated test data; a feature which makes any updating attempt much more taxing. Finally, it was decided to use a sensitivity-based updating algorithm rather than a response-based one but this choice is rather arbitrary and work is on the way to repeat the same updating exercise using the latter technique. 2. BASIC THEORY The Inverse Eigensensitivity Method (IESM) has been the subject of numerous research papers on model updating 131 and only a summary will be given here. Let [MA] and [KA] be the mass and stiffness matrices of the FE model, and [Mx] and [Kx] those of the real structure. One can now write [Mxl = [MAI + [AMI Kxl = [&I + [AK1 (1) where matrices [AM] and [AK] represent the mass and stiffness error matrices between the two models. These can be defined as follows: [AM]= 2 pmc [M,] e=, To this end, it is proposed to have three different finite element models of increasing mesh density to represent a channel-like structure and to try to update each of these models using the same updating algorithm. Unlike the third one, the first two models are not fully-converged in the finite [AK1 = 2 P& [K, 1 e=, (2) 1373

3 Ne is the number of (super) elements in the FE being estimated to be E=206 GN/m* and p =7860 model, and pme, pde and pke are the element kg/m3. The structure is not continuous but is mass and stiffness correction factors for made of several (sometimes overlapping) plates individual finite elements. It is also possible to joined together by numerous nominally-identical include damping in a similar fashion. spot welds. (Fig. 1). By defining a number of design variables and computing the sensitivity of the eigensolution to changes in these parameters together with an objective function to minimise the differences between the current and target eigensolutions, it is possible to write a matrix sensitivity equation of the form: the iterative solution of which yields the required correction factors. One problem associated with equation (3) is the inherent ill-conditioning of the sensitivity matrix since the magnitudes of the eigenvector derivatives are usually much smaller than those of the eigenvalue derivatives. Therefore the sensitivity matrix must be balanced prior to the solution, a procedure which may well lead to diminish the effect of the eigenvalues which are suppressed numerically. In addition, some sensitivity algorithms introduce constraint equations of the form: (3) Cti pi = 0 (4) where CXis the penalty coefficient which expresses the level of confidence in the i* element variable pi. 3. MODAL ANALYSIS AND CORRELATION The structure was tested in a free-free condition using impact testing. Various support conditions were tried to minimise the suspension and it was eventually decided to suspend the test piece from one comer by drilling a very small hole. A total of 120 FRFs were measured along the channel length in both cross-section axes directions in the O-1600 Hz frequency range with a 2 Hz frequency increment. The measured FRFs were analysed using a global curve-fitter and the mode shapes were identified directly. Most of the modes were found to be real but some exhibited highly complex behaviour, probably due to several spot welds joining the various plates together (Fig. 2). The structure was found to be lightly damped, the structural damping values ranging between 0.5 and 1.7% for the first ten modes. 3.2 Finite Element Modelling The channel structure was modelled using the finite element program ANSYS and 4-node quadratic plate elements (STIF63) were used throughout, The joint model was restricted to the overlap of the two plates which were assumed to be connected in a perfectly rigid fashion. The discontinuity in the structure was also included in the model by two overlapping plates. Table 1 Details of the FE models used 3.1 Modal Test and Analysis The overall dimensions of the channel structure are 570 x 90 x 1 mm, the material properties 1374

4 Table 2 The predicted natural frequencies (Hz) of the M 1 Description 1 Small 1 MAC 1 Medium 1 MAC I Law? Correlation of Finite Element and Modal Test Results The degree of correlation between the experimental model and the three finite element models was investigated in several stages: (i) Comparison of natural frequencies (Table 3), (ii) MAC calculations (Fig. 3), (iii) 45O mode shape plots (Fig. 4), and (iv) FRF overlays (Figs. 5). I I I 10 I , After experimenting with different mesh sizes, it was decided to use three different meshes with an increasing number of nodes and elements (Table 1). To avoid errors due to matrix reduction, the individual finite element mass and stiffness matrices were read from ANSYS and a full eigensolution was performed by reassembling these matrices, The results are summarised in Table 2 which also includes MAC values for the large-medium and the medium-small FE model pairs. As can be seen from Table 2, the natural frequency agreement between the models is not very good although the basic vibratory behaviour is predicted reasonably well in the sense that the first two close modes are followed by a cluster of again close modes with higher natural frequencies. This behaviour is mainly due to the lack of symmetry because of the discontinuity along the channels length which is modelled by two overlapping plates. It should also be pointed out that almost all mode shapes involve a large amount of torsional motion and hence the natural frequencies are very sensitive to changes in the cross-sectional properties. As one would expect, the agreement between the FE and experimental models increases with increasing mesh density and the small FE model is not a particularly good representation of the test structure. 4. UPDATING THE FE MODELS 4. 1 Numerical Considerations Program MODULATE 141 was used for updating the three finite element models described above. Individual element mass and stiffness matrices, together with the connectivity information, are read into the program and the global mass and stiffness matrices are assembled internally at each iteration step. In this particular study, the correction factors were computed using both eigenvalue and eigenvector sensitivities and it was noted that the IESM updated model was not unique since the results depended on: (i) the number of modes kept in the sensitivity analysis, (ii) the balancing of the sensitivity matrix, (iii) the selection of elements to be included in the formulation, and (iv) the use of penalty functions. The convergence of the algorithm for all three cases was found to be dependent on the number 1375

5 of modes kept in the analysis. Generally speaking, the convergence was found to be faster and more stable when a small number of modes was used. In all cases considered, it was found necessary to balance the sensitivity matrix as this contains elements spanning several orders of magnitude. However, as there are many ways to balance a matrix (by columns, by rows or by using a combination) there will be at least as many updated models, each corresponding to a particular balancing case. Here, it is proposed to focus attention on the best results only since matrix balancing is very case-dependent and general conclusions cannot easily be drawn. It was noted that some of the elements contributed little to the variation of the correction factors and so these were excluded from the analysis for numerical reasons. However, it should be noted that there is little physical or modelling basis for adopting this approach since there is no reason why modelling errors should not exist at insensitive parts of the structure. In any case, the convergence of the sensitivity algorithm was found to be very dependent on the number of elements kept in the analysis and, in all the cases studied, no convergence was achieved when all the elements were kept in the analysis. Once an acceptable combination of parameters (i) to (iv) was determined, it was possible to compute iteratively a set of modification factors which minim&d the difference between the experimental and theoretical models. The main parameters are summarised in Table 4. Table 4 Key parameters for IESM updating small Medium Large Model Model Model Modes used Balancing Column cohlmn Cd& row Elements kept 70% 60% 60% Penalty func. YES NO NO 4.2 Comparing the Updated Models The success of the updated models was assessed by defining and comparing two parameters. Global frequency error. M0dC.S IIAfll = [ z((target fi - Input fi)/target fj )2] O.5 :-I Also, the modification factors were constrained to Global eigenvector error be as small as possible by using a penalty function Modes Co-ords technique, a safeguard which prevents them from becoming unrealistically large and which reflects IIA~II = [ X ) C - Input ~ij 2 ] 0.5 the degree of confidence in the original FE model: i=l j=l in general terms, the more weight that is given to the penalty constraints, the more trust is placed in The percentage improvement of the updated the FE model. In cases where the very first model was defined by yf and j $, where iteration yields excessively large modification factors and the resulting system matrices are not positive-definite, the use of penalty constraints y = (Ilh II initial - IIA II updated) / lla 1 initial x 100 may shift the minimisation process to a different solution path with faster convergence, an and the results are listed in Table 5. acceptable course of action since the solution is not expected to be unique. 1376

6 The natural frequencies corresponding to all three cases above are listed in Table 6. The MAC values between the experimental model and the three updated models are given in Table 7 As the final and most taxing comparison, the measured, initially-predicted and updated FRFs are plotted in Fig. 6 for all three finite element models. At this stage it must be remembered that the IESM actually updates the synthesised FRF which is regenerated using identified modal parameters rather than measured (raw) data and much depends on the quality of modal analysis which produces the reference (or target) set of modal parameters. 4.3 Computational Requirements The updating exercise was conducted on a variety of computers ranging from 486.based PCs to a Cray Y-MP8, and including DEC 5100 and IBM RS/6000 workstations. The main two requirements of in-core memory and dedicated CPU time are summarised below in Table 8 where the latter has been normalised with respect to the IBM RS/6000 model 530 processor. Four iterations were needed for the small and medium models while five iterations were used for the large model. It should also be noted that Table 8 does not include the CPU effort required for the trial runs in order to determine the key parameters such as the number of modes, the number of elements kept in the analysis and the balancing method to be used. Table 8 Computational requirements 1 Small 1 Medium 1 Large 1 5. CONCLUDING REMARKS (i) An inspection of the updated FRFs reveals that the large FE model produces the best updating results, closely followed by the medium model. Also, The small model results are not disappointing at all and they certainly exhibit a very marked improvement over those obtained from the initial FE model. The immediate conclusion is that it is possible to update any FE model, including those with coarser meshes, provided there is a basic initial agreement. (ii) When both eigenvalue and eigenvector sensitivities are used, the computational requirement rapidly becomes prohibitive with increasing model size. There are many instances for which it is not possible to update existing models unless the mesh size is reduced significantly. The present feasibility study originates from the need to explore such an approach. (iii) An immediate application of (ii) can be found in the common practice of using the same finite element model for both static and dynamic calculations. In the former type of analysis it is important to capture stress concentrations and hence particular emphasis is placed on sudden geometry changes. However, these features are not all that important in a dynamic analysis where the overall stiffness and mass play a much more important role and hence there is a case for simplifying the static analysis model (iv) The question of which model to choose as the starting point invites a much more fundamental question: the purpose for which the updated model is going to be used. As there are so many different applications, it is not possible to generalise or to make recommendations, However, the coarse mesh route may well be an attractive option for a number of cases. 1377

7 6. ACKNOWLEDGEMENTS The authors gratefully acknowledge the contribution of the Nissan Motor Co. for sponsoring some of the work and for providing the test specimen. The authors also thank Dr. A. S. Nobari and Mr. N. Imamovic for providing most of the numerical data presented in this paper. 7. REFERENCES 1. Ewins, D. J. and Imregun, M. State-of-the-Art Assessment of Structural Dynamic Response Analysis Methods (DYNAS), Shock and Vibration Bulletin, Vol. 56 (I), Imregun M., and Visser W. J. A Review of Model Updating Techniques, Shock and Vibration Digest., Vol. 23, No 1, ~~9-20, Jan Zhang, Q. and Lallement, G. A Complete Procedure for the Adjustment of a Mathematical Model From Identified Complex Modes, Proc. IMAC 5, April MODULATE Users Guide to Version 1.0, Imperial College Analysis, Testing and Software, September 1993 Table 3 Measured and predicted natural frequencies (Hz) of the channel structure Mode Exp. Small Error Medium Error Large Error Mode, m \ ^rl^l I% l,.-l.., 0% 7, I I UGI, 7, I I szi, I ; :.tl - l j-23.4 ) ) c l ~ I, l-23.9 I I I I-12.8 ( (-30.7 ( I j-12.9 c OAA 1 J 0 U.I ( ~ : 3.3 ( r; - I - I I I 24.2% I 14.3% I 5.5% I Table 5 Updating parameters for the three finite element models III IlAfll l1al$ll Stlltdl Medium Large Initial FE Updated y Initial FE Updated y Initial FE updated y % % % % % % 1378

8 Table 6 Natural frequencies (Hz) predicted by the updated models Mode Exper. Small Error Medium Error Large Error I I - 10 / Average error 5.1% (Initial error) (24.2%) 4.4 Yb0.l I I 0.U 1.9% 0.0% (14.3%) (2.9%) Table 7 MAC values (%) between experimental and updated models (The last row is the average value for that column. The correlating mode pairs are shown in brackets) Small MC :dium Large I -- \.,.,, 41 (8,ll) 146 (8,9) I43 ~~ i8.9, (28 (8,9) 42 (- ~,t(, -, 4b *- (9,8) j 72 (10.11) I65 (10.11) I I 50 I 59 I b-l I I Fig 1. The test structure 1379

9 I Fig 2. The experimentally-determined mode shapes 1380

10 (a) 03) (cl Fig. 3 The MAC values for (a) small (h) medium and (c) large models Fig. 5 Measured and initially-predicted point inertances 1381

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