A Toolbox for Validation of Nonlinear Finite Element Models

Transcription

1 A oolbox for Validation of Nonlinear Finite Element Models Dr.MarkC.Anderson ACA Inc. 279 Skyark Drive, Suite 3 orrance, CA Phone: (3) anderson@actainc.com imothy K. Hasselman ACA Inc. John E. Crawford Karagozian & Case Structural Engineers Abbreviations DRA Defense hreat Reduction Agency HFPB High fidelity, hysics-based PCD Princial comonents decomosition SVD Singular value decomosition Keywords Model-test correlation, Model udating, Modeling uncertainty, Predictive accuracy, Surrogate model 5-

2 ABSRAC With increasingly owerful comutational resources at our disosal, it is becoming commonlace to use analytical redictions in lieu of exerimentation for characterization of hysical systems and events. his trend, with its erceived otential for reducing costs, is the basis for the simulation-based rocurement initiatives currently gaining momentum within the Government and industry. However, a simulation-based aroach is often sold to decision-makers via lausible visualizations of model simulation results; the simulations themselves having only been validated in an ad hoc manner using anecdotal comarisons with real events. Exerience has shown that, while simulation using hysics-based models may lead to qualitatively correct results, there can be large quantitative discreancies between simulation and exerimental results for a given hysical event, and between simulation results from different analysts for the same event. Obviously, for simulation-based rocurement to be a viable alternative to more traditional test-based rocurement, the quantitative accuracy of the simulations must be insured. o do this requires at least a modicum of exerimental data (erhas at the comonent or subsystem level) to serve as a yardstick with which the accuracy of simulation results can be measured. And it requires minimizing the differences between corresonding analytical and exerimental results in hysically meaningful ways, as well as characterizing the ability of the models to redict future events. his aer describes a toolbox for the validation of nonlinear finite element models. he toolbox includes tools for quantitatively udating model arameters based on the differences between test results and analytical redictions, as well as estimating the redictive accuracy of the model based on generically similar comarisons. Use of the toolbox is illustrated for a DYNA model of a reinforced concrete wall subjected to blast loading. INRODUCION he Defense hreat Reduction Agency (DRA) has romoted the verification and validation of nonlinear dynamic codes and models for many years. Numerous recision tests have been conducted over the years to suort this effort, leading to codes such as SHARC (Hikida, et al., 988), AUODYN (Century Dynamics, 989), and DYNA3D (Whirley and Engleman, 993) that generate high fidelity hysics-based (HFPB) models of exlosive loads on structures and of structural resonse to those loads in terms of structural damage and residual strength. he urose of verification and validation is to confirm the stability and accuracy of numerical algorithms and the behavior of material models under controlled conditions, so that the codes may be used with greater confidence to extraolate limited test exerience to a range of ractical alications. he difficulty with this aroach has been the lack of a coherent methodology and comutational tools for its imlementation, esecially tools for modeltest correlation, model udating, and redictive accuracy assessment. he organization of the tools develoed under this roject is shown in Figure. he modeltest correlation ortion includes tools for statistical analysis of the differences between model redictions and test measurements based on difference analysis or rincial comonent comarisons. he model udating section includes tools for arameter sensitivity analysis, arameter effects analysis, and resonse surface modeling. It also includes tools for the generation of surrogate models, as well as various continuous and discrete arameter estimation algorithms. he redictive accuracy assessment ortion includes a tool for evaluating modeling uncertainty based on rincial comonents derived from analysis and test data, and a tool for roagating these statistics through models to evaluate their redictive accuracy. ools for rearation of data for these analyses are also included. 5-2

3 Iteration Data Prearation Model-est Correlation Model Udating Predictive Accuracy Assessment Difference Analysis Parameter Analysis Uncertainty Analysis Princial Comonents Comarison Surrogate Model Develoment Uncertainty Proagation Parameter Estimation Figure. Design for Nonlinear Model Validation oolbox APPROACH he tools for model validation are based on rincial comonents analysis of model redictions and exerimental measurements. Princial comonents analysis facilitates comarisons of data useful for model-test correlation and uncertainty analysis. It also rovides a simle means of generating local, surrogate models useful for arameter estimation with comutationally intensive finite element models. Princial Comonents Analysis Princial comonents analysis is based on the singular value decomosition (SVD) of a collection of time-histories (Klema and Laub, 98). Let xt () denote a resonse time-history, where x may be dislacement, velocity, or any time-deendent quantity of interest. A resonse matrix, X, is a collection of discretized time-histories, X ( ) L ( ) x t x t é n ù ê ú = ê M O M ú êxm( t ) L xm( tn) ú ë û () where each row corresonds to either a different measurement location or set of hysical arameters, and each column corresonds to resonse at a secific time. he SVD of X may be written X U V = Σ (2) where U is an orthonormal m m matrix whose columns are the left singular vectors of X, Σ is an m n matrix containing the singular values of X along the main diagonal and zeros elsewhere, and V is an n n orthonormal matrix whose columns corresond to the right singular vectors of X. 5-3

4 he matrices on the right hand side of (2) may be artitioned so that éd ù éη ù X = éφ φù ê ú ê ú η ë û ê ë ú û ê ë ú û where D is the diagonal matrix of nonzero singular values, d i ( i =,, min( m, n) ), φ and η are the matrices of left and right rincial vectors, resectively, corresonding to the nonzero singular values, and φ and η san the orthogonal comlements of the resective subsaces sanned by φ and η. By (3), X = φ Dη (4) he columns (rows) of φ (η ) are airwise orthonormal, i.e., K (3) φφ= ηη = I (5) where I is the -dimensional identity matrix. he factorization given by (4) is called the rincial comonents decomosition (PCD) of the resonse matrix (Hasselman, Anderson, and Gan, 998). Model-est Correlation Princial comonents methods are useful for nonlinear model-test correlation for the same reason that modal roerties are useful in linear structural dynamics. In nonlinear models, however, the interretation of these modal roerties deends on the selection of resonse data included in Xt ( ). Nevertheless, there are certain roerties of the PCD that can be exloited for uroses of model-test correlation. hese roerties are suggested by the following equations: ψ = φ φ (6a) D = D D (6b) v = η η (6c) where φ, D,and η reresent modal arameters derived from analysis for comarison with the corresonding modal arameters φ, D, and η derived from exerimental data. Model Udating he PCD furnishes a comact reresentation of the resonse of a nonlinear model. he scaled right rincial vectors, diη i, reresent the resonse time-histories of the rincial comonents. Each row of the left rincial vector matrix, φ, denotes the secific linear combination of the rincial comonent resonse time-histories which reroduces the total resonse timehistory at the corresonding value of the arameter vector, θ, and satial location of the resonse. For examle, when each row of the resonse matrix corresonds to the resonse at a single location and a unique set of arameters, then each left rincial vector can be considered as a function of the arameter vector only. If the left rincial vectors are considered as functions of the arameter vector, θ,then xtθ (; ) may be aroximated by 5-4

5 ,and is q ( θ) ˆ φ( θ) η( ) ˆ φ ( θ) η ( ) xt ˆ ; = D t = å d t (7) i= i i i where the columns of φ ˆ( θ ), φˆ j ( θ ), are reresented by individual resonse surfaces (Hasselman, Anderson, and Zimmerman, 998). Consequently, one may define a Bayesian objective function of the form ( ) εε ( ) ( θ θo) θθ ( θ θo) ˆ ˆ J = x x S x x + S (8) where θ and θ o reresent the current and initial estimates of the variable arameter vectors, resectively, and x and ˆx reresent the measured and currently redicted resonses, resectively. he covariance matrices S θθ and S εε reresent uncertainties in the initial arameter estimates and resonse measurements, resectively. Bayesian estimation rovides a revised covariance matrix of the udated arameter estimates given by ( ) x x * Sθθ Sθθ θ Sεε θ = + (9) where x θ is the sensitivity matrix, x / θ, relating the resonse vector to the arameter vector. Uncertainty Analysis When the rincial comonents aroach is used to reresent a nonlinear model, the arameters are φ, D, andη, and modeling uncertainty is defined in terms of these arameters. Once a covariance matrix of the modal arameters is obtained, it can be transformed to obtain a covariance matrix of the resonse variables. he redictive accuracy of the model is thereby determined (Hasselman, Chrostowski, and Ross, 992, and Anderson, Gan, and Hasselman, 998). A first order aroximation of the exerimental resonse matrix is given by N m X ij X = ij å rk r k = k () where the arameter, r k, is taken to reresent any of the elements of the matrices ψ, D,or ν, D = D/ race( D) Nm denotes the number of modal arameters collectively contained in ψ, D,andν.Here rk an element of the vector r with ( ψ ) ìvec ü ï ï r = ívec ( D) ý () ï ï ïîvec( ν ) ïþ where ψ = ψ I, resect to r k is o D = ( D D) / race( D),and ν = ν I. he derivative of X ij with 5-5

6 X ij æ D e ψ i D D D ν ö = φ ç + + η j rk è rk rk rk ø (2) Equation () has a articularly simle form when the matrix, u = vec( X), ur X, is also vectorized as u = r (3) where the elements of ur are oulated by the scalar values given in (2). he covariance matrix of the vector r is given by N rr = E é ù = i ë û å i N i = S r r r r (4) where the index, i, is on a articular data set consisting ofcorresondinganalysis-testairs,n is the total number of data sets in the samle, and the analytical model is assumed to redict mean resonse. When the analytical model contains bias-tye error, as may be indicated when relicate test measurements are available, then where µ r is S r r å r r N rr = E é( µ r)( µ r) ù = é( i µ r)( i µ r) ù ë û N ë û i= (5) the mean of the vector r. S rr reresents the generic modeling uncertainty inherent in analytical redictions of the resonse matrix, X, based on normalized comarisons of revious analysis and test data. In order to evaluate the redictive accuracy of a new resonse rediction, Equation (3) is used, with the understanding that ur is evaluated with resect to the new model, i.e., the values of φ, D,and η reresenting the modal arameters of the new model, rather than those of the models that have been correlated with revious test data. hen uu ur ë û rr ur S = E é u u ù = S (6) DISCUSSION OF RESULS o illustrate these concets the arameters of a comlex, nonlinear system model were udated using available test data. he hysical scenario for the examle is deicted in Figure 2a and consists of a three-room, buried, reinforced concrete structure subjected to blast loading in the center room. Exerimental measurements included blast overressures inside the room and accelerations of the two interior walls. Accelerations were integrated to obtain dislacements. he corresonding model shown if Figure 2b was a quarter-symmetry, nonlinear finite element model. his model consisted of aroximately 8, continuum elements for the concrete and surrounding soil, and roughly 2, structural beam elements for the steel reinforcement. A cold joint near the bottom of the interior wall was exlicitly modeled. he material models contained dozens of arameters, many of which were candidates for estimation. Since the loading was distributed and available inut measurements were few, an intermedi- 5-6

7 ate inut model was required. his model was a comutational fluid dynamics model of the interior room ressure. A reliminary validation of the inut model was erformed by a third arty using available ressure data. (a) Physical scenario (without soil) (b) HFPB quarter-symmetry model Figure 2. Examle Problem he re-udate redictive accuracy of the model is illustrated in Figure 3. he redicted and measured resonses at the center of the wall are shown, along with ± 2σ uncertainty bands. hese bands were based on an unbiased estimate of the measurement uncertainty for the center wall resonse using rincial comonents analysis. Note that the measured resonse generally falls within the uncertainty bands, but that the bands fail to cature the exerimental behavior in the neighborhood of.2 seconds..5 Dislacement (in).5 Model Model +/ 2 σ est ime (sec) Figure 3. Pre-Udate Predictive Accuracy he estimation roblem consisted of using measurements of the dislacement of the interior wall to udate some of the concrete and steel material arameters so that the redicted resonse matched the test data. Measurements used for estimation included samled dislacement histories at four locations on the wall. Estimation arameters were chosen by a threestage rocess comrised of engineering judgement, sensitivity analysis, and arameter effects analysis. he results of this rocess indicated that the concrete strain rate enhancement and shear dilatancy, the steel reinforcement tensile strength, and the cold joint friction were the 5-7

8 most significant arameters. Since the inut model was tentatively validated, inut arameters were excluded from consideration for the initial estimation. Numerous attemts to roduce meaningful arameter estimates were made. he results of a tyical attemt are shown for the center wall resonse in Figure 4. Figure 4a indicates that the revised model is noticeably worse than the nominal model with resect to the details of the resonse rediction. Figure 4b illustrates the quality of the arameter estimates. Dislacement(in) Data Nominal Model Revised Model ime (sec) Percent Increase in Confidence, (σ/σ * ) % 5 4 Reject Revise Confirm Revise Reject 3 θ 2 2 θ 4 θ 3 θ Relative Change in Parameter Estimate, θ/σ (a) ime history comarison (b) Quality of the estimates Figure 4. Results of the Initial Estimation he statistics of the initial arameter estimates are given in able. Even though the osterior variances of the estimates are smaller than the rior variances, the osterior correlation matrix contains some large off-diagonal elements. Using an absolute correlation of.25 as a measure of significance imlies that the concrete strain rate enhancement is correlated with the tensile strength of the steel reinforcement and the cold joint friction. An even higher degree of correlation can be noted between the concrete shear dilatancy and the steel reinforcement strength. hese results are clearly unaccetable. able. Initial Prior and Posterior Estimate Statistics Parameter Symbol Concrete Strain Rate Enhance. Steel Rein. ensile Strength Concrete Shear Dilatancy θ θ2 θ3 θ4 Cold Joint Friction Prior Est ,.5.2 Posterior Est , Prior Var..6x -4.6x x -2.89x -2 Posterior Var. 6.5x -5.53x x x -3 Posterior Corr. Matrix θ θ θ θ 4 (Sym.) After initial arameter estimation attemts failed, all available auxiliary information was reviewed to determine if it could be used to eliminate one or more of the arameters. his information included tensile test results for the steel reinforcement, as well as ost-test measurements of the sli across the cold joint. wenty-five indeendent measurements of the 5-8

9 strength of steel reinforcing bars from the same batch used to construct the test structure were available. he results of these tests indicated that the rior value of the reinforcement strength was accurate, and this arameter was eliminated from further consideration. he measured cold joint sli was matched by the nominal model rediction, indicating that the cold joint friction may be a candidate for elimination. Before roceeding with additional estimation attemts, the inut model was reviewed to determine if any adjustments were required. Figure 5 comares the ressure and cumulative imulse measurements and redictions at one of the ressure gage locations. With the excetion of the eak of the initial ulse, the redicted ressure history matched the data well, as indicated in the uer lot of the figure. However, the cumulative imulse data differed significantly from the model rediction, as shown in the lower lot. Simly scaling the redicted initial ulse by a factor of.6 rectified the roblem at all locations for which data were available, as indicated in the figure. Similar scaling was alied to all inuts to the finite element model. Pressure (si) Cumulative Imulse (si sec) Data Nominal Inut Model Revised Inut Model ime (sec) x 3 2 Data.5 Nominal Inut Model Revised Inut Model ime (sec) x 3 Figure 5. Inut Model Udate he final estimation rocess began by using the revised inut model to generate a new model rediction. his indicated that the cold joint friction had to be reduced to a lower level (.5) so that the cold joint sli was accurately modeled. he finite element model was then used to generate the new nominal rediction using the revised inuts and cold joint friction. Parameter estimates were generated using the two remaining arameters, the concrete strain rate enhancement and shear dilatancy. he results of these efforts indicated that the shear dilatancy should not be revised even though the osterior correlation between the two arameters was high. he final estimation attemt used only the strain rate enhancement, with the dilatancy fixed at its nominal value. he final estimated value of the strain rate enhancement arameter was.58, or about 55 ercent of the original nominal value. he osterior variance estimate was two orders of magnitude less than the rior estimate. he results of the final estimation rocess are illustrated in Figure 6. Figure 6a comares the measured dislacement history at the center of the wall with that redicted by the model with the original nominal, modified nominal, and revised value of the arameters. Similar comarisons were made at other locations where data were available. he results clearly indicated that the revised model not only matched the data very well in a mean square sense, but also catured the character of the data significantly better than the original nominal model. Figure 6b shows the high quality of the arameter 5-9

10 estimate. herefore, one is led to conclude that the estimated arameter values are likely to rovide accurate redictions for future analyses using the same materials. Dislacement(in) Data.2 Nominal Model (Original) Nominal Model (After Inut Udate) Revised Model ime (sec) Percent Increase in Confidence, (σ/σ * ) % Reject Revise Confirm Revise Reject Relative Change in Parameter Estimate, θ/σ θ (a) ime history comarison (b) Quality of the estimate Figure 6. Results of the Final Estimation Figure 7 deicts the re-udate redictive accuracy of the model. As before, the redicted and measured resonses at the center of the wall are shown, along with ± 2σ uncertainty bands. Note that the measured resonse falls comletely within the uncertainty bands..4.2 Dislacement (in) Model.2 Model +/ 2 σ est ime (sec) Figure 7. Post-Udate Predictive Accuracy SUMMARY he rincial comonents-based nonlinear model validation methodology summarized in this aer rovides a means of systematically comaring model redictions with available data and udating model arameters to increase the fidelity of resonse redictions. his is a vast imrovement over traditional ad hoc techniques. Included in the methodology are tools for evaluating the statistical significance and consistency of the arameter estimates, and the redictive accuracy of the udated model. hese tools enable the analyst to confirm that the estimated arameter values are statistically meaningful, a rerequisite for true model imrovement, and to quantify the degree of uncertainty associated with model simulations, based on 5-

11 structure-secific recision test data if relicate measurements are available, or historical data from generically similar structures and tests if they are not. Alication of the methodology to the air blast resonse of a reinforced concrete wall demonstrated statistical arameter estimation and redictive accuracy evaluation of a nonlinear HFPB model. Princial comonents analysis was instrumental in generating the fast-running aroximate model used for function aroximation in the nonlinear Bayesian arameter estimation, and as a means for quantifying modeling uncertainty in the evaluation of redictive accuracy. he same comutational tools are currently being alied to other roblems involving weaon-target interaction. ACKNOWLEDGEMENS he work reorted in this aer was erformed under contract to DRA. he authors wish to acknowledge the encouragement and suort of the COR, Michael Giltrud. he nonlinear finite element models used in the numerical examle were develoed by Karagozian & Case, Structural Engineers. he assistance of David Bogosian and Brian Dunn in the rearation execution of these models is also acknowledged. REFERENCES ANDERSON, M. C., GAN, W., and HASSELMAN,. K. (998), Statistical Analysis of Modeling Uncertainty and Predictive Accuracy for Nonlinear Finite Element Models, Proceedings of the 69 th Shock and Vibration Symosium, Minneaolis/St. Paul, MN. CENURY DYNAMICS (989), AUODYN Users Manual, Century Dynamics, Oakland, CA. HASSELMAN,. K., ANDERSON, M. C., and GAN, W. (998), Princial Comonents Analysis for Nonlinear Model Correlation, Udating and Uncertainty Evaluation, Proceedings of the 6 th International Modal Analysis Conference, Santa Barbara, CA. HASSELMAN,. K., ANDERSON, M. C., and ZIMMERMAN, D. C. (998), Fast Running Aroximations of High Fidelity Physics Based Models, Proceedings of the 69 th Shock and Vibration Symosium, Minneaolis/St. Paul, MN. HASSELMAN,., CHROSOWSKI, J., and ROSS,. (992), Interval Prediction in Structural Dynamic Analysis, AIAA , Proceedings of the 33 rd AIAA/ASME/ASCE /AHS/ASC Structures, Structural Dynamics and Materials Conference, Dallas, X. HIKIDA, S., BELL, R., and NEEDHAM, C. (988), he SHARC Codes: Documentation and Samle Problems, echnical Reort SS-R , S-Cubed Division of Maxwell Laboratories, Albuquerque, NM. KLEMA, V. C., and LAUB, A. J. (98), he Singular Value Decomosition; Its Comutation and Some Alications, IEEE rans. on Automatic Control, Vol. AC25, No. 2, WHIRLEY, R. G., and ENGLEMAN, B. E. (993), DYNA3D: A Nonlinear Exlicit hreedimensional Finite Element Code for Solid and Structural Mechanics, User Manual, Reort UCRL-MA-7254 Rev., Lawrence Livermore National Laboratory, Livermore, CA. 5-

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