Because damage prognosis solutions rely on the deployment of a predictive capability, the credibility of numerical simulations must be established. This is accomplished through various activities collectively referred to as Verification and Validation (V&V). A hierarchy of three validation experiments is defined to validate various aspects of the modeling of multilayered composite plates. The experiments are modal testing, quasistatic loading, and impact testing. Modal testing is used to extract lowfrequency resonant modes, from which the linear response can be validated. Quasistatic loading is used to validate the large displacement response, although without fully exercising the fiber splitting and delamination damage modes nor the Cohesive Zone Model algorithm. Impact testing is used to validate the modeling of the highfrequency, transient wave propagation coupled with all aspects (constitutive and damage models, numerical algorithms) of the composite damage models.
The discussion presented here is restricted to the validation of the modal response of eightply laminated composite plates. Other aspects of the modeling are currently being validated, pending the completion of the quasistatic and impact experiments. After verifying some implementation aspects of the code, mesh convergence studies are conducted. Effect screening is performed to restrict the varying input parameters to the most significant ones. Polynomial metamodels are developed to replace the potentially expensive finite element simulations. Uncertainty is propagated to estimate the variability of predictions given input uncertainty. Test measurements are compared to predictions of modal frequencies. A final statement is made about the predictive accuracy of the composite model and the level of confidence with which modal frequency predictions can be made for potentially different multilayered configurations.
The modal response of a population of eight nominally identical composite plates is tested. Replicate experiments are needed to estimate the variability of the modal response due to manufacturing and testing uncertainties. Each plate is 152.0 mm (6.0 inch) square, 1.0 mm (0.04 inch) thick, and made of eight orthotropic carbon fiber plies. Each ply is 0.127 mm (0.005 inch) thick. The ply orientation angles are [0; 45; 90; 45; 45; 90; 45; 0] degrees. The first five modes are identified with mean frequency values of 107.4, 191.8, 274.1, 315.3, and 398.9 Hertz. It is observed that the standard deviations of identified frequencies are less than 1¼% of mean values, which indicates that the measurements are very repeatable.
The Finite Element Model (FEM) developed to simulate the modal response of the plates is meshed with 20node quadratic hexahedra, with ten elements in each inplane direction and one element through the thickness of each ply. The nominal orthotropic material properties are provided by the manufacturer and confirmed through coupon testing. Using the nominal material properties, natural frequencies are predicted at 107.5, 205.7, 278.1, 334.0, and 411.8 Hertz, which provides a good qualitative agreement with the identified frequencies.
The FEM is implemented and analyzed using the generalpurpose finite element package HKS/ABAQUSTM [1]. Computer code verification is the first step of any V&V activity. It verifies that the code is errorfree by comparing analytical, closedform solutions to numerical results. It is verified that the predicted frequencies for a square, freefree, isotropic plate are less than 0.5% different from the closedform solution [2]. Similar results are obtained for a square, simply supported, orthotropic plate [3]. The conclusion is that the FEM software package and composite material module solve the equations correctly for this problem.
Calculation verification assesses the convergence of the numerical solution for the application of interest. It requires several calculations using successively refined mesh sizes and/or time steps, from which the order of convergence is estimated and the truebutunknown solution is extrapolated. The Richardson extrapolation and Grid Convergence Index (GCI) are brought to bear to verify that the mesh size yields converged frequencies [4].
The average order of convergence for frequencies 15 is found equal to 1.8, close to the value of two that should be obtained because quadratic elements are used. Small GCI values (less than 1%) also suggest asymptotic convergence. Errors obtained between predictions and the Richardson estimation of the truebutunknown solutions are less than 1%. These results indicate that the FEM and its spatial discretization yield converged natural frequencies.
We seek next to better understand the relationship between the variability of input parameters, such as the ply orientation angles and material properties, and response feature variability. Two designs of computer experiments are performed to explore the values of frequencies for various combinations of input parameters [5]. The most significant effects are screened using methods such as the analysisofvariance that performs multiple regression analyses and estimates the correlation between input effects and output features [6].
The analysis demonstrates that the variability of the five frequencies is controlled for the most part by four ply orientation angles, two material coefficients, and the accelerometer mass.Because the other factors do not produce significant frequency variability, they are kept constant and equal to their nominal values.The number of input parameters is hence reduced from fifteen to seven only, which greatly simplifies metamodeling and the forward and backward propagations of uncertainty.
Based on screening results, metamodels are developed to replace the FEM simulations. Polynomial metamodels provide fastrunning surrogates that accurately capture the relationship between input parameters and natural frequencies without including details of the geometry or material modeling.
The data needed to train metamodels are provided by a design of computer experiments. Confirmatory screening and model fitting are performed using a Bayesian effect screening method [7]. It provides a population of metamodels that can be characterized statistically by establishing distributions of polynomial coefficients. Model fitting errors are assessed by segregating the available data into training and validation sets, and found to be in the order of 1%. The metamodeling study concludes that the FEM simulations can be replaced by polynomial metamodels with little loss of accuracy.
Variability of the ply orientation angles and material properties is propagated through the family of metamodels to estimate the natural frequency variability. Model fitting uncertainty is also accounted for by sampling the coefficients of the metamodel polynomials. A Monte Carlo simulation based on 10+6 runs yields a population of frequencies from which statistics can be extracted. The effects of input distribution and correlation structure on output statistics are also investigated [8]. The statistics of frequency values are 107.5 +/ 6.7, 206.6 +/ 15.8, 281.2 +/ 19.8, 329.0 +/ 24.8, and 408.0 +/ 27.9 Hertz (mean +/ standard deviation).
The sources of uncertainty originating from experimental variability, coding mistakes, calculation verification, model fitting, and parametric variability are summarized in Table 1. For simplicity, statistics are averaged for the five frequencies. Aggregating the different sources of uncertainty leads to a total prediction error (mean +/ standard deviation) of 3.64 +/ 19.02 Hertz.
Table 1. Summary of the quantification of testing and modeling uncertainty.
Source of Uncertainty 
Statistics 


Mean (μ) 
Standard Deviation (s) 

Code Error, Solution Convergence 
μ_{1} = 1.49 Hertz 
s_{1} = 0.87 Hertz 
Metamodel Fitting Error 
μ_{2} = 2.15 Hertz 
s_{2} = 0.18 Hertz 
Model Parameter Variability 
μ_{3} = 0.00 Hertz 
s_{3} = 19.00 Hertz 
Experimental Variability 
μ_{4} = 0.00 Hertz 
s_{4} = 2.49 Hertz 
Testanalysis Correlation 
μ_{C} = 8.97 Hertz 
s_{C} = 19.18 Hertz 
Direct comparison between experimental measurements and numerical predictions leads to a prediction error of 8.97 +/ 19.18 Hertz. The discrepancy between the two error models can be attributed to the residual modeling error, that is, the error made when the composite model is substituted to reality.
The conclusion is twofold. First, better characterization of the composite layup and material properties would further constraint the variability of input parameters which, in turn, could reduce some of the residual prediction error. Reference [8] also shows that calibration can be taken advantage of to infer the values of input parameters that cannot be obtained through direct measurement or coupon testing. Second, we are confident that natural frequencies for different multilayered configurations can be predicted with a similar degree of accuracy as long as the same modeling rules are followed.
[1] Abaqus™/Explicit, User’s Manual, Version 6.2, Hibbitt, Karlsson & Sorensen, Pawtucket, Rhode Island, 2001.
[2] Leissa, A.W., “The Free Vibration of Rectangular Plates,” Journal of Sound and Vibration, Vol. 31, No. 3, 1973, pp. 257293.
[3] Srinivas, S., Rao, A.K., “Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates,” International Journal of Solids and Structures, Vol. 6, 1970, pp. 14631481.
[4] Hemez, F.M., Doebling, S.W., Anderson, M.C., “A Brief Tutorial on Verification and Validation,” 22nd SEM International Modal Analysis Conference, Dearborn, Michigan, January 2629, 2004. Available as unlimited, public release LAUR038491 of the Los Alamos National Laboratory, Los Alamos, New Mexico.
[5] Myers, R.H., Montgomery, D.C., Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Wiley Interscience, New York, 1995.
[6] Saltelli, A., Chan, K., Scott, M., Editors, Sensitivity Analysis, John Wiley & Sons, 2000.
[7] Kerschen, G., Golinval, J.C., Hemez, F.M., “Bayesian Model Screening for the Identification of Nonlinear Mechanical Structures,” ASME Journal of Vibration and Acoustics, Vol. 125, July 2003, pp. 389397.
[8] Hemez, F.M., Tippetts, “Verification and Validation of a Composite Model,” 23rd SEM International Modal Analysis Conference, Orlando, Florida, January 31February 3, 2005. Available as unlimited, public release LAUR048195 of the Los Alamos National Laboratory, Los Alamos, New Mexico.