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The development of a manufactured solution for enclosure radiation in an infinitely long circular cylinder with a nonparticipating medium is presented. This solution is then used to verify the correct implementation of the commonly used discrete enclosure equations. The circular cross section is approximated by a faceted geometry; the numbers of facets used are 4, 8, 16, 32, 64, and 128. The crossed-string method, which is exact in this application, is used to compute the view factors. Computational results using six levels of grid refinement suggest that the error norm between the integral equation solution and the discrete equation solution behaves as h 2 where h is a characteristic mesh size.

This special issue presents the results of the Sandia organized Model Validation Challenge Workshop, held May 2006. The workshop brought together researchers from different fields to present various approaches to model validation, and focused on the methodological elements of model validation rather than on model building. Three problems were defined in the disciplines of structural statics, structural dynamics, and heat transfer, all with a uniform structure. The workshop was specifically designed to investigate the relative merits of different approaches to hierarchal model validation through application to these problems. This paper describes a hierarchal approach in the challenge problems, presents the uniform conceptual framework that was used for the challenge problem definitions, and provides an overview of the organization of this special issue.

This paper summarizes the approaches used to address the thermal validation challenge problem. The approaches differ in their characterization of the thermal properties and uncertainty, the definitions and use of validation metrics, the use of validation experimental data to characterize or improve the model predictions, and the assessment of regulatory compliance. All approaches estimated regulatory failure with the resulting estimated probabilities varying by an order of magnitude.

This paper presents an engineering approach to the thermal challenge problem defined by Dowding et al. (this issue). This approach to model validation is based on a multivariate validation metric that accounts for model parameter uncertainty and correlation between multiple measurement/prediction differences. The effect of model parameter uncertainty is accounted for through first-order sensitivity analysis for the ensemble/validation tests, and first-order sensitivity analysis and Monte-Carlo analysis for the regulatory prediction. While sensitivity based approaches are less computational expensive than Monte-Carlo approaches, they are less likely to capture the far tail behavior of even mildly nonlinear models. The application of the sensitivity based validation metric provided strong evidence that the tested model was not consistent with the experimental data. The use of a temperature dependent effective conductivity with the linear model resulted in model predictions that were consistent with the data. The correlation structure of the model was used to pool the prediction/measurement differences to evaluate the corresponding cumulative density function (CDF). Both the experimental CDF and the predicted CDFs indicated that the regulatory criterion was not met.

This paper describes the thermal problem and presents the experimental data for validation. The thermal problem involves validating a model for heat conduction in a solid. The mathematical model is based on one-dimensional, linear heat conduction in a solid slab, with heat flux boundary conditions. Experimental data from a series of material characterization, validation, and accreditation experiments related to the mathematical model are provided. The objective is to use the series of experiments to assess the model, and then use the model to predict regulatory performance relative to a regulatory requirement. The regulatory requirement is defined in terms of the probability that a surface temperature not exceed a specified temperature at the regulatory conditions.

Verification of the codes that provide numerical heat transfer solutions obtained by finite difference and other methods is important. One way to verify these solutions is to compare the values with exact solutions. However, these exact solutions should also be verified. Fortunately, intrinsic verification methods are possible. Intrinsic verification utilizes at least two independent exact solutions to obtain accurate numerical values. Three different types of intrinsic verification for transient and steady state heat conduction are developed and illustrated by examples.

To estimate parameters from experiments requires the specification of models and each model will exhibit different degrees of sensitivity to the parameters sought. Although experiments can be optimally designed without regard to the experimental data actually realized, the precision of the estimated parameters is a function of the sensitivity and the statistical characteristics of the data. The precision is affected by any correlation in the data, either auto or cross, and by the choice of the model used to estimate the parameters. An informative way of looking at an experiment is by using the concept of Information. An analysis of an actual experiment is used to show how the information, the optimal number of sensors, the optimal sampling rates, and the model are affected by the statistical nature of the signals. The paper demonstrates that one must differentiate between the data needed to specify the model and the precision in the estimated parameters provided by the data.

An analytical solution is provided to the nonlinear diffusion equation, with the thermal conductivity given as a linear function of temperature. The derivation of the solution, and implications of it, are presented. The boundary and initial conditions associated with the solution provide applicability to specific cases. The solution is useful for verifying numerical (computer) solutions to thermal diffusion with temperature-dependent thermal conductivity. The (nonlinear) analytical solution is compared to a numerical solution from a finite element code to verify the accuracy of the code and to establish the order of convergence for the spatial discretization error

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.

This paper reports the evaluation of a spectral technique for estimating thermophysical properties. It demonstrates that one can construct a virtual quasi-steady periodic experiment from a limited but properly selected set of transient non-periodic data. In the spectral domain, the phase angles of the responses at different locations relative to a periodic input signal depend on the thermophysical properties. For the purpose of this evaluation, the transient temperature responses to a surface heat flux input are analytically obtained at pre-selected sensor locations. The transient data are converted to periodic data, phase angles are computed, and thermophysical properties are estimated. All deviations from known property values due to numerical errors are reported.

Methods are discussed for computing the sensitivity of the temperature field to changes in material properties and initial boundary condition parameters for heat conduction problems. The most general method is to derive sensitivity equations by differentiating the energy equation with respect to the parameter of interest and solving the resulting sensitivity equations numerically. An example problem in which there are 12 parameters of interest is presented and the resulting sensitivity equations and associated boundary initial conditions are derived. The sensitivity equations are implemented in a general-purpose unstructured-grid control-volume finite-element code. Numerical results are presented for thermal conductivity and volumetric heat capacity sensitivity coefficients for heat conduction in a 2-D orthotropic body. The numerical results are compared with the analytical solution to demonstrate that the numerical sensitivity method is second-order accurate as the mesh is refined spatially.

In parameter estimation, considerable insight is provided by examining sensitivity coefficients. This article focuses on the use of sensitivity coefficients in connection with estimating thermal properties in the heat conduction equation. It is demonstrated how sensitivity coefficients are used in the design analysis of parameter estimation experiments. A general methodology for computing sensitivity coefficients can be an important design tool in many other regards also. A control-volume, finite-element program is used to implement numerical sensitivity coefficient calculations. In this approach, general problems can be studied. Several example cases are presented to demonstrate the insight gained from sensitivity coefficients. The cases are selected from experimental studies to characterize the thermal properties of carbon carbon composite. However, no experimental data are reported. Sensitivity coefficients show that in an experiment that is not well designed, additional materials in the experimental c...

Parameter estimation techniques are applied to estimate temperature-dependent thermal properties from a series of transient experiments. Several experiments with one and two-dimensional heat flow that cover a range from room temperature to 500°C are analyzed. Temperature-dependent thermal properties are estimated by connecting the independent experiments in the series during the analysis. The techniques are applied to estimate effective properties of carbon-carbon composite. The temperature dependence of two components of thermal conductivity and volumetric heat capacity are estimated to characterize the assumed orthotropic material. The techniques can be equally applied to other homogenous materials. Combining experiments during the analysis is referred to as a sequential analysis, which uses the concepts of regularization and prior information. Regularization controls variations in the estimated parameters. Prior information carries information from a previous analysis into a subsequent analysis

Fracture or tearing of ductile metals is a pervasive engineering concern, yet accurate prediction of the critical conditions of fracture remains elusive. Sandia National Laboratories has been developing and implementing several new modeling methodologies to address problems in fracture, including both new physical models and new numerical schemes. The present study provides a double-blind quantitative assessment of several computational capabilities including tearing parameters embedded in a conventional finite element code, localization elements, extended finite elements (XFEM), and peridynamics. For this assessment, each of four teams reported blind predictions for three challenge problems spanning crack initiation and crack propagation. After predictions had been reported, the predictions were compared to experimentally observed behavior. The metal alloys for these three problems were aluminum alloy 2024-T3 and precipitation hardened stainless steel PH13-8Mo H950. The predictive accuracies of the various methods are demonstrated, and the potential sources of error are discussed.

Thermophysical properties including density, specific heat, and thermal diffusivity of a poly (diallyl phthalate) inert filler composite material were characterized over a wide temperature range from room temperature to 800 °C. Over this temperature range, the material decomposition was approximated by a one-step process with first-order kinetics. Thermal kinetics data were obtained by thermal gravimetric analysis with Fourier transform infrared spectroscopy (TGA-FTIR) and thermophysical properties were obtained from differential scanning calorimetry (DSC) and laser flash diffusivity experiments. The response of the material to radiant heating was simulated with a computational heat transfer, multidimensional, finite element code. Additionally, the experimental uncertainty in the measurements was quantified to estimate the uncertainty in the reaction parameters due to heating rate and variability in inert filler-polymer composition in large sample sizes. Hence, the thermal response and the uncertainty were quantified for a complex decomposing material in a practical geometry for technologically important applications.Copyright © 2009 by ASME

Coupled fire-environment/thermal-response models were validated using data for an object engulfed in a JP8 hydrocarbon fuel fire. Fire model predictions of heat flux were used as boundary conditions in the thermal response calculations of the object. Predictions of transient external shell temperatures as well as the surface temperatures of the embedded mass were averaged spatially and compared to data. The solution sensitivity to mesh size, time step, nonlinear iterations, and radiation rays were assessed and the uncertainties in the predictions were quantified using a Latin Hypercube Sampling (LHS) technique. The comparisons showed that the response variable was more sensitive to fire model parameters than to thermal model parameters. The observed relative difference in measurements and model predictions was also compared to the model uncertainty. The comparisons showed that the model plus uncertainty bounded the experimental data. I. Introduction Sandia National Laboratories has been engaged in testing weapon system safety in fire environments since the 1950s. Due to the high consequences involved, system safety has traditionally been demonstrated through full scale system tests, albeit with a limited number of tests. Historically developed standardized tests include the placement of a system in a fully engulfing fire for 1 hour. Systems are declared qualified and ready for production based on passage of these standardized tests and with reference to the testing and analysis during development. Beginning in the early to mid 1990’s, the DOE began a program of Science Based Stockpile Stewardship. A significant part of this program is the Advanced Simulation and Computing (ASC) program, in which modeling and simulation, through high performance computing has been applied to system development and qualification. As part of the ASC program, Sandia engaged in developing the capability to model fire environments coupled to system response in those environments. An important thrust area within the ASC program includes the advancement of the verification and validation (V&V) methodologies and uncertainty quantification techniques. Sandia National Laboratories has made strides in developing new capabilities in this area and applying them to current applications. A best estimate plus uncertainty approach has been fully adopted and incorporated into safety themes for system qualification. Providing uncertainty estimates along with deterministic results has provided value to Sandia programs and gives more insight into predictive capability. The direct contribution of this study to current and future systems is an understanding of the uncertainties in predicting internal system temperatures when an object is engulfed in a JP8 fire environment. The uncertainty in input parameters can be used with other scenarios and configurations to evaluate situations that challenge safety themes. Confidence gained in validation processes such as discussed in the current work is crucial when evaluating system qualification activities that include modeling and simulation. II. Numerical Modeling

Numerical models of complex phenomena often contain approximations due to our inability to fully model the underlying physics, the excessive computational resources required to fully resolve the physics, the need to calibrate constitutive models, or in some cases, our ability to only bound behavior. Here we illustrate the relationship between approximation, calibration, extrapolation, and model validation through a series of examples that use the linear transient convective/dispersion equation to represent the nonlinear behavior of Burgers equation. While the use of these models represents a simplification relative to the types of systems we normally address in engineering and science, the present examples do support the tutorial nature of this document without obscuring the basic issues presented with unnecessarily complex models.

A case study is reported to document the details of a validation process to assess the accuracy of a mathematical model to represent experiments involving thermal decomposition of polyurethane foam. The focus of the report is to work through a validation process. The process addresses the following activities. The intended application of mathematical model is discussed to better understand the pertinent parameter space. The parameter space of the validation experiments is mapped to the application parameter space. The mathematical models, computer code to solve the models and its (code) verification are presented. Experimental data from two activities are used to validate mathematical models. The first experiment assesses the chemistry model alone and the second experiment assesses the model of coupled chemistry, conduction, and enclosure radiation. The model results of both experimental activities are summarized and uncertainty of the model to represent each experimental activity is estimated. The comparison between the experiment data and model results is quantified with various metrics. After addressing these activities, an assessment of the process for the case study is given. Weaknesses in the process are discussed and lessons learned are summarized.

A Chemical-structure-based PolyUrethane Foam (CPUF) decomposition model has been developed to predict the fire-induced response of rigid, closed-cell polyurethane foam-filled systems. The model, developed for the B-61 and W-80 fireset foam, is based on a cascade of bondbreaking reactions that produce CO2. Percolation theory is used to dynamically quantify polymer fragment populations of the thermally degrading foam. The partition between condensed-phase polymer fragments and gas-phase polymer fragments (i.e. vapor-liquid split) was determined using a vapor-liquid equilibrium model. The CPUF decomposition model was implemented into the finite element (FE) heat conduction codes COYOTE and CALORE, which support chemical kinetics and enclosure radiation. Elements were removed from the computational domain when the calculated solid mass fractions within the individual finite element decrease below a set criterion. Element removal, referred to as ?element death,? creates a radiation enclosure (assumed to be non-participating) as well as a decomposition front, which separates the condensed-phase encapsulant from the gas-filled enclosure. All of the chemistry parameters as well as thermophysical properties for the CPUF model were obtained from small-scale laboratory experiments. The CPUF model was evaluated by comparing predictions to measurements. The validation experiments included several thermogravimetric experiments at pressures ranging from ambient pressure to 30 bars.more » Larger, component-scale experiments were also used to validate the foam response model. The effects of heat flux, bulk density, orientation, embedded components, confinement and pressure were measured and compared to model predictions. Uncertainties in the model results were evaluated using a mean value approach. The measured mass loss in the TGA experiments and the measured location of the decomposition front were within the 95% prediction limit determined using the CPUF model for all of the experiments where the decomposition gases were vented sufficiently. The CPUF model results were not as good for the partially confined radiant heat experiments where the vent area was regulated to maintain pressure. Liquefaction and flow effects, which are not considered in the CPUF model, become important when the decomposition gases are confined.« less

Enhanced software methodology and improved computing hardware have advanced the state of simulation technology to a point where large physics-based codes can be a major contributor in many systems analyses. This shift toward the use of computational methods has brought with it new research challenges in a number of areas including characterization of uncertainty, model validation, and the analysis of computer output. It is these challenges that have motivated the work described in this report. Approaches to and methods for model validation and (model-based) prediction have been developed recently in the engineering, mathematics and statistical literatures. In this report we have provided a fairly detailed account of one approach to model validation and prediction applied to an analysis investigating thermal decomposition of polyurethane foam. A model simulates the evolution of the foam in a high temperature environment as it transforms from a solid to a gas phase. The available modeling and experimental results serve as data for a case study focusing our model validation and prediction developmental efforts on this specific thermal application. We discuss several elements of the ''philosophy'' behind the validation and prediction approach: (1) We view the validation process as an activity applying to the use of a specific computational model for a specific application. We do acknowledge, however, that an important part of the overall development of a computational simulation initiative is the feedback provided to model developers and analysts associated with the application. (2) We utilize information obtained for the calibration of model parameters to estimate the parameters and quantify uncertainty in the estimates. We rely, however, on validation data (or data from similar analyses) to measure the variability that contributes to the uncertainty in predictions for specific systems or units (unit-to-unit variability). (3) We perform statistical analyses and hypothesis tests as a part of the validation step to provide feedback to analysts and modelers. Decisions on how to proceed in making model-based predictions are made based on these analyses together with the application requirements. Updating modifying and understanding the boundaries associated with the model are also assisted through this feedback. (4) We include a ''model supplement term'' when model problems are indicated. This term provides a (bias) correction to the model so that it will better match the experimental results and more accurately account for uncertainty. Presumably, as the models continue to develop and are used for future applications, the causes for these apparent biases will be identified and the need for this supplementary modeling will diminish. (5) We use a response-modeling approach for our predictions that allows for general types of prediction and for assessment of prediction uncertainty. This approach is demonstrated through a case study supporting the assessment of a weapons response when subjected to a hydrocarbon fuel fire. The foam decomposition model provides an important element of the response of a weapon system in this abnormal thermal environment. Rigid foam is used to encapsulate critical components in the weapon system providing the needed mechanical support as well as thermal isolation. Because the foam begins to decompose at temperatures above 250 C, modeling the decomposition is critical to assessing a weapons response. In the validation analysis it is indicated that the model tends to ''exaggerate'' the effect of temperature changes when compared to the experimental results. The data, however, are too few and to restricted in terms of experimental design to make confident statements regarding modeling problems. For illustration, we assume these indications are correct and compensate for this apparent bias by constructing a model supplement term for use in the model-based predictions. Several hypothetical prediction problems are created and addressed. Hypothetical problems are used because no guidance was provided concerning what was needed for this aspect of the analysis. The resulting predictions and corresponding uncertainty assessment demonstrate the flexibility of this approach.

Validation is a process to compare a mathematical model with a set of physical experiment to quantify the accuracy of the model to represent the physical world (experiment). Because the goal is to use experiments to quantify the accuracy of the mathematical model, the interaction of the model and experiment must be carefully studied. Advancing the comparison beyond a qualitative nature requires consideration of the errors in the process and the effect of these errors on the comparison. The mathematical model, in conjunction with sensitivity analysis, uncertainty analysis, and statistical analysis are tools for studying the interaction of the model and experiment and quantifying the effect of errors. A model for steady state heat conduction is used to discuss issues associated with the errors in the validation process and demonstrate a quantitative process to study validation of mathematical models.

The thermal conductivity of 304 stainless steel has been estimated from transient temperature measurements and knowing the volumetric heat capacity. Sensitivity coefficients were used to guide the design of this experiment as well as to estimate the confidence interval in the estimated thermal conductivity. The uncertainty on the temperature measurements was estimated by several means, and its impact on the estimated conductivity is discussed. The estimated thermal conductivity of 304 stainless steel is consistent with results from other sources.

The response and operation of a heat flux gage is studied using sensitivity analysis. Sensitivity analysis is the process by which one determines the sensitivity of a model output to changes in the model parameters. This process uses sensitivity coefficients, which are defined as partial derivatives of field variables (e.g. temperature) with respect to model parameters (e.g. thermal properties and boundary conditions). Computing sensitivity coefficients, in addition to the response of a heat flux gage, aids in identifying model parameters that significantly impact the temperature response. A control volume finite element based code is used to implement numerical sensitivity coefficient calculations, allowing general problems to be studied. Sensitivity coefficients are discussed for the well known Gardon gage.

Equations are presented to describe the sensitivity of the temperature field in a heat conducting body to changes in the volumetric heat source and volumetric heat capacity. These sensitivity equations, along with others not presented, are applied to a thermal battery problem to compute the sensitivity of the temperature field to 19 model input parameters. Sensitivity coefficients, along with assumed standard deviation in these parameters, are used to estimate the uncertainty in the temperature prediction. From the 19 parameters investigated, the battery cell heat source and volumetric heat capacity were clearly identified as being the major contributors to the overall uncertainty in the temperature predictions. The predicted operational life of the thermal battery was shown to be very sensitive to uncertainty in these parameters.