Capabilities and Applications of Probabilistic Methods in ... - CiteSeerX

Probabilistic Engineering Analysis Using the NESSUS Software Ben H. Thackera, David S. Rihaa, Simeon H.K. Fitchb, Luc J. Huysea a

Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78228, USA b

Mustard Seed Software, San Antonio, TX

ABSTRACT The development of reliability-based design methods requires the use of general-purpose engineering analysis tools that predict the uncertainty in a response due to uncertainties in the model formulation and input parameters. Barriers that have prevented the full acceptance of probabilistic analysis methods in the engineering design community include availability of tools, ease of use, robust and accurate probabilistic analysis methods, and the ability to perform probabilistic analyses for large-scale problems. The goal of the reported work has been to develop a software tool that fully addresses these three aspects (availability, robustness and efficiency) to enable the designer to efficiently and accurately account for uncertainties as they might affect structural reliability and risk assessment. The paper discusses the NESSUS probabilistic engineering analysis software with specific sections on the reliability modeling and analysis process in NESSUS, the robust and accurate solution strategies incorporated in the available probabilistic analysis methods, and several application examples to demonstrate the applicability of probabilistic analysis to large-scale engineering problems. Keywords: NESSUS, Probabilistic, Software, Reliability, Uncertainty, Stochastic

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INTRODUCTION AND BACKGROUND

As performance requirements and testing costs for engineered systems continue to increase, computational simulation is being increasingly relied upon to serve as a predictive tool. To meet these requirements, analysts are developing higher fidelity models in an attempt to accurately 1

represent the behavior of the physical system. It is not uncommon nowadays for these models to involve multiple physics, complex interfaces, and several million finite elements. Despite the recent extraordinary increase in computer power, analyses performed with these high fidelity models continue to take hours or even days to complete for a single deterministic analysis. Structural performance is directly affected by uncertainties associated with models or in physical parameters and loadings. The traditional design approach has been to adopt safety factors to ensure that the risk of failure is sufficiently small, albeit not quantified. However, probabilistic analysis permits a more rigorous quantification of the various uncertainties, and ultimately will facilitate a more efficient design process. Areas in which probabilistic methods are being successfully applied include engineered components and systems with high consequences of failure driven by safety or cost concerns. Some of these areas include aircraft propulsion systems, airframes, biomechanical systems and prosthetics, nuclear and conventional weapon systems, space vehicles, pipelines, nuclear waste disposal, offshore structures and automobiles. In general, probabilistic analysis requires multiple solutions of the underlying (deterministic) performance model. Consequently, the development of efficient and accurate probabilistic analysis methods and software tools are critically needed. Southwest Research Institute (SwRI) has been addressing the need for efficient probabilistic analysis methods for over twenty years. Much of the reliability technology developed and implemented by SwRI researchers is available in the NESSUS probabilistic analysis software. [1] NESSUS (Numerical Evaluation of Stochastic Structures Under Stress) is a general-purpose tool for computing the probabilistic response or reliability of engineered systems. NESSUS can be used to simulate uncertainties in loads, geometry, material behavior, and other user-defined random variables to predict the probabilistic response, reliability and probabilistic sensitivity measures of engineered systems. The software was originally developed by a team led by SwRI as part of the NASA project entitled “Probabilistic Structural Analysis Methods (PSAM) for Select Space Propulsion Components.” [2] Since the inception of the NASA program, SwRI has continued to conduct research, development and implementation of probabilistic methods in NESSUS. Through automatic 2

downloads from the NESSUS web site (www.nessus.swri.org), over a thousand copies of the software have been distributed to a wide range of users around the world. Many of these users include professors, researchers and students who are incorporating probabilistic design methodologies into their teaching and research projects. Users from commercial industry and government are applying NESSUS to a wide range of problems. NESSUS allows the user to perform probabilistic analysis with analytical models, external computer programs such as commercial finite element codes, and general combinations of the two. As an example, consider the problem of estimating the damage to the high-strength steel used in a containment vessel that confines high explosive experiments. In NESSUS the user can define a simulation to include 1) an explosive burn calculation to compute the pressure history at the containment wall boundary, 2) a finite element stress analysis using the computed pressure history as a load input, and 3) an analytical cumulative damage life calculation based on the computed stresses. Each model in the simulation can include random variables. This sequentially linked hierarchy of models allows the user to quickly and easily create complex multi-physics based probabilistic simulations. The NESSUS graphical user interface (GUI) is highly configurable and allows tailoring to specific applications. This GUI provides a capability for commercial or in-house developed codes to be easily integrated into the NESSUS framework. Eleven probabilistic algorithms are available in NESSUS including methods such as Monte Carlo simulation, first order reliability method, advanced mean value method and adaptive importance sampling. [3] NESSUS is available on a wide variety of computer platforms including Windows (NT, 2000, XP), IRIX, Solaris, HP-UX and Linux. The user selects the desired platform before downloading from the NESSUS web site. Recent work in NESSUS has been based on reducing the time required to define complex probabilistic problems, improving support for large-scale numerical models (greater than one million elements), and improving the robustness of the low-level probability integration routines. In the area of robustness, research is underway to improve most probable point (MPP) search 3

algorithms, develop solution strategies for identifying and solving problems that have multiple MPP’s, and implement adaptive algorithms that can detect numerical difficulties and automatically switch to alternative solution strategies. [4,5] Work is also underway to allow uncertainty due to vague or non-specific input such as expert opinion to also be considered in the probabilistic analysis. [6,7] In the following sections, the capabilities and approach to reliability modeling using the NESSUS software is described. As appropriate, references are made to considerations for largescale complex models. Three application problems are presented at the end of the paper to illustrate the application of NESSUS to real-world problems. 2 2.1

OVERVIEW OF NESSUS Component Reliability Analysis

In NESSUS, component reliability analysis denotes the reliability of a component considering a single failure mode, where reliability is simply one minus the probability of failure, p f . NESSUS can compute a single failure probability corresponding to a specific performance value, or multiple failure probabilities such that the complete cumulative distribution function (CDF) can be constructed. Alternatively, NESSUS can compute a single performance value corresponding to a specific failure probability. The choice of analysis type depends on the problem being solved. Traditional reliability analysis involves computing the probability of stress, S, exceeding strength, R, Pr [ R ≤ S ] or Pr [ g ≤ 0] , where g = R − S is referred to as the limit state function. In general, g will be more complex than g = R − S and will be given by g = g ( X ) , where X are the input random variables. In addition to the failure probability, NESSUS computes probabilistic importance factors, ∂β / ∂u , where β is inversely related to p f and u are the input random variables transformed into standard normal space, and probabilistic sensitivity factors,

∂β / ∂θ , where θ are the parameters of the input random variables, e.g., mean value and standard deviation.

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2.2

System Reliability Analysis

Most engineering structures can fail in more than one way. System reliability considers the possible failure of multiple components of a system, or multiple failure modes of a component. In NESSUS, system reliability problems are formulated and solved using a probabilistic fault tree analysis (PFTA) method. [8] A fault tree is constructed in NESSUS by connecting “bottom events” with “AND” and “OR” gates. Each bottom event models a separate failure event, which can be a complex multi-physics simulation as described earlier in the paper. The topology of the fault tree is defined by the failure modes being simulated. Once defined, several options are available for solving the system reliability problem. First, direct Monte Carlo simulation is available but may be cost prohibitive if the limit state functions of the bottom events are computationally expensive. Alternatively, NESSUS can compute the probability of system failure using the Advanced Mean Value (AMV+) method [9] or Adaptive Importance Sampling (AIS) [10]. Because the NESSUS PFTA uses a limit state function to represent each bottom event, correlations due to common random variables between the bottom events is fully accounted for regardless of the probabilistic method used. In addition to quantifying the system reliability, NESSUS also computes probabilistic sensitivities of the system probability of failure with respect to the each random variable’s mean value and standard deviation. [3] These results provide a ranking based on the relative contribution of each variable to the total probability of failure. The sensitivities are also useful in design optimization, test planning and resource allocation. Probabilistic fault trees for system problems are defined in NESSUS using a graphical editor. Once the system is defined in the GUI, the corresponding Boolean algebraic statement is transferred to the problem statement window, where the user then defines each event. An example fault tree and problem statement for a two gate, three-event system is shown in Figure 1.

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2.3

Reliability Modeling Process

The steps needed to solve a reliability problem in NESSUS include: 1) Develop the functional relationships that define the model, 2) define the random variable inputs, 3) define the numerical models needed in the functional relationship, 4) perform parameter variation studies to check and understand the deterministic behavior of the model, 5) perform the probabilistic analysis, and 6) visualize the results. NESSUS uses an outline structure to define the problem, as shown in the left hand side of Figure 2. The user navigates through the nodes of the outline from top to bottom to define the problem and perform the analysis. Each of these steps is described in more detail in the following sections. 2.3.1

Problem Statement Definition

The problem statement window in NESSUS is where the functional relationships are entered to define the model. In the problem statement window, each model is defined only in terms of input and output variables and mathematical operators. This improves readability, conveys the essential flow of the analysis, and allows complex reliability assessments to be defined when more than one model is required to define the system performance. An example problem statement is shown in the upper right hand portion of Figure 2. A powerful feature of NESSUS is the ability to create complex probabilistic simulations by linking models together in a sequential fashion. In this example, the performance measure is life (given by number of cycles to failure), which requires input from other models. Two stress quantities from an ABAQUS (ABAQUS, Inc.) finite element analysis are used in the analytical life model. Many other finite element codes are interfaced with NESSUS and will be described in a subsequent section. Finally, the ABAQUS model requires input from several independent variables. The problem statement parser in NESSUS identifies all of the independent variables in the problem statement window and transfers these variables to the random variable input window for further definition. 2.3.2

Random Variable Input and Probabilistic Database

The random variable inputs are defined in the random variable definition window in NESSUS. A graphical input editor is provided for distributions requiring parameters other than the mean and standard deviation, such as upper and lower bounds for truncated distributions. The probability 6

density function (PDF) and cumulative distribution function (CDF) plotting capability in NESSUS provides a quick visual inspection of the random variables. Random variables allowed in NESSUS include normal, lognormal, Weibull, extreme value type I, chi-square, maximum entropy, curve-fit, Frechet, truncated normal and truncated Weibull. The maximum entropy and curve-fit distributions can be used for distributions not directly supported. NESSUS maintains a library of relevant PDFs in a probabilistic database. Random variables can be defined and stored using a distribution type and associated parameters. Distribution fitting functions are provided to determine the best fit from raw data. The entries can be grouped and multiple databases are supported. This allows users to develop their own, possibly proprietary, databases for use in NESSUS. Random variable definitions from the database contents can be inserted directly in the random definition table in NESSUS using a right mouse click as shown in Figure 3. 2.3.3

Response Model Definition

Functions defined in the problem statement window (Figure 2) are assigned in the response model definition. The available function types are selected from the model type drop down menu and include analytical, regression, numerical, and predefined as shown in Figure 4. The analytical function type allows models to be defined with standard mathematical operators, using a format identical to definitions in the problem statement window. The numerical model type allows the use of interfaced codes or a user-defined code. Codes currently interfaced to NESSUS include ABAQUS, ANSYS (ANSYS, Inc.), DYNA3D (Lawrence Livermore National Laboratory), LS-DYNA (LSTC, Inc.), NASA_GRC_FEM (NASA Glenn Research Center), MADYMO (TNO Automotive), MSC.NASTRAN (MSC.Software), PRONTO (Sandia National Laboratories), and USER_DEFINED. The regression model type allows the user to input function coefficients or raw perturbation data that can be fit to linear or quadratic functions using linear regression. Finally, the predefined model type allows linking user-written Fortran subroutines with NESSUS, which requires the user to have access to a Fortran compiler. The user-defined numerical model allows the user to link the NESSUS probabilistic engine with any stand-alone analysis code.

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Figure 4 shows an example using the ABAQUS finite element program. The execution command window provides the command or commands required to execute ABAQUS. The input and output files are also defined on this input screen. Default execution options for the supported codes are inserted automatically by NESSUS from a configurable template file, and can be modified by the user as needed. A batch processing option is provided to allow processing on different computers. This allows NESSUS to run on a local workstation while the analysis codes run on a different workstation, cluster, supercomputer, etc. Related to the batch processing feature, NESSUS also provides an automatic restart option. The restart capability provides probabilistic solution refinement, recovering from abnormal solver termination, and evaluating additional performance measures without rerunning previous steps of the solver analyses. 2.3.3.1 Mapping Random Variables to Numerical Models When performing probabilistic analysis using a numerical model, a realization of a random variable must be reflected in the numerical model’s input. The variable may be a random variable or a computed variable from another code or analytical equation. In general, the variable can map to a single value in the code’s input or to a vector of values such as nodal coordinates in a finite element model. Typical examples of single value mappings include Young's modulus or a concentrated point load. Examples of vector mappings are a pressure field acting on a set of elements or a geometric parameter that effects multiple node locations. Mapping variables to the numerical model input in NESSUS is achieved by graphically identifying the lines and columns that are changed when the variable changes as shown in Figure 5. The visual feedback of the mapping between the random variable and the finite element input greatly reduce the chance for error. The mapping capability in NESSUS has been optimized to support model input files in excess of several million lines in length.

Vector mappings require a functional relationship between the input random variable and the analysis program input. Because different realizations of these variables are required, a general approach is used in NESSUS to relate a change in the input random variable value to the code’s

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input. For example, if the random variable is the radius of a hole, changes to a set of nodal coordinate values will be required each time the radius is changed. A “delta vector,” ∆x , is defined that relates how the coordinates change with a change in the variable. The vector of perturbed nodal coordinates, xˆ , is related to the mean value of the coordinates, µ x , plus a shift

factor, s, times the amount of change for the coordinates, ∆x , or in equation form, xˆ = µx + s ⋅ ∆x .

The delta vector is the normalized difference between the mean value of the random variable and the perturbed value. One approach to generating ∆x is to perturb the nominal mesh, subtract the nominal from the perturbed, and then normalize. This procedure, performed only once at the beginning of the analysis, is then used by NESSUS to create a finite element mesh for any value of the random variable. Several other approaches are available for defining vector variables. Some analysis codes allow the finite element model to be parametrically defined. In this case, the variables can be mapped directly without defining the delta vector. Another option is to include a finite element preprocessor using the linked model capability. The variables can be mapped to the preprocessor input and the resulting model used for the analysis. 2.3.3.2 Selecting Responses for Numerical Models The final step in defining the numerical model is to identify the response quantity or quantities that are to be returned to NESSUS. The approach used in NESSUS is to read the analysis results for a given set of node, element and time steps directly from the analysis code’s results file. Figure 6 shows the response selection for the ABAQUS finite element software. NESSUS supports automated extraction for most engineering quantities of interest including displacements, velocities, accelerations, stresses, strains, etc. When multiple response quantities are requested, NESSUS provides further options to reduce the results down to a single value using functions such as maximum, minimum, average, etc. For dynamic codes, selection of the response from a result time series is provided across multiple times such as maximum, last, and user specified. In some cases, the response time series can be filtered to smooth the response before use in the probabilistic analysis. 9

A flexible user defined numerical model capability is provided in NESSUS. This capability allows users to link in-house developed codes with NESSUS. The response of interest is selected by defining a specific location in the analysis code’s results file. A user subroutine for extracting responses is also available for more complex situations such as results extraction from a binary database file. 2.3.4

Deterministic and Parameter Variation Analysis

NESSUS’ deterministic analysis option provides a useful tool to verify the problem statement definition. Any computed value (on the left of the equal sign) in the problem statement will be evaluated at the mean values of the input random variables. Parameter variation analysis is another useful tool to understand how the performance varies with changes in the random variables. NESSUS provides several methods for defining variable perturbations including, backward, central, and forward differences as well as variable sweeps. In addition, specific perturbation values can be input directly to define experimental designs. Visualization of the response variation is provided in predefined XY scatter plots. 2.3.5

Probabilistic Analysis Definitions

Many efficient probabilistic analysis methods have been devised to alleviate the need for Monte Carlo simulation, which is impractical for large-scale high-fidelity problems. [11] The traditional methods include, for example, the first and second-order reliability methods (FORM and SORM) [12], the response surface method (RSM) [13], and Latin hypercube simulation (LHS) [14]. Methods tailored for complex probabilistic finite element analysis include, for example, the advanced mean value family of methods (AMV+) [9] and adaptive importance sampling (AIS). [10] Further details on these methods and their implementation in NESSUS are given in Ref. [3]. NESSUS has a suite of probabilistic analysis methods as listed in Table 1 for both component and system probabilistic analysis. The range of methods allows the analyst to obtain probabilistic solutions with different levels of fidelity based on the requirements of the analysis. NESSUS provides complete control of each of the available probabilistic methods. As an example, Figure 7 shows the probabilistic analysis definition screen for the AMV+ method. Default parameters 10

for the different methods are supplied based on experience with the method on previous problems. In addition to defining the probabilistic method, several other options can be selected: parameter correlations, confidence bounds, and analysis type. Linear correlation between any two input variables is defined by entering the correlation coefficient. By default the input variables are assumed to be statistically independent, i.e., zero correlation. If any non-zero correlations are entered, NESSUS will perform a numerical transformation during the probability integration to account for the correlation. NESSUS computes confidence bounds on the computed probabilities from statistical uncertainty on the mean or standard deviation of each input random variable. To define statistical uncertainty, the user enters a coefficient of variation (COV) on the mean and standard deviation for each of the input random variables. All COV values are zero by default. The analysis type definition indicates that the probabilistic method will compute 1) the full CDF of the response, 2) the probability associated with a specified performance or list of performance values, or 3) the performance given a specified probability or set of probability values. 2.3.6

Results Visualization

NESSUS includes a powerful post processing capability. After completing the probabilistic analysis, the user can visualize the CDF in several formats (Figure 8). In addition, the various probabilistic sensitivity measures computed by NESSUS can be viewed as shown in Figure 8. Multiple analyses can be compared on a single plot as a means of comparing different analysis methods (e.g., Monte Carlo and AMV+) or random variable changes (design “what-if” analyses). Finally, the user has control over all plot formats such as line styles, titles, and number format. All plots are easily exported for inclusion in reports or presentations. Three-dimensional model viewing with color probability contouring is another highly useful visualization output of NESSUS. The failure probability is computed at different locations in the model (e.g. all of some of the nodes in a finite element mesh) and visualized by contouring isoprobability values. Contours of probability can reveal regions of high risk that may not be apparent from the contours of model response quantities. Consider the spatial thickness fluctuations of a part induced by the rolling or stamping process. Figure 9 shows that even

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though the mean stress at point B is higher than at point A, the probability of failure is lower due to the larger uncertainty at point A. An example of a large-scale analysis utilizing probability contouring is shown in Figure 10. The probability contours identify regions where there is significant probability that the equivalent plastic strain exceeds the design limit. As shown, these high-probability regions are not identified by the equivalent plastic strain contours. 3

APPLICATION EXAMPLES

The NESSUS software has been used to predict the reliability and probabilistic response for a wide range of problems. [15-23] Three problems are presented in this section to demonstrate the application and flexibility of NESSUS and to illustrate the current developments to support efficient probabilistic model development and support for large-scale problems. 3.1

Stochastic Crashworthiness

The NESSUS probabilistic analysis software was used to compute the system reliability of a Sport utility vehicle to small vehicle frontal offset impact event. The analysis was designed to identify important variables contributing to the crashworthiness reliability and use this information to improve the design and manufacturing processes. The ultimate goal of the analysis is to improve vehicle reliability using a computational approach to reduce expensive crash testing. Additional details about this analysis can be found in Ref. [24]. 3.1.1

Problem Description

An LS-DYNA finite element model of a vehicle frontal offset impact and a MADYMO model of a 50th percentile male Hybrid III dummy were integrated with NESSUS to comprise the crashworthiness characteristics (Figure 11). A number of different response quantities from the models were used to define four occupant injury acceptance criteria and six compartment intrusion criteria. The NESSUS problem statement for the Head Injury Criteria (HIC) is shown in Figure 12. An acceleration history from the LS-DYNA vehicle model is used as the crash pulse input to the occupant injury model in MADYMO. The other three occupant injury criteria are modeled in the same fashion. The compartment intrusion criteria are determined from relative displacements of the points in the small vehicle model. These ten acceptance criteria 12

were used as events in a probabilistic fault tree to compute the overall system reliability of the impact scenario. Uncertainty inputs to the model consist of 16 random variables. These random variables include parameters that define key energy absorbing components of the vehicles such as material properties for bumpers and rails, test environment uncertainties such as impact velocity and angle, manufacturing variations in the form of rail and bumper installation parameters, and inherent uncertainty of material characteristics. Each of these random variables is characterized by a statistical distribution defined from manufacturing data, literature and/or expert opinion. The distributions for parameters that affect the geometry are based on design/manufacturing tolerances. A response surface model was developed for each acceptance criteria to facilitate the probabilistic analysis and vehicle design tradeoff studies. The parameter variation analysis capability in NESSUS was utilized to develop the response surface models. A vehicle redesign was performed based on the probabilistic sensitivity information to improve the reliability. 3.1.2

Results

The system reliability was computed using the Monte Carlo simulation method in NESSUS with 100,000 samples. The computed system reliability for the original design is 23%. A Monte Carlo analysis was performed for each criterion and the results are shown in Table 2. The femur axial load acceptance criteria event has the lowest reliability followed by the HIC event and the door aperture closure event. All other acceptance criteria have relatively high reliability. The computed probabilistic sensitivity factors are shown in Figure 13. From the figure, the nominal value of the yield strength of the small vehicle rail material can be most influential in increasing the reliability. The objective of the redesign analysis is to provide a recommendation to improve the reliability of the small vehicle in a vehicle-to-vehicle frontal offset impact. The approach used is to rely on

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the probabilistic sensitivity factors to identify the dominant parameters (random variable mean and standard deviation) that will improve system reliability. The reliability for each acceptance criteria in the new design is listed in Table 2. The dominant event for the original design was the femur axial load acceptance criteria. The femur axial load also shows the lowest reliability for the final design but increased from a reliability of 46% to 93%. The reliability improvements are shown in Figure 14 along with a description of the parameter changes to achieve the improvement. The system reliability for the final design is 86%. A system reliability analysis is critical to the correct evaluation of the vehicle performance especially for evaluating the probabilistic sensitivity factors at the system level for redesign analysis. Certain parameters such as stiffness/strength parameters can improve reliability for compartment intrusion performance measures but may be detrimental to the crash pulse attenuated to the vehicle occupant. The system model correctly accounts for events with common variables (correlated events) and thus correctly identifies the important variables on the system level. 3.2

Blast Containment Vessel

Over the past 30 years, Los Alamos National Laboratory (LANL), under the auspices of DOE, has been conducting confined high explosion experiments utilizing large, spherical, steel pressure vessels. These experiments are performed in a containment vessel to prevent the release of explosion products to the environment. Design of these spherical vessels was originally accomplished by maintaining that the vessel’s kinetic energy, developed from the detonation impulse loading, be equilibrated by the elastic strain energy inherent in the vessel. Within the last decade, designs have been accomplished utilizing sophisticated and advanced 3D computer codes that address both the detonation hydrodynamics and the vessel’s highly nonlinear structural response. Additional details about this analysis can be found in Refs. [22,25]. 3.2.1

Problem Description

The containment vessel, shown on the left side in Figure 15, is a spherical vessel with three access ports: two 16-inch ports aligned in one axis on the sides of the vessel and a single 22-inch 14

port at the top of the vessel. The vessel has an inside diameter of 72 inches and a 2 inch nominal wall thickness. The vessel is fabricated from HSLA-100 steel, chosen for its high strength, high fracture toughness, and no requirement for post weld heat treatment. The vessel’s three ports must maintain a seal during use to prevent any release of reaction product gases or material to the external environment. Each door is connected to the vessel with 64 high strength bolts, and four separate seals at each door ensure a positive pressure seal. A series of hydrodynamic and structural analyses of the spherical containment vessel were performed using a combination of two numerical techniques. Using an uncoupled approach, the transient pressures acting on the inner surface of the vessel were computed using the Eulerian hydrodynamics code, CTH (Sandia National Laboratories), which simulated the high explosive (HE) burn, the internal gas dynamics, and shock wave propagation. The HE was modeled as spherically symmetric with the initiating burn taking place at the center of the sphere. The vessel’s structural response to these pressures was then analyzed using the DYNA3D explicit finite element structural dynamics code. The simulation required the use of a large, detailed mesh to accurately represent the dynamic response of the vessel and to adequately resolve the stresses and discontinuities caused by various engineering features such as the bolts connecting the doors to their nozzles. Taking advantage of two planes of symmetry, one quarter of the structure was meshed using approximately one million hex elements. Six hex elements were used through the 2-inch wall thickness to accurately simulate the bending behavior of the vessel wall. The one-quarter symmetry model is shown on the right hand side of Figure 15. The structural response simulation used an explicit finite element code called PARADYN (Lawrence Livermore National Laboratory), which is a massively parallel version of DYNA3D, a nonlinear, explicit Lagrangian finite element analysis code for three-dimensional transient structural mechanics. PARADYN was run on 504 processors of LANL’s “Blue Mountain,” massively parallel computer, which is an interconnected array of independent SGI (Silicon Graphics, Inc.) computers. The containment vessel model can be solved on the Blue Mountain computer with approximately 2.5 hours of run time. The same analysis would have taken about 35 days when run on a single processor.

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The four random variables considered are radius of the vessel wall (radius), thickness of the vessel wall (thickness), modulus of elasticity (E), and yield stress (Sy) of the HSLA steel. A summary of the probabilistic inputs is included in Table 3. The properties for radius and thickness are based on a series of quality control inspection tests that were performed by the vessel manufacturer. The coefficients of variation for the material properties are based on engineering judgment. In this case, the material of the entire vessel, excluding the bolts, is taken to be a random variable. When the thickness and radius random variables are perturbed, the nodal coordinates of the finite element model change with the exception of the three access ports in the vessel, which remain constant in size and move only to accommodate the changing wall dimensions. This was accomplished in NESSUS by defining a set of scale factors that defined how much and in what direction each nodal coordinate was to move for a given perturbation in both thickness and radius. The NESSUS mapping procedure allows the perturbations in radius and thickness to be cumulative so these variables can be perturbed simultaneously. Once the scale factors are defined and input to NESSUS, the probabilistic analysis, whether by simulation or using AMV+, can be performed without further user intervention. The response metric for the probabilistic analysis is the maximum equivalent plastic strain occurring over all times at the bottom of the vessel finite element model. This maximum value occurred well after the initial pulse and was caused by bending modes created by the ports. 3.2.2

Results

The AMV+ method in NESSUS was used to calculate the CDF of equivalent plastic strain. Also, LHS was performed with 100 samples to verify the correctness of the AMV+ solution near the mean value. The CDF is plotted on the left in Figure 16 on a standard normal probability scale. As shown, the LHS and AMV+ results are in excellent agreement. However, in contrast to the LHS solution, the AMV+ solution predicts accurate probabilities in the extreme tail regions with far fewer PARADYN model evaluations. Probabilistic sensitivities are shown in on the right in Figure 16. The sensitivities are multiplied by σ i to nondimensionalize the values and facilitate a relative comparison between parameters. 16

The values are also normalized such that the maximum value is equal to one. It can be concluded that the reliability is most sensitive to the mean and standard deviation of the thickness of the containment vessel wall. 3.3

Cervical Spine Impact Injury

Cervical spine injuries occur as a result of impact or from large inertial forces such as those experienced by military pilots during ejections, carrier landings, and ditchings. Other examples include motor vehicle, diving, and athletic-related accidents. Reducing the likelihood of injury by identifying and understanding the primary injury mechanisms and the important factors leading to injury motivates research in this area. [26] Because of the severity associated with most cervical spine injuries, it is of great interest to design occupant safety systems to minimize probability of injury. To do this, the designer must have quantified knowledge of the probability of injury due to different impact scenarios, and also know which model parameters contribute the most to the injury probability. Finite element stress analysis plays a critical role in understanding the mechanics of injury and the effects of degeneration as a result of disease on the structural performance of spinal segments. However, in many structural systems, there is a great deal of uncertainty associated with the environment in which the structure is required to function. This variability or uncertainty has a direct effect on the structural response of the system. Biological systems are a textbook example: uncertainty and variability exist in the physical and mechanical properties and geometry of the bone, ligaments, cartilage, as well as uncertainty in joint and muscle loads. Hence, the broad objective of this investigation is to explore how uncertainties influence the performance of an anatomically accurate, three-dimensional, non-linear, experimentally validated finite element model of the human lower cervical spine. 3.3.1

Problem Description

A validated three-dimensional ABAQUS finite element model of the C4-C5-C6 spinal segment developed at the Medical College of Wisconsin [27] was used to calculate the structural response of the lower cervical spine and to quantify the effect of uncertainties on the performance of the biological system. The load-deflection response was validated against experimental results from eight cadaver specimens [28]. The moment–rotation response of the finite element model was 17

validated against experimental results reported in the literature [29]. The model is shown in Figure 17. Additional details about this analysis can be found in Ref. [30]. Biological variability was accounted for by modeling material properties and spinal segment loading as random variables. Where available, experimental data was used to generate the random variable definitions (e.g. the spinal ligaments load-deflection behavior). The probabilistic finite element model was exercised under flexion (chin down) loading by applying a pure bending moment of 2 N-m to the superior surface of the C4 vertebra. The inferior surface of the C6 vertebra was fixed in all directions and rotation was measured between the superior aspect of C4 and the fixed boundary of C6. Computing the rotation and monitoring the reaction forces at the fixed boundary quantified the moment-rotation behavior. Cumulative probability distribution functions, probability distribution functions, and probabilistic sensitivity factors were determined. The random variables considered in the analysis are given in Table 4. The mean values were set equal to the nominal values used in Ref. [27]. Thus, the original deterministic solution will be obtained when the random variables are set equal to their mean values. Since data were not available to characterize all of the model inputs, it was decided to model all random variables with a coefficient of variation of ten percent. Default distributions were assigned based on experience, i.e., a lognormal distribution was used to model modulus variables and a normal distribution was used otherwise. 3.3.2

Results

Probabilistic results were computed using the coupled NESSUS – ABAQUS model and the AMV+ probabilistic method. In the analysis, 41 function (ABAQUS) evaluations were required. The probabilistic rotation response had an approximate mean of 3.82 degrees and a standard deviation of 0.38 degrees resulting in a coefficient of variation of 10%. The CDF and PDF of rotation is shown on the left in Figure 18. The CDF is used to determine probabilities directly, e.g., the probability that the rotation will be less than or equal to 4.2o is 82%.

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The probabilistic sensitivity factors indicate that the loading (FLEXLOAD) is the dominant variable. The bar graph on the right in Figure 18 shows the sensitivity information for the eight most significant random variables with FLEXLOAD removed so that the other variables can be more clearly seen. Not including FLEXLOAD, the most important variables are the: 1) annulus C45 and C56 Young’s modulus, 2) interspinous ligament nonlinear spring force-deflection relationship, and 3) ligamentum flavum nonlinear spring force-deflection relationship. These results can be used eliminate unimportant variables from the random variable vector and to focus further characterization efforts on those variables that are most significant. 4

CONCLUSIONS

Although NESSUS was initially developed for aerospace applications, the methods are broadly applicable and their use warranted in situations where uncertainty is known or believed to have a significant impact on the structural response. The framework of NESSUS allows the user to link advanced probabilistic algorithms with analytical equations, commercial finite element analysis programs and “in-house” stand-alone deterministic analysis codes to compute the probabilistic response or reliability of a system. For probabilistic methods to be accepted for use in design, probabilistic tools must be robust, easy to use, and interfaced with widely used commercial analysis packages. This integration with commercially available analysis software leverages the investment made in learning and becoming proficient with the software. The graphical user interface in NESSUS makes defining and executing the probabilistic analysis straightforward and efficient for simple problems as well as problems involving extremely large multi-physics models. Several applications were presented that demonstrated the flexibility of the NESSUS software. The advanced probabilistic analysis methods in NESSUS allow for using high-fidelity models to define the structure or system even when each function evaluation may take several hours to run. In the application problems presented, the probabilistic results revealed additional information that would not have been available if deterministic approaches were used.

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Future progress in probabilistic mechanics relies strongly on the development of validated analysis models, systematic data collection and synthesis to resolve probabilistic inputs, and identification and classification of failure modes. Research and development in this area is needed to improve the robustness of the underlying probability integration methods, to develop alternative uncertainty modeling approaches and integrate these approaches with established probabilistic tools, and to apply probabilistic methods to model verification and validation, system certification and prognosis, component life assessment and integrity, and structural system health monitoring and management. 5

ACKNOWLEDGEMENTS

The authors wish to acknowledge the support of the NASA Glenn Research Center and the Los Alamos National Laboratory for their significant support of the NESSUS software. The 2000 DaimlerChrysler Challenge Fund, Naval Air Warfare Center Aircraft Division, and Los Alamos National Laboratory are also acknowledged for their support for the applications problems summarized in the paper. 6

REFERENCES

1. NESSUS User’s manual, Version 8, Southwest Research Institute, www.nessus.swri.org, 2004. 2. Southwest Research Institute, “Probabilistic Structural Analysis Methods (PSAM) for Select Space Propulsion System Components,” Final Report NASA Contract NAS3-24389, NASA Lewis Research Center, Cleveland, Ohio, 1995. 3. NESSUS Theory manual, Version 8, Southwest Research Institute, 2004. 4. Thacker, B.H., Riha, D.S., Millwater, H.R., Enright, M.P., “Errors and Uncertainties in Probabilistic Engineering Analysis,” Proceedings AIAA/ASME/ASCE/AHS/ASC 42nd Structures, Structural Dynamics, and Materials (SDM) Conference, AIAA 2001-1239, Seattle, WA, 16-19 April 2001. 5. Riha, D.S., Thacker, B.H.,Fitch, S.H.K “NESSUS Capabilities for Ill-Behaved Performance Functions,” Proceedings AIAA/ASME/ASCE/AHS/ASC 45th Structures, Structural Dynamics, and Materials (SDM) Conference, AIAA 2004-1832, Palm Springs, CA, 19-22 April 2004.

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6. Huyse, L. and B.H. Thacker, “Treatment of Conflicting Expert Opinion in Probabilistic Analysis,” Proc. 11th IFIP WG7.5 Working Conference on Reliability and Optimization of Structural Systems, M.A. Maes and L. Huyse, Eds, A.A. Balkema Publishers, Banff, Canada, November 2003. 7. Thacker, B.H. and L. Huyse, A Framework to Estimate Uncertain Random Variables, AIAA Paper 2004-1828, 45th AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics and Materials Conference, 2004. 8. Torng, T.Y., Wu, Y.-T., and Millwater, H.R., "Structural System Reliability Calculation Using a Probabilistic Fault Tree Analysis Method," Proc. 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Paper No. AIAA-92-2410, Dallas, TX, April 13-15, 1992. 9. Wu, Y.-T., H. R. Millwater, and T. A. Cruse, “Advanced Probabilistic Structural Analysis Methods for Implicit Performance Functions,” AIAA Journal, 28(9), 1990. 10. Wu, Y.-T., “Computational Method for Efficient Structural Reliability and Reliability Sensitivity Analysis,” AIAA Journal, Vol. 32, 1994. 11. Ang, A. H.-S. and Tang, W. H., Probabilistic Concepts in Engineering Planning and Design, Volume II: Decision, Risk, and Reliability, New York: John Wiley & Sons, Inc., 1984. 12. Madsen, H. O., Krenk, S., and Lind, N. C., Methods of Structural Safety, Prentice-Hall, Inc., New Jersey, 1986 13. Faravelli, L., “Response Surface Approach for Reliability Analysis,” J. of Eng. Mech., 115(12), 1989. 14. McKay, M.D. and R.J. Beckman, 1979, “A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code.” Technometrics, Vol. 21, No. 2, pp. 239-245. 15. Riha, D.S., B.H. Thacker, D.A. Hall, T.R. Auel, and S.D. Pritchard, “Capabilities and Applications of Probabilistic Methods in Finite Element Analysis,” Int. J. of Materials & Product Technology, Vol. 16, Nos. 4/5, 2001. 16. Millwater, H., Griffin, K., Wieland, D., West, A., Smith, H., Holly, M., and Holzwarth, R., “Probabilistic Analysis of an Advanced Fighter/Attack Composite Wing Structure,” Proc. 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conf., Paper No. 2000-1567, Atlanta, GA, April 3-6, 2000. 21

17. Shah, C.R., Sui, P., Wang, W., and Wu, Y.-T., "Probabilistic Reliability Analysis of an Engine Crankshaft," Proceedings 8th Int. ANSYS Conf., August 1998. 18. Thacker, B.H., Oswald, C.J., Wu, Y.-T., Patterson, B.C., Senseny, P.E., and Riha, D.S., "A Probabilistic Multi-Mode Damage Model for Tunnel Vulnerability Assessment," Proc. 8th Annual Symposium on the Interaction of the Effects of Munitions with Structures, Volume II, pg. 137-148, McClean, VA, April 22-25, 1997. 19. Millwater, H. R. and Wu, Y.-T., 1993, “Computational Structural Reliability Analysis of a Turbine Blade,” Proc. of the International Gas Turbine And Aeroengine Congress and Exposition, Cincinnati, OH, May 24-27. 20. Thacker, B. H., Wu, Y.-T., Nicolella, D. P. and Anderson, R. C., "Probabilistic Injury Analysis of the Cervical Spine," Proc. AIAA/ASME/ASCE/AHS/ASC 38th Structures, Structural Dynamics, and Materials (SDM) Conf., AIAA 97-1135, Kissimmee, Florida, 7-10 April 1997. 21. Thacker, B.H., Rodriguez, E.A., Pepin, J.E., Riha, D.S., “Application of Probabilistic Methods to Weapon Reliability Assessment,” Proc. AIAA/ASME/ASCE/AHS/ASC 42nd Structures, Structural Dynamics, and Materials (SDM) Conference, AIAA 2001-1458, Seattle, WA, 16-19 April 2001. 22. Rodriguez, E.A., Pepin, J.W., Thacker, B.H., Riha, D.S., “Uncertainty Quantification of A Containment Vessel Dynamic Response Subjected to High-Explosive Detonation Impulse Loading”, Proc. AIAA/ASME/ASCE/AHS/ASC 43rd Structures, Structural Dynamics, and Materials (SDM) Conference, AIAA 2002-1567, Denver, CO, April 2002. 23. Pepin, J.E., Thacker, B.H., Rodriguez, E.A., Riha, D.S., “A Probabilistic Analysis of a Nonlinear Structure Using Random Fields to Quantify Geometric Shape Uncertainties”, Proc. AIAA/ASME/ASCE/AHS/ASC 43rd Structures, Structural Dynamics, and Materials (SDM) Conference, AIAA 2002-1641, Denver, CO, April 2002. 24. Riha, D.S., Hassan, J.E., Forrest, M.D., Ding, K., “Stochastic Approach for Vehicle Crash Models,” Proc. SAE 2004 World Congress & Exhibition, 2003-01-0460, Detroit, MI., March 2004. 25. Thacker, B.H., Rodriguez, E.A., Pepin, J.E., Riha, D.S., “Uncertainty Quantification of a Containment Vessel Dynamic Response Subjected to High-Explosive Detonation Impulse

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Loading,” IMAC-XXI: Conference & Exposition on Structural Dynamics, No. 261, Kissimmee, FL, 3-6 February 2003. 26. Thacker, B.H., Y.-T. Wu, and Nicolella, D.P., Frontiers in Head and Neck Trauma: Clinical and Biomechanical, Probabilistic Model of Neck Injury, N. Yoganandan, F.A. Pintar, S.J. Larson and A. Sances, Jr. (eds.), IOS Press, Harvard, MA, 1998. 27. Kumaresan, S., Yoganandan, N., Pintar, F.A., and Maiman D., “Finite Element Modeling of the Lower Cervical Spine; Role of Intervertebral Disc under Axial and Eccentric Loads,” Medical Engineering and Physics, Vol. 21, pp. 689-700, 2000. 28. Pintar, F.A., Yoganandan, N., Pesigan, M., Reinartz, J.M., Sances, A., and Cusik, J.F., “Cervical vertebral strain measurements under axial and eccentric loading,” ASME Journal of Biomechanical Engineering, Vol. 117, pp. 474-478, 1995. 29. Shea, M., Edwards, W.T., White, A.A., and Hayes, W.C., “Variations of stiffness and strength along the human cervical spine,” Journal of Biomechanics, Vol. 24(2) pp. 95-107, 1991. 30. Thacker, B.H., D. P. Nicolella, S. Kumaresan, N. Yoganandan, F. A. Pintar, “Probabilistic Finite Element Analysis of the Human Lower Cervical Spine,” Math. Modeling and Sci. Computing, Vol. 13, No. 1-2, pp. 12-21, 2001.

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Table 1: Probabilistic analysis methods in NESSUS. Probabilistic Method First order reliability method (FORM) Advance first order reliability method Second order reliability method (SORM) Importance sampling with radius reduction factor Monte Carlo simulation Importance sampling with user-defined radius Plane-based adaptive importance sampling Curvature-based adaptive importance sampling Mean value Advanced mean value Advanced mean value with iterations Latin hypercube simulation Response surface method with Monte Carlo simulation

Component X X X X X X X X X X X X X

System

X X X

Table 2: Original and final design reliability for the stochastic car crash example. Acceptance Criteria Description NESSUS Variable

Reliability Original Design Final Design

HIC

g_hic

57.7910%

94.0120%

Chest acceleration

g_cg

92.2970%

98.8240%

Chest deflection

g_chestd

99.9752%

99.9999%

Femur axial load

g_femurl

46.4020%

92.9330%

Footrest intrusion

g_fri

99.9623%

100.0000%

Toepan deflection

g_tpd

100.0000%

100.0000%

Brake pedal location

g_bpd

100.0000%

100.0000%

Instrument panel def.

g_ipd

99.6870%

99.9719%

Door aperture closure

g_dac

72.6750%

98.7460%

Engine location

g_engd

99.6000%

99.9997%

Table 3: Probabilistic inputs for the containment vessel example problem. Variable Radius (in) Thick (in) E (lb/in2) Sy (lb/in2)

PDF Normal Lognormal Lognormal Normal

σ 37.0 2.0 29.0E+06 106.0E+03

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µ 0.0521 0.08667 1.0E+06 4.0E+03

COV 0.00141 0.04333 0.03448 0.03774

Table 4: Probabilistic inputs for the cervical spine impact injury example problem. Name

Description Anterior longitudinal ligament nonlinear MALLRV spring force-deflection relationship. Posterior longitudinal ligament MPLLRV nonlinear spring force-deflection relationship. Interspinous ligament nonlinear spring MISRV force-deflection relationship. Capsular ligament nonlinear spring MCLRV force-deflection relationship. Ligamentum flavum nonlinear spring MLFRV force-deflection relationship. 45 disc cross-sectional area (rebar C45AREA element) inner/middle/outer & in/out. 56 disc cross-sectional area (rebar C56AREA element) inner/middle/outer & in/out. C4PE C4 posterior Young’s modulus. C5PE C5 posterior Young’s modulus. C6PE C6 posterior Young’s modulus. C4CE C4 cancellous Young’s modulus. C5CE C5 cancellous Young’s modulus. C6CE C6 cancellous Young’s modulus. A45E Annulus C45 Young’s modulus. Lusckha membrane C45 Young’s LM45E modulus. A56E Annulus C56 Young’s modulus. Lusckha membrane C56 Young’s LM56E modulus. Synovial membrane C45/C56 facet SM456E Young’s modulus. NP45FD Nucleus pulposus C45 fluid density. NP56FD Nucleus pulposus C45 fluid density. FSF45FD Facet synovial fluid C45 fluid density. FSF56FD Facet synovial fluid C56 fluid density. LJ45FD Luschka’s joint C45 fluid density. LJ56FD Luschka’s joint C56 fluid density. CORTE Cortical Young’s modulus. ENDPE Endplate Young’s modulus. CARTE Cartilage Young’s modulus. FIBRE Fiber Young’s modulus. FLEXLOAD Flexion loading.

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Mean

Standard Deviation

Distribution

0

1.0

Normal

0

1.0

Normal

0

1.0

Normal

0

1.0

Normal

0

1.0

Normal

0.195

0.0195

Lognormal

0.1985

0.01985

Lognormal

3500 3500 3500 100 100 100 4.7

735 735 735 30 30 30 0.705

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

12

1.2

Lognormal

4.7

0.705

Lognormal

12

1.2

Lognormal

12

1.2

Lognormal

0.000001 0.000001 0.000001 0.000001 0.000001 0.000001 12000 600 10.4 500 1

0.0000001 0.0000001 0.0000001 0.0000001 0.0000001 0.0000001 2520 126 1.04 50 0.01

Normal Normal Normal Normal Normal Normal Lognormal Lognormal Lognormal Lognormal Normal

Figure 1: Multiple limit states are combined in a probabilistic fault tree that is created graphically by the user (left) and entered into the problem statement window in equation form by NESSUS (right).

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Figure 2: NESSUS outline structure guides the probabilistic problem setup and analysis.

27

Figure 3: NESSUS allows random variables to be defined from the probabilistic database via a right mouse click.

Figure 4: The numerical model definition screen in NESSUS defines the execution command and required input/output files for executing the numerical model. 28

Figure 5: NESSUS provides a graphical mapping tool to identify the portions of the code’s input that change when the random variable changes. The mapping can include multiple lines and columns in the code’s input.

29

Figure 6: NESSUS result selection screen for ABAQUS.

30

Figure 7: Options for the AMV+ probabilistic method in NESSUS.

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Figure 8: NESSUS computed cumulative distribution function (left) and probabilistic importance factors (right).

Stress Contours A

Point B: Higher mean stress Lower probability

Point A: Lower mean stress Higher probability

Failure Stress

B Probability Contours Stress

Figure 9: Stress and probability contours illustrating how the failure probability can be higher at a low stress point than at a higher stress point.

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Equivalent Plastic Strain

Probability of Failure pf=P{eps>0.005}

Figure 10: Probability of failure contours (right) indicate critical design regions not identified from the mean equivalent plastic strain contours (left).

Figure 11: Vehicle-to-vehicle frontal offset crash simulation model.

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Figure 12: NESSUS problem statement for the head injury criterion (HIC).

Figure 13: Probabilistic sensitivity factors for the original design indicate that changing the mean value of the rail yield strength will have the largest impact on the overall reliability.

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100 90

Tighten rail and bumper installation tolerances Reduce foam properties COV Reduce yield stress COV

System Reliability (%)

80 70 60

Reduce front weld stiff. COV Reduce rail thickness COV Reduce front weld stiff. COV Reduce yield stress COV

50

Increase weld stiffness (front) 40

Increase weld stiffness

30

Reduce yield stress COV Increase yield stress

20 10 0

2

4

6

8

10

Redesign Iteration Figure 14: Vehicle system reliability improvement study performed with NESSUS.

Figure 15: Containment vessel (left) and one quarter symmetry mesh used for the structural analysis (right).

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Figure 16: Cumulative distribution function of equivalent plastic strain plotted on standard normal scale (left) and probabilistic sensitivity factors (u = 3) (right).

Figure 17: Probabilistic cervical motion segment model (C4-C6).

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0.1 0.05 0 -0.05 -0.1

56

V CL R V FI B R C E 56 A R EA C 45 A R EA

Rotation (Degrees)

M

6

IS R

5

M

4

M

3

A

45

2

A

1

V

-0.15 E

PDF

E

CDF

0.2 0.15

LF R

Probabilistic Sensitivity

Probability

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 18: Cumulative distribution function and probability density function of the rotation of the lower cervical spine segment subjected to pure flexion loading (left). The eight most influential random variables (normalized scale on ordinate) are shown (variable FLEXLOAD removed for clarity).

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