Probe Ng Analysis Nessus

www.elsevier.com/locate/strusafe

Structural Safety 28 (2006) 83–107

STRUCTURAL

SAFETY

Probabilistic engineering analysis using the NESSUS software

Ben H. Thacker a,*, David S. Riha a, Simeon H.K. Fitch b, Luc J. Huyse a,Jason B. Pleming a

a Southwest Research Institute, Reliability and Materials Integrity, 6220 Culebra Road, San Antonio, TX 78228, USAb Mustard Seed Software, San Antonio, TX, USA

Received 8 April 2004; accepted 4 November 2004

Available online 17 February 2005

Abstract

The development of reliability-based design methods requires the use of general-purpose engineering anal-

ysis 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 engineer-

ing 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 andefficiency) 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 applica-

tion examples to demonstrate the applicability of probabilistic analysis to large-scale engineering problems.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: NESSUS; Probabilistic; Software; Reliability; Uncertainty; Stochastic

0167-4730/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.strusafe.2004.11.003

* Corresponding author. Tel.: +1 210 522 3896; fax: +1 208 460 3808.

E-mail address: [email protected] (B.H. Thacker).

mailto:[email protected]

84 B.H. Thacker et al. / Structural Safety 28 (2006) 83–107

1. Introduction and background

Computational simulation is being increasingly used as performance requirements for engineer-ing structures increase. To meet these requirements, analysts are developing higher fidelity modelsin an attempt to more accurately represent the true behavior of the physical system. It is notuncommon nowadays for these models to involve multiple physics and several million finite ele-ments. Despite extraordinary increases in computer power, analyses performed with these highfidelity 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 physicalparameters and loadings. The traditional design approach has been to adopt safety factors to ensurethat the risk of failure is sufficiently small, albeit not quantified. However, probabilistic analysis per-mits amore rigorous quantification of the various uncertainties, and ultimately will facilitate amoreefficient design process. Areas in which probabilistic methods are being successfully applied includeengineered components and systems with high consequences of failure driven by safety or cost con-cerns. Some of these areas include aircraft propulsion systems, airframes, biomechanical systemsand prosthetics, nuclear and conventional weapon systems, space vehicles, pipelines, nuclear wastedisposal, offshore structures and automobiles. In general, probabilistic analysis requires multiplesolutions of the underlying (deterministic) performance model. Consequently, the developmentof efficient and accurate probabilistic analysis methods and software tools are critically needed.

Southwest Research Institute (SwRI) has been addressing the need for efficient probabilistic anal-ysis methods for over 20 years. Much of the reliability technology developed and implemented bySwRI researchers is available in the NESSUS probabilistic analysis software [1]. Numerical evalu-ation of stochastic structures under stress (NESSUS) is a general-purpose tool for computing theprobabilistic response or reliability of engineered systems. NESSUS can be used to simulate uncer-tainties in loads, geometry, material behavior, and other user-defined random variables to predictthe 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 AnalysisMethods (PSAM) for Select Space Propulsion Components’’ [2].

Since the inception of the NASA program, SwRI has continued to conduct research, develop-ment and implementation of probabilistic methods in NESSUS. Through automatic downloadsfrom the NESSUS web site (www.nessus.swri.org), hundreds of copies of the software has beendistributed to a large number of users around the world. Many of these users include universityprofessors and researchers who are incorporating probabilistic design methodologies into theirteaching and research projects.

NESSUS allows the user to perform probabilistic analysis with analytical models, external com-puter programs such as commercial finite element codes, and general combinations of the two. Asan example, consider the problem of estimating the damage to the high-strength steel used in acontainment vessel that confines high explosive experiments. In NESSUS the user can define asimulation to include: (1) an explosive burn calculation to compute the pressure history at thecontainment wall boundary, (2) a finite element stress analysis using the computed pressure his-tory as a load input, and (3) an analytical cumulative damage life calculation based on the com-puted stresses. Each model in the simulation can include random variables. This sequentiallylinked hierarchy of models allows the user to quickly and easily create complex multi-physicsbased probabilistic simulations.

http://www.nessus.swri.org

B.H. Thacker et al. / Structural Safety 28 (2006) 83–107 85

The NESSUS graphical user interface (GUI) is highly configurable and allows tailoring to spe-cific applications. This GUI provides a capability for commercial or in-house developed codes tobe easily integrated into the NESSUS framework. Eleven probabilistic algorithms are available inNESSUS including methods such as Monte Carlo simulation, first-order reliability method, ad-vanced mean value method and adaptive importance sampling [3].

Recent work in NESSUS has been based on reducing the time required to define complex prob-abilistic problems, improving support for large-scale numerical models (greater than one millionelements), and improving the robustness of the low-level probability integration routines. In thearea of robustness, research is underway to improve most probable point (MPP) search algorithms,develop solution strategies for identifying and solving problems that have multiple MPP�s, andimplement adaptive algorithms that can detect numerical difficulties and automatically switch toalternative solution strategies [4,5]. Work is also underway to allow uncertainty due to vague ornon-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 NES-SUS software is described. As appropriate, references are made to considerations for large-scalecomplex models. Three application problems are presented at the end of the paper to illustrate theapplication of NESSUS to real-world problems.

2. Overview of NESSUS

2.1. Component reliability analysis

In NESSUS, component reliability analysis denotes the reliability of a component considering asingle failure mode, where reliability is simply one minus the probability of failure, pf.

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) canbe constructed. Alternatively, NESSUS can compute a single performance value corresponding to aspecific failure probability. The choice of analysis type depends on the problem being solved.

Traditional reliability analysis involves computing the probability of stress, S, exceedingstrength, R, Pr[R 6 S] or Pr[g 6 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 arethe input random variables. In addition to the failure probability, NESSUS computes probabilis-tic importance factors, ob/ou, where b is inversely related to Pf and u are the input random vari-ables transformed into standard normal space, and probabilistic sensitivity factors, ob/oh, where hare the parameters of the input random variables, e.g., mean value and standard deviation.

2.2. System reliability analysis

Most engineering structures can fail in more than one way. System reliability considers the pos-sible failure of multiple components of a system, or multiple failure modes of a component. InNESSUS, system reliability problems are formulated and solved using a probabilistic fault treeanalysis (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 bythe failure modes being simulated. Once defined, several options are available for solving thesystem reliability problem. First, direct Monte Carlo simulation is available but may be cost pro-hibitive if the limit state functions of the bottom events are computationally expensive. Alterna-tively, 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 PFTAuses a limit state function to represent each bottom event, correlations due to common randomvariables between the bottom events is fully accounted for regardless of the probabilistic methodused.

In addition to quantifying the system reliability, NESSUS also computes probabilistic sensitiv-ities of the system probability of failure with respect to the each random variable�s mean value andstandard deviation [3]. These results provide a ranking based on the relative contribution of eachvariable 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 trans-ferred to the problem statement window, where the user then defines each event. An example faulttree and problem statement for a two gate, three-event system is shown in Fig. 1.

Fig. 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).


2.3. Reliability modeling process

The steps needed to solve a reliability problem in NESSUS include: (1) develop the functionalrelationships that define the model, (2) define the random variable inputs, (3) define the numericalmodels needed in the functional relationship, (4) perform parameter variation studies to checkand 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 shownin the left-hand side of Fig. 2. The user navigates through the nodes of the outline from top to

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


bottom to define the problem and perform the analysis. Each of these steps is described in moredetail in the following sections.

2.3.1. Problem statement definitionThe 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 inputand output variables and mathematical operators. This improves readability, conveys the essentialflow of the analysis, and allows complex reliability assessments to be defined when more than onemodel is required to define the system performance.

An example problem statement is shown in the upper right-hand portion of Fig. 2. A powerfulfeature of NESSUS is the ability to create complex probabilistic simulations by linking modelstogether in a sequential fashion. In this example, the performance measure is life (given by numberof cycles to failure), which requires input from other models. Two stress quantities from an ABA-QUS (ABAQUS, Inc.) finite element analysis are used in the analytical life model. Many otherfinite 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 problemstatement parser in NESSUS identifies all of the independent variables in the problem statementwindow and transfers these variables to the random variable input window for further definition.

2.3.2. Random variable input and probabilistic databaseThe 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 meanand standard deviation, such as upper and lower bounds for truncated distributions. The proba-bility density function (PDF) and cumulative distribution function (CDF) plotting capability inNESSUS provides a quick visual inspection of the random variables. Random variables allowedin NESSUS include normal, lognormal, Weibull, extreme value type I, chi-square, maximum en-tropy, curve-fit, Frechet, truncated normal and truncated Weibull. The maximum entropy andcurve-fit distributions can be used for distributions not directly supported.

NESSUS maintains a library of relevant PDFs in a probabilistic database. Random variablescan be defined and stored using a distribution type and associated parameters. Distribution fittingfunctions are provided to determine the best fit from raw data. The entries can be grouped andmultiple 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 beinserted directly in the random definition table in NESSUS using a right mouse click as shownin Fig. 3.

2.3.3. Response model definitionFunctions defined in the problem statement window (Fig. 2) are assigned in the response

model definition. The available function types are selected from the model type drop down menuand include analytical, regression, numerical, and predefined as shown in Fig. 4. The analyticalfunction type allows models to be defined with standard mathematical operators, using a for-mat identical to definitions in the problem statement window. The numerical model type allowsthe use of interfaced codes or a user-defined code. Codes currently interfaced to NESSUS in-

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

Fig. 4. The numerical model definition screen in NESSUS defines the execution command and required input/output

files for executing the numerical model.


clude ABAQUS, ANSYS (ANSYS, Inc.), DYNA3D (Lawrence Livermore National Labora-tory), LS-DYNA (LSTC, Inc.), NASA_GRC_FEM (NASA Glenn Research Center), MAD-YMO (TNO Automotive), MSC.NASTRAN (MSC.Software), PRONTO (Sandia National


Laboratories), and USER_DEFINED. The NASA_GRC_FEM finite element program is in-cluded with the NESSUS software. The regression model type allows the user to inputfunction coefficients or raw perturbation data that can be fit to linear or quadratic functionsusing linear regression. Finally, the predefined model type allows linking user-written Fortransubroutines with NESSUS, which requires the user to have access to a Fortran compiler. Theuser-defined numerical model allows the user to link the NESSUS probabilistic engine withany stand-alone analysis code.

Fig. 4 shows an example using the ABAQUS finite element program. The execution commandwindow provides the command or commands required to execute ABAQUS. The input and out-put files are also defined on this input screen. Default execution options for the supported codesare inserted automatically by NESSUS from a configurable template file, and can be modified bythe user as needed.

A batch processing option is provided to allow processing on different computers. This allowsNESSUS 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 anautomatic restart option. The restart capability provides probabilistic solution refinement, recov-ering from abnormal solver termination, and evaluating additional performance measures with-out rerunning previous steps of the solver analyses. The batch and restart capabilities can becombined to perform distributed processing of the function evaluations either manually by theanalyst or automatically using simple scripts.

2.3.3.1. Mapping random variables to numerical models. When performing probabilistic analysisusing a numerical model, a realization of a random variable must be reflected in the numericalmodel�s input. The variable may be a random variable or a computed variable from another codeor analytical equation. In general, the variable can map to a single value in the code�s input or to avector of values such as nodal coordinates in a finite element model. Typical examples of singlevalue mappings include Young�s modulus or a concentrated point load. Examples of vector map-pings are a pressure field acting on a set of elements or a geometric parameter that effects multiplenode locations.

Mapping variables to the numerical model input in NESSUS is achieved by graphically iden-tifying the lines and columns that are changed when the variable changes as shown in Fig. 5.The mapping capability in NESSUS has been optimized to support model input files in excessof several million lines in length.

Vector mappings require a functional relationship between the input random variable and theanalysis program input. Because different realizations of these variables are required, a generalapproach is used in NESSUS to relate a change in the input random variable value to the code�sinput. For example, if the random variable is the radius of a hole, changes to a set of nodal coor-dinate values will be required each time the radius is changed. A ‘‘delta vector,’’ Dx, is defined thatrelates how the coordinates change with a change in the variable. The vector of perturbed nodalcoordinates, x̂, is related to the mean value of the coordinates, lx, plus a shift factor, s, times theamount of change for the coordinates, Dx, or in equation form, x̂ ¼ lx þ s � Dx.

The delta vector is the normalized difference between the mean value of the random variableand the perturbed value. One approach to generating Dx is to perturb the nominal mesh, subtractthe nominal from the perturbed, and then normalize. This procedure, performed only once at the

Fig. 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.


beginning of the analysis, is then used by NESSUS to create a finite element mesh for any value ofthe random variable.

Several other approaches are available for defining vector variables. Some analysis codes allowthe finite element model to be parametrically defined. In this case, the variables can be mappeddirectly without defining the delta vector. Another option is to include a finite element preproces-sor using the linked model capability. The variables can be mapped to the preprocessor input andthe resulting model used for the analysis.

2.3.3.2. Selecting responses for numerical models. The final step in defining the numerical model isto identify the response quantity or quantities that are to be returned to NESSUS. The approachused in NESSUS is to read the analysis results for a given set of node, element and time steps di-rectly from the analysis code�s results file. Fig. 6 shows the response selection for the ABAQUSfinite element software. NESSUS supports automated extraction for most engineering quantities

Fig. 6. NESSUS result selection screen for ABAQUS.


of interest including displacements, velocities, accelerations, stresses, strains, etc. When multiplequantities are requested, NESSUS provides further options to reduce the results down to a singlevalue using functions such as maximum, minimum, average, etc. For dynamic codes, selection ofthe response from a result time series is provided across multiple times such as maximum, last, anduser specified. In some cases, the response time series can be filtered to smooth the response beforeuse in the probabilistic analysis.

A flexible user defined numerical model capability is provided in NESSUS. This capability al-lows users to link in-house developed codes with NESSUS. The response of interest is selected bydefining a specific location in the analysis code�s results file. A user subroutine for extracting re-sponses is also available for more complex situations such as results extraction from a binarydatabase file.

2.3.4. Deterministic and parameter variation analysis

NESSUS� deterministic analysis option provides a useful tool to verify the problem statementdefinition. Any computed value (on the left of the equal sign) in the problem statement will beevaluated at the mean values of the input random variables.


Parameter variation analysis is another useful tool to understand how the performance varieswith changes in the random variables. NESSUS provides several methods for defining variableperturbations including, backward, central, and forward differences as well as variable sweeps.In addition, specific perturbation values can be input directly to define experimental designs. Visu-alization 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 MonteCarlo simulation, which is impractical for large-scale high-fidelity problems [11]. The traditionalmethods include, for example, the first- and second-order reliability methods (FORM andSORM) [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 AIS [10]. Further details on thesemethods and their implementation in NESSUS are given in [3].

NESSUS has a suite of probabilistic analysis methods as listed in Table 1 for both compo-nent and system probabilistic analysis. The range of methods allows the analyst to obtain prob-abilistic 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 exam-ple, Fig. 7 shows the probabilistic analysis definition screen for the AMV+ method. Defaultparameters for the different methods are supplied based on experience with the method on pre-vious problems.

In addition to defining the probabilistic method, several other options can be selected: param-eter correlations, confidence bounds, and analysis type. Linear correlation between any two inputvariables is defined by entering the correlation coefficient. By default the input variables are as-sumed to be statistically independent, i.e., zero correlation. If any non-zero correlations are en-tered, NESSUS will perform a numerical transformation during the probability integration toaccount for the correlation. NESSUS computes confidence bounds on the computed probabilities

Table 1

Probabilistic analysis methods in NESSUS

Probabilistic method Component System

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 ·

Fig. 7. Options for the AMV+ probabilistic method in NESSUS.


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 meanand standard deviation for each of the input random variables. All COV values are zero by de-fault. The analysis type definition indicates that the probabilistic method will compute: (1) thefull CDF of the response, (2) the probability associated with a specified performance or listof performance values, or (3) the performance given a specified probability or set of probabilityvalues.

2.3.6. Results visualizationNESSUS includes a powerful post processing capability. After completing the probabilistic

analysis, the user can visualize the CDF in several formats (Fig. 8). In addition, the various prob-

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


abilistic sensitivity measures computed by NESSUS can be viewed as shown in Fig. 8. Multipleanalyses 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 plotsare easily exported for inclusion in reports or presentations.

Probability contouring is another highly useful visualization output. The failure probability iscomputed at different locations in the model (e.g., nodes in a finite element mesh) and visualizedby contouring iso-probability values. Contours of probability can reveal regions of high risk thatmay not be apparent from the contours of model response quantities. Consider the spatial thick-ness fluctuations of a part induced by the rolling or stamping process. Fig. 9 shows that eventhough the mean stress at point B is higher than at point A, the probability of failure is lowerdue to the larger uncertainty at point A.

An example of a large-scale analysis utilizing probability contouring is shown in Fig. 10. Theprobability contours identify regions where there is significant probability that the equivalentplastic strain exceeds the design limit. These regions are not identified by the mean valuecontours.

3. Application examples

The NESSUS software has been used to predict the reliability and probabilistic response for awide range of problems [15–23]. Three problems are presented in this section to demonstrate the

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

a higher stress point.

Fig. 10. Probability of failure contours (right) indicate critical design regions not identified from the mean equivalent

plastic strain contours (left).


application and flexibility of NESSUS and to illustrate the current developments to support effi-cient 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 aSport utility vehicle to small vehicle frontal offset impact event. The analysis was designed to iden-tify important variables contributing to the crashworthiness reliability and use this information toimprove the design and manufacturing processes. The ultimate goal of the analysis is to improvevehicle reliability using a computational approach to reduce expensive crash testing. Additionaldetails about this analysis can be found in [24].


3.1.1. Problem description

An LS-DYNA finite element model of a vehicle frontal offset impact and a MADYMO modelof a 50th percentile male Hybrid III dummy were integrated with NESSUS to comprise the crash-worthiness characteristics (Fig. 11). A number of different response quantities from the modelswere used to define four occupant injury acceptance criteria and six compartment intrusion crite-ria. The NESSUS problem statement for the head injury criteria (HIC) is shown in Fig. 12. An

Fig. 11. Vehicle-to-vehicle frontal offset crash simulation model.

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


acceleration history from the LS-DYNA vehicle model is used as the crash pulse input to theoccupant injury model in MADYMO. The other three occupant injury criteria are modeled inthe same fashion. The compartment intrusion criteria are determined from relative displacementsof the points in the small vehicle model. These ten acceptance criteria were used as events in aprobabilistic 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 in-clude parameters that define key energy absorbing components of the vehicles such as materialproperties 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 inherentuncertainty of material characteristics. Each of these random variables is characterized by a sta-tistical distribution defined from manufacturing data, literature and/or expert opinion. The distri-butions 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 probabi-listic analysis and vehicle design tradeoff studies. The parameter variation analysis capability inNESSUS was utilized to develop the response surface models. A vehicle redesign was performedbased 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 NESSUSwith 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 eventand the door aperture closure event. All other acceptance criteria have relatively high reliability.The computed probabilistic sensitivity factors are shown in Fig. 13. From the figure, the nominalvalue of the yield strength of the small vehicle rail material can be most influential in increasingthe reliability.

The objective of the redesign analysis is to provide a recommendation to improve the reliabilityof the small vehicle in a vehicle-to-vehicle frontal offset impact. The approach used is to rely on

Table 2

Original and final design reliability for the stochastic car crash example

Acceptance criteria Reliability (%)

Description NESSUS variable 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 definition g_ipd 99.6870 99.9719

Door aperture closure g_dac 72.6750 98.7460

Engine location g_engd 99.6000 99.9997

Fig. 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.


the probabilistic sensitivity factors to identify the dominant parameters (random variable meanand standard deviation) that will improve system reliability.

The reliability for each acceptance criteria in the new design is listed in Table 2. The dominantevent for the original design was the femur axial load acceptance criteria. The femur axial loadalso shows the lowest reliability for the final design but increased from a reliability of 46–93%.The reliability improvements are shown in Fig. 14 along with a description of the parameterchanges 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 espe-cially for evaluating the probabilistic sensitivity factors at the system level for redesign analysis.Certain parameters such as stiffness/strength parameters can improve reliability for compartmentintrusion performance measures but may be detrimental to the crash pulse attenuated to the vehi-cle occupant. The system model correctly accounts for events with common variables (correlatedevents) 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 pressurevessels. These experiments are performed in a containment vessel to prevent the release of explosion

0 2 4 6 8 10

Redesign Iteration

10

20

30

40

50

60

70

80

90

100

Sys

tem

Rel

iabi

lity

(%)

Increase yield stressReduce yield stress COV

Increase weld stiffness

Increase weld stiffness

Reduce yield stress COV

Reduce front weld stiff. COV

Reduce rail thickness COVReduce front weld stiff. COV

Reduce yield stress COV Reduce foam properties COV

Tighten rail and bumper installation tolerances

Fig. 14. Vehicle system reliability improvement study performed with NESSUS.


products to the environment. Design of these spherical vessels was originally accomplished bymaintaining that the vessel�s kinetic energy, developed from the detonation impulse loading, beequilibrated by the elastic strain energy inherent in the vessel. Within the last decade, designs havebeen accomplished utilizing sophisticated and advanced 3D computer codes that address both thedetonation hydrodynamics and the vessel�s highly non-linear structural response. Additional detailsabout this analysis can be found in [22,25].

3.2.1. Problem descriptionThe containment vessel, shown on the left side in Fig. 15, is a spherical vessel with three access

ports: two 16-in. ports aligned in one axis on the sides of the vessel and a single 22-in. port at thetop of the vessel. The vessel has an inside diameter of 72 in. and a 2 in. nominal wall thickness.The vessel is fabricated from HSLA-100 steel, chosen for its high strength, high fracture tough-ness, and no requirement for post weld heat treatment. The vessel�s three ports must maintaina seal during use to prevent any release of reaction product gases or material to the external envi-ronment. Each door is connected to the vessel with 64 high strength bolts, and four separate sealsat each door ensure a positive pressure seal.

A series of hydrodynamic and structural analyses of the spherical containment vessel were per-formed using a combination of two numerical techniques. Using an uncoupled approach, thetransient pressures acting on the inner surface of the vessel were computed using the Eulerianhydrodynamics code, CTH (Sandia National Laboratories), which simulated the high explosive(HE) burn, the internal gas dynamics, and shock wave propagation. The HE was modeled asspherically symmetric with the initiating burn taking place at the center of the sphere. The vessel�sstructural response to these pressures was then analyzed using the DYNA3D explicit finite ele-ment structural dynamics code.

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


The simulation required the use of a large, detailed mesh to accurately represent the dy-namic response of the vessel and to adequately resolve the stresses and discontinuities causedby various engineering features such as the bolts connecting the doors to their nozzles. Tak-ing advantage of two planes of symmetry, one quarter of the structure was meshed usingapproximately one million hex elements. Six hex elements were used through the 2-in. wallthickness to accurately simulate the bending behavior of the vessel wall. The one-quartersymmetry model is shown on the right-hand side of Fig. 15. The structural response simula-tion used an explicit finite element code called PARADYN (Lawrence Livermore NationalLaboratory), which is a massively parallel version of DYNA3D, a non-linear, explicitLagrangian finite element analysis code for three-dimensional transient structural mechanics.PARADYN was run on 504 processors of LANL�s ‘‘Blue Mountain,’’ massively parallel com-puter, which is an interconnected array of independent SGI (Silicon Graphics, Inc.) comput-ers. The containment vessel model can be solved on the Blue Mountain computer withapproximately 2.5 h of run time. The same analysis would have taken about 35 days whenrun on a single processor.

The four random variables considered are radius of the vessel wall (radius), thickness of thevessel wall (thickness), modulus of elasticity (E), and yield stress (Sy) of the HSLA steel. A sum-mary of the probabilistic inputs is included in Table 3. The properties for radius and thickness

Table 3

Probabilistic inputs for the containment vessel example problem

Variable PDF r l COV

Radius (in.) Normal 37.0 0.0521 0.00141

Thickness (in.) Lognormal 2.0 0.08667 0.04333

E (lb/in.2) Lognormal 29.0E + 06 1.0E + 06 0.03448

Sy (lb/in.2) Normal 106.0E + 03 4.0E + 03 0.03774


are based on a series of quality control inspection tests that were performed by the vessel man-ufacturer. The coefficients of variation for the material properties are based on engineering judg-ment. In this case, the material of the entire vessel, excluding the bolts, is taken to be a randomvariable.

When the thickness and radius random variables are perturbed, the nodal coordinates of thefinite element model change with the exception of the three access ports in the vessel, which re-main constant in size and move only to accommodate the changing wall dimensions. This wasaccomplished in NESSUS by defining a set of scale factors that defined how much and in whatdirection each nodal coordinate was to move for a given perturbation in both thickness and ra-dius. The NESSUS mapping procedure allows the perturbations in radius and thickness to becumulative so these variables can be perturbed simultaneously. Once the scale factors are definedand input to NESSUS, the probabilistic analysis, whether by simulation or using AMV+, can beperformed without further user intervention.

The response metric for the probabilistic analysis is the maximum equivalent plastic strainoccurring over all times at the bottom of the vessel finite element model. This maximum valueoccurred 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 nearthe mean value. The CDF is plotted on the left in Fig. 16 on a standard normal probability scale.As shown, the LHS and AMV+ results are in excellent agreement. However, in contrast to theLHS solution, the AMV+ solution predicts accurate probabilities in the extreme tail regions withfar fewer PARADYN model evaluations.

Probabilistic sensitivities are shown in on the right in Fig. 16. The sensitivities are multiplied byri to non-dimensionalize the values and facilitate a relative comparison between parameters. The

Fig. 16. Cumulative distribution function of equivalent plastic strain plotted on standard normal scale (left) and

probabilistic sensitivity factors (u = 3) (right).


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

3.3. Cervical spine impact injury

Cervical spine injuries occur as a result of impact or from large inertial forces such as thoseexperienced by military pilots during ejections, carrier landings, and ditchings. Other examples in-clude motor vehicle, diving, and athletic-related accidents. Reducing the likelihood of injury byidentifying and understanding the primary injury mechanisms and the important factors leadingto injury motivates research in this area [26].

Because of the severity associated with most cervical spine injuries, it is of great interest todesign occupant safety systems to minimize probability of injury. To do this, the designer musthave quantified knowledge of the probability of injury due to different impact scenarios, and alsoknow which model parameters contribute the most to the injury probability. Finite elementstress analysis plays a critical role in understanding the mechanics of injury and the effects ofdegeneration 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 environmentin which the structure is required to function. This variability or uncertainty has a direct effecton the structural response of the system. Biological systems are a textbook example: uncertaintyand variability exist in the physical and mechanical properties and geometry of the bone, liga-ments, cartilage, as well as uncertainty in joint and muscle loads. Hence, the broad objective ofthis investigation is to explore how uncertainties influence the performance of an anatomicallyaccurate, three-dimensional, non-linear, experimentally validated finite element model of the hu-man lower cervical spine.

3.3.1. Problem description

A validated three-dimensional ABAQUS finite element model of the C4–C5–C6 spinal segmentdeveloped at the Medical College of Wisconsin [27] was used to calculate the structural responseof the lower cervical spine and to quantify the effect of uncertainties on the performance of thebiological system. The load–deflection response was validated against experimental results fromeight cadaver specimens [28]. The moment–rotation response of the finite element model was val-idated against experimental results reported in the literature [29]. The model is shown in Fig. 17.Additional details about this analysis can be found in [30].

Biological variability was accounted for by modeling material properties and spinal segmentloading as random variables. Where available, experimental data was used to generate the randomvariable definitions (e.g., the spinal ligaments load–deflection behavior).

The probabilistic finite element model was exercised under flexion (chin down) loading byapplying a pure bending moment of 2 N m to the superior surface of the C4 vertebra. The inferiorsurface of the C6 vertebra was fixed in all directions and rotation was measured between the supe-rior aspect of C4 and the fixed boundary of C6. Computing the rotation and monitoring the reac-tion forces at the fixed boundary quantified the moment–rotation behavior. Cumulative

Fig. 17. Probabilistic cervical motion segment model (C4–C6).


probability distribution functions, probability distribution functions, and probabilistic sensitivityfactors were determined.

3.3.2. Results

The probabilistic rotation response had an approximate mean of 3.82� and a standard deviationof 0.38� resulting in a coefficient of variation of 10%. The CDF and PDF of rotation is shown onthe left in Fig. 18. The CDF is used to determine probabilities directly, e.g., the probability thatthe rotation will be less than or equal to 4.2� is 82%.

The probabilistic sensitivity factors indicate that the loading (FLEXLOAD) is the domi-nant variable. The bar graph on the right in Fig. 18 shows the sensitivity information forthe eight most significant random variables with FLEXLOAD removed so that the othervariables can be more clearly seen. Not including FLEXLOAD, the most important variablesare the: (1) annulus C45 and C56 Young�s modulus, (2) interspinous ligament non-linearspring force–deflection relationship, and (3) ligamentum flavum non-linear spring force–deflec-tion relationship. These results can be used eliminate unimportant variables from the randomvariable vector and to focus further characterization efforts on those variables that are mostsignificant.

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6Rotation (Degrees)

Pro

bab

ility

CDF

PDF

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

A45E

A56E

MLFRV

MISRV

MCLRV

FIBRE

C56AREA

C45AREA

Pro

bab

ilist

ic S

ensi

tivi

ty

Fig. 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).


4. Conclusions

Although NESSUS was initially developed for aerospace applications, the methods are broadlyapplicable and their use warranted in situations where uncertainty is known or believed to have asignificant impact on the structural response. The framework of NESSUS allows the user to linkadvanced probabilistic algorithms with analytical equations, commercial finite element analysisprograms and ‘‘in-house’’ stand-alone deterministic analysis codes to compute the probabilisticresponse 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 withcommercially available analysis software leverages the investment made in learning and becomingproficient with the software. The graphical user interface in NESSUS makes defining and execut-ing the probabilistic analysis straightforward and efficient for simple problems as well as problemsinvolving extremely large multi-physics models.

Several applications were presented that demonstrated the flexibility of the NESSUS soft-ware. The advanced probabilistic analysis methods in NESSUS allow for using high-fidelitymodels to define the structure or system even when each function evaluation may take sev-eral hours to run. In the application problems presented, the probabilistic results revealedadditional information that would not have been available if deterministic approaches wereused.

Future progress in probabilistic mechanics relies strongly on the development of validated anal-ysis models, systematic data collection and synthesis to resolve probabilistic inputs, and identifi-cation and classification of failure modes. Research and development in this area is needed toimprove the robustness of the underlying probability integration methods, to develop alternativeuncertainty modeling approaches and integrate these approaches with established probabilistictools, and to apply probabilistic methods to model verification and validation, system certificationand prognosis, component life assessment and integrity, and structural system health monitoringand management.


Acknowledgements

The authors acknowledge the support of the NASA Glenn Research Center and the Los Ala-mos National Laboratory for their significant support of the NESSUS software. The 2000 Daim-lerChrysler Challenge Fund, Naval Air Warfare Center Aircraft Division, and Los AlamosNational Laboratory are also acknowledged for their support for the applications problems sum-marized in the paper.

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