Probabilistic Engineering Analysis Using the NESSUS Software

Ben H. Thackera, David S. Rihaa, Simeon H.K. Fitchb, Luc J. Huysea

aSouthwest Research Institute, 6220 Culebra Road, San Antonio, TX 78228, USA

bMustard 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

1 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

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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

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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

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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 large-

scale complex models. Three application problems are presented at the end of the paper to

illustrate the application 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 a

single failure mode, where reliability is simply one minus the probability of failure, fp .

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 0g ≤ , where is referred to as the limit state function. In

general, will be more complex than and will be given by , where are

the input random variables. In addition to the failure probability, NESSUS computes

probabilistic importance factors,

g R S= −

S−g g R= ( )g g= X X

/β∂ ∂u , where β is inversely related to fp and u are the input

random variables transformed into standard normal space, and probabilistic sensitivity factors,

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

standard deviation.

θ

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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

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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,” ∆ , is

defined that relates how the coordinates change with a change in the variable. The vector of

perturbed nodal coordinates, x , is related to the mean value of the coordinates, , plus a shift

factor, s, times the amount of change for the coordinates, ∆ , or in equation form,

x

ˆ xµ

x

ˆ sµ= + ⋅∆xx x .

The delta vector is the normalized difference between the mean value of the random variable and

the perturbed value. One approach to generating 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.

x∆

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.

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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

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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 iso-

probability 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

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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

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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 to nondimensionalize the values and facilitate a relative comparison between parameters. iσ

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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%.

18

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.

19

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.

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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

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22. Rodriguez, E.A., Pepin, J.W., Thacker, B.H., Riha, D.S., “Uncertainty Quantification of A

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Models,” Proc. SAE 2004 World Congress & Exhibition, 2003-01-0460, Detroit, MI., March

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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

22

Loading,” IMAC-XXI: Conference & Exposition on Structural Dynamics, No. 261,

Kissimmee, FL, 3-6 February 2003.

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Larson and A. Sances, Jr. (eds.), IOS Press, Harvard, MA, 1998.

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the Lower Cervical Spine; Role of Intervertebral Disc under Axial and Eccentric Loads,”

Medical Engineering and Physics, Vol. 21, pp. 689-700, 2000.

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23

Table 1: Probabilistic analysis methods in NESSUS.

Probabilistic Method Component System First order reliability method (FORM) X Advance first order reliability method X Second order reliability method (SORM) X Importance sampling with radius reduction factor X X Monte Carlo simulation X X Importance sampling with user-defined radius X Plane-based adaptive importance sampling X Curvature-based adaptive importance sampling X X Mean value X Advanced mean value X Advanced mean value with iterations X Latin hypercube simulation X Response surface method with Monte Carlo simulation X

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 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 PDF σ µ COV Radius (in) Normal 37.0 0.0521 0.00141 Thick (in) Lognormal 2.0 0.08667 0.04333 E (lb/in2) Lognormal 29.0E+06 1.0E+06 0.03448 Sy (lb/in2) Normal 106.0E+03 4.0E+03 0.03774

24

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

Name Description Mean Standard Deviation Distribution

MALLRV Anterior longitudinal ligament nonlinear spring force-deflection relationship. 0 1.0 Normal

MPLLRV Posterior longitudinal ligament nonlinear spring force-deflection relationship.

0 1.0 Normal

MISRV Interspinous ligament nonlinear spring force-deflection relationship. 0 1.0 Normal

MCLRV Capsular ligament nonlinear spring force-deflection relationship. 0 1.0 Normal

MLFRV Ligamentum flavum nonlinear spring force-deflection relationship. 0 1.0 Normal

C45AREA 45 disc cross-sectional area (rebar element) inner/middle/outer & in/out. 0.195 0.0195 Lognormal

C56AREA 56 disc cross-sectional area (rebar element) inner/middle/outer & in/out. 0.1985 0.01985 Lognormal

C4PE C4 posterior Young’s modulus. 3500 735 Lognormal C5PE C5 posterior Young’s modulus. 3500 735 Lognormal C6PE C6 posterior Young’s modulus. 3500 735 Lognormal C4CE C4 cancellous Young’s modulus. 100 30 Lognormal C5CE C5 cancellous Young’s modulus. 100 30 Lognormal C6CE C6 cancellous Young’s modulus. 100 30 Lognormal A45E Annulus C45 Young’s modulus. 4.7 0.705 Lognormal

LM45E Lusckha membrane C45 Young’s modulus. 12 1.2 Lognormal

A56E Annulus C56 Young’s modulus. 4.7 0.705 Lognormal

LM56E Lusckha membrane C56 Young’s modulus. 12 1.2 Lognormal

SM456E Synovial membrane C45/C56 facet Young’s modulus. 12 1.2 Lognormal

NP45FD Nucleus pulposus C45 fluid density. 0.000001 0.0000001 Normal NP56FD Nucleus pulposus C45 fluid density. 0.000001 0.0000001 Normal FSF45FD Facet synovial fluid C45 fluid density. 0.000001 0.0000001 Normal FSF56FD Facet synovial fluid C56 fluid density. 0.000001 0.0000001 Normal LJ45FD Luschka’s joint C45 fluid density. 0.000001 0.0000001 Normal LJ56FD Luschka’s joint C56 fluid density. 0.000001 0.0000001 Normal CORTE Cortical Young’s modulus. 12000 2520 Lognormal ENDPE Endplate Young’s modulus. 600 126 Lognormal CARTE Cartilage Young’s modulus. 10.4 1.04 Lognormal FIBRE Fiber Young’s modulus. 500 50 Lognormal

FLEXLOAD Flexion loading. 1 0.01 Normal

25

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).

26

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.

31

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

factors (right).

Stress Contours

Probability Contours

A

B

Point B:Higher mean stressLower probability

Stress

Failure Stress

Point A:Lower mean stressHigher probability

Stress Contours

Probability Contours

A

B

Point B:Higher mean stressLower probability

Stress

Failure Stress

Point A:Lower mean stressHigher probability

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.

32

Equivalent Plastic Strain

Probabilityof Failurepf=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.

33

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.

34

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 (front)

Reduce yield stress COV

Reduce front weld stiff. COVReduce rail thickness COV

Reduce front weld stiff. COV

Reduce yield stress COVReduce foam properties COV

Tighten rail and bumper installation tolerances

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).

35

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).

36

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6Rotation (Degrees)

Prob

abili

tyCDFPDF

-0.15-0.1

-0.050

0.050.1

0.150.2

A45E

A56E

MLFRV

MISRV

MCLRV

FIBRE

C56

AREA

C45

AREA

Pro

babi

listic

Sen

sitiv

ity

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).

37