PSI - Issue 57

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Procedia Structural Integrity 57 (2024) 1–3

Fatigue Design 2023 (FatDes 2023) Fatigue Design Preface Fabien Lefebvre* Cetim, 52 avenue Félix Louat, 60304, Senlis

On behalf of the Fatigue Design 2023 International Scientific Committee and Organizing Committee, we would like to thnak you all to the 10 th edition of Fatigue Design 2023 taking place at Cetim, Senlis, France on November 29 & 30 2023. This anniversary edition provides an opportunity to reflect on the origins of this conference, which has been held biennially since 2005. Indeed, the current relevance of this conference prompts us to reevaluate its title, which unites two terms that have historically been overlooked: ‘Fatigue’ and ‘Design’. In the realm of Mechanics, ‘Fatigue’ refers to ‘Fatigue Resistance’, which denotes a material’s ability to endure damage and, more broadly, a component or mechanical structure’s capacity to withstand repeated loads during service. With the contemporary demands for weight reduction, enhanced performance, and the assessment of remaining service life, mechanical stress levels have increased, and the risk of fatigue failure has become a critical concern in the design

* Corresponding author. Tel.: +33.687. 188. 942 ; E-mail address: fabien.lefebvre@cetim.fr

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers

2452-3216 © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers 10.1016/j.prostr.2024.03.001

Fabien Lefebvre et al. / Procedia Structural Integrity 57 (2024) 1–3 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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of mechanical components. Consequently, in recent years, the concept of ‘Fatigue Design’ has supplanted that of ‘Fatigue Verification’ post -design. Organized by Cetim and his partners, the 10 th Fatigue Design 2023 International Conference aim to present the most innovative approaches, scientific and technological advances in design methodologies, testing methods and tools to evaluate and extend the fatigue lifetime of the industrial equipment. The papers published in PROCEDIA STRUCTURAL INTEGRITY are mostly focusing on the industrial applications. The Nordic Countries have been designated as the ‘partner country’ for Fatigue Design 2023, particularly , in the context of cutting-edge research on fatigue and fracture mechanics in Northern Europe. Over 30% of the conference speakers hail from the Northern European region. This edition has placed a special emphasis on Fatigue Life Extension and Residual Life Assessment, featuring four keynotes on these subjects and multiple dedicated sessions. Building on the introduction of Big Data and Artificial Intelligence concepts in the previous edition, these themes will continue to be part of the conference. Additionally, a new topic related to fatigue in transmission systems will be introduced as a new point of discussion. The following scientific and technical topics are covered: • contact fatigue and fatigue in transmission system • damage tolerance and fatigue life, • experimental and numerical design and validation methods, • fatigue of assemblies (mechanical, welded, adhesive-bonding, multimaterial...), • reliability-based approaches and probabilistic methods, • influence of manufacturing process in fatigue analysis (effect of microstructure, welding, stress relief techniques...), • vibration fatigue behaviour. The papers focus on the latest development and most recent experimental, numerical simulation techniques and the associated engineering tools applied to the large domain of the industrial applications. The 10 th edition of Fatigue Design International conference is being organized in close collaboration with the proceedings’ publisher, Elsevier, and will be made available online through Structural Integrity Procedia on ScienceDirect, ensuring global accessibility for better dissemination and maximum exposure. The selection and peer review of the papers have been conducted in cooperation with the International Scientific and Organizing Committees with a minimum of two reviewers by paper. • • • • additive manufacturing, big data and Artificial Intelligence, complex loading composite and elastomers,

Fabien Lefebvre et al. / Procedia Structural Integrity 57 (2024) 1–3 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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“Fatigue Design” has evolved into the leading conference for addressing industrial concerns related to the fatigue design of structures and components, serving as a pivotal intersection between industry and academia. With 104 oral presentations, approximately 50% of which are presented by industry experts, this edition features 20 new oral presentations, along with a new dedicated room for the conference. In total, 93 papers are published for the 10 th edition. All speakers and delegates have engaged in fruitful exchanges and discussions on technical and scientific developments and issues throughout the conference. Poster sessions and exhibition booths have provided additional opportunities for fostering interactions between scientists, industry professionals, PhD students, and solution providers. I extend my sincere gratitude to the dedicated members of the International Scientific and Organizing Committees for their invaluable scientific guidance in the selection and review of papers. I also express my appreciation to the authors, delegates, exhibitors, and sponsors for their meaningful contributions. Special thanks go to the SF2M Fatigue commission and our colleagues at Cetim for their tireless efforts in organizing and ensuring the success of this conference. The 11 th edition of Fatigue Design conference will be held at Senlis the 19 & 20 November 2025.

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Procedia Structural Integrity 57 (2024) 718–730 Structural Integrity Procedia 00 (2023) 000–000 Structural Integrity Procedia 00 (2023) 000–000

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© 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0 ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers. Keywords: Fatigue; Reliability; Monte Carlo; Design Of Experiments (DOE); Reduced Order Modelling (ROM); Weibull Abstract Across all industries, it is becoming increasingly necessary to qualify components based on their reliability and risk of failure. This paper considers how the existing design, simulation, verification, and validation process, may be enhanced to o ff er significant improvements in predicted confidence. Particular attention is paid to simulating the variability and uncertainty of fatigue life predictions. This allows simulation to be better verified using evidence from fewer qualification tests. The paper highlights the need for additional low-cost measurements during qualification testing, and a need to test to failure. It considers how Reduced Order Modelling, Design of Experiments, and statistical reliability analysis, are used in simulating the variability and uncertainty of fatigue failure. It demonstrates how simulation and physical tests mutually benefit one another, and concludes with a case study comparing simulated variability and uncertainty with the values measured in a typical qualification test. © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers. Keywords: Fatigue; Reliability; Monte Carlo; Design Of Experiments (DOE); Reduced Order Modelling (ROM); Weibull Fatigue Design 2023 (FatDes 2023) Achieving high confidence in fatigue reliability by quantifying the e ff ects of uncertainty Andrew Halfpenny a , Amaury Chabod b, ∗ , Balaje Thumati c , TudorMiu a a Hottinger Bruel & Kjaer UK Ltd. AMP Technology Centre, Brunel Way, Catcli ff e, S60 5WG, UK. b Hottinger Bruel & Kjaer France, 2-4 rue Benjamin Franklin, 94370 Sucy-en-brie, France c Hottinger Bruel & KjaerUSA Abstract Across all industries, it is becoming increasingly necessary to qualify components based on their reliability and risk of failure. This paper considers how the existing design, simulation, verification, and validation process, may be enhanced to o ff er significant improvements in predicted confidence. Particular attention is paid to simulating the variability and uncertainty of fatigue life predictions. This allows simulation to be better verified using evidence from fewer qualification tests. The paper highlights the need for additional low-cost measurements during qualification testing, and a need to test to failure. It considers how Reduced Order Modelling, Design of Experiments, and statistical reliability analysis, are used in simulating the variability and uncertainty of fatigue failure. It demonstrates how simulation and physical tests mutually benefit one another, and concludes with a case study comparing simulated variability and uncertainty with the values measured in a typical qualification test. Fatigue Design 2023 (FatDes 2023) Achieving high confidence in fatigue reliability by quantifying the e ff ects of uncertainty Andrew Halfpenny a , Amaury Chabod b, ∗ , Balaje Thumati c , TudorMiu a a Hottinger Bruel & Kjaer UK Ltd. AMP Technology Centre, Brunel Way, Catcli ff e, S60 5WG, UK. b Hottinger Bruel & Kjaer France, 2-4 rue Benjamin Franklin, 94370 Sucy-en-brie, France c Hottinger Bruel & KjaerUSA

1. Introduction 1. Introduction

In the aerospace, automotive, and power generation industries, it is becoming increasingly necessary to qualify components based on their reliability or risk of failure. For example, all new EVs (Electric Vehicles) sold in the US require an 8-year or 100,000 mile (160,000 km) battery warranty. EV batteries are complex mechanical structures that support dynamic masses, and transmit loads and vibration through thousands of joints and components. Fatigue failure therefore presents a significant risk to the overall reliability of the battery system. The design requirement over a population of EVs is expressed in terms of a statistical reliability target, for example: In the aerospace, automotive, and power generation industries, it is becoming increasingly necessary to qualify components based on their reliability or risk of failure. For example, all new EVs (Electric Vehicles) sold in the US require an 8-year or 100,000 mile (160,000 km) battery warranty. EV batteries are complex mechanical structures that support dynamic masses, and transmit loads and vibration through thousands of joints and components. Fatigue failure therefore presents a significant risk to the overall reliability of the battery system. The design requirement over a population of EVs is expressed in terms of a statistical reliability target, for example:

∗ Corresponding author. Tel.: + 33-6134-04974 E-mail address: amaury.chabod@hbkworld.com ∗ Corresponding author. Tel.: + 33-6134-04974 E-mail address: amaury.chabod@hbkworld.com

2452-3216 © 2024 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers 10.1016/j.prostr.2024.03.078 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers. 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under responsibility of the scientific committee of the Fatigue Design 2023 organizers.

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“Over a warranty period of 8 years or 100,000 miles, I require greater than 95% reliability (i.e. fewer than 5% failures), with no less than 90% confidence in my estimation.”

The statistical reliability target comprises three parts:

1. Period : a minimum ‘age’ over which the component is expected to function properly and reliably. This can be expressed in di ff erent units depending on the application, for example: time, distance, number of ground-air ground cycles, or number start-up / shut-down cycles. 2. Reliability : a minimum reliability target, or conversely, the maximum permissible number of failures that are acceptable over the specified period. This will vary depending on the criticality of the failure in terms of both safety and economic risk. 3. Confidence : the required accuracy of the reliability estimation. This is a measure of how confident we are in the estimation of the statistical parameters, for example, the mean and standard deviation. It is dependent on sample size. The larger the sample, the higher the confidence. For small sample sizes, a high additional safety margin is required to achieve the desired confidence target. This paper considers how digital simulation, physical component testing, uncertainty analysis, and statistical re liability analysis, are used to support the fatigue design requirement, and thereby reduce a company’s exposure to unacceptable safety and warranty risks. Fatigue design of mechanical systems has historically followed a deterministic process. That means, for a given set of input loads and component strength parameters, it will return a consistent fatigue life estimate with no variation. In reality, the input loads and component strengths are statistically variable and uncertain. They have a mean expected value, a statistical variability, and uncertainty associated with them. Deterministic design methods take no implicit account of variability or uncertainty. In practice, the designer applies a safety factor greater than one to each input load, and a factor less than one to the strength, in order to account for these uncertainties. Often an additional safety factor is then applied to the final result to allow for ‘modelling errors’. In most cases, the engineer is fairly certain that the qualification is conservative, but cannot state with any confidence what the expected safety margin, reliability, or failure rate will be. Furthermore, uncertainty in the safety margin makes it almost impossible to validate the simulation against a physical durability or reliability test. In comparison, a stochastic design propagates the e ff ects of variability and uncertainty throughout the analysis. That means input loads and strength parameters are expressed statistically by their probability distributions as illus trated in Fig. 1. A stochastic simulation model is therefore verified on its ability to accurately estimate the mean expected fatigue life, and the uncertainty and variability in that life. Such verification yields considerable additional confidence in the simulation. A brief introduction to stochastic design and uncertainty quantification is given by NAFEMS (2018). The x axis in Fig. 1 is represented by a parameter called ‘Damage’. This generic term accounts for damage weighted aging of the component. For example, in a simple constant-amplitude pressure test, the age may be expressed in terms of the number of pressure cycles to failure. However, if the test were to use variable-amplitude pressure cycles, then the damage term must consider the relative e ff ect of the di ff erent pressure amplitudes. It is normal to select a damage parameter which scales linearly with respect to the cumulative exposure to damaging events of di ff erent amplitudes. A common example of this is seen in the Plamgren-Miner linear damage accumulation rule Palmgren (1924), Miner (1945). This linear assumption permits the subsequent use of linear statistical analysis and linear algebra methods in the design process. In reality this assumption is not strictly correct. Damage is found to accumulate non-linearly and is therefore dependent on the sequence of loading events. However, when applied over long periods, to a large population of users with statistically di ff erent usage characteristics, the principal of the central limit theorem ensures the damage parameter is su ffi ciently accurate for design purposes. This is especially true of fatigue because fatigue cracks initiate as a microstructural phenomenon and propagate at a rate that is exponentially proportional to the applied loads. This unavoidable source of uncertainty usually outweighs the error associated with 1.1. Deterministic and Stochastic design

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Fig. 1. Modelling variability and uncertainty

linearization. Linearization errors can therefore be modelled as a source of uncertainty whose influence is usually less than that of material variability. Fig. 1 illustrates the probability of damage at the end of the target reliability period. The variability and uncertainty in customer loading is expressed as a probability curve titled ‘Stress’. Similarly, the distribution of variability and uncertainty in fatigue strength is expressed as a probability curve title ‘Strength’. The area under the intersection of the curves represents the probability of failure. For example, in the automotive industry, this could represent the likelihood of the most aggressive users being in possession of the statistically worst-performing product. The general approach to design qualification in the automotive industry is illustrated in Fig. 2. A similar approach is taken in the aerospace industry where the ‘proving ground’ is e ff ectively replaced with a ‘flight profile’, (which is also known as a ‘duty schedule’, or ‘mission profile’). The four stages of design qualification are discussed below. 1.2.1. Statistical target customer analysis This involves a measurement campaign to characterise the variability of usage throughout the customer base. Historically this was based on measurements taken several decades ago; however, these have become engrained in national, international, and company qualification test standards such as, MIL-STD-810G (2008), RTCA / DO-160G (2010), and Def Stan 00-35 (2021). Recently, standards have evolved to represent new technological advances such as EV battery systems, for example, SAE J2380 (2013), ISO (2020), and IEC (2018). OEMs (Original Equipment Manufacturers) have also become interested in re-characterising their own product usage. This drive is prompted in the automotive industry by economic requirements to develop a ‘world car’ that is suitable for all markets, and accounts for evolving performance and driving characteristics. A similar phenomenon occurs in the aerospace industry, where many aircraft are expected to survive long service periods under evolving operational profiles. The intention is to transform a representative customer vehicle into an unbiased transducer for measuring usage severity. A range of instrumentation is available in both aerospace and automotive applications. These instruments are often suboptimal for design purposes because they are compromised by the requirements of real-world customers. This means they cannot adequately measure the individual component loads required for design purposes, but are suitable for characterising usage. (Component loads are better measured using proving ground (or flight test) data where more appropriate instrumentation options are available.) A detailed study of target customer analysis in the automotive industry is given by Halfpenny and Pompetzki (2011). 1.2. Simulation, Verification and Validation (SV & V)

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Fig. 2. Simulation Verification and Validation (SV&V)

1.2.2. Proving ground (or flight test) profiling The proving ground (or flight test) profile represents the real-world usage in the form of a series of discrete events. Automotive proving ground examples include road profiles, such as, Belgian block, city, urban, o ff -road, cross-country, potholes, and corrugations. These may be traversed at various speeds and vehicle weight conditions. Aerospace examples include flight manoeuvres, such as, taxi, take-o ff , climb, level flight, turns, decent, and landing events. These may be undertaken at various rates, configurations, and conditions. The intention is to provide a scientifically reproducible set of conditions to allow detailed measurements to be made which are otherwise impracticable under real-world conditions. For example, advanced instrumentation are used on proving ground vehicles and flight test aircraft to better measure and understand component loading. Such extensive instrumentation campaigns are impractical on real roads (or aircraft) by real customers. The statistical target customer is characterised by a given mix (or profile) of proving ground or flight events. This ensures that accelerated qualification tests are produced using scientifically reproducible data over a statistically representative usage profile. A detailed study of proving ground correlation in the automotive industry is given by Halfpenny and Pompetzki (2011). 1.2.3. Accelerated qualification tests These are derived from proving ground (or flight test) measurements which are then suitably scaled by the target user profile. Loads are edited to reduce test times whilst ensuring damage retention and avoiding changes in the failure mode. This process is known as ‘Accelerated Testing’, ‘Vibration Profile Design’, or ‘Fatigue Editing’. An introduc tion to accelerated testing is given by ReliaSoft (2017). Specific application to fatigue qualification testing of structural systems is given by Halfpenny and Thumati (2021). Further applications applied to automotive and aerospace vibrat ing system is given by Halfpenny et al (2007), and Halfpenny and Walton (2010) respectively. The methods referenced are in compliance with international qualification testing standards MIL-STD-810G (2008), RTCA / DO-160G (2010), and Def Stan 00-35 (2021).

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1.2.4. Simulation Computer simulation involves CAE (Computer Aided Engineering) analysis of systems, subsystems, or relevant components under test. It is often applied twice during the design process:

1. Simulation of the component as mounted in the qualification test fixture. 2. Simulation of the component as mounted in the final vehicle.

The first analysis supports the verification process and ensures the simulation is properly representative of the real component. Once verified, the simulation model may be integrated into the overall structural model to simulate the overall performance of the component in the real-world. In the automotive industry, this is know as the ‘body-in-white’ model.

2. Complementary simulation and qualification testing

In an e ffi cient design process, simulation and qualification testing are mutually complementary. This is not always true for historical qualification tests because many of these predate the introduction of simulation. However, it is relatively simple and cost-e ff ective to enhance a qualification test to incorporate the benefits of simulation. Only two enhancements are necessary:

1. Test instrumentation is added in order to verify the simulation results. 2. Tests must be continued to failure to verify the fatigue simulation.

The benefits of simulation and qualification testing are discussed next. 2.1. Benefits of simulation prior to physical testing

Firstly, simulation is used to determine the probability of passing the qualification test. It can estimate the expected safety margin and therefore facilitates optimisation to reduce cost and weight at the early stages of design. It can also help to justify confidence in the design prior to expensive prototype testing. Secondly, simulation is used to predict failure modes and locations. This is useful in identifying if the failure mode is obvious or subtle. Additional instrumentation may be required to detect subtle failures, whilst safety precautions may be required to protect people and equipment from the most catastrophic failure modes. It also allows early confirmation that the failure modes correlate with expectation based on previous experience, or that gained through a Failure Modes and E ff ects Analysis (FMEA). Finally, simulation also models the extent of any beneficial or detrimental e ff ects resulting from test simplifications. For example, a component that is vibrated simultaneously in three directions is often tested on a uniaxial shaker table where each direction is applied sequentially. This often leads to an underestimation of the damage on test. Simulation can therefore help to improve the test by suggesting it be extended slightly in each axis thereby more closely representing the target usage and failure mode without adding significant test complexity. 2.2. Benefits of simulation during physical testing Historically, many qualification tests were sparsely instrumented. With the advent of simulation it became bene ficial to enhance the value of the test by adding further instrumentation. Simulation is used to determine the optimal positioning of measurement transducers such as accelerometers, strain gauges, displacement transducers, and thermo couples, in order to better characterise performance. Additional instrumentation permits a detailed verification of the simulation to be made. The use of virtual trans ducers allows the simulated response to be compared directly with test data. Furthermore, additional simulation parameters may be obtained using measurements from the physical test. For example, modal analysis of test data are used to calculate accurate damping properties. The use of additional instru mentation, therefore, significantly enhances the value of a qualification test which leads to an overall reduction in costs as well as improving model verification.

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2.3. Benefits of simulation after physical testing

A properly verified simulation model significantly enhances the value of the physical test. For example, if the test article were to fail prematurely, the simulation model may be used to help rectify the fault. In many cases, di ff erences between the simulation and test article are attributable to the modelling of fixtures, boundary constraints, and modal properties. Rectifying these issues permits the simulation engineer to simulate the observed failure mode and then refine the design to avoid subsequent failures. A properly verified simulation model o ff ers reassurance on reliability risk. For example, a qualification test may run to 16 hours with no signs of failure. However, how many tests are required to ensure 95% reliability with a certainty of 90%? Simulation helps by o ff ering a clear estimation of the safety margin. For example, is this a factor of 2, 10, 1000, or 100,000? With continual pressure to reduce test budgets, it is important to prioritise tests which are demonstrably important to safety and reliability, over those that are performed for historical purposes. Another common question is: should the test be stopped at the target duration, or run to failure? If the test is run to failure, then how long will that take? For verification purposes, it is important that qualification tests are run to failure and this is becoming more common. For example, a qualification test is scheduled to run for 16 hours. After this time the test article is found to be unbroken and has o ffi cially ‘passed the test’. However, it is now increasingly common for the test engineer to increase the loading amplitude and continue to run the test to failure in order to gain correlation with the simulation. The simulation is therefore useful in estimating a reasonable amplification factor in order to reach failure in an acceptable period without altering the failure mode. Note, in the absence of a simulation model, it is becoming common for test engineers to increase the loading by approximately 10% every 1 4 test cycle until failure. This progressive load increase is easily modelled in simulation and is used to verify the simulation. A properly verified simulation model also permits the investigation of many additional design scenarios and load combinations. For example, simulation can identify certain demographics or usage roles that are more sensitive to reliability issues. This knowledge can help to develop predictive maintenance schedules, or encourage fleet balancing, to avoid increased safety concerns and warranty costs. As stated previously, it is important to test to failure when verifying fatigue simulation models. However, the fatigue failure mechanism results in significant variability in the measured lives. For example, a sample of five apparently identical material coupons are likely to exhibit a factor of 2 di ff erence in fatigue life. This is not because fatigue theory is mathematically uncertain, it is because fatigue initiates as a microstructural phenomenon, and no two coupons are the same at this scale. Furthermore, fatigue is exponentially proportional to stress, and, despite our attempts to linearize it, damage accumulates throughout its life in a non-linear, sequence-dependent fashion. Progressing from material coupons to real-world applications typically increases the variability from a factor of 2, to a factor of between 3 and 5, and it is not uncommon to observe factors of 10 in more complex components or applications. If simulation is to be properly verified through qualification testing, uncertainties in the input loads, material prop erties, and the analysis model must be properly estimated. There are two types of uncertainty which are attributable to these parameters, these are: 3. Simulating uncertainty and variability

1. Reducible (or epistemic ) uncertainties 2. Irreducible (or aleatoric ) uncertainties

Epistemic uncertainties are reduceable through better knowledge; for example, better characterisation of the usage environment, more accurate FE (Finite Element) analysis, or improved measurements of residual stresses. Where epistemic uncertainties prevail, a cost-benefit study will often reveal whether it is prudent to invest more money in improved measurement and simulation, or absorb the costs through over-design. Aleatoric uncertainties are irreducible and are attributable to the inherent, or natural variability in a system; they cannot be reduced through improved measurements. Examples include the inherent scatter in a material fatigue curve.

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Fig. 3. Simulation of Uncertainty and Variability in Design

Even though aleatoric uncertainties are irreducible, it is important that accurate measurements are taken in order to characterise them. In the case of fatigue tests, for example, accurate loadcells and strain values are required to minimise the epistemic uncertainties such that the aleatoric uncertainties are accurately quantified.

3.1. Monte Carlo simulation and Reduced Order Models (ROM)

Stochastic simulation is performed using a Monte Carlo approach as illustrated in Fig. 3. The deterministic fa tigue analysis is run repeatedly with input parameters varied statistically with each run according to their individual probability distributions. This creates a sample of simulated fatigue life results which are processed using statistical reliability analysis. Stochastic fatigue simulation often requires a large number of simulation runs. In the case of large FE models this can lead to excessive run times and significant file storage. This is addressed through the use of a ‘Reduced Order Model’ (ROM), (also known as a ’Surrogate Model’). The ROM is e ff ectively a linear transfer function. It transforms input loads into resulting stress responses at the critical failure locations. It is based on the principal of linear superposition. In most cases the ROM is processed directly by the Monte Carlo loop and it is unnecessary to perform expensive FE analysis within the Monte Carlo process. An additional benefit with fatigue analysis is that fatigue is exponentially proportional to stress. This tends to minimise the number of critical failure modes and reduces the complexity of the ROM. 3.1.1. Linear-static FE analysis For linear static FE analysis, the ROM is simply a scaling factor that relates an input load case to the stress tensor at a particular failure site. The resulting stress tensors are then summed over all the load cases using linear static superposition as illustrated in eq. 1. σ ( t ) = k P k ( t ) · s k (1)

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Where σ ( t ) is the stress tensor time history at a particular critical node, P k ( t ) is the load time history for load case k , and s k is the stress tensor result for load case k obtained from FE analysis under a static unit input load.

3.1.2. Harmonic analysis Where the structure responds dynamically and steady-state, then the ROM is represented by a harmonic frequency response function H ( ω ) as illustrated in eq. 2. σ ( ω ) = k H k ( ω ) · F k ( ω ) (2) Where σ ( ω ) is the Fourier transform of the complex stress tensor, F k ( ω ) is the Fourier transform of the load time history for load case k , and H k ( ω ) is the harmonic transfer function obtained from FE analysis. ω is the frequency expressed in rad s . 3.1.3. Random PSD loading Where the structure is analysed using random dynamic loads expressed as PSD (Power Spectral Density) functions, the ROM is also represented by the harmonic frequency response function H ( ω ) as illustrated in eq. 3 and described in Halfpenny (1999).

k b = 1

k a = 1

H a ( ω ) · W ab ( ω ) · H b ( ω )

(3)

G ( ω ) =

Where G ( ω ) is the single-sided PSD stress tensor, W ab ( ω ) is the complex cross-power spectral density function of loading between load cases a and b , and H a ( ω ) and H a ( ω ) are the harmonic transfer function and its complex conjugate respectively.

3.1.4. Non-linear FEA In the case of non-linear FE models, the ROM may become more di ffi cult to define. In the worst case the entire FE model, along with the fatigue model, must be solved repeatedly within the Monte Carlo loop. This might necessitate having to reduce the number of simulations because of runtime / cost implications. However, it should be remembered that simulations are still much more cost-e ff ective than physical prototyping. In other cases, the non-linear ROM may be modelled using a non-linear regression technique. These are beyond the scope of this paper. For more information on nonlinear FE analysis, refer to Hinton (1992). For information on non linear structural dynamics and response, refer to Worden and Tomlinson (2001), Masri et al (2005) and Mohammad et al (1992). Statistical sampling techniques, known as DOE, are used to optimize the size and quality of the design space matrix. The benefit of DOE is to significantly reduce the number of simulations required. Halfpenny et al (2019) describes three di ff erent DOE methods depending on their application: 1. Design for reliability – this is concerned with exploring the statistical variability of design space. It is particularly useful for identifying the 50 th to 99 th percentile cases, such as quantifying warranty exposure. This application favours a DOE based on Latin Hypercube sampling. 2. Design for robustness – this is concerned with exploring the extremities of design space. It is particularly use ful for exploring abusive loads and safety events. This application favours a DOE based on either ‘Factorial sampling,’ or an iterative ‘Response Surface Model’. See ReliaSoft (2015a). 3.2. Design of Experiments (DOE)

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Fig. 4. Weibull regression and contour plots

3.3. Statistical Reliability Analysis

Monte Carlo simulation produces a sample of simulated fatigue life results. Statistical Reliability Analysis is then performed for each failure mode. As fatigue damage is exponentially proportional to stress, it is common to use an A Priori assumption that the PDF (Probability Density Function) of life follows either a log-normal or Weibull distribution. The Weibull distribution is given in eq. 4.

β η

η

β − 1

η

e −

x − γ

β

x − γ

(4)

p ( x ) =

Where p ( x ) is the Weibull PDF of life x in the range ( x ≥ γ ). η is known as the characteristic life. β is the shape parameter (or slope). γ is the location parameter (or, where γ > 0, the failure-free life ). (This is commonly referred to as a 3-parameter Weibull curve. In the case of a 2-parameter Weibull curve, the location parameter γ = 0 ) A typical Weibull plot for 5 experimental data points is shown in Fig. 4. Fig. 4 a) shows the life data plotted on Weibull probability paper. The solid line represents the regression line and the dotted lines the 2-sided 90% confidence bounds. This implies, with a confidence of 90%, that the true regression line will lie somewhere between these two extremities. Fig. 4 b) shows an alternative view of the Weibull parameters. The contour plot describes the possible range of parameters β and η within the 90% confidence bounds. Knowledge of the confidence bounds are especially important for small sample sizes. The cumulative density function Q ( x ) is given as eq. 5. This is also known as the Weibull unreliability function. Q ( x ) = x γ p ( t ) dt = 1 − e − x − γ η β (5) The Weibull reliability function R ( x ), is simply one minus the cumulative density function and is given as eq. 6. R ( x ) = 1 − Q ( x ) = e − x − γ η β (6) The Weibull failure rate function λ ( x ) is given as eq. 7.

β η

η

β − 1

x − γ

p ( x ) R ( x )

(7)

λ ( x ) =

=

Thee ff ectsof β , η and γ are illustrated in Fig. 5. In the case of a 2-parameter Weibull distribution, the characteristic life is taken as η , whereas, in the case of a 3-parameter Weibull distribution, it is taken as θ = η + γ . The shape parameter β describes the type of failure. For a 2-parameter Weibull distribution, these are assumed as:

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Fig. 5. The e ff ects of Weibull β , η and γ on the life of a component

• β < 1 : infant mortality – the failure rate, represented by the green curve in Fig. 5 c), shows significant early failures diminishing rapidly with respect to time. • β = 1 : constant failure rate – the failure rate, represented by the blue curve in Fig. 5 c), shows a constant failure rate with respect to time. (Note: as β → 1, the Weibull distribution tends to an exponential distribution when γ = 0.) • β > 1 : fatigue / wear out failure – the failure rate, represented by the red curve in Fig. 5 c), shows failure rates increasing with respect to time.

Note: in the case of a 3-parameter Weibull curve, the values of β can be smaller on account of the failure free life given by γ .

Estimation of the statistical parameters is based on curve fitting. There are two general methods available as de scribed by Nelson (1990) and ReliaSoft (2015b). The performance characteristics of both methods make them suitable to the following situations: 1. Rank Regression Estimation (RRE) : is preferred for small sample sizes such as those found in physical quali fication tests. Special measures may be taken for tests stopped prior to failure (known as run-out, suspended, or right-censored tests), these are discussed in Halfpenny et al (2019) and ReliaSoft (2015a). 2. Maximum Likelihood Estimation (MLE) : is preferred for larger sample sizes such as those found in simulation tests. These methods take implicit account of suspensions. Halfpenny et al (2019) further recommends that fa tigue simulations that return ‘life beyond endurance’, (where the maximum stress is less than the fatigue strength of the material), be modelled as suspensions. Conversely, in cases where Monte Carlo simulation returns a result of ‘static failure’, (where the maximum stress exceeds the static strength limitations of the material), Halfpenny et al (2019) recommends running additional simulations using the ‘Design for Robustness’ approach to deter mine if this is a statistically likely result. Such failures are often indicative of failure through anticipated abusive loading, or raise safety concerns, and therefore warrant further consideration.

4. Case study: A statistical comparison of simulation with physical test

A typical comparison of simulated fatigue results with qualification test data was presented by Halfpenny et al (2019) Five qualification tests were performed to failure and are represented as black diamonds in Fig. 6. A factor of two was observed between the minimum and maximum values. A number of simulated runs were performed to model the variability due to fatigue parameter uncertainty (an aleatoric uncertainty), and FE modelling error (an epistemic uncertainty). The statistical reliability of the simulated runs are shown as red points in Fig. 6.

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Fig. 6. Weibull Comparison of Simulated Data with Reliability Test Data

• The mean simulated fatigue life (50% unreliability or 0.5 reliability) is seen to correlate well with the mean qualification test (both curves intersect at the mean). • At the 95% reliability level (5% unreliability or 0.95 reliability), the simulation is seen to overestimate the expected test damage. This results in a shorter predicted life. However, this prediction is still within the 90% confidence bounds of the qualification test data, so the null hypothesis (the simulation is truly representative of the test) is not rejected. The fact that damage is slightly overestimated by the simulation is often welcomed by the design engineer. • At the 5% reliability level (95% unreliability or 0.05 reliability), the simulation is seen to underestimate the expected test damage. This results in a longer than observed predicted life. In this case the simulation lies outside of the 90% confidence bounds so we must conclude that the simulation is not representative of the test at the 90% confidence level. This situation is not alarming to the design engineer because these lives are significantly beyond the warranty life (100,000 cycles in this case). A further simulation was undertaken to compare the uncertainty resulting from Material performance (an aleatoric uncertainty) and that resulting from FE modelling errors (an epistemic uncertainty). The results are shown in Fig. 7. At the 90% confidence level all contours intersect which implies that the null hypothesis (the simulation is truly representative of the test) is not rejected. It also shows that the largest variability is due to material fatigue performance. This further implies that any additional improvements in FE simulation will not reduce the overall uncertainty as this is largely attributable to the fatigue performance of the material.

5. Conclusion

In the aerospace, automotive, and power generation industries, it is becoming increasingly necessary to qualify components based on their reliability and risk of failure. This paper considers how the existing design, simulation, verification, and validation process used by the industries may be enhanced to o ff er significant improvements in predicted confidence. These enhancements are cost-e ff ective and fairly easy to implement. They consist of:

1. Additional test instrumentation applied in the qualification test to verify the simulation results. 2. Qualification tests that must be continued to failure to verify the fatigue simulation.

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Fig. 7. Statistical comparison between sources of uncertainty

Due to the extent of variability and uncertainty in the physical fatigue process, a direct comparison between simu lated and measured fatigue lives is problematic. A solution was proposed based on stochastic analysis using the Monte Carlo method to simulate variability and uncertainty. Research into Reduced Order Models showed that existing methods based on linear static superposition o ff er significant e ffi ciency savings in the Monte Carlo calculation by avoiding the need for repeatedly running expensive FE simulation. Research into Design Of Experiments highlighted three methods that were suitable for di ff erent design tasks. In the case of Design for Reliability, where the 50 th to 99 th percentile results are required, a method based on Latin Hypercube sampling was preferred. In the case of safety and extreme event modelling, methods based on Factorial sampling or Response Surface models were preferred. Research into statistical reliability analysis concluded that a priori assumptions on the life curve could significantly reduce the number of Monte Carlo simulations. The log-normal and Weibull distributions were found to o ff er excellent correlation with the variability and uncertainties associated with fatigue life. Weibull parameter estimation techniques based on Rank Regression methods were recommended for characterising test data with relatively few data points, and Maximum Likelihood techniques were preferable for characterising simulation test data. A case study showed how statistical reliability results from test and simulation could be compared directly in order to verify the simulation model to a specified confidence level. Further examples on the use of properly verified simulation models were proposed. Def Stan 00-35, 2021: Environmental handbook for defence materiel, part 3: Environmental test methods. UK Ministry of Defence. Halfpenny, A., 1999. A frequency domain approach for fatigue life estimation from finite element analysis, in: Key Engineering Materials. Trans Tech Publ, pp. 401–410. Halfpenny, A., Bonato, M., Chabod, A., Czapski, P., Aldred, J., Munson, K., 2019. Probabilistic fatigue and reliability simulation, in: NAFEMS World Congress 2019, 17-20 June, Quebec City, Canada. Halfpenny, A., Heyes, P., Kim, S.H., 2007. Statistically representative psd spectra for vibration induced fatigue analysis and testing, in: NAFEMS World Congress 2007. NAFEMS; NAFEMS. Halfpenny, A., Pompetzki, M., 2011. Proving ground optimization and damage correlation with customer usage. SAE Technical Paper 2011-01 0484. Halfpenny, A., Thumati, B., 2021. Accelerating fatigue qualification tests, in: Annual Symposium on Reliability and Maintainability (RAMS) 2021. Halfpenny, A., Walton, T., 2010. New techniques for vibration qualification of vibrating equipment on aircraft, in: Aircraft Airworthiness & Sustainment Conference, 2010. Hinton, E., 1992. Introduction to non-linear finite element analysis. NAFEMS. IEC62660-2, 2018. IEC 62660-2: Secondary lithium-ion cells for the propulsion of electric road vehicles - part 2: Reliability and abuse testing. International Electrotechnical Commission (IEC). ISO 19453-6, 2020. ISO 19453-6: Road vehicles — environmental conditions and testing for electrical and electronic equipment for drive system of electric propulsion vehicles — part 6: Traction battery packs and systems. International Organization for Standardization (ISO). References

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