Model Validation Efforts Research Articles

Imputation of plasma lipid species to facilitate integration of lipidomic datasets

Recent advancements in plasma lipidomic profiling methodology have significantly increased specificity and accuracy of lipid measurements. This evolution, driven by improved chromatographic and mass spectrometric resolution of newer platforms, has made it challenging to align datasets created at different times, or on different platforms. Here we present a framework for harmonising such plasma lipidomic datasets with different levels of granularity in their lipid measurements. Our method utilises elastic-net prediction models, constructed from high-resolution lipidomics reference datasets, to predict unmeasured lipid species in lower-resolution studies. The approach involves (1) constructing composite lipid measures in the reference dataset that map to less resolved lipids in the target dataset, (2) addressing discrepancies between aligned lipid species, (3) generating prediction models, (4) assessing their transferability into the targe dataset, and (5) evaluating their prediction accuracy. To demonstrate our approach, we used the AusDiab population-based cohort (747 lipid species) as the reference to impute unmeasured lipid species into the LIPID study (342 lipid species). Furthermore, we compared measured and imputed lipids in terms of parameter estimation and predictive performance, and validated imputations in an independent study. Our method for harmonising plasma lipidomic datasets will facilitate model validation and data integration efforts.

Assessing the release, transport, and retention of radioactive aerosols from hypothetical breaches in spent fuel storage canisters

Interim dry storage of spent nuclear fuel involves storing the fuel in welded stainless-steel canisters. Under certain conditions, the canisters could be subjected to environments that may promote stress corrosion cracking leading to a risk of breach and release of aerosol-sized particulate from the interior of the canister to the external environment through the crack. Research is currently under way by several laboratories to better understand the formation and propagation of stress corrosion cracks, however little work has been done to quantitatively assess the potential aerosol release. The purpose of the present work is to introduce a reliable generic numerical model for prediction of aerosol transport, deposition, and plugging in leak paths similar to stress corrosion cracks, while accounting for potential plugging from particle deposition. The model is dynamic (changing leak path geometry due to plugging) and it relies on the numerical solution of the aerosol transport equation in one dimension using finite differences. The model’s capabilities were also incorporated into a Graphical User Interface (GUI) that was developed to enhance user accessibility. Model validation efforts presented in this paper compare the model’s predictions with recent experimental data from Sandia National Laboratories (SNL) and results available in literature. We expect this model to improve the accuracy of consequence assessments and reduce the uncertainty of radiological consequence estimations in the remote event of a through-wall breach in dry cask storage systems.

Improved modelling of SEP event onset within the WSA–Enlil–SEPMOD framework

Multi-spacecraft observations of solar energetic particle (SEP) events not only enable a deeper understanding and development of particle acceleration and transport theories but also provide important constraints for model validation efforts. However, because of computational limitations, a given physics-based SEP model is usually best suited to capture a particular phase of an SEP event, rather than its whole development from onset through decay. For example, magnetohydrodynamic (MHD) models of the heliosphere often incorporate solar transients only at the outer boundary of their so-called coronal domain – usually set at a heliocentric distance of 20–30 R⊙. This means that particle acceleration at coronal mass ejection (CME)-driven shocks is also computed from this boundary onwards, leading to simulated SEP event onsets that can be many hours later than observed, since shock waves can form much lower in the solar corona. In this work, we aim to improve the modelled onset of SEP events by inserting a “fixed source” of particle injection at the outer boundary of the coronal domain of the coupled WSA–Enlil 3D MHD model of the heliosphere. The SEP model that we employ for this effort is Solar Energetic Particle MODel (SEPMOD), a physics-based test-particle code based on a field line tracer and adiabatic invariant conservation. We apply our initial tests and results of SEPMOD’s fixed-source option to the 2021 October 9 SEP event, which was detected at five well-separated locations in the inner heliosphere – Parker Solar Probe, STEREO-A, Solar Orbiter, BepiColombo, and near-Earth spacecraft.

2012–2016年三江源垂穗披碱草人工草地碳水热通量观测数据集

In the Sanjiangyuan Area known as the “Chinese Water Tower”, planting perennial artificial grassland has become an important measure for the restoration of degraded grassland in this area. The spatial-temporal pattern of the carbon, water, and heat fluxes of Elymus nutans artificial grasslands are crucial for comprehending the ecological service functions of the Elymus nutans artificial grasslands in the Sanjiangyuan Area. Sanjiangyuan Grassland Ecosystem National Observation and Research Station (referred to as Sanjiangyuan Sation) has used the eddy covariance technique to monitor the carbon, water, and heat fluxes over an Elymus nutans artificial grassland for 5 years since 2009. In order to advance the research on carbon, water, and heat fluxes and other related researches in the Sanjiangyuan Area and the broader Qinghai-Tibetan Plateau alpine ecosystems, we plan to publish the routine meteorological data of carbon, water and heat fluxes over an Elymus nutans artificial grassland observed from 2012 to 2016. This dataset includes routine meteorological data (i.e. air temperature, air relative humidity, water vapor pressure, soil temperature, soil moisture, photosynthetically active radiation, and precipitation) and carbon, water, and heat fluxes data (net ecosystem CO2 exchange, ecosystem CO2 respiration, gross ecosystem CO2 exchange, latent heat flux, and sensible heat flux) on half-hourly, daily, monthly, and yearly scales. The dataset used the flux data processing method recommended by ChinaFLUX. It is expected to provide field observational data support for scientific understanding, remote sensing retrieval processes, and model validation efforts in exploring the spatiotemporal patterns of carbon, water, and heat exchanges in artificial Elymus nutans grasslands.

Small Field Plots Can Cause Substantial Uncertainty in Gridded Aboveground Biomass Products from Airborne Lidar Data

Emerging satellite radar and lidar platforms are being developed to produce gridded aboveground biomass (AGB) predictions that are poised to expand our understanding of global carbon stocks and changes. However, the spatial resolution of AGB map products from these platforms is often larger than the available field plot data underpinning model calibration and validation efforts. Intermediate-resolution/extent remotely sensed data, like airborne lidar, can serve as a bridge between small plots and map resolution, but methods are needed to estimate and propagate uncertainties with multiple layers of data. Here, we introduce a workflow to estimate the pixel-level mean and variance in AGB maps by propagating uncertainty from a lidar-based model using small plots, taking into account prediction uncertainty, residual uncertainty, and residual spatial autocorrelation. We apply this workflow to estimate AGB uncertainty at a 100 m map resolution (1 ha pixels) using 0.04 ha field plots from 11 sites across four ecoregions. We compare uncertainty estimates using site-specific models, ecoregion-specific models, and a general model using all sites. The estimated AGB uncertainty for 1 ha pixels increased with mean AGB, reaching 7.8–33.3 Mg ha−1 for site-specific models (one standard deviation), 11.1–28.2 Mg ha−1 for ecoregion-specific models, and 21.1–22.1 Mg ha−1 for the general model for pixels in the AGB range of 80–100 Mg ha−1. Only 3 of 11 site-specific models had a total uncertainty of <15 Mg ha−1 in this biomass range, suitable for the calibration or validation of AGB map products. Using two additional sites with larger field plots, we show that lidar-based models calibrated with larger field plots can substantially reduce 1 ha pixel AGB uncertainty for the same range from 18.2 Mg ha−1 using 0.04 ha plots to 10.9 Mg ha−1 using 0.25 ha plots and 10.1 Mg ha−1 using 1 ha plots. We conclude that the estimated AGB uncertainty from models estimated from small field plots may be unacceptably large, and we recommend coordinated efforts to measure larger field plots as reference data for the calibration or validation of satellite-based map products at landscape scales (≥0.25 ha).

Simulated Ecosystem and Farm-Level Economic Impacts of Conservation Tillage in a Northeastern Iowa County

While the ecological benefits of no-till are largely indisputable, the economic impacts are less certain, and the latter may be partly to blame for lower-than-expected adoption of no-till. In this study, we contribute to a better understanding of the ecosystem and farm-level economic impacts of no-till, with Buchanan County in the northeastern region of the U.S. State of Iowa as the backdrop due to previously established data and model validation efforts in that region. Using the Agricultural Policy Environmental eXtender (APEX) and Farm Economic Model (FEM), we simulated two tillage scenarios—a conservation tillage baseline and no-till—for continuous corn and corn–soybean rotations in Buchanan County using gridded historical climate data. We find that no-till provides clear ecosystem benefits, except that soluble nutrient losses might actually rise. We also find that under current commodity prices for corn and soybeans, no-till is not as profitable as the conservation tillage baseline. For no-till to be at least as profitable as the baseline under current commodity prices, the yield penalty associated with no-till cannot be higher than 1.5% for corn and 0.8% for soybeans, or similar combinations that entail a revenue penalty of about $24,000 for an 809-hectare continuous corn or corn–soybean operation. Given the simulated yield penalties associated with no-till, corn and soybean prices would have to be substantially lower in order for no-till to break even. Consequently, incentives for conservation practice implementation may need to be tied to commodity prices and yield penalties in order to elicit greater adoption rates.

Hyper-resolution PCR-GLOBWB: opportunities and challenges from refining model spatial resolution to 1 km over the European continent

Abstract. The quest for hydrological hyper-resolution modelling has been on-going for more than a decade. While global hydrological models (GHMs) have seen a reduction in grid size, they have thus far never been consistently applied at a hyper-resolution (<=1 km) at the large scale. Here, we present the first application of the GHM PCR-GLOBWB at 1 km over Europe. We thoroughly evaluated simulated discharge, evaporation, soil moisture, and terrestrial water storage anomalies against long-term observations and subsequently compared results with the established 10 and 50 km resolutions of PCR-GLOBWB. Subsequently, we could assess the added value of this first hyper-resolution version of PCR-GLOBWB and assess the scale dependencies of model and forcing resolution. Eventually, these insights can help us in understanding the current challenges and opportunities from hyper-resolution models and in formulating the model and data requirements for future improvements. We found that, for most variables, epistemic uncertainty is still large, and issues with scale commensurability exist with respect to the long-term yet coarse observations used. Merely for simulated discharge, we can confidently state that model output at hyper-resolution improves over coarser resolutions due to better representation of the river network at 1 km. However, currently available observations are not yet widely available at hyper-resolution or lack a sufficiently long time series, which makes it difficult to assess the performance of the model for other variables at hyper-resolution. Here, additional model validation efforts are needed. On the model side, hyper-resolution applications require careful revisiting of model parameterization and possibly the implementation of more physical processes to be able to resemble the dynamics and spatial heterogeneity at 1 km. With this first application of PCR-GLOBWB at 1 km, we contribute to meeting the grand challenge of hyper-resolution modelling. Even though the model was only assessed at the continental scale, valuable insights could be gained which have global validity. As such, it should be seen as a modest milestone on a longer journey towards locally relevant model output. This, however, requires a community effort from domain experts, model developers, research software engineers, and data providers.

Data needs to be a priority

Findability, Accessibility, Interoperability, and Reusability (FAIR) data are essential to heliophysics and all scientific research. The principles of FAIR data ensure the reusability and findability of data, as well as its long-term care. The goal is that data are accessible for the ongoing discovery and verification process and can be used on their own or with newly generated data in future studies leading to innovations. With the onset in the previous decades of NASA and other agencies requiring mission data to be open to the public, heliophysics has already made great strides toward FAIR data and benefited from these efforts. Continued improvements in our metadata, data archives, and data portals and the addition of DOIs for data citation will ensure data will be FAIR, enabling further scientific discoveries, reproducibility of results, longitudinal studies, and verification and validation of models. Currently, not all the data collected are findable and on open networks or archives, and not all data on archives have DOIs. Within this study, we make recommendations to prioritize resources needed to satisfy FAIR data principles, treating them as a fundamental research infrastructure rather than a simple research product.• Data collection, preparation, archiving, and accessibility need to be a priority.• Data collection, preparation, archiving, and accessibility need dedicated and sustained funding support.• Data need to be accessible through investment in infrastructure: tools to access and read the data and personnel to maintain these data and IT infrastructure.• Data need to be collected in sustained ways to enable further science and, specifically, model validation efforts.

A Review on Rain Signal Attenuation Modeling, Analysis and Validation Techniques: Advances, Challenges and Future Direction

Radio waves are attenuated by atmospheric phenomena such as snow, rain, dust, clouds, and ice, which absorb radio signals. Signal attenuation becomes more severe at extremely high frequencies, usually above 10 GHz. In typical equatorial and tropical locations, rain attenuation is more prevalent. Some established research works have attempted to provide state-of-the-art reviews on modeling and analysis of rain attenuation in the context of extremely high frequencies. However, the existing review works conducted over three decades (1990 to 2022), have not adequately provided comprehensive taxonomies for each method of rain attenuation modeling to expose the trends and possible future research directions. Also, taxonomies of the methods of model validation and regional developmental efforts on rain attenuation modeling have not been explicitly highlighted in the literature. To address these gaps, this paper conducted an extensive literature survey on rain attenuation modeling, methods of analyses, and model validation techniques, leveraging the ITU-R regional categorizations. Specifically, taxonomies in different rain attenuation modeling and analysis areas are extensively discussed. Key findings from the detailed survey have shown that many open research questions, challenges, and applications could open up new research frontiers, leading to novel findings in rain attenuation. Finally, this study is expected to be reference material for the design and analysis of rain attenuation.

Single–bubble dynamics in histotripsy and high–amplitude ultrasound: Modeling and validation

A variety of approaches have been used to model the dynamics of a single, isolated bubble nucleated by a microsecond length high–amplitude ultrasound pulse (e.g. a histotripsy pulse). Until recently, the lack of single–bubble experimental radius vs. time data for bubble dynamics under a well–characterized driving pressure has limited model validation efforts. This study uses radius vs. time measurements of single, spherical histotripsy–nucleated bubbles in water to quantitatively compare and validate a variety of bubble dynamics modeling approaches, including compressible and incompressible models as well as different thermal models. A strategy for inferring an analytic representation of histotripsy waveforms directly from experimental radius vs. time and cavitation threshold data is presented. We compare distributions of a calculated validation metric obtained for each model applied to 88 experimental data sets. There is minimal distinction (<1%) among the modeling approaches for compressibility and thermal effects considered in this study. These results suggest that our proposed strategy to infer the waveform, combined with simple models minimizing parametric uncertainty and computational resource demands accurately represent single–bubble dynamics in histotripsy, including at and near the maximum bubble radius. Remaining sources of parametric and model–based uncertainty are discussed.

Estimation of Inflow Uncertainties in Laminar Hypersonic Double-Cone Experiments

This paper proposes a probabilistic framework for assessing the consistency of an experimental dataset, i.e., whether the stated experimental conditions are consistent with the measurements provided. In case the dataset is inconsistent, our framework allows one to hypothesize and test sources of inconsistencies. This is crucial in model validation efforts. The framework relies on Bayesian inference to estimate experimental settings deemed uncertain, from measurements deemed accurate. The quality of the inferred variables is gauged by its ability to reproduce held-out experimental measurements. The correctness of the framework is tested on three double-cone experiments conducted in the Calspan-University at Buffalo Research Center’s Large Energy National Shock Tunnel-I (LENS-I), which have also been numerically simulated successfully. Thereafter, the framework is used to investigate two double-cone experiments (executed in the LENS-XX), which have encountered difficulties when used in model validation exercises. An inconsistency is detected with one of the LENS-XX experiments. In addition, two causes are hypothesized for our inability to simulate LEXS-XX experiments accurately and are tested using our framework. It is found that there is no single cause that explains all the discrepancies between model predictions and experimental data, but different causes explain different discrepancies, to larger or smaller extent. This paper proposes that uncertainty quantification methods be used more widely to understand experiments and characterize facilities, and three different methods are cited to do so, the third of which is presented in this paper.

Long-term Data from the Swedish National Environmental Monitoring Program of Pesticides in Surface Waters.

Environmental monitoring is essential for assessing the impact of human activities on the environment. Monitoring data are used to ascertain that environmental standards are met, to inform policy making, to determine trends, and to provide parameterization data for prediction models. The design of monitoring programs depends on what is being monitored, for what purpose, and available resources. Here we describe the strategy and design of the Swedish environmental monitoring program for chemical pesticides in surface waters and provide data generated within this program since 2002 (). We include examples of how the data can be used for toxicity assessments, trend analyses, and comparison between sampling strategies. Our goal is to increase awareness of this dataset and provide detailed information about the data so that it may be incorporated into meta-analytical research, comparison studies, model validation, and other scientific efforts.

A Year‐Long Comparison of GPS TEC and Global Ionosphere‐Thermosphere Models

AbstractThe prevalence of GPS total electron content (TEC) observations has provided an opportunity for extensive global ionosphere‐thermosphere model validation efforts. This study presents a year‐long data‐model comparison using the Global Ionosphere‐Thermosphere Model (GITM) and the Thermosphere‐Ionosphere‐Electrodynamics General Circulation Model (TIE‐GCM). For the entire year of 2010, each model was run and compared to GPS TEC observations. The results were binned according to season, latitude, local time, and magnetic local time. GITM was found to overestimate the TEC everywhere, except on the midlatitude nightside, due to high O/N2 ratios. TIE‐GCM produced much less TEC and had lower O/N2 ratios and neutral wind speeds. Seasonal and regional biases in the models are discussed along with ideas for model improvements and further validation efforts.

Wavelet ANOVA bisection method for identifying simulation model bias

High-resolution computer models can simulate complex systems and processes in order to evaluate a solution quickly and inexpensively. Many simulation models produce dynamic functional output, such as a set of time-series data generated during a process. These computer models require verification and validation (V&V) to assess the correctness of these simulations. In particular, the model validation effort evaluates if the model is an appropriate representation of the real-world system that it is meant to simulate. However, when assessing a model capable of generating functional output, it is useful to learn more than simply whether the model is valid or invalid. Specifically, if the model is deemed invalid, then what aspects of the model are incorrect? Is it possible to identify over what range the model data are a poor representation of the system data? Current V&V methods cannot identify these ranges. This paper proposes a wavelet analysis of variance (WANOVA) bisection method that first assesses model validity and can also identify the interval(s) over which the model is biased. The technique is illustrated using several simulation studies. Ultimately, this new method supports and expands the efficacy of model validation efforts.

On the role of visualisation in fisheries management

Environmental change has focused the attention of scientists, policy makers and the wider public on the uncertainty inherent in interactions between people and the environment. Governance in fisheries is required to involve stakeholder participation and to be more inclusive in its remit, which is no longer limited to ensuring a maximum sustainable yield from a single stock but considers species and habitat interactions, as well as social and economic issues. The increase in scope, complexity and awareness of uncertainty in fisheries management has brought methodological and institutional changes throughout the world. Progress towards comprehensive, explicit and participatory risk management in fisheries depends on effective communication. Graphic design and data visualisation have been underused in fisheries for communicating science to a wider range of stakeholders. In this paper, some of the general aspects of designing visualisations of modelling results are discussed and illustrated with examples from the EU funded MYFISH project. These infographics were tested in stakeholder workshops, and improved through feedback from that process. It is desirable to convey not just modelling results but a sense of how reliable various models are. A survey was developed to judge reliability of different components of fisheries modelling: the quality of data, the quality of knowledge, model validation efforts, and robustness to key uncertainties. The results of these surveys were visualized for ten different models, and presented alongside the main case study.

Crystal Plasticity Model Validation Using Combined High-Energy Diffraction Microscopy Data for a Ti-7Al Specimen

High-Energy Diffraction Microscopy (HEDM) is a 3-d X-ray characterization method that is uniquely suited to measuring the evolving micro-mechanical state and microstructure of polycrystalline materials during in situ processing. The near-field and far-field configurations provide complementary information; orientation maps computed from the near-field measurements provide grain morphologies, while the high angular resolution of the far-field measurements provides intergranular strain tensors. The ability to measure these data during deformation in situ makes HEDM an ideal tool for validating micro-mechanical deformation models that make their predictions at the scale of individual grains. Crystal Plasticity Finite Element Models (CPFEM) are one such class of micro-mechanical models. While there have been extensive studies validating homogenized CPFEM response at a macroscopic level, a lack of detailed data measured at the level of the microstructure has hindered more stringent model validation efforts. We utilize an HEDM dataset from an alpha-titanium alloy (Ti-7Al), collected at the Advanced Photon Source, Argonne National Laboratory, under in situ tensile deformation. The initial microstructure of the central slab of the gage section, measured via near-field HEDM, is used to inform a CPFEM model. The predicted intergranular stresses for 39 internal grains are then directly compared to data from 4 far-field measurements taken between ~4 and ~80 pct of the macroscopic yield strength. The evolution of the elastic strain state from the CPFEM model and far-field HEDM measurements up to incipient yield are shown to be in good agreement, while residual stress at the individual grain level is found to influence the intergranular stress state even upon loading. Implications for application of such an integrated computational/experimental approach to phenomena such as fatigue are discussed.

Unremarked or Unperformed? Systematic Review on Reporting of Validation Efforts of Health Economic Decision Models in Seasonal Influenza and Early Breast Cancer.

BackgroundTransparent reporting of validation efforts of health economic models give stakeholders better insight into the credibility of model outcomes. In this study we reviewed recently published studies on seasonal influenza and early breast cancer in order to gain insight into the reporting of model validation efforts in the overall health economic literature.MethodsA literature search was performed in Pubmed and Embase to retrieve health economic modelling studies published between 2008 and 2014. Reporting on model validation was evaluated by checking for the word validation, and by using AdViSHE (Assessment of the Validation Status of Health Economic decision models), a tool containing a structured list of relevant items for validation. Additionally, we contacted corresponding authors to ask whether more validation efforts were performed other than those reported in the manuscripts.ResultsA total of 53 studies on seasonal influenza and 41 studies on early breast cancer were included in our review. The word validation was used in 16 studies (30 %) on seasonal influenza and 23 studies (56 %) on early breast cancer; however, in a minority of studies, this referred to a model validation technique. Fifty-seven percent of seasonal influenza studies and 71 % of early breast cancer studies reported one or more validation techniques. Cross-validation of study outcomes was found most often. A limited number of studies reported on model validation efforts, although good examples were identified. Author comments indicated that more validation techniques were performed than those reported in the manuscripts.ConclusionsAlthough validation is deemed important by many researchers, this is not reflected in the reporting habits of health economic modelling studies. Systematic reporting of validation efforts would be desirable to further enhance decision makers’ confidence in health economic models and their outcomes.Electronic supplementary materialThe online version of this article (doi:10.1007/s40273-016-0410-3) contains supplementary material, which is available to authorized users.

Measurement of electron temperature fluctuations using a tunable correlation electron cyclotron emission system on Alcator C-Mod.

A tunable correlation electron cyclotron (CECE) system was recently installed on the Alcator C-Mod tokamak to provide local, quantitative measurement of electron temperature fluctuations in the tokamak core. This system represents a significant upgrade from the original CECE system, expanding the measurement capabilities from 4 to 8 total channels, including 2 remotely tunable YIG filters (6-18 GHz; 200 MHz bandwidth). Additional upgrades were made to the optical system to provide enhanced poloidal resolution and allow for measurement of turbulent fluctuations below kθρs < 0.3. These expanded capabilities allow for single shot measurement of partial temperature fluctuation profiles in the region ρ = 0.7 - 0.9 (square root of normalized toroidal flux) in a wide variety of plasma conditions. These measurements are currently being used to provide stringent tests of the gyrokinetic model in ongoing model validation efforts. Details of the hardware upgrades, turbulent fluctuation measurements, and ongoing comparisons with simulations are presented.

A demonstration of ABM validation techniques by applying docking to the Epstein civil violence model

The increased focus of the United States Department of Defense (DoD) on irregular warfare and counterinsurgency has served to identify the lack of credible models and simulations to represent the relevant civilian populations – the centers of gravity of such operations. While agent-based models (ABMs) have enjoyed widespread use in the social science community, many senior DoD officials are skeptical that agent-based models can provide useful tools to underpin DoD analysis, training, and acquisition needs mainly because of validation concerns. This paper uses docking and other forms of alignment that enable the linking of the Epstein civil violence agent-based model results to other models. These examples of model-to-model analysis could serve to assist and encourage DoD ABM human domain model validation efforts.

Maximizing Design Confidence in Sequential Simulation-Based Optimization

Computational simulation models support a rapid design process. Given model approximation and operating conditions uncertainty, designers must have confidence that the designs obtained using simulations will perform as expected. The traditional approach to address this need consists of model validation efforts conducted predominantly prior to the optimization process. We argue that model validation is too daunting of a task to be conducted with meaningful success for design optimization problems associated with high-dimensional space and parameter spaces. In contrast, we propose a methodology for maximizing confidence in designs generated during the simulation-based optimization process. Specifically, we adopt a trust-region-like sequential optimization process and utilize a Bayesian hypothesis testing technique to quantify model confidence, which we maximize by calibrating the simulation model within local domains if and when necessary. This ensures that the design iterates generated during the sequential optimization process are associated with maximized confidence in the utilized simulation model. The proposed methodology is illustrated using a cantilever beam design subject to vibration.