International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Illustrating a Bayesian approach to seismic collapse risk assessment Gokkaya, Beliz U.; Baker, Jack W.; Deierlein, Gregory G. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Illustrating a Bayesian Approach to Seismic Collapse RiskAssessmentBeliz U. GokkayaGraduate Student, Dept. of Civil and Environmental Engineering, Stanford University,Stanford, CA, USAJack W. BakerProfessor, Dept. of Civil and Environmental Engineering, Stanford University, Stanford,CA, USAGregory G. DeierleinProfessor, Dept. of Civil and Environmental Engineering, Stanford University, Stanford,CA, USAABSTRACT: In this study, we present a Bayesian method for efficient collapse response assessmentof structures. The method facilitates integration of prior information on collapse response with datafrom nonlinear structural analyses in a Bayesian setting to provide a more informed estimate of thecollapse risk. The prior information on collapse can be obtained from a variety of sources, includinginformation on the building design criteria and simplified linear dynamic analysis or nonlinear static(pushover) analysis. The proposed method is illustrated on a four-story reinforced concrete momentframe building to assess its seismic collapse risk. The method is observed to significantly improve thestatistical and computational efficiency of collapse risk predictions compared to alternative methods.1. INTRODUCTIONBuilding codes achieve seismic performance goalsrelated to life safety of building occupants by con-trolling collapse risk of structures to acceptable lev-els. Modeling of structural collapse is challeng-ing due to highly nonlinear structural response un-der extreme ground shaking, and its simulation re-quires nonlinear structural analysis tools and mod-els that can capture various sources of cyclic and in-cycle degradation in structural components. More-over, robust estimation of collapse risk should con-sider uncertainties in the earthquake ground mo-tions and structural modeling, and propagate theseeffects from component through to system level re-sponse. These uncertainties affect the statistical ef-ficiency of the estimated collapse risk parametersand add to the computational demand associatedwith collapse risk assessment of structures.Bayesian statistics facilitate the incorporation ofany prior knowledge to inform statistical inference.Singhal and Kiremidjian (1998) proposed usingBayesian statistics to update fragility functions withobservational building damage data. Jaiswal et al.(2011) also used a Bayesian approach for comput-ing empirical collapse fragility functions combin-ing expert opinion and field data for global buildingtypes. Jalayer et al. (2010) used a Bayesian frame-work for assessing the effects of structural model-ing uncertainty. They incorporated test and inspec-tion results of structures in order to update the priorinformation on the modeling uncertainties.In this study, we present a Bayesian methodfor efficient collapse response assessment of struc-tures combining analysis data and judgment. Themethod facilitates integration of prior informationto estimate collapse fragility parameters with datafrom nonlinear structural analyses. The combina-tion of nonlinear analysis simulations with priorcollapse fragility information aims to improve com-putational and statistical efficiency.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152. BAYESIAN APPROACHIn this section, we present a Bayesian method tocollapse risk assessment. We discuss backgroundto development of the method, and present themethodology along with the analysis rule.2.1. Collapse Risk MetricsCollapse fragility functions define probability ofcollapse as a function of ground motion intensity(IM). It is common to use lognormal distributionto represent collapse fragility curves (Bradley andDhakal, 2008). Using a lognormal distribution, theprobability of collapse given a ground motion in-tensity, P(C|IM = im), is defined asP(C|IM = im) = Φ(ln(im)− ln(θˆ)βˆ)(1)where Φ() is the standard normal cumulative distri-bution function, and θˆ and βˆ represent median col-lapse capacity and logarithmic standard deviation(dispersion), respectively.Collapse risk is often quantified using mean an-nual frequency of collapse (λc), which is defined asλc =∫ ∞0P(C|IM = im) |dλIM(im)| (2)where λIM is the mean annual frequency of ex-ceedance of IM. This metric, by augmenting struc-tural collapse response with site seismic hazardcharacteristics, provides a site-specific measure ofcollapse risk. In this study, our goal is to reliablyestimate λc for collapse risk assessment.2.2. Proposed MethodIn this section, we provide a step-by-step procedurefor conducting collapse risk assessment usingthe Bayesian approach. Figure 1 illustrates thesteps. The essence of this approach is to transforman initial estimate of the collapse response toan informed estimate using nonlinear structuralanalyses data with the goal of efficiently estimatingλc. The steps are listed as follows:a) Define an initial estimate of the collapsefragility curve and estimate the uncertainty inthe median collapse capacity (Figure 1a).b) Select two or more IM levels at which to scaleand conduct nonlinear dynamic analyses. Quan-tify the prior distributions at these IM levels(Figure 1b).c) Select ground motion suites consistent with con-ditional spectra at the chosen IM levels (Figure1c).d) Conduct nonlinear time history analyses usingthe selected ground motions scaled to the IMlevels of interest. Incorporate modeling uncer-tainty by sampling model realizations (Figure1d).e) Obtain posterior distributions at the IM levels byupdating the prior distributions with data fromstructural analyses (Figure 1e).f) Obtain a final estimate of the collapse fragilityfunction using the maximum likelihood method.For this method, likelihood is obtained using theposterior distributions at the IM levels (Figure1f).2.3. Details of the Proposed MethodUsers of this method are expected to provide priorinformation on the collapse response of the struc-ture. The prior information on collapse responsecan be informed by a variety of sources, includinginformation on the building design criteria and sim-plified linear dynamic analysis or nonlinear static(pushover) analysis. Using a lognormal assump-tion, we expect the users to provide an estimateof median collapse capacity (θ ) and dispersion (β )defining the initial collapse fragility function. Totreat epistemic and aleatory uncertainties in col-lapse fragility functions, median collapse capacity(Θ) is defined to be a Bayesian random variable. Θis modeled using a lognormal distribution with me-dian θ and dispersion βθ . In addition to θ and β ,users are also expected to provide βθ . Figure 1aillustrates an initial fragility curve and uncertaintyin median collapse capacity using by a probabilitydensity function. In cases where information on βθis not available, users can make judgment-based as-sumptions for estimating βθ considering the lim-itations in structural idealizations, calibration of212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 0.5 1 1.5 2 2.5 300.20.40.60.81Sa (T1, 5%)P collapse  Initial FragilityUncertainty in MedianBound5%Bound95%(a) Initial collapse fragility curveand uncertainty in median capacity0 IM1 IM200.20.40.60.81Sa (T1, 5%)P collapse  Initial FragilityPrior Dist.(b) Selected IM levels and prior dis-tributions at these levels10−1 100 10110−210−1100T (s)Sa (g)(c) Ground motions selected tomatch conditional spectra0 IM1 IM200.20.40.60.81Sa (T1, 5%)P collapse  Initial FragilityPrior Dist.Analyses Data(d) Data from nonlinear time historyanalyses at selected IM levels0 IM1 IM200.20.40.60.81Sa (T1, 5%)P collapse  Initial FragilityPrior Dist.Analyses DataPosterior Dist.(e) Posterior distributions at selectedIM levels0 IM1 IM200.20.40.60.81Sa (T1, 5%)P collapse  Initial FragilityPrior Distr.Analyses DataPosterior Dist.Bayesian Est.(f) Final estimate of the collapsefragility functionFigure 1: Steps of the proposed methodmodel parameters, number of analyses, and soft-ware used for structural analyses such as: “Mediancollapse capacity is estimated as θ within±∆% cer-tainty with (1−α)% confidence." This leads to themedian collapse capacity being defined using a log-normal distribution having median at θ and disper-sion βθ as given in the equation below (Ellingwoodand Kinali, 2009).βθ =√√√√ln([∆Φ−1(1−α/2)]2+1)(3)An example of such an assumption is as fol-lows: “Median collapse capacity is estimated as 1 gwithin± 50% certainty with 90% confidence." Thisstatement is translated into a lognormal distributionhaving a median of 1 g (θ = 1) and dispersion of ap-proximately 0.3 (βθ = 0.3) . The 90% confidenceinterval for this distribution is 0.61 g to 1.64 g.Two or more IM levels should be used to se-lect ground motions and conduct structural analy-ses. A simulation-based grid search is conductedto identify the combinations of IM levels that leadto minimum error of λc estimates. Due to spaceconstraints, we do not present the results of thesimulation-based grid search. Based on search re-sults, we recommend that IM1 is selected corre-sponding to probability of collapse of 10% or loweron the initial collapse fragility curve. Select IM2such that it corresponds to probability of collapsebetween 30% and 80% on the initial fragility curveas an increasing function of ∆.The prior distribution at the ith IM level (IMi) de-fines the probability of collapse at IMi (P(τi)). Itis characterized using a beta distribution P(τi) ∼Beta(αi,βi). In this method, the parameters of thebeta distribution, namely αi, βi, are calibrated asfollows:We define two curves, namely Bound5% andBound95%, which are lognormal cumulative distri-butions. They have dispersions of β . The mediansof Bound5% and Bound95% correspond to 5% and312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 201595% quantiles of Θ, respectively. Figure 1a illus-trates Bound5% and Bound95%.For any IMi, the 5% quantile of P(τi) is obtainedas the point corresponding to IMi on Bound5%.Similarly, the 95% quantile of P(τi) is obtained asthe point corresponding to IMi on Bound95%. Thepoint on the initial fragility function correspondingto IMi denotes the mode of P(τi). Using these con-straints, one can obtain the parameters, αi and βi,defining P(τi) ∼ Beta(αi,βi). Figure 1b illustratesselected IM levels along with the prior distributionsat these levels.Using Bayes theorem, the posterior distribution,P(τi|Xi) is defined as follows:P(τi|Xi) ∝ P(τi)P(Xi|τi) (4)where P(Xi|τi) defines the likelihood function. Inthis method, we use data from structural analy-ses to define the likelihood function. Nonlineartime history analyses should be conducted for thispurpose, preferably incorporating modeling uncer-tainty and record-to-record variability. To accountfor record-to-record variability, hazard consistentground motion suites are selected at each IM level.An example ground motion suite is selected match-ing the conditional spectra, and is shown in Fig-ure 1c. For ground motion selection, readers arereferred to Jayaram et al. (2011). Previous re-search has shown that neglecting modeling uncer-tainty results in inconservative estimates of collapsecapacity (Liel et al., 2009; Dolsek, 2009; Ugurhanet al., 2014). For robust estimates of collapserisk, we recommend characterizing the uncertaintyin modeling parameters, and obtaining samples ofmodel realizations from the characterized proba-bility distributions. Nonlinear time history analy-ses should be conducted using the ground motionsuites scaled to IM levels of interest matched withthe sampled model realizations. Analyses resultsare collected as binomial data (Xi) in the form ofnumber of collapses and no-collapses, P(Xi|τi) ∼Binomial(τi,Xi). Figure 1d illustrates the resultsfrom structural analyses in terms of proportions ofcollapses at the two selected IM levels.In this method, prior distributions are defined us-ing beta distributions, and likelihood data is col-lected using a binomial distribution. Since beta andbinomial distributions form a conjugate pair, poste-rior distributions are also defined to have beta dis-tributions P(τi|Xi) ∼ Beta(αˆi, βˆi). Posterior distri-butions are illustrated for the selected IM levels inFigure 1e.Using the two posterior distributions at differentIM levels as the probability distributions defining alikelihood function, we use a maximum likelihoodestimator to find the final parameters of a collapsefragility function, namely θˆ , βˆ (Figure 1f).2.4. Validation of the MethodOur current work explores to validate the proposedmethod. In the interest of computational efficiency,instead of analyzing real structural models we usesimulated structural analyses data, which is bino-mial data drawn from an assumed target fragilityfunction. We apply the Bayesian method to ob-tain estimates the collapse fragility function and λc.This procedure is repeated in a Monte-Carlo simu-lation framework, which allows us to study the sta-tistical efficiency of the Bayesian approach in termsof the variance and bias of the estimators.In the interest of space, we will not be presentingthe results from the validation analyses. Key ob-servations from the validation analyses are as fol-lows: The variability in collapse risk metrics issignificantly reduced using the Bayesian approachcompared to an alternative method of maximumlikelihood estimate. If used with an initial un-biased estimate of the collapse fragility function,Bayesian method provides an unbiased estimate ofλc. However, a biased initial guess produces biasin Bayesian estimates. The amount of bias intro-duced in the prior information heavily affects theamount of bias that will be observed in the Bayesianestimates. Posterior predictive checking (Gelmanet al., 2013) is a model checking method to assessthe plausibility of the posterior distributions. Thismethod is well suited to provide a check for the es-timates obtained using the Bayesian approach.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153. APPLICATION OF BAYESIAN AP-PROACH3.1. Structural Modeling and AnalysisCollapse risk assessment of a 4-story reinforcedconcrete special moment frame is conducted to il-lustrate the application of the Bayesian approach.The structure was designed by Haselton and Deier-lein (2007) in accordance with 2003 InternationalBuilding Code and ASCE7-02 provisions. It is lo-cated at a seismically active site in downtown LosAngeles, CA with Vs,30 = 285 m/s.We use concentrated plasticity modeling toidealize the structural system. Frame membersare modeled as elastic elements with zero-lengthrotational springs at both ends. The hystereticbehavior and in-cycle and cyclic deteriorationrules are governed by Ibarra et al. (2005). Thefundamental period of the structure is T1 = 0.94s and a Rayleigh damping of 3% is applied toit. The structure is modeled and analyzed usingOpen System for Earthquake Engineering Platform(McKenna et al., 2014).3.2. Prior Information on Collapse ResponseNonlinear static analysis is a common structuralanalysis strategy, and is generally conducted tocheck the nonlinear structural analysis model.Vamvatsikos and Cornell (2005) provide a fastmethod, SPO2IDA, which uses pushover analy-sis results to infer nonlinear dynamic response byestablishing connections between pushover curvesand incremental dynamic analyses (IDA) curves. Inthis study, we use this method to obtain an initialestimate of the collapse response of the structure.Since nonlinear static analysis is not computation-ally demanding and is generally conducted beforeany dynamic analyses, we assume that it does notadd to the computational demand of the Bayesianmethod.The 4-story frame structure is analyzed usingnonlinear static analysis. Using the software forSPO2IDA, we obtain estimates of 16, 50 and 84%fractal IDA curves as shown in Figure 2. Dynamicinstability, which is characterized by the zero slopeof an IDA curve, is observed at ductility valuesaround 10. These estimated fractals of IDA curves0 5 10 1500.511.522.533.5Sa(T 1,5%)µ=δ/δyield  IDA−16%IDA−50%IDA−84%Static PushoverFigure 2: Pushover and 16, 50 and 84% fractal IDAcurves obtained using the method by Vamvatsikos andCornell (2005)lead to a lognormal collapse fragility function hav-ing a median collapse capacity of 2.19 g and dis-persion of 0.43. We use ∆ = 0.4 to characterize theuncertainty in median collapse capacity estimate.3.3. Ground Motion and Modeling UncertaintyTo represent record-to-record variability, groundmotion suites are selected consistent with condi-tional spectra at two different ground motion inten-sity levels. IM is used as 5% damped spectral ac-celeration at T1 = 0.94 s, Sa(T1,5%). IM1 is recom-mended to be selected corresponding to or below10% probability of collapse on the initial fragilitycurve. Probability of collapse of 4% on the ini-tial fragility function corresponds to Sa(T1,5%) of1.05 g. This is the ground motion intensity having aprobability of exceedance of 2% in 50 years for thesite where the structure is located. Similarly, IM2is recommended to be selected such that it corre-sponds to probability of collapse between 30% and80% on the initial fragility curve as an increasingfunction of ∆. Probability of collapse of 40% on theinitial fragility function corresponds to Sa(T1,5%)of 1.96 g. This is the ground motion intensity hav-ing probability of exceedance of 1% in 200 yearsfor this site. We selected 30 ground motions at eachIM level matching the conditional spectra.Modeling uncertainty is propagated using LatinHypercube sampling. The parameters that are iden-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015MyMcChord RotationMomentθpcθcap,plEIstfFigure 3: Monotonic behavior of a concentrated plas-ticity hinge modeltified as random variables are six parameters defin-ing the monotonic backbone and hysteretic rulesof a structural component. These parameters areflexural strength (My), ratio of maximum momentand yield moment capacity (Mc/My), effective ini-tial stiffness which is defined by the secant stiff-ness to 40% of yield force (EIst f ,40/EIg), plas-tic rotation capacity (θcap,pl), post-capping rotationcapacity (θpc) and energy dissipation capacity forcyclic stiffness and strength deterioration (γ). Themonotonic backbone curve as a function of theseparameters is shown in Figure 3. The variabil-ity in the modeling parameters is represented us-ing logarithmic standard deviations of 0.73, 0.59,0.5, 0.31, 0.27 and 0.1 for θpc, θcap,pl , γ , My,EIst f ,40/EIg and Mc/My, respectively. These valuesare computed using the beam-column element cal-ibration database and the predictive equations de-veloped by Haselton et al. (2008). In total, twodifferent components are assumed to exist in thestructural model, namely, beam and column com-ponents. Equivalent viscous damping ratio, columnfooting rotational stiffness and joint shear strengthare also treated as random with logarithmic stan-dard deviations of 0.6, 0.3 and 0.1 (Haselton andDeierlein, 2007), respectively.Correlation structure among the random vari-ables are adopted from Ugurhan et al. (2014). Inthis correlation model, beam-to-beam and column-to-column correlations are idealized as perfect cor-relation, whereas beam-to-column and within com-ponent correlations are idealized by correlation co-efficients that are derived using random effects re-gression. In total 15 random variables are used inthis study and 30 realizations of these variables areobtained using Latin Hypercube sampling.3.4. ResultsGround motions in each suite are scaled to the cor-responding IM level and are matched with modelrealizations in order to conduct nonlinear time his-tory analyses. The number of collapses observedare 6 and 13 out of 30 analyses at IM1 = 1.05 g andIM2 = 1.96 g, respectively.The initial collapse fragility function is definedto have a median collapse capacity of 2.19 g anddispersion of 0.43. Median collapse capacity is es-timated within ±40% with 90% confidence.At IM1, prior information leads to a distribu-tion of Beta(α1 = 1.75,β1 = 17.42). Using theBayesian approach along with the observed data of6 collapses out of 30 analyses, this distribution isupdated to Beta(αˆ1 = 7.75, βˆ1 = 41.42).Similarly, at IM2, prior information leads to adistribution of Beta(α2 = 2.55,β2 = 3.34). UsingBayesian approach along with the observed data of13 collapses out of 30 analyses, this distribution isupdated to Beta(αˆ2 = 15.55, βˆ2 = 20.34).Using the maximum likelihood estimationmethod, where the posterior distributions at thetwo IM levels are used as probability distributionsdefining the likelihood function, the final collapsefragility curve is obtained to have a median collapsecapacity of 2.22 g and dispersion of 0.7.We see that Bayesian method increases the dis-persion of the initial fragility function by 64%. Italso increases the median collapse capacity of theinitial fragility function by 1.5%.3.5. DiscussionsTo benchmark the collapse response of the casestudy structure, we conduct an extensive collapseresponse analysis of the structure incorporatingground motion and modeling uncertainties. Weselect 200 ground motions consistent with condi-tional spectra at each IM level. We use five IMlevels correspond to probabilities of exceedanceof 5% in 50 years, 2% in 50 years, 1% in 50years, 1% in 100 years and 1% in 200 years. 200612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Sa (T1, 5%)0 0.5 1 1.5 2 2.5 3 3.5 4P collapse00.20.40.60.81Initial FragilityPrior Dist.Analyses DataPosterior Dist.Bayesian Est.Target FragilityMLE Est.Figure 4: Collapse fragility functions obtained usingdifferent methodsmodel realization are obtained using Latin Hyper-cube sampling. Each ground motion is matchedwith a model realization, and in total 1000 nonlin-ear time history analyses are conducted. Using amaximum likelihood estimator, we obtain the col-lapse fragility curve of the structure having a me-dian of 2.09 g and a dispersion of 0.61. Figure 4shows the collapse fragility functions obtained us-ing different approaches. The red line shows theresult from 1000 analyses, and is indicated as thetarget fragility function. The fragility curve ob-tained using the Bayesian approach is shown inblue, whereas the estimate obtained by applyingmaximum likelihood estimation on observed struc-tural analyses data only (MLE) is in green. Table1 summarizes the results obtained using the afore-mentioned approaches in terms of different collapsemetrics. Listed in Table 1 are the collapse fragilityparameters θ and β along with λc, which is ob-tained using the site hazard curve, and the probabil-ity of collapse in 50 years, which is obtained usinga Poisson assumption. Collapse risk estimates ob-tained using the median model parameters for thecase study structure are also given in Table 1.Table 1 shows that the Bayesian method shiftsthe median collapse capacity away from the targetvalue. Although the initial estimate overestimatesthe median collapse capacity by 5%, updating theinitial curve increases this difference to 6.6%. WeTable 1: Collapse risk estimatesEstimator θ (g) β λc(10−5) Pcol50yearsInitial 2.19 0.43 10.61 0.0053Bayesian 2.22 0.7 31.76 0.0158MLE 2.29 0.93 71.99 0.0354Target 2.09 0.61 25.37 0.0126Median 2.17 0.49 13.9 0.0069also see that dispersion of the target fragility func-tion is overestimated by 15.6% by the Bayesian ap-proach.Although median collapse capacities and disper-sions of the target and the Bayesian estimate differ,Figure 4 shows that the lower portion of the fragilityfunction, up until 25%, is well-constrained by theBayesian method. Table 1 shows that the Bayesianmethod provides a good estimate of λc. This is be-cause of the good match in the lower portion of thefragility function, since the lower tail of the fragilityfunction is an important contributor to λc.The Bayesian method starts with an initial esti-mate of λc which underestimates the target λc by58%. After applying by the method, the final esti-mate of λc differs from the target response by 25%.This difference is 184% using an MLE approach.It is also observed that by neglecting model uncer-tainty and using median model parameters, λc is un-derestimated by 45%.4. CONCLUSIONSThis study presents a Bayesian method for collapserisk assessment of structures. The method usesprior information on collapse response of struc-tures and augments this information with data froma small number of structural analyses. The ap-proach enables propagating ground motion vari-ability and model uncertainty through efficient sam-pling of model realizations. An illustration of themethod is provided through a collapse risk assess-ment of a 4-story frame structure. The Bayesianmethod is observed to significantly improve statisti-cal efficiency of collapse risk predictions comparedto alternative methods, and provide considerable re-duction in computational demand for probabilisticcollapse risk assessment of structures.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20155. ACKNOWLEDGEMENTSWe thank Dimitrios Vamvatsikos for providing thesoftware for SPO2IDA. This project is financiallysupported by NSF CMMI-1031722. Any opinions,findings and conclusions or recommendations ex-pressed in this material are those of the authors anddo not necessarily reflect the views of the NationalScience Foundation.6. REFERENCESBradley, B. A. and Dhakal, R. P. (2008). “Error estima-tion of closed-form solution for annual rate of struc-tural collapse.” Earthquake Engineering & StructuralDynamics, 37(15), 1721–1737.Dolsek, M. (2009). “Incremental dynamic analysis withconsideration of modeling uncertainties.” EarthquakeEngineering & Structural Dynamics, 38(6), 805–825.Ellingwood, B. and Kinali, K. (2009). “Quantifyingand communicating uncertainty in seismic risk as-sessment.” Structural Safety, 31(2), 179–187.Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B.,Vehtari, A., and Rubin, D. B. (2013). Bayesian DataAnalysis, Third Edition. CRC Press (November).Haselton, C. B. and Deierlein, G. G. (2007). “Assessingseismic collapse safety of modern reinforced concretemoment frame buildings.” Report No. 156, The JohnA. Blume Earthquake Engineering Center, Stanford,CA.Haselton, C. B., Liel, A. B., Lange, S., and Deier-lein, G. G. 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