crystals
Article Inverse Estimation Method of Material Randomness Using Observation
Dae-Young Kim 1, Pawel Sikora 2,3 , Krystyna Araszkiewicz 3 and Sang-Yeop Chung 1,*
1 Department of Civil and Environmental Engineering, Sejong University, 209 Neungdong-ro, Gwangjin-gu, Seoul 05006, Korea; [email protected] 2 Building Materials and Construction Chemistry, Technische Universität Berlin, Gustav-Meyer-Allee 25, 13355 Berlin, Germany; [email protected] 3 Faculty of Civil Engineering and Architecture, West Pomeranian University of Technology Szczecin, Al. Piastow 50, 70-311 Szczecin, Poland; [email protected] * Correspondence: [email protected]; Tel.: +82-2-6935-2471
Received: 16 April 2020; Accepted: 15 June 2020; Published: 16 June 2020
Abstract: This study proposes a method for inversely estimating the spatial distribution characteristic of a material’s elastic modulus using the measured value of the observation data and the distance between the measurement points. The structural factors in the structural system possess temporal and spatial randomness. One of the representative structural factors, the material’s elastic modulus, possesses temporal and spatial randomness in the stiffness of the plate structure. The structural factors with randomness are typically modeled as having a certain probability distribution (probability density function) and a probability characteristic (mean and standard deviation). However, this method does not consider spatial randomness. Even if considered, the existing method presents limitations because it does not know the randomness of the actual material. To overcome the limitations, we propose a method to numerically define the spatial randomness of the material’s elastic modulus and confirm factors such as response variability and response variance.
Keywords: Bayesian updating; spatial randomness; uncertainty; correlation distance; stochastic field
1. Introduction Research toward the development and incorporation of new building materials in modern engineering structures in order to meet sustainability goals has gathered substantial attention in recent years. Various new cement-based composites have been investigated including lightweight materials (foamed concrete and lightweight aggregates concretes) [1], pervious concretes [2], nano-modified, and self-cleaning materials [3,4]. However, due to the relatively higher production costs of modern building materials than in the case of conventional ones, it is still imperative to find a solution to support the simulating techniques and their accuracy toward decreasing the number of site trials, thus reducing the costs and environmental impact of material. In general, structural material analysis can be classified into a deterministic or a probabilistic method [5]. The deterministic method uses finite element analysis (FEA) considering the material properties, geometry, and forces. In other words, the various internal and external factors that can exist in a structure are represented by constants. However, these factors are all assumptions, and in the case of actual structures, it would be more reasonable to assume that the factors have different values depending on the position vector in the structural domain. The probabilistic method assumes that the structure possesses arbitrary material properties, loads, and geometries, and generates a random sample with some statistical characteristics. A deterministic FEA is repeatedly performed on the generated samples to obtain the characteristic behavior of
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deterministic FEA is repeatedly performed on the generated samples to obtain the characteristic thebehavior structure of the [6, 7structure]. Structural [6,7]. uncertainty, Structural inuncertai termsnty, of numerical in terms of considerations, numerical considerations, can be classified can asbe intrinsic,classified measurement, as intrinsic, measurement, and statistical and uncertainty, statistical or uncertainty, uncertainty or in theuncertainty mathematical in the model mathematical [8,9]. In eachmodel case, [8,9]. the In measurement each case, the uncertainty measurement is that uncertainty involved is in that establishing involved uncertainty in establishing factors uncertainty through experiments.factors through Experimental experiments. and Experimental statistical uncertainty and statisti iscal that uncertainty obtained is due that to obtained the lack due of data to the due lack to limitedof data timedue andto limited space information.time and space Uncertainty informatio inn. the Uncertainty mathematical in modelthe mathematical implies uncertainty model implies due to theuncertainty difference due between to the the difference actual model between of the the uncertainty actual model coeffi ofcient the anduncertainty the simulated coefficient mathematical and the model.simulated Intrinsic mathematical uncertainty model. is an Intrinsic uncertainty uncertainty in the structural is an uncertainty material, geometric in the structural factors expressing material, thegeometric shape of factors structures, expressing and applied the shape loads of [10structur,11]. Thesees, and uncertainties applied loads are [10,11]. generally These considered, uncertainties both inare the generally actual behavior considered, of the both structure in the andactual in thebehavior reliability of the analysis. structure Among and in them, the thereliability stochastic analysis. finite elementAmong methodthem, the (SFEM) stochastic is mainly finite focused element on the method intrinsic (SFEM) uncertainty is mainly with the focused greatest influence.on the intrinsic SFEM isuncertainty a combination with ofthe a greatest stochastic influence. method SFEM and FEM. is a combination The purpose of of a SFEM stochastic is to estimatemethod and the uncertainFEM. The responsepurpose of variation SFEM is of to the estimate structure the with uncertain respect response to the spatial variation and temporalof the structure randomness with respect of the factors to the inspatial the structural and temporal system randomness [12,13]. SFEM of the is divided factors in into the statistical structural and system non-statistical [12,13]. SFEM methods is divided in terms into of analyticalstatistical methodology.and non-statistical methods in terms of analytical methodology. ThereThere areare manymany analyticalanalytical methodsmethods suchsuch asas thethe K-LK-L ExpansionExpansion [[14],14], PolynomialPolynomial ChaosChaos ExpansionExpansion [[15],15], perturbationperturbation methodsmethods [[16],16], andand thethe weightedweighted integrationintegration methodmethod thatthat isis basedbased onon non-statisticalnon-statistical methodsmethods [ 17[17–22].–22]. However, However, the the non-statistical non-statistical method method is mainly is mainly based based on the on first-order the first- expansionorder expansion or the second-orderor the second-order series expansionseries expansio for then for main the variablesmain variables and is and applicable is applicable only when only thewhen coe thefficient coefficient of variation of variation (COV) (COV) of the of stochastic the stochastic fields fields is low is [low23]. [23]. In fact, In fact, when when COV COV (=σ /(=µσ/μ) of) theof the stochastic stochastic field field is large,is large, it isit is accurate accurate and and shows shows a a significant significant di differencefference from fromthe the Monte-CarloMonte-Carlo simulationsimulation (MCS).(MCS). TheThe MCS,MCS, whichwhich isis aa representative statisticalstatistical method,method, hashas thethe advantageadvantage ofof providingproviding aa solutionsolution toto mostmost stochasticstochastic problems.problems. However, toto obtainobtain a MCS with high eefficiencyfficiency and accuracy, aa properproper algorithmalgorithm and considerable time time is is also also required required for for analysis. analysis. This This study study focuses focuses on onestimating estimating the thespatial spatial randomness randomness of materials of materials through through observation observation (partial (partial elastic elastic modulus) modulus) and and overcoming overcoming the thelimitations limitations of the of theexisting existing statistical statistical methods. methods. For Forthisthis purpose, purpose, a method a method to numerically to numerically define define the thespatial spatial randomness randomness of ofthe the material’s material’s elastic elastic modu moduluslus was was proposed, proposed, and and the the obtained obtained results werewere demonstrateddemonstrated usingusing factorsfactors suchsuch asas responseresponse variabilityvariability andand responseresponse variance.variance.
2. Disadvantage of Current Statistical Methods 2. Disadvantage of Current Statistical Methods 2.1. Statistical Methods without Considering Spatial Randomness 2.1. Statistical Methods without Considering Spatial Randomness In the current statistical method, the target random variable, which is the elastic modulus, is In the current statistical method, the target random variable, which is the elastic modulus, is assumed to be a normal distribution (N(µ, σ)). In this case, a random process satisfying a specific normal assumed to be a normal distribution (N(μ, σ)). In this case, a random process satisfying a specific normal distribution is generated as a pseudo time process (here, the random variable satisfies rt = r(t), t (0, T) distribution is generated as a pseudo time process (here, the random variable satisfies 𝑟 =𝑟∈(𝑡),𝑡 ∈ and an analysis is performed for generated N constant to satisfy the normal distribution (N (µ, σ)). (0, 𝑇) and an analysis is performed for generated N constant to satisfy the normal distribution (N(μ, σ)). However, as shown in Figure1, the factor of the actual material’s elastic modulus does not have However, as shown in Figure 1, the factor of the actual material’s elastic modulus does not have the same value according to the position vector. Therefore, the current method of using a constant field the same value according to the position vector. Therefore, the current method of using a constant field is different from the actual material, and a random sample of the statistical method should satisfy both is different from the actual material, and a random sample of the statistical method should satisfy both functions simultaneously. functions simultaneously.
Figure 1. DifferentDifferent elastic modulus in didifferentfferent positionspositions duedue toto materialmaterial heterogeneity.heterogeneity.
2.2. Statistical Methods Considering Spatial Randomness
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The Crystalsspatial2020 randomness, 10, 512 distribution of the uncertainty factor is represented by 𝑓3 of(𝑥 17) in the stochastic field function, stochastic field 𝑓(𝑥 ), and at any point 𝑥 ∈Ω , belonging to the structural Ω domain 2.2. , Statistical which must Methods satisfy Considering the probability Spatial Randomness density function (PDF). At the same time, it must also satisfy the spectral density function (SDF) (or autocorrelation function) representing the aspect existing The spatial randomness distribution of the uncertainty factor is represented by f (x) in the stochastic in the structural domain [24]. Even in the case of a stochastic field with the same mean and standard field function, stochastic field f (xi), and at any point xi Ωstr, belonging to the structural domain ∈ 𝜇 =𝜇 deviation,Ω thestr, whichactual must shape satisfy of the probabilitystochastic density field can function exist (PDF). in various At the sameforms time, ( it must ). also This satisfy variance is due to thethe characteristics spectral density of function the stochastic (SDF) (or autocorrelationfield based on function) the ensemble representing concept, the aspect which existing is a statistical in characteristicthe structural in a direction domain orth [24].ogonal Even into the the case stochastic of a stochastic plane. field with the same mean and standard deviation, the actual shape of the stochastic field can exist in various forms (µk = µk). This variance is In general, the distribution characteristic of the stochastic field is determinedi j by the correlation due to the characteristics of the stochastic field based on the ensemble concept, which is a statistical 𝑓(𝑥; 𝑑) =0 𝑓(𝑥; 𝑑) =∞ distance. characteristicThe stochastic in a directionfield can orthogonal have two to extreme the stochastic conditions: plane. and . In the former case, Inthe general, stochastic the distribution field appears characteristic in the of form the stochastic of a white field is noise determined field by and the correlationcontains all the theoreticallydistance. possible The spectra. stochastic In field the canlatter, have the two stocha extremestic conditions:field is a constantf (x; d = 0field) and withf (x; done= value,). In which ∞ is called thethe random former case, variable the stochastic state [25]. field For appears example, in the d is form infinite of a when white noisespatial field randomness and contains is allnot taken into consideration.the theoretically possible spectra. In the latter, the stochastic field is a constant field with one value, which is called the random variable state [25]. For example, d is infinite when spatial randomness is The notdistribution taken into consideration.of the stochastic field using the correlation distance can simulate (imitate) the spatial randomnessThe distribution of the material. of the stochastic However, field it is using difficult the correlation to determine distance the canspatial simulate randomness (imitate) of the actual materialthe spatial with randomness this approach. of the material. Various However, methods it is are diffi availablecult to determine to compute the spatial the randomness elastic modulus of of concrete, thebut actual the spatial material randomness with this approach. is difficult Various to methods identify. are To available investigate to compute the spatial the elastic randomness modulus of a target objective,of concrete, the butstochastic the spatial field randomness with a iscorrelat difficultion to identify.distance To (d) investigate ranging the from spatial zero randomness to infinity of needs a target objective, the stochastic field with a correlation distance (d) ranging from zero to infinity needs to be generated.to be generated. In addition, In addition, this spatial this spatial characteristic characteristic can can be examined be examined by by the the COV COV trendtrend obtained obtained from the Monte-Carlofrom the simulation, Monte-Carlo simulation,as shown asin shownFigure in 2 Figure [26,27].2[26 ,27].
Figure 2. The schematic of the response variability according to the correlation distance (d). Figure 2. The schematic of the response variability according to the correlation distance (d). However, since the actual correlation distance (d) of the target elastic modulus cannot be known, However,the exact since result the is actual unknown correlation (which is notdistance a meaningful (d) of the result). target Therefore, elastic modulus we aimed tocannot overcome be known, the problem (disadvantages) of the existing statistical methods. the exact result is unknown (which is not a meaningful result). Therefore, we aimed to overcome the problem (disadvantages)2.3. Objective and Methods of the existing statistical methods. In this study, we proposed a method of inversely estimating the correlation distance using 2.3. Objectivethe observed and Methods values and the distance between the observation points. In particular, the population can be predicted using some samples. Then, the obtained real correlation distance d1 (from the population) In this study, we proposed a method of inversely estimating the correlation distance using the observed values and the distance between the observation points. In particular, the population can be predicted using some samples. Then, the obtained real correlation distance d1 (from the population) is compared with the d2 (predicted from samples). Once the correlation distance of the actual material is found using the proposed method, the exact behavior of structures by the actual material can be described instead of using the tendency of COV. The process is described in the following three steps. First, generate the stochastic field using the spectral representation method [28,29]. The purpose of creating a sample by applying the algorithm is because the exact d value
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Crystals 2020is compared, 10, x FOR with PEER the REVIEW d2 (predicted from samples). Once the correlation distance of the actual material 4 of 16 is found using the proposed method, the exact behavior of structures by the actual material can be correspondingdescribed to instead the samples of using the through tendency the of COV.algorithm The process is known. is described If the in theproposed following inverse three steps. estimation methodFirst, correctly generate traces the stochastic d, it starts field from using the the assump spectraltion representation that some method data of [28 the,29]. actual The purpose structure of can be creating a sample by applying the algorithm is because the exact d value corresponding to the samples used to obtain an actual d corresponding to the structure. Second, we assumed values at some through the algorithm is known. If the proposed inverse estimation method correctly traces d, it locationsstarts to be from used the assumptionas sample thatdata, some as datashown of the in actualFigure structure 5, and can compared be used to the obtain estimated an actual correlation d distancecorresponding (using only to some the structure. observation) Second, and we assumed the correla valuestion at some distance locations of tothe be stochastic used as sample field. data, Third, the probabilityas shown characteristics in Figure3, and are compared obtained the using estimated the Ba correlationyesian method. distance (using This onlyis a method some observation) of obtaining new probabilityand thecharacteristics correlation distance (posterior) of the stochastic by updating field. Third,the prior the probability probability characteristics characteristics are obtained based on new using the Bayesian method. This is a method of obtaining new probability characteristics (posterior) test data [30]. by updating the prior probability characteristics based on new test data [30].
Figure 3. Observation in the stochastic field with spatial randomness. Figure 3. Observation in the stochastic field with spatial randomness. 3. Investigation Procedures and Their Examples
3. Investigation3.1. Consideration Procedures Method ofand Spatial Their Randomness Examples of Material’s Elastic Modulus
For every real-valued 2D-1V homogeneous stochastic field f0(x1, x2) with the mean value equal 3.1. Consideration Method of Spatial Randomness( ) of Material’s Elastic Modulus ( ) to zero and a bi-quadrant SDF as G f0 f0 κ1, κ2 , two mutually orthogonal real processes u κ1, κ2 and v(κ1, κ2) with orthogonal increments du(κ1, κ2) and dv(κ1, κ2) can be assigned so that: For every real-valued 2D-1V homogeneous stochastic field 𝑓 (𝑥 ,𝑥 ) with the mean value equal Z ∞ ( ) ( ) to zero and a bi-quadrant SDF as 𝐺 𝜅 ,𝜅 , two mutually orthogonal real processes 𝑢 𝜅 ,𝜅 and f0(x1, x2) = [cos(κ1x1 + κ2x2)du(κ1, κ2) + sin(κ1x1 + κ2x2)dv(κ1, κ2)] (1) 𝑣(𝜅 ,𝜅 ) with orthogonal increments−∞ 𝑑𝑢(𝜅 ,𝜅 ) and 𝑑𝑣(𝜅 ,𝜅 ) can be assigned so that: ∞ E(x, y) = E(1 + fi(x, y)) (2) 𝑓 (𝑥 ,𝑥 ) = 𝑐𝑜𝑠(𝜅 𝑥 +𝜅 𝑥 ) 𝑑𝑢(𝜅 ,𝜅 ) +𝑠𝑖𝑛(𝜅 𝑥 +𝜅 𝑥 ) 𝑑𝑣(𝜅 ,𝜅 ) ( ! ) (1) ∞ 2 ∆ξ1 ∆ξ2 R(ξ1, ξ2) = σ0 exp (3) − d1 − d2 In the generation of homogeneous𝐸(𝑥, random 𝑦) fields, =𝐸(1+ the𝑓 (𝑥, spectral 𝑦)) representation method [31–33] (2) can be employed, which takes advantage of fast Fourier transform. In the random fields of elastic modulus, the stiffness of the elastic modulus is assumed to beΔ𝜉 homogeneous Δ𝜉 Gaussian and represented 𝑅(𝜉 ,𝜉 ) =𝜎 𝑒𝑥𝑝 − − (3) by Equation (2), where E is the mean value of the elastic𝑑 modulus 𝑑 and fi(x, y) is a homogeneous random field. The auto-correlation functions for the respective random field fi(x, y) are assumed by In theEquation generation (3) [34]. of In homogeneous the spectral representation random scheme,fields, the the numericalspectral representation generation of a homogeneous method [31–33] can be employed, which takes advantage of fast Fourier transform. In the random fields of elastic modulus, the stiffness of the elastic modulus is assumed to be homogeneous Gaussian and represented by Equation (2), where 𝐸 is the mean value of the elastic modulus and 𝑓 (𝑥,) 𝑦 is a homogeneous random field. The auto-correlation functions for the respective random field 𝑓 (𝑥,) 𝑦 are assumed by Equation (3) [34]. In the spectral representation scheme, the numerical generation of a homogeneous uni-variate i-th random sample 𝑓 (𝑥,) 𝑦 with a zero mean in two dimensions can be generated via the summation of the cosine functions, as shown in Equation (4) [35,36].