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 Crystals 2020, 10, 512; doi:10.3390/cryst10060512 www.mdpi.com/journal/crystals CrystalsCrystals2020 2020,,10 10,, 512x FOR PEER REVIEW 22 ofof 1716 deterministic FEA is repeatedly performed on the generated samples to obtain the characteristic thebehavior structure of the [6, 7structure]. 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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