Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
14
3
2016
PREFACE: UNCERTAINTY MODELING AND PROPAGATION TECHNIQUES IN ENGINEERING MECHANICS: AMULTISCALE PERSPECTIVE
v
George
Deodatis
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY
Ioannis A.
Kougioumtzoglou
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY
Pol
Spanos
Rice University
This special issue of the International Journal of Multiscale Computational Engineering (IJMCE) comprises seven
papers and aims to present recent advances and emerging cross-disciplinary approaches in the broad field of uncertainty
modeling and propagation in engineering mechanics with a focus on multiscale techniques.
UNCERTAINTY QUANTIFICATION OF MANUFACTURING PROCESS EFFECTS ON MACROSCALE MATERIAL PROPERTIES
191-213
Guowei
Cai
Vanderbilt University, Nashville, TN
Sankaran
Mahadevan
Civil and Environmental Engineering Department, Vanderbilt University, Nashville, Tennessee 37235, USA
This paper presents a methodology to propagate the uncertainties in the manufacturing process parameters to bulk material properties through multiscale modeling. Randomness of material initial condition and uncertainties in the manufacturing process lead to variability in the microstructure, which in turn leads to variability in the macrolevel properties of the material. In this paper, 2D dual-phase polycrystalline microstructure is simulated based on the initial condition of the grain cores and the manufacturing environment, instead of Voronoi tessellation, which assumes equal grain growth velocities for different phases and therefore is unable to link variability in grain growth velocity to the manufacturing process variability. Then a homogenization method is applied to predict macrolevel properties. The cooling schedule of a dual-phase alloy is used to illustrate the methodology, and Young's modulus is the prediction quantity of interest. Even with a given cooling schedule, spatial variation of temperature affects the microstructure and properties; this variability is also incorporated in this paper through a random field representation. The uncertainty quantification methodology uses Gaussian process surrogate modeling for computational efficiency. The relative contributions of both aleatory and epistemic sources to the overall bulk property uncertainty are quantified using an innovative global sensitivity analysis approach; this provides guidance for manufacturing process control in order to meet the desired uncertainty bounds in the bulk property estimates.
STOCHASTIC DYNAMIC RESPONSE ANALYSIS OF NONLINEAR STRUCTURES WITH GENERAL NONUNIFORM RANDOM PARAMETERS BY MINIMIZING GL2-DISCREPANCY
215-235
Jianbing
Chen
State Key Laboratory of Disaster Reduction in Civil Engineering and School of Civil Engineering, Tongji University, Shanghai, PRC
Pengyan
Song
College of Civil Engineering and Architecture, Hebei University, Baoding, Hebei, PRC
Xiaodan
Ren
School of Civil Engineering, Tongji University, Shanghai, PRC
The impact of randomness of structural parameters on structural responses and performance is of paramount importance. In engineering practice, the distributions of most random parameters are nonuniform and non-Gaussian. The probability density evolution method is capable in these cases of capturing the probability density functions of the responses. To optimally select the representative point set, the L2-discrepancy in cubature formulae, which is only applicable to uniform distributions with equal weights, is generalized to: (i) consider general nonuniform, non-Gaussian distributions; and (ii) involve the impact of the assigned probabilities as unequal weights. The extended Koksma-Hlawka inequality is proved rigorously, and the explicit expression for the generalized L2-discrepancy (GL2-discrepancy) is derived. A point selection strategy by minimizing the GL2-discrepancy is proposed. In particular, a two-step approach is suggested, and the existence and uniqueness of optimal assigned probabilities are proved. Numerical examples are illustrated, showing the fair accuracy and efficiency of the proposed method. Particularly, in obvious contrast to most existing random vibration analyses of nonlinear structures where the component-structure two-level models are employed, stochastic response of a structure with a refined model incorporating the stochastic damage constitutive law of concrete material is implemented. Problems to be further studied are discussed.
MATERIAL RESPONSE AT MICRO-, MULTI-, AND MACROSCALES
237-254
Mircea
Grigoriu
Cornell University, Ithaca, NY
Three examples are used to illustrate relationships between material responses at different scales, practical use of these relationships, and differences between microscale and multiscale solutions. In the first example, force-displacement constitutive laws are developed for rods with random, linear/nonlinear, small-scale stress-strain relationships. It is shown that the continuum mechanics constitutive law may or may not match, on average, the corresponding microscale-based law depending on material properties at small scale. In the second example, it is assumed that a spatial correlation parameter of the microscale conductivity field is unknown. Because this parameter cannot be measured directly, measurements of the apparent conductivity on laboratory-scale specimens are used to identify this microscale model parameter. In the third example, one-dimensional conductivity problems are used to quantify differences between microscale and multiscale solutions that relate solely to approximate representations of the random field describing the microstructure conductivity.
NONLINEAR SYSTEM RESPONSE EVOLUTIONARY POWER SPECTRAL DENSITY DETERMINATION VIA A HARMONIC WAVELETS BASED GALERKIN TECHNIQUE
255-272
Fan
Kong
School of Civil Engineering and Achitecture, Wuhan University of Technology, Wuhan, China
Ioannis A.
Kougioumtzoglou
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY
Pol
Spanos
Rice University
Shujin
Li
School of Civil Engineering and Achitecture, Wuhan University of Technology, Wuhan, China
A generalized harmonic wavelets (GHWs-) based statistical linearization technique is developed for determining the response evolutionary power spectral density (PSD) of nonlinear time-varying oscillators. Specifically, a recently derived GHWs-based input-output relationship for linear systems is utilized that circumvents the assumption/restriction of local stationarity inherent in earlier treatments of the problem. Next, this excitation-response relationship is extended via statistical linearization to account for nonlinear systems as well. This involves the concept of determining optimal equivalent linear elements corresponding to specific time and frequency bands, whereas the response evolutionary PSD is determined via an iterative scheme. Pertinent numerical examples and Monte Carlo simulation data are included as well for demonstrating the reliability of the technique.
PERTURBATION-BASED SURROGATE MODELS FOR DYNAMIC FAILURE OF BRITTLE MATERIALS IN A MULTISCALE AND PROBABILISTIC CONTEXT
273-290
Junwei
Liu
Department of Civil Engineering, Johns Hopkins University, Baltimore, MD
Lori
Graham-Brady
Department of Civil Engineering, Johns Hopkins University, Baltimore, MD
Localization of failure in many materials is associated with the heterogeneity in the material microstructure. Multiscale models often address this heterogeneity by passing field variables back and forth between a macroscale model and subscale analyses at each integration point. Although this technique is often effective, it can be extremely costly to perform distinct microscale analyses for every integration point in the domain. The proposed work uses a perturbation-based approach, conceptually similar to in situ adaptive tabulation, which provides a straightforward surrogate model that can be orders of magnitude more efficient than the microscale model. The approach is demonstrated specifically for models of dynamic brittle failure, in which crack populations are tracked from one load step to the next. Furthermore, following an approach similar to that used in perturbation-based stochastic finite elements, this technique streamlines the process of probabilistic characterization of the instantaneous stress and the uniaxial compressive strength. Numerical examples show that the approach is accurate and highly efficient when considering random perturbations in both the underlying flaw population and the strain history in these brittle materials.
MODELING HETEROGENEITY IN NETWORKS USING POLYNOMIAL CHAOS
291-302
Karthikeyan
Rajendran
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ
Andreas C.
Tsoumanis
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ
Constantinos I.
Siettos
School of Applied Mathematics and Physical Sciences, NTUA, Athens, Greece
Carlo R.
Laing
Institute for Natural and Mathematical Sciences, Massey University, Auckland, New Zealand
Ioannis G.
Kevrekidis
Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ; Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ
Using the dynamics of information propagation on a network as our illustrative example, we present and discuss a systematic approach to quantifying heterogeneity and its propagation that borrows established tools from uncertainty quantification, specifically, the use of polynomial chaos. The crucial assumption underlying this mathematical and computational "technology transfer" is that the evolving states of the nodes in a network quickly become correlated with the corresponding node identities: features of the nodes imparted by the network structure (e.g., the node degree, the node clustering coefficient). The node dynamics thus depend on heterogeneous (rather than uncertain) parameters, whose distribution over the network results from the network structure. Knowing these distributions allows one to obtain an efficient coarse-grained representation of the network state in terms of the expansion coefficients in suitable orthogonal polynomials. This representation is closely related to mathematical/computational tools for uncertainty quantification (the polynomial chaos approach and its associated numerical techniques). The polynomial chaos coefficients provide a set of good collective variables for the observation of dynamics on a network and, subsequently, for the implementation of reduced dynamic models of it. We demonstrate this idea by performing coarse-grained computations of the nonlinear dynamics of information propagation on our illustrative network model using the Equation-Free approach.
MULTISCALE STOCHASTIC STRUCTURAL ANALYSIS TOWARD RELIABILITY ASSESSMENT FOR LARGE COMPLEX REINFORCED CONCRETE STRUCTURES
303-321
Hao
Zhou
School of Civil Engineering, Tongji University, Shanghai, China
Jie
Li
State Key Laboratory of Disaster Reduction in Civil Engineering, and School of Civil Engineering, Tongji University, Shanghai 200092, China
Xiaodan
Ren
School of Civil Engineering, Tongji University, Shanghai, PRC
This paper focuses on a multiscale methodology to model and analyze large high-rise buildings subject to disastrous dynamic excitations. Starting from the material randomness and the nonlinear behavior of concrete, a mesoscopic stochastic damage model (SDM) is recommended in which the fracture strain of concrete at the microlevel is modeled as a Gaussian random field. By integrating the SDM and the refined structural elements into the finite element analysis, the structural dynamic responses can be comprehensively investigated using the explicit integration algorithm to solve the dynamic equations. To represent the probability information of structural responses, the probability density evolution method (PDEM) is employed. Also, the randomness propagation across different levels can be readily addressed via PDEM. The absorbing boundary condition corresponding to the failure criterion of structures is introduced to assess the dynamic reliability. As a case study, the stochastic dynamic analysis and the reliability assessment are illustratively carried out in terms of a prototype reinforced concrete structure. The simulated results show that the randomness of concrete materials plays a critical role in the stochastic response and dynamic reliability of reinforced concrete structures.