Download or read book Reduced Order Multiscale Modeling of Nonlinear Processes in Heterogeneous Materials written by Satyaki Bhattacharjee and published by . This book was released on 2017 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Multiscale Modeling of Heterogeneous Structures written by Jurica Sorić and published by Springer. This book was released on 2017-11-30 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of multiscale approaches and homogenization procedures as well as damage evaluation and crack initiation, and addresses recent advances in the analysis and discretization of heterogeneous materials. It also highlights the state of the art in this research area with respect to different computational methods, software development and applications to engineering structures. The first part focuses on defects in composite materials including their numerical and experimental investigations; elastic as well as elastoplastic constitutive models are considered, where the modeling has been performed at macro- and micro levels. The second part is devoted to novel computational schemes applied on different scales and discusses the validation of numerical results. The third part discusses gradient enhanced modeling, in particular quasi-brittle and ductile damage, using the gradient enhanced approach. The final part addresses thermoplasticity, solid-liquid mixtures and ferroelectric models. The contents are based on the international workshop “Multiscale Modeling of Heterogeneous Structures” (MUMO 2016), held in Dubrovnik, Croatia in September 2016.
Download or read book Towards Optimal Design of Multiscale Nonlinear Structures written by Liang Xia and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: High-performance heterogeneous materials have been increasingly used nowadays for their advantageous overall characteristics resulting in superior structural mechanical performance. The pronounced heterogeneities of materials have significant impact on the structural behavior that one needs to account for both material microscopic heterogeneities and constituent behaviors to achieve reliable structural designs. Meanwhile, the fast progress of material science and the latest development of 3D printing techniques make it possible to generate more innovative, lightweight, and structurally efficient designs through controlling the composition and the microstructure of material at the microscopic scale. In this thesis, we have made first attempts towards topology optimization design of multiscale nonlinear structures, including design of highly heterogeneous structures, material microstructural design, and simultaneous design of structure and materials. We have primarily developed a multiscale design framework, constituted of two key ingredients : multiscale modeling for structural performance simulation and topology optimization forstructural design. With regard to the first ingredient, we employ the first-order computational homogenization method FE2 to bridge structural and material scales. With regard to the second ingredient, we apply the method Bi-directional Evolutionary Structural Optimization (BESO) to perform topology optimization. In contrast to the conventional nonlinear design of homogeneous structures, this design framework provides an automatic design tool for nonlinear highly heterogeneous structures of which the underlying material model is governed directly by the realistic microstructural geometry and the microscopic constitutive laws. Note that the FE2 method is extremely expensive in terms of computing time and storage requirement. The dilemma of heavy computational burden is even more pronounced when it comes to topology optimization : not only is it required to solve the time-consuming multiscale problem once, but for many different realizations of the structural topology. Meanwhile we note that the optimization process requires multiple design loops involving similar or even repeated computations at the microscopic scale. For these reasons, we introduce to the design framework a third ingredient : reduced-order modeling (ROM). We develop an adaptive surrogate model using snapshot Proper Orthogonal Decomposition (POD) and Diffuse Approximation to substitute the microscopic solutions. The surrogate model is initially built by the first design iteration and updated adaptively in the subsequent design iterations. This surrogate model has shown promising performance in terms of reducing computing cost and modeling accuracy when applied to the design framework for nonlinear elastic cases. As for more severe material nonlinearity, we employ directly an established method potential based Reduced Basis Model Order Reduction (pRBMOR). The key idea of pRBMOR is to approximate the internal variables of the dissipative material by a precomputed reduced basis computed from snapshot POD. To drastically accelerate the computing procedure, pRBMOR has been implemented by parallelization on modern Graphics Processing Units (GPUs). The implementation of pRBMOR with GPU acceleration enables us to realize the design of multiscale elastoviscoplastic structures using the previously developed design framework inrealistic computing time and with affordable memory requirement. We have so far assumed a fixed material microstructure at the microscopic scale. The remaining part of the thesis is dedicated to simultaneous design of both macroscopic structure and microscopic materials. By the previously established multiscale design framework, we have topology variables and volume constraints defined at both scales.
Download or read book Multiscale Modeling in Solid Mechanics written by Ugo Galvanetto and published by Imperial College Press. This book was released on 2010 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique volume presents the state of the art in the field of multiscale modeling in solid mechanics, with particular emphasis on computational approaches. For the first time, contributions from both leading experts in the field and younger promising researchers are combined to give a comprehensive description of the recently proposed techniques and the engineering problems tackled using these techniques. The book begins with a detailed introduction to the theories on which different multiscale approaches are based, with regards to linear Homogenisation as well as various nonlinear approaches. It then presents advanced applications of multiscale approaches applied to nonlinear mechanical problems. Finally, the novel topic of materials with self-similar structure is discussed. Sample Chapter(s). Chapter 1: Computational Homogenisation for Non-Linear Heterogeneous Solids (808 KB). Contents: Computational Homogenisation for Non-Linear Heterogeneous Solids (V G Kouznetsova et al.); Two-Scale Asymptotic Homogenisation-Based Finite Element Analysis of Composite Materials (Q-Z Xiao & B L Karihaloo); Multi-Scale Boundary Element Modelling of Material Degradation and Fracture (G K Sfantos & M H Aliabadi); Non-Uniform Transformation Field Analysis: A Reduced Model for Multiscale Non-Linear Problems in Solid Mechanics (J-C Michel & P Suquet); Multiscale Approach for the Thermomechanical Analysis of Hierarchical Structures (M J Lefik et al.); Recent Advances in Masonry Modelling: Micro-Modelling and Homogenisation (P B Louren o); Mechanics of Materials with Self-Similar Hierarchical Microstructure (R C Picu & M A Soare). Readership: Researchers and academics in the field of heterogeneous materials and mechanical engineering; professionals in aeronautical engineering and materials science.
Download or read book Fundamentals of Multiscale Modeling of Structural Materials written by Wenjie Xia and published by Elsevier. This book was released on 2022-11-26 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of Multiscale Modeling of Structural Materials provides a robust introduction to the computational tools, underlying theory, practical applications, and governing physical phenomena necessary to simulate and understand a wide-range of structural materials at multiple time and length scales. The book offers practical guidelines for modeling common structural materials with well-established techniques, outlining detailed modeling approaches for calculating and analyzing mechanical, thermal and transport properties of various structural materials such as metals, cement/concrete, polymers, composites, wood, thin films, and more.Computational approaches based on artificial intelligence and machine learning methods as complementary tools to the physics-based multiscale techniques are discussed as are modeling techniques for additively manufactured structural materials. Special attention is paid to how these methods can be used to develop the next generation of sustainable, resilient and environmentally-friendly structural materials, with a specific emphasis on bridging the atomistic and continuum modeling scales for these materials. Synthesizes the latest cutting-edge computational multiscale modeling techniques for an array of structural materials Emphasizes the foundations of the field and offers practical guidelines for modeling material systems with well-established techniques Covers methods for calculating and analyzing mechanical, thermal and transport properties of various structural materials such as metals, cement/concrete, polymers, composites, wood, and more Highlights underlying theory, emerging areas, future directions and various applications of the modeling methods covered Discusses the integration of multiscale modeling and artificial intelligence
Download or read book Multiscale Modeling of Complex Materials written by Tomasz Sadowski and published by Springer. This book was released on 2014-10-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this volume deal with materials science, theoretical mechanics and experimental and computational techniques at multiple scales, providing a sound base and a framework for many applications which are hitherto treated in a phenomenological sense. The basic principles are formulated of multiscale modeling strategies towards modern complex multiphase materials subjected to various types of mechanical, thermal loadings and environmental effects. The focus is on problems where mechanics is highly coupled with other concurrent physical phenomena. Attention is also focused on the historical origins of multiscale modeling and foundations of continuum mechanics currently adopted to model non-classical continua with substructure, for which internal length scales play a crucial role.
Download or read book Uncertainty Quantification in Multiscale Materials Modeling written by Yan Wang and published by Woodhead Publishing Limited. This book was released on 2020-03-12 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty Quantification in Multiscale Materials Modeling provides a complete overview of uncertainty quantification (UQ) in computational materials science. It provides practical tools and methods along with examples of their application to problems in materials modeling. UQ methods are applied to various multiscale models ranging from the nanoscale to macroscale. This book presents a thorough synthesis of the state-of-the-art in UQ methods for materials modeling, including Bayesian inference, surrogate modeling, random fields, interval analysis, and sensitivity analysis, providing insight into the unique characteristics of models framed at each scale, as well as common issues in modeling across scales.
Download or read book Stochastic Models for Nonlinear Transport in Multiphase and Multiscale Heterogeneous Media written by Farzaneh Rajabi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Elucidating multiscale, multiphase and multiphysics phenomena of flow and transport processes in porous media is the cornerstone of numerous environmental and engineering applications. Several factors including spatial and temporal heterogeneity on a continuity of scales, the strong coupling of processes at such different scales at least at a localized region within the domain, combined with the nonlinearity of processes calls for a new modeling paradigm called multiscale models, which are able to properly address all such issues while presenting an accurate descriptive model for processes occurring at field scale applications. Furthermore, the typical temporal resolution used in modern simulations significantly exceeds characteristic time scales at which the system is driven and a solution is sought. This is especially so when systems are simulated over time scales that are much longer than the typical temporal scales of forcing factors. In addition to spatial and temporal heterogeneity, mixing and spreading of contaminants in the subsurface is remarkably influenced by oscillatory forcing factors. While the pore-scale models are able to handle the experimentally-observed phenomena, they are not always the best choice due to the high computational burden. Although handling across-scale coupling in environments with several simultaneous physical mechanisms such as advection, diffusion, reaction, and fluctuating boundary forcing factors complicates the theoretical and numerical modeling capabilities at high resolutions, multiscale models come to rescue. To this end, we investigate the impact of space-time upscaling on reactive transport in porous media driven by time-dependent boundary conditions whose characteristic time scale is much smaller than that at which transport is studied or observed at the macroscopic level. We first introduce the concept of spatiotemporal upscaling in the context of homogenization by multiple-scale expansions, and demonstrate the impact of time-dependent forcings and boundary conditions on macroscopic reactive transport. Proposing such a framework, we scrutinize the behavior of porous media for ``quasisteady stage time'' (thousands of years), where there is an interplay between signal frequency and the three physical underlying mechanisms; advection, molecular diffusion and heterogeneous reaction. To this end, we demonstrate that the transient forcing factors augment the solute mixing as they are combined with diffusion at the pore-scale. We then derive the macroscopic equation as well as the corresponding applicability criteria based on the order of magnitude of the dimensionless Peclet and Damkohler numbers. Also, we demonstrate that the dynamics at the continuum scale is strongly influenced by the interplay between signal frequency at the boundary and transport processes at the pore level. We validate such a framework for reactive transport in a planar fracture in which the single-component solute particle is undergoing nonlinear first-order heterogeneous reaction at the solid-liquid interface, while the medium is episodically influenced by time-dependent boundary conditions at the inlet. We also present the alternative effective transport model at a much lower cost, albeit at the regions where the corresponding applicability criteria are satisfied. We perform direct numerical simulations to study several test cases with different controlling parameters i.e. Peclet and Damkohler numbers and the space/time scale separation parameters; the ratio of characteristic transversal and longitudinal lengths $\varepsilon$ and $\omega$; the ratio of period of time-fluctuating boundary conditions to the observation time scale. A rigorous justification of the effective transport model for the given applicability conditions is demonstrated, essentially by comparing the local vertically averaged microscopic simulations with their corresponding macroscopic counterparts. Moreover, as a special case, we employ a singular perturbation technique to look at the effective model for vertical mixing through a narrow and long two-dimensional pore. We obtain explicit expressions for dispersion tensor as well as the other effective coefficients in the coarse-scale homogenized equation. Our analysis manifests robustness of the sufficient and necessary applicability constraints which validate the upscaled model as a solid replacement of the pore-scale one within the accuracy prescribed by homogenization theory. While a deterministic model is sufficiently robust for a plethora of subsurface applications, a more realistic setting is often required when dealing with other scopes of engineering applications, e.g. reservoir engineering and enhanced oil recovery. Rigorous modeling of these systems calls for sophisticated strategies for uncertainty quantification and stochastic treatment of the system under study. Such an uncertainty is inherent to, and critical for any physical modeling, essentially due to the incomplete knowledge of state of the world, noisy observations, and limitations in systematically recasting physical processes in a suitable mathematical framework. To this end, accurate predictions of outputs (e.g. saturation fields) from reservoir simulations guarantee precise oil recovery forecasts. These quantitative predictions rely on the quality of the input measurements/data, such as the reservoir permeability and porosity fields as well as forcings, such as initial and boundary conditions. However, the available information about a particular geologic formation, e.g. from well logs and seismic data of an outcrop, is usually sparse and inaccurate compared to the size of the natural system and the complexity arising from multiscale heterogeneity of the underlying system. Eventually, the uncertainty in the flow prediction can have a huge impact on the oil recovery. Consequently, we also develop a probabilistic approach to map the parametric uncertainty to the output state uncertainty in first-order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two-phase transport (Buckley-Leverett displacement) in heterogeneous porous media in the presence of a stochastic velocity field, where the uncertainty in the velocity field can arise from the incomplete description of either porosity field, injection flux, or both. Such uncertainty leads to the spatiotemporal uncertainty in the outputs of the problem. Given information about the spatial/temporal statistics of the correlated heterogeneity, we leverage method of distributions (MD) to derive deterministic equations that govern the evolution of single-point CDF of saturation in the form of linear hyperbolic conservation laws. We first derive the semi-analytical solution of the raw CDF of saturation at a given point, for the cases in which two shocks are present due to the gravitational forces. Then, we describe development of the partial differential equation that governs the evolution of the raw CDF of saturation, subject to uniquely specified boundary conditions in the phase space, wherein no closure approximations are required. Hereby, we give routes to circumventing the computational cost of Monte Carlo scheme while obtaining the full statistical description of saturation. This derivation is followed by conducting a set of numerical experiments for horizontal reservoirs and more complex scenarios in which gravity segregation takes place. We then compare the CDFs as well as the first two moments of saturation computed with the method of distributions, against those obtained using the statistical moment equations (SME) approach and kernel density estimation post-processing of exhaustive high-resolution Monte Carlo simulations (MCS). This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, i.e. standard deviation and correlation length of the underlying random fields, while the corresponding low-order statistical moment equations significantly deviate from Monte Carlo results, unless for very small values of standard deviation and correlation length.
Download or read book Materials with Internal Structure written by Patrizia Trovalusci and published by Springer. This book was released on 2015-10-17 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a series of concise papers by researchers specialized in various fields of continuum and computational mechanics and of material science. The focus is on principles and strategies for multiscale modeling and simulation of complex heterogeneous materials, with periodic or random microstructure, subjected to various types of mechanical, thermal, chemical loadings and environmental effects. A wide overview of complex behavior of materials (plasticity, damage, fracture, growth, etc.) is provided. Among various approaches, attention is given to advanced non-classical continua modeling which, provided by constitutive characterization for the internal and external actions (in particular boundary conditions), is a very powerful frame for the gross mechanical description of complex material behaviors, able to circumvent the restrictions of classical coarse–graining multiscale approaches.
Download or read book From Multiscale Modeling to Meso Science written by Jinghai Li and published by Springer Science & Business Media. This book was released on 2013-03-22 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiscale modeling is becoming essential for accurate, rapid simulation in science and engineering. This book presents the results of three decades of research on multiscale modeling in process engineering from principles to application, and its generalization for different fields. This book considers the universality of meso-scale phenomena for the first time, and provides insight into the emerging discipline that unifies them, meso-science, as well as new perspectives for virtual process engineering. Multiscale modeling is applied in areas including: multiphase flow and fluid dynamics chemical, biochemical and process engineering mineral processing and metallurgical engineering energy and resources materials science and engineering Jinghai Li is Vice-President of the Chinese Academy of Sciences (CAS), a professor at the Institute of Process Engineering, CAS, and leader of the EMMS (Energy-minimizing multiscale) Group. Wei Ge, Wei Wang, Ning Yang and Junwu Wang are professors at the EMMS Group, part of the Institute of Process Engineering, CAS. Xinhua Liu, Limin Wang, Xianfeng He and Xiaowei Wang are associate professors at the EMMS Group, part of the Institute of Process Engineering, CAS. Mooson Kwauk is an emeritus director of the Institute of Process Engineering, CAS, and is an advisor to the EMMS Group.
Download or read book Multiscale Model Reduction written by Eric Chung and published by Springer Nature. This book was released on 2023-06-07 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.
Download or read book Heterogeneous Materials written by Muhammad Sahimi and published by Springer Science & Business Media. This book was released on 2006-05-31 with total page 650 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph describes and discusses the properties of heterogeneous materials, comparing two fundamental approaches to describing and predicting materials’ properties. This multidisciplinary book will appeal to applied physicists, materials scientists, chemical and mechanical engineers, chemists, and applied mathematicians.
Download or read book Multiscale Structural Topology Optimization written by Liang Xia and published by Elsevier. This book was released on 2016-04-27 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multiscale Structural Topology Optimization discusses the development of a multiscale design framework for topology optimization of multiscale nonlinear structures. With the intention to alleviate the heavy computational burden of the design framework, the authors present a POD-based adaptive surrogate model for the RVE solutions at the microscopic scale and make a step further towards the design of multiscale elastoviscoplastic structures. Various optimization methods for structural size, shape, and topology designs have been developed and widely employed in engineering applications. Topology optimization has been recognized as one of the most effective tools for least weight and performance design, especially in aeronautics and aerospace engineering. This book focuses on the simultaneous design of both macroscopic structure and microscopic materials. In this model, the material microstructures are optimized in response to the macroscopic solution, which results in the nonlinearity of the equilibrium problem of the interface of the two scales. The authors include a reduce database model from a set of numerical experiments in the space of effective strain. Presents the first attempts towards topology optimization design of nonlinear highly heterogeneous structures Helps with simultaneous design of the topologies of both macroscopic structure and microscopic materials Helps with development of computer codes for the designs of nonlinear structures and of materials with extreme constitutive properties Focuses on the simultaneous design of both macroscopic structure and microscopic materials Includes a reduce database model from a set of numerical experiments in the space of effective strain
Download or read book Multiscale Modeling and Analysis for Materials Simulation written by Weizhu Bao and published by World Scientific. This book was released on 2012 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Institute for Mathematical Sciences at the National University of Singapore hosted a two-month research program on "Mathematical Theory and Numerical Methods for Computational Materials Simulation and Design" from 1 July to 31 August 2009. As an important part of the program, tutorials and special lectures were given by leading experts in the fields for participating graduate students and junior researchers. This invaluable volume collects four expanded lecture notes with self-contained tutorials. They cover a number of aspects on multiscale modeling, analysis and simulations for problems arising from materials science including some critical components in computational prediction of materials properties such as the multiscale properties of complex materials, properties of defects, interfaces and material microstructures under different conditions, critical issues in developing efficient numerical methods and analytic frameworks for complex and multiscale materials models. This volume serves to inspire graduate students and researchers who choose to embark into original research work in these fields.
Download or read book Multiscale Modeling and Simulation of Composite Materials and Structures written by Young Kwon and published by Springer Science & Business Media. This book was released on 2007-12-04 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the state-of-the-art in multiscale modeling and simulation techniques for composite materials and structures. It focuses on the structural and functional properties of engineering composites and the sustainable high performance of components and structures. The multiscale techniques can be also applied to nanocomposites which are important application areas in nanotechnology. There are few books available on this topic.
Download or read book Modeling and Simulation of Large Scale Nonlinear Processes written by Nikolaos I. Xiros and published by Wiley. This book was released on 2023-03-13 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Offers a comprehensive resource that presents nonlinearity within a multi-physics context that can be applied to a wide range of engineering problems Modeling and Simulation of Large-Scale, Nonlinear Processes fills a gap in the literature for a resource that explores the formal analytical and systematic methods in the analysis of large-scale, distributed and continuous nonlinear processes. Written by experts in the field, this vital text develops and proposes techniques for dealing with nonlinearity in the same way as systems with infinite dynamical order: by reducing the nonlinearity degree to a finite level while avoiding full linearization which often leads to oversimplification which render crucial features elusive. Formulation of the dynamics of multi-physics, large-scale systems is a necessary first step towards reduced-order modeling techniques, robust against parametric uncertainty and neglected dynamics, suitable for analysis, synthesis, control and monitoring of linear and nonlinear multi-physics, and can be applied to large-scale systems such as those encountered among others in mechatronics, fluid-structure interactions, ship propulsion, marine machinery, and power-plants. This important resource: Examines nonlinearity within a multi-physics context that can be applied to engineering problems Presents the material in an integrated way and combines theory with practical applications Helps to establish a solid foundation in the theoretical aspects before looking at applications of the methods Offers a consolidated overview of systematic methods for the analysis of large-scale and continuous nonlinear processes Contains applications spanning a broad range of topics in diverse areas of engineering Designed for engineers, especially those who may be unfamiliar with nonlinearities, Modeling and Simulation of Large-Scale, Nonlinear Processes is the essential text filled with useful material for courses on dynamics, nonlinear vibrations and control.
Download or read book Adaptive Multiscale Modeling of Polymeric Materials Using Goal oriented Error Estimation Arlequin Coupling and Goals Algorithms written by Paul Thomas Bauman and published by . This book was released on 2008 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scientific theories that explain how physical systems behave are described by mathematical models which provide the basis for computer simulations of events that occur in the physical universe. These models, being only mathematical characterizations of actual phenomena, are obviously subject to error because of the inherent limitations of all mathematical abstractions. In this work, new theory and methodologies are developed to quantify such modeling error in a special way that resolves a fundamental and standing issue: multiscale modeling, the development of models of events that transcend many spatial and temporal scales. Specifically, we devise the machinery for a posteriori estimates of relative modeling error between a model of fine scale and another of coarser scale, and we use this methodology as a general approach to multiscale problems. The target application is one of critical importance to nanomanufacturing: imprint lithography of semiconductor devices. The development of numerical methods for multiscale modeling has become one of the most important areas of computational science. Technological developments in the manufacturing of semiconductors hinge upon the ability to understand physical phenomena from the nanoscale to the microscale and beyond. Predictive simulation tools are critical to the advancement of nanomanufacturing semiconductor devices. In principle, they can displace expensive experiments and testing and optimize the design of the manufacturing process. The development of such tools rest on the edge of contemporary methods and high-performance computing capabilities and is a major open problem in computational science. In this dissertation, a molecular model is used to simulate the deformation of polymeric materials used in the fabrication of semiconductor devices. Algorithms are described which lead to a complex molecular model of polymer materials designed to produce an etch barrier, a critical component in imprint lithography approaches to semiconductor manufacturing. Each application of this so-called polymerization process leads to one realization of a lattice-type model of the polymer, a molecular statics model of enormous size and complexity. This is referred to as the base model for analyzing the deformation of the etch barrier, a critical feature of the manufacturing process. To reduce the size and complexity of this model, a sequence of coarser surrogate models is generated. These surrogates are the multiscale models critical to the successful computer simulation of the entire manufacturing process. The surrogate involves a combination of particle models, the molecular model of the polymer, and a coarse-scale model of the polymer as a nonlinear hyperelastic material. Coefficients for the nonlinear elastic continuum model are determined using numerical experiments on representative volume elements of the polymer model. Furthermore, a simple model of initial strain is incorporated in the continuum equations to model the inherit shrinking of the A coupled particle and continuum model is constructed using a special algorithm designed to provide constraints on a region of overlap between the continuum and particle models. This coupled model is based on the so-called Arlequin method that was introduced in the context of coupling two continuum models with differing levels of discretization. It is shown that the Arlequin problem for the particle-tocontinuum model is well posed in a one-dimensional setting involving linear harmonic springs coupled with a linearly elastic continuum. Several numerical examples are presented. Numerical experiments in three dimensions are also discussed in which the polymer model is coupled to a nonlinear elastic continuum. Error estimates in local quantities of interest are constructed in order to estimate the modeling error due to the approximation of the particle model by the coupled multiscale surrogate model. The estimates of the error are computed by solving an auxiliary adjoint, or dual, problem that incorporates as data the quantity of interest or its derivatives. The solution of the adjoint problem indicates how the error in the approximation of the polymer model inferences the error in the quantity of interest. The error in the quantity of interest represents the relative error between the value of the quantity evaluated for the base model, a quantity typically unavailable or intractable, and the value of the quantity of interest provided by the multiscale surrogate model. To estimate the error in the quantity of interest, a theorem is employed that establishes that the error coincides with the value of the residual functional acting on the adjoint solution plus a higher-order remainder. For each surrogate in a sequence of surrogates generated, the residual functional acting on various approximations of the adjoint is computed. These error estimates are used to construct an adaptive algorithm whereby the model is adapted by supplying additional fine-scale data in certain subdomains in order to reduce the error in the quantity of interest. The adaptation algorithm involves partitioning the domain and selecting which subdomains are to use the particle model, the continuum model, and where the two overlap. When the algorithm identifies that a region contributes a relatively large amount to the error in the quantity of interest, it is scheduled for refinement by switching the model for that region to the particle model. Numerical experiments on several configurations representative of nano-features in semiconductor device fabrication demonstrate the effectiveness of the error estimate in controlling the modeling error as well as the ability of the adaptive algorithm to reduce the error in the quantity of interest. There are two major conclusions of this study: 1. an effective and well posed multiscale model that couples particle and continuum models can be constructed as a surrogate to molecular statics models of polymer networks and 2. an error estimate of the modeling error for such systems can be estimated with sufficient accuracy to provide the basis for very effective multiscale modeling procedures. The methodology developed in this study provides a general approach to multiscale modeling. The computational procedures, computer codes, and results could provide a powerful tool in understanding, designing, and optimizing an important class of semiconductormanufacturing processes. The study in this dissertation involves all three components of the CAM graduate program requirements: Area A, Applicable Mathematics; Area B, Numerical Analysis and Scientific Computation; and Area C, Mathematical Modeling and Applications. The multiscale modeling approach developed here is based on the construction of continuum surrogates and coupling them to molecular statics models of polymer as well as a posteriori estimates of error and their adaptive control. A detailed mathematical analysis is provided for the Arlequin method in the context of coupling particle and continuum models for a class of one-dimensional model problems. Algorithms are described and implemented that solve the adaptive, nonlinear problem proposed in the multiscale surrogate problem. Large scale, parallel computations for the base model are also shown. Finally, detailed studies of models relevant to applications to semiconductor manufacturing are presented.
