In this thesis, a framework is laid for holistic uncertainty management for simulation-based design of multiscale systems. The work is founded on uncertainty management for microstructure mediated design (MMD) of material and product, which is a representative example of a system over multiple length and time scales, i.e., a multiscale system. The characteristics and challenges for uncertainty management for multiscale systems are introduced context of integrated material and product design. This integrated approach results in different kinds of uncertainty, i.e., natural uncertainty (NU), model parameter uncertainty (MPU), model structure uncertainty (MSU) and propagated uncertainty (PU). We use the Inductive Design Exploration Method to reach feasible sets of robust solutions against MPU, NU and PU. MMD of material and product is performed for the product autonomous underwater vehicle (AUV) employing the material in-situ metal matrix composites using IDEM to identify robust ranged solution sets. The multiscale system results in decision nodes for MSU consideration at hierarchical levels, termed as multilevel design. The effectiveness of using game theory to model strategic interaction between the different levels to facilitate decision making for mitigating MSU in multilevel design is illustrated using the compromise decision support problem (cDSP) technique. Information economics is identified as a research gap to address holistic uncertainty management in simulation-based multiscale systems, i.e., to address the reduction or mitigation of uncertainty considering the current design decision and scope for further simulation model refinement in order to reach better robust solutions. It necessitates development of an improvement potential (IP) metric based on value of information which suggests the scope of improvement in a designer's decision making ability against modeled uncertainty (MPU) in simulation models in multilevel design problem. To address the research gap, the integration of robust design (using IDEM), information economics (using IP) and game theoretic constructs (using cDSP) is proposed. Metamodeling techniques and expected value of information are critically reviewed to facilitate efficient integration. Robust design using IDEM and cDSP are integrated to improve MMD of material and product and address all four types of uncertainty simultaneously. Further, IDEM, cDSP and IP are integrated to assist system level designers in allocating resources for simulation model refinement in order to satisfy performance and robust process requirements. The approach for managing MPU, MSU, NU and PU while mitigating MPU is presented using the MMD of material and product. The approach presented in this article can be utilized by system level designers for managing all four types of uncertainty and reducing model parameter uncertainty in any multiscale system.

Many engineering and scientific applications of flow in porous media are characterized by transport phenomena at multiple spatial scales, including pollutant transport, groundwater remediation, and acid injection to enhance well production. Carbon sequestration in particular is a multiscale problem, because the trapping and leakage mechanisms of CO2 in the subsurface occur from the sub-pore level to the basin scale. Quantitative and predictive pore-scale modeling has long shown to be a valuable tool for studying fluid-rock interactions in porous media. However, due to the size limitation of the pore-scale models (10-4-10-2m), it is impossible to model an entire reservoir at the pore scale. A straightforward multiscale approach would be to upscale macroscopic parameters (e.g. permeability) directly from pore-scale models and then input them into a continuum-scale simulator. However, it has been found that the large-scale models do not predict in many cases. One possible reason for the inaccuracies is oversimplified boundary conditions used in this direct upscaling approach. The hypothesis of this work is that pore-level flow and upscaled macroscopic parameters depends on surrounding flow behavior manifested in the form of boundary conditions. The detailed heterogeneity captured by the pore-scale models may be partially lost if oversimplified boundary conditions are employed in a direct upscaling approach. As a result, extracted macroscopic properties may be inaccurate. Coupling the model to surrounding media (using finite element mortars to ensure continuity between subdomains) would result in more realistic boundary conditions, and can thus improve the accuracy of the upscaled parameters. To test the hypothesis, mortar coupling is employed to couple pore-scale models and also couple pore-scale models to continuum models. Flow field derived from mortar coupling and direct upscaling are compared, preferably against a true solution if one exists. It is found in this dissertation that pore-scale flow and upscaled parameters can be significantly affected by the surrounding media. Therefore, using arbitrary boundary conditions such as constant pressure and no-flow boundaries may yield misleading results. Mortar coupling captures the detailed variation on the interface and imposes realistic boundary conditions, thus estimating more accurate upscaled values and flow fields. An advanced upscaling tool, a Super Permeability Tensor (SPT) is developed that contains pore-scale heterogeneity in greater detail than a conventional permeability tensor. Furthermore, a multiscale simulator is developed taking advantage of mortar coupling to substitute continuum grids directly with pore-scale models where needed. The findings from this dissertation can significantly benefit the understanding of fluid flow in porous media, and, in particular, CO2 storage in geological formations which requires accurate modeling across multiple scales. The fine-scale models are sensitive to the boundary conditions, and the large scale modeling of CO2 transport is sensitive to the CO2 behavior affected by the pore-scale heterogeneity. Using direct upscaling might cause significant errors in both the fine-scale and the large-scale model. The multiscale simulator developed in this dissertation could integrate modeling of CO2 physics at all relevant scales, which span the sub-pore or pore level to the basin scale, into one single simulator with effective and accurate communication between the scales. The multiscale simulator provides realistic boundary conditions for the fine scales, accurate upscaled information to continuum-scale, and allows for the distribution of computational power where needed, thus maintaining high accuracy with relatively low computational cost. / text

Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields. / text

In this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.

The soft matter field considers a wide class of objects such as liquids, polymers, gels, colloids, liquid crystals and biological macromolecules, which have complex internal structure and conformational flexibility leading to phenomena and properties having multiple spacial and time scales. Existing computer simulation methods are able to cover these scales, but with different resolutions, and ability to link them together performing a multiscale simulation is highly desirable. The present work addresses systematic multiscaling approach for soft matter studies, using structure-based coarse-graining (CG) methods such as iterative Boltzmann inversion and inverse Monte Carlo. A new software package MagiC implementing these methods is introduced. The software developed for the purpose of effective CG potential derivation is applied for ionic water solution and for water solution of DMPC lipids. A thermodynamic transferability of the obtained potentials is studied. The effective inter-ionic solvent mediated potentials derived for NaCl successfully reproduce structural properties obtained in explicit solvent simulation, which indicates the perspectives of using the structure-based coarse-graining for studies of ion-DNA and other polyelectrolytes systems. The potentials have temperature dependence, dominated mostly by the electrostatic long-range part which can be described by temperature dependent effective dielectric permittivity, leaving the short-range part of the potential thermodynamically transferable. For CG simulations of lipids a 10-bead water-free model of dimyristoylphosphatidylcholine is introduced. Four atomistic reference systems, having different lipid/water ratio are used to derive the effective bead-bead potentials, which are used for subsequent coarse-grained simulations of lipid bilayer. A significant influence of lipid/water ratio in the reference system on the properties of the simulated bilayers is noted, however it can be softened by additional angle-bending interactions. At the same time the obtained bilayers have stable structure with correct density profiles. The model provides acceptable agreement between properties of coarse-grained and atomistic bilayer, liquid crystal - gel phase transition with temperature change, as well as realistic self-aggregation behavior, which results in formation of bilayer, bicell or vesicle from a dispersed lipid solution in a large-scale simulation. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Submitted. </p><p> </p>

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Numerous efforts have contributed to the study of phase-change problems for over a century|both analytical and numerical. Among those numerical approximations applied to solve phase-transition problems, phase-field models attract more and more attention because they not only capture two important effects, surface tension and supercooling, but also enable explicitly labeling the solid and liquid phases and the position of the interface. In the research of this dissertation, a phase-field model has been employed to simulate 2-D dendrite growth of pure nickel without a flow, and 2-D ice crystal growth in a high-Reynolds-number lid-driven-cavity flow. In order to obtain the details of ice crystal structures as well as the flow field behavior during freezing for the latter simulation, it is necessary to solve the phase-field model without convection and the equations of motion on two different scales. To accomplish this, a heterogeneous multiscale method is implemented for the phase-field model with convection such that the phase-field model is simulated on a microscopic scale and the equations of motion are solved on a macroscopic scale. Simulations of 2-D dendrite growth of pure nickel provide the validation of the phase-field model and the study of dendrite growth under different conditions, e.g., degree of supercooling, interface thickness, kinetic coefficient, and shape of the initial seed. In addition, simulations of freezing in a lid-driven-cavity flow indicate that the flow field has great effect on the small-scale dendrite structure and the flow eld behavior on the large scale is altered by freezing inside it.

Superplastic forming (SPF) is a valuable near net shape fabrication method, used to produce very complex, contoured and monolithic structures that are often lighter, stronger and safer than the assemblies they replace. However, the widespread industrial use of Superplastic (SP) alloys is hindered by a number of issues including low production rate and limited predictive capabilities of stability during deformation and failure. Failure during SPD may result from geometrical macroscopic instabilities and/or microstructural aspects. However, the available failure criteria are either based on geometrical instabilities or microstructural features and do not account for both failure modes. The present study presents a generalized multi-scale stability criterion for SP materials, accounting for both aspects of failure under various loading conditions. A combined model accounting for cavity nucleation and plasticity controlled cavity growth along with a grain growth model and a modified microstructure based constitutive equation for SP materials is incorporated into Harts stability analysis to develop the proposed stability criterion for different loading conditions. Effects of initial grain size, initial levels of cavitation, nucleation strain, strain-rate sensitivity, and grain-growth exponent on the optimum forming curves of different SP alloys are investigated, for different loading conditions.

Tremendous advances in nanoscience during the past decades have drawn a new horizon for the future of science. Many biological and structural elements such as DNA, bio-membranes, nanotubes, nanowires and thin films have been studied carefully in the past decades. In this work we target to speed up the computational methods by incorporating the structural symmetries that nanostructures have. In particular, we use the Objective Structures (OS) framework to speed up molecular dynamics (MD), lattice dynamics (phonon analysis) and multiscale methods. OS framework is a generalization of the standard idea for crystal lattices of assuming periodicity of atomic positions with a large supercell. OS not only considers the translational periodicity of the structure, but also other symmetries such as rotational and screw symmetries. In addition to the computational efficiency afforded by Objective Structures, OS provides us with more flexibility in the shape of the unit cell and the form of the external deformation and loading, comparing to using the translational periodicity. This is because the deformation and loading should be consistent in all cells and not all deformations keep the periodicity of the structures. For instance, bending and twisting cannot be modeled with methods using the structure's periodicity. Using OS framework we then carefully studied carbon nanotubes under non-equilibrium deformations. We also studied the failure mechanism of pristine and twisted nanotubes under tensile loading. We found a range of failure mechanisms, including the formation of Stone-Wales defects, the opening of voids, and the motion of atoms out of the cross-section. We also used the OS framework to make concrete analogies between crystalline phonons and normal modes of vibration in non-crystalline but highly symmetric nanostructures.

Ultra-High-Performance Concretes (UHPCs) are a promising class of cementitious materials possessing mechanical properties superior to those of Normal Strength Concretes (NSCs). However, UHPCs have been slow to transition from laboratory testing to insertion in new applications, partly due to an intuitive trial-and-error materials development process. This research seeks to addresses this problem by implementing a materials design process for the design of UHPC materials and structures subject to blast loads with specific impulses between 1.25- and 1.5-MPa-ms and impact loads resulting from the impact of a 0.50-caliber bullet travelling between 900 and 1,000 m/s. The implemented materials design process consists of simultaneous bottom-up deductive mappings and top-down inductive decision paths through a set of process-structure-property-performance (PSPP) relations identified for this purpose. The bottom-up deductive mappings are constructed from a combination of analytical models adopted from the literature and two hierarchical multiscale models developed to simulate the blast performance of a 1,626-mm tall by 864-mm wide UHPC panel and the impact performance of a 305-mm tall by 305-mm wide UHPC panel. Both multiscale models employ models at three length scales – single fiber, multiple fiber, and structural – to quantify deductive relations in terms of fiber pitch (6-36 mm/revolution), fiber volume fraction (0-2%), uniaxial tensile strength of matrix (5-12 MPa), quasi-static tensile strength of fiber-reinforced matrix (10-20 MPa), and dissipated energy density (20-100 kJ/m²). The inductive decision path is formulated within the Inductive Design Exploration Method (IDEM), which determines robust combinations of properties, structures, and processing steps that satisfy the performance requirements. Subsequently, the preferred material and structural designs are determined by rank order of results of objective functions, defined in terms of mass and costs of the UHPC panel.