In numerical simulations of geomechanics problems, a grand challenge consists of overcoming the difficulties in making accurate and robust predictions by revealing the true mechanisms in particle interactions, fluid flow inside pore spaces, and hydromechanical coupling effect between the solid and fluid constituents, from microscale to mesoscale, and to macroscale. While simulation tools incorporating subscale physics can provide detailed insights and accurate material properties to macroscale simulations via computational homogenizations, these numerical simulations are often too computational demanding to be directly used across multiple scales. Recent breakthroughs of Artificial Intelligence (AI) via machine learning have great potential to overcome these barriers, as evidenced by their great success in many applications such as image recognition, natural language processing, and strategy exploration in games. The AI can achieve super-human performance level in a large number of applications, and accomplish tasks that were thought to be not feasible due to the limitations of human and previous computer algorithms. Yet, machine learning approaches can also suffer from overfitting, lack of interpretability, and lack of reliability. Thus the application of machine learning into generation of accurate and reliable surrogate constitutive models for geomaterials with multiscale and multiphysics is not trivial. For this purpose, we propose to establish an integrated modeling process for automatic designing, training, validating, and falsifying of constitutive models, or "metamodeling". This dissertation focuses on our efforts in laying down step-by-step the necessary theoretical and technical foundations for the multiscale metamodeling framework. The first step is to develop multiscale hydromechanical homogenization frameworks for both bulk granular materials and granular interfaces, with their behaviors homogenized from subscale microstructural simulations. For efficient simulations of field-scale geomechanics problems across more than two scales, we develop a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate path-dependent cohesive laws for localized physical discontinuities at both grain and specimen scales. To enable AI in taking over the trial-and-error tasks in the constitutive modeling process, we introduce a novel “metamodeling” framework that employs both graph theory and deep reinforcement learning (DRL) to generate accurate, physics compatible and interpretable surrogate machine learning models. The process of writing constitutive models is simplified as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive models generated through self-playing. To overcome the obstacle of limited information in geomechanics, we improve the efficiency in utilization of experimental data by a multi-agent cooperative metamodeling framework to provide guidance on database generation and constitutive modeling at the same time. The modeler agent in the framework focuses on evaluating all modeling options (from domain experts’ knowledge or machine learning) in a directed multigraph of elasto-plasticity theory, and finding the optimal path that links the source of the directed graph (e.g., strain history) to the target (e.g., stress). Meanwhile, the data agent focuses on collecting data from real or virtual experiments, interacts with the modeler agent sequentially and generates the database for model calibration to optimize the prediction accuracy. Finally, we design a non-cooperative meta-modeling framework that focuses on automatically developing strategies that simultaneously generate experimental data to calibrate model parameters and explore weakness of a known constitutive model until the strengths and weaknesses of the constitutive law on the application range can be identified through competition. These tasks are enabled by a zero-sum reward system of the metamodeling game and robust adversarial reinforcement learning techniques.

The present work presents a framework for multiscale modeling of multimaterial flows using surrogate modeling techniques in the particular context of shocks interacting with clusters of particles. The work builds a framework for bridging scales in shock-particle interaction by using ensembles of resolved mesoscale computations of shocked particle laden flows. The information from mesoscale models is “lifted” by constructing metamodels of the closure terms - the thesis analyzes several issues pertaining to surrogate-based multiscale modeling frameworks. First, to create surrogate models, the effectiveness of several metamodeling techniques, viz. the Polynomial Stochastic Collocation method, Adaptive Stochastic Collocation method, a Radial Basis Function Neural Network, a Kriging Method and a Dynamic Kriging Method is evaluated. The rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied. For sufficiently large number of training points, Stochastic Collocation methods generally converge faster than the other metamodeling techniques, while the DKG method converges faster when the number of input points is less than 100 in a two-dimensional parameter space. Because the input points correspond to computationally expensive micro/meso-scale computations, the DKG is favored for bridging scales in a multi-scale solver. After this, closure laws for drag are constructed in the form of surrogate models derived from real-time resolved mesoscale computations of shock-particle interactions. The mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of Mach Number and particle volume fraction. Two Kriging-based methods, viz. the Dynamic Kriging Method (DKG) and the Modified Bayesian Kriging Method (MBKG) are evaluated for their ability to construct surrogate models with sparse data; i.e. using the least number of mesoscale simulations. It is shown that unlike the DKG method, the MBKG method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments. In macroscale models for shock-particle interactions, Subgrid Particle Reynolds’ Stress Equivalent (SPARSE) terms arise because of velocity fluctuations due to fluid-particle interaction in the subgrid/meso scales. Mesoscale computations are performed to calculate the SPARSE terms and the kinetic energy of the fluctuations for different values of Mach Number and particle volume fraction. Closure laws for SPARSE terms are constructed using the MBKG method. It is found that the directions normal and parallel to those of shock propagation are the principal directions of the SPARSE tensor. It is also found that the kinetic energy of the fluctuations is independent of the particle volume fraction and is 12-15% of the incoming shock kinetic energy for higher Mach Numbers. Finally, the thesis addresses the cost of performing large ensembles of resolved mesoscale computations for constructing surrogates. Variable fidelity techniques are used to construct an initial surrogate from ensembles of coarse-grid, relative inexpensive computations, while the use of resolved high-fidelity simulations is limited to the correction of initial surrogate. Different variable-fidelity techniques, viz the Space Mapping Method, RBFs and the MBKG methods are evaluated based on their ability to correct the initial surrogate. It is found that the MBKG method uses the least number of resolved mesoscale computations to correct the low-fidelity metamodel. Instead of using 56 high-fidelity computations for obtaining a surrogate, the MBKG method constructs surrogates from only 15 resolved computations, resulting in drastic reduction of computational cost.

In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales effect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more effective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a finite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coefficients. For linear coefficients, the multiscale finite volume element method is viewed as a perturbation of multiscale finite element method. The analysis uses substantially the existing finite element results and techniques. The multiscale method for nonlinear coefficients will be analyzed in the finite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the flow in porous media. Problems that we will consider are multiphase immiscible flow, multicomponent miscible flow, and soil infiltration in saturated/unsaturated flow.

A novel multiscale approach is introduced for determining the thermal conductivity of polymer nanocomposites (PNCs) reinforced with single-walled carbon nanotubes (SWCNTs), which accounts for their intrinsic uncertainties associated with dispersion, distribution, and morphology. Heterogeneities in PNCs on nanoscale are identified and quantified in a statistical sense, for the calculation of effective local properties. A finite element method computes the overall macroscale properties of PNCs in conjunction with the Monte Carlo simulations. This Monte Carlo Finite Element Approach (MCFEA) allows for acquiring the randomness in spatial distribution of the nanotubes throughout the composite. Furthermore, the proposed MCFEA utilizes the nanotube content, orientation, aspect ratio and diameter inferred from their statistical information. Local SWCNT volume or weight fractions are assigned to the finite elements (FEs), based on various spatial probability distributions. Multi-phase homogenization techniques are applied to each FE to calculate the local thermal conductivities. Then, the Monte Carlo simulations provide the statistics on the overall thermal conductivity of the PNCs. Subsequently, dispersion characteristics of the nanotubes are assessed by incorporating nanotube agglomerates. In this regard, a multi-phase homogenization method is developed for enhanced accuracy and effectiveness. The effect of the nanotube orientation in a polymer is studied for the cases where the SWCNTs are randomly oriented as well as longitudinally aligned. The influence of voids existing in the polymer is investigated on the thermal conductivity, to capture the uncertainties in PNCs more extensively. Further, a unique damage evaluation model is proposed to assess the degradation of PNCs when subjected to thermal cycling. The growth in void content is represented with a Weibull-based equation, to quantify the deterioration of the thermal and mechanical properties of PNCs under thermal fatigue. In addition, the MCFEA considers the interface resistance of the carbon nanotubes as one of the key factors in the thermal conductivity of nanocomposites. Parametric studies are performed comprehensively. The numerical results obtained are compared with available analytical techniques at hand and with the data from pertinent independent experimental studies. It is found that the proposed MCFEA is capable of estimating the thermal conductivity with good accuracy.

In order to create informed predictive models that capture artery dependent responses during atherosclerosis progression and the long term response to hypertension, one needs to know the structural, biochemical and mechanical properties as a function of time in these diseased states. In the case of hypertension more is known about the mechanical changes; while, less is known about the structural changes over time. For atherosclerotic plaques, more is known about the structure and less about the mechanical properties. We established a congruent multi-scale model to predict the adapted salient arterial geometry, structure and biochemical response to an increase in pressure. Geometrical and structural responses to hypertension were then quantified in a hypertensive animal model. Eventually this type of model may be used to predict mechanical changes in complex disease such as atherosclerosis. Thus for future verification and implementation we experimentally tested atherosclerotic plaques and quantified composition, structure and mechanical properties. Using the theoretical models we can now predict arterial changes in biochemical concentrations as well as salient features such as geometry, mass of elastin, smooth muscle, and collagen, and circumferential stress, in response to hemodynamic loads. Using an aortic coarctation model of hypertension, we found structural arterial responses differ in the aorta, coronary and cerebral arteries. Effects of elevated pressure manifest first in the central arteries and later in distal muscular arteries. In the aorta, there is a loss and then increase of cytoskeleton actin fibers, production of fibrillar collagen and elastin, hyperplasia or hypertrophy with nuclear polypoid, and recruitment of hemopoeitic progenitor cells and monocytes. In the muscular coronary, we see similar changes albeit it appears actin fibers are recruited and collagen production is only increased slightly in order to maintain constant the overall ratio of ~55 percent. In the muscular cerebral artery, despite a temporary loss in actin fibers there is little structural change. Contrary to hypertensive arteries, characterizing regional stiffness in atherosclerotic plaques has not been done before. Therefore, experimental testing on atherosclerotic plaques of Apolipoprotein E Knockout mice was performed and revealed nearly homogenously lipidic plaques with a median axial compressive stiffness value of 1.5 kPa.

In this dissertation we develop and analyze numerical method to solve general elliptic boundary value problems with many scales. The numerical method presented is intended to capture the small scales effect on the large scale solution without resolving the small scale details, which is done through the construction of a multiscale map. The multiscale method is more effective when the coarse element size is larger than the small scale length. To guarantee a numerical conservation, a finite volume element method is used to construct the global problem. Analysis of the multiscale method is separately done for cases of linear and nonlinear coefficients. For linear coefficients, the multiscale finite volume element method is viewed as a perturbation of multiscale finite element method. The analysis uses substantially the existing finite element results and techniques. The multiscale method for nonlinear coefficients will be analyzed in the finite element sense. A class of correctors corresponding to the multiscale method will be discussed. In turn, the analysis will rely on approximation properties of this correctors. Several numerical experiments verifying the theoretical results will be given. Finally we will present several applications of the multiscale method in the flow in porous media. Problems that we will consider are multiphase immiscible flow, multicomponent miscible flow, and soil infiltration in saturated/unsaturated flow.

Ni-base superalloys are widely used in hot sections of gas turbine engines due to the high resistance to fatigue and creep at elevated temperatures. Due to the demands for improved performance and efficiency in applications of the superalloys, new and improved higher temperature alloy systems are being developed. Constitutive relations for these materials need to be formulated accordingly to predict behavior of cracks at notches in components under cyclic loading with peak dwell periods representative of gas turbine engine disk materials. Since properties are affected by microstructure at various length scales ranging from 10 nm tertiary γ' precipitates to 5-30 μm grains, hierarchical multiscale modeling is essential to address behavior at the component level. The goal of this work is to develop a framework for hierarchical multiscale modeling network that features linkage of several fine scale models to incorporate relevant microstructure attributes into the framework to improve the predictability of the constitutive model. This hierarchy of models is being developed in a collaborative research program with the Ohio State University. The fine scale models include the phase field model which addresses dislocation dissociation in the γ matrix and γ' precipitate phases, and the critical stresses from the model are used as inputs to a grain scale crystal plasticity model in a bottom-up fashion. The crystal plasticity model incorporates microstructure attributes by homogenization. A major task of the present work is to link the crystal plasticity model, informed by the phase field model, to the macroscale model and calibrate models in a top-down fashion to experimental data for a range of microstructures of the improved alloy system by implementing a hierarchical optimization scheme with a parameter clustering strategy. Another key part of the strategy to be developed in this thesis is the incorporation of polycrystal plasticity simulations to model a large range of virtual microstructures that have not been experimentally realized (processed), which append the experimentally available microstructures. Simulations of cyclic responses with dwell periods for this range of virtual (and limited experimental) polycrystalline microstructures will be used to (i) provide additional data to optimize parameter fitting for a microstructure-insensitive macroscopic internal state variable (ISV) model with thermal recovery and rate dependence relevant to the temperatures of interest, and (ii) provide input to train an artificial neural network that will associate the macroscopic ISV model parameters with microstructure attributes for this material. Such microstructure sensitive macroscopic models can then be employed in component level finite element studies to model cyclic behavior with dwell times at smooth and cracked notched specimens.

Two multiscale continuum models for simulating protein dynamics are developed which allow for resolution of protein peptide planes in a beam-like finite element. A curvature and strain based finite element formulation is utilized. This formulation is advantageous in simulating proteins since amino acid chains may be described by a single element, even when the protein segment considered exhibits large curvature and twist such as the alpha-helical shapes prominent in many proteins. Specifically, concurrent and hierarchical multiscale models are developed for the curvature and strain based beam formulation. The hierarchical multiscale continuum model utilizes a novel shooting method to calculate the deformed configuration of the protein. An optimization algorithm determines the requisite stiffness parameters by varying the beam stiffness used in the shooting method until deformed configurations of test cases correspond to those produced by the LAMMPS molecular dynamics software. Additionally, a concurrent multiscale method is detailed for evaluating protein inter-atomic potential parameters from the curvature and strain degrees of freedom employed in the model. This allows internal forces and moments to be calculated using nonlinear protein potentials. Proof of concept testing and model verification for both models includes comparing the multiscale techniques to all-atom molecular dynamics solutions. Specifically, the models are verified by simulating a polypeptide in a vacuum and comparing the predicted results to those computed using LAMMPS.

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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