Physical simulations have influenced the advancements in engineering, technology, and science more rapidly than ever before. However, it remains challenging for effective and efficient modeling of complex linear and nonlinear material systems based on phenomenological approaches which require predefined functional forms. The goal of this dissertation is to enhance the predictivity and efficiency of physical simulations by developing thermodynamically consistent data-driven computing and reduced-order materials modeling methods based on emerging machine learning techniques for manifold learning, dimensionality reduction, sequence learning, and system identification. For reversible mechanical systems, we first develop a new data-driven material solver built upon local convexity-preserving reconstruction to capture anisotropic material behaviors and enable data-driven modeling of nonlinear anisotropic elastic solids. A material anisotropic state characterizing the underlying material orientation is introduced for the manifold learning projection in the material solver. To counteract the curse of dimensionality and enhance the generalization ability of data-driven computing, we employ deep autoencoders to discover the underlying low-dimensional manifold of material database and integrate a convexity-preserving interpolation scheme into the novel autoencoder-based data-driven solver to further enhance efficiency and robustness of data searching and reconstruction during online data-driven computation. The proposed approach is shown to achieve enhanced efficiency and generalization ability over a few commonly used data-driven schemes. For irreversible mechanical systems, we develop a thermodynamically consistent machine learned data-driven constitutive modeling approach for path-dependent materials based on measurable material states, where the internal state variables essential to the material path-dependency are inferred automatically from the hidden state of recurrent neural networks. The proposed method is shown to successfully model soil behaviors under cyclic shear loading using experimental stress-strain data. Lastly, we develop a non-intrusive accurate and efficient reduced-order model based on physics-informed adaptive greedy latent space dynamics identification (gLaSDI) for general high-dimensional nonlinear dynamical systems. An autoencoder and dynamics identification models are trained simultaneously to discover intrinsic latent space and learn expressive governing equations of simple latent-space dynamics. To maximize and accelerate the exploration of the parameter space for optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for optimal training samples on the fly, which outperforms the conventional predefined uniform sampling. Compared with the high-fidelity models of various nonlinear dynamical problems, gLaSDI achieves 66 to 4,417x speed-up with 1 to 5% relative errors.

Physics-based numerical simulation remains challenging as the complexity of today's high-fidelity models has dramatically increased. Model order reduction (MOR) and data-driven modeling, based on the emerging techniques of data learning and physical modeling, present a promising way to tackle the computational bottleneck related to the computational intensity and model complexity. Nevertheless, MOR has proven to be significantly more difficult for parameterized mechanics systems that exhibit a wide variety of parameter-dependent nonlinear behaviors or that involve localized essential features. The first objective of this work is to develop robust, physics-preserving MOR methods. As constructing a low-dimensional MOR model can be considered as the hybrid data-physics approach, one can optimize it through a learning process using both data and physical models. As such, we first propose a MOR method based on decomposed reduced-order projections that well preserve the essential near-tip characteristic for fracture mechanics. Moreover, we develop an enhanced reduced-order basis to construct a low-dimensional subspace, deriving from a generalized manifold learning framework that allows the employment of local information in the data structure during the learning phase. This approach can yield a robust reduced-order model against noise and outliers and is well suited for parameterized nonlinear physical systems. Finally, a nonlinear MOR for a meshfree Galerkin formulation with the stabilized conforming nodal integration (SCNI) scheme is developed to yield a pure node based MOR that is particularly effective for hyper-reduction techniques. A numerical example of two-phase hyperelastic solid with perturbed loading conditions is used to validate the effectiveness of the proposed reduction method. The second goal of the dissertation is to develop a robust data-driven computational framework, which provides an alternative to conventional scientific computing for complex materials. This framework aims at performing physical simulation by directly interacting with material data via machine learning procedures instead of employing phenomenological constitutive models, and especially addressing the robustness issue associated with noisy and scarce data. To this end, we propose to search data solutions from a locally reconstructed convex hull associated with the k-nearest neighbor points, which leads to robustness to noisy data and ensures convergence stability. The accuracy and robustness of the proposed data-driven approach are demonstrated in the modeling of linear and nonlinear elasticity problems. In addition, we present a preliminary result of data-driven modeling of biological tissue using material data collected from laboratory testing on heart valve tissue, showing the potential of data-driven simulation by integrating physical modeling and machine learning techniques.

This lecture presents research on a general framework for perceptual organization that was conducted mainly at the Institute for Robotics and Intelligent Systems of the University of Southern California. It is not written as a historical recount of the work, since the sequence of the presentation is not in chronological order. It aims at presenting an approach to a wide range of problems in computer vision and machine learning that is data-driven, local and requires a minimal number of assumptions. The tensor voting framework combines these properties and provides a unified perceptual organization methodology applicable in situations that may seem heterogeneous initially. We show how several problems can be posed as the organization of the inputs into salient perceptual structures, which are inferred via tensor voting. The work presented here extends the original tensor voting framework with the addition of boundary inference capabilities; a novel re-formulation of the framework applicable to high-dimensional spaces and the development of algorithms for computer vision and machine learning problems. We show complete analysis for some problems, while we briefly outline our approach for other applications and provide pointers to relevant sources.

Most problems in science involve many scales in time and space. An example is turbulent ?ow where the important large scale quantities of lift and drag of a wing depend on the behavior of the small vortices in the boundarylayer. Another example is chemical reactions with concentrations of the species varying over seconds and hours while the time scale of the oscillations of the chemical bonds is of the order of femtoseconds. A third example from structural mechanics is the stress and strain in a solid beam which is well described by macroscopic equations but at the tip of a crack modeling details on a microscale are needed. A common dif?culty with the simulation of these problems and many others in physics, chemistry and biology is that an attempt to represent all scales will lead to an enormous computational problem with unacceptably long computation times and large memory requirements. On the other hand, if the discretization at a coarse level ignoresthe?nescale informationthenthesolutionwillnotbephysicallymeaningful. The in?uence of the ?ne scales must be incorporated into the model. This volume is the result of a Summer School on Multiscale Modeling and S- ulation in Science held at Boso ¤n, Lidingo ¤ outside Stockholm, Sweden, in June 2007. Sixty PhD students from applied mathematics, the sciences and engineering parti- pated in the summer school.

Reduced order models (ROMs) are a highly efficient alternative to full-order finite element models (FEM) of geometrically nonlinear structures. Many non-intrusive reduced order modeling methods have been developed over the decades to serve as a digital twin of geometrically nonlinear structures, providing accurate dynamic simulations with dramatically reduced computational cost. However, the ROM methods pose some critical issues. The existing methods are sometimes not reliable, and so expensive simulations must be run to check the accuracy and optimality of the ROMs before they can be used confidently. Also, a ROM is typically only valid for a single FEM and does not account for variations in the FEM. Thus, if the design of the structure changes so that the FEM changes in some way, one must recompute the corresponding ROM with a new set of static load-displacement solutions. This also greatly increases the cost of analysis using ROMs, making them less attractive. This dissertation proposes a new data-driven reduced order modeling method for geometrically nonlinear structures, which can resolve these issues while keeping the computational cost reasonable. The first chapter presents an application of quasi-static modal analysis (QSMA) to reduced order modeling of geometrically nonlinear structures, to investigate the effect of static versus dynamic modal coupling on the nonlinear dynamic behaviors and how that can be used to create an efficient single-mode ROM. The second study proposes a new data-driven reduced order model based on Gaussian process regression (GPR), which accurately captures how the ROM coefficients change as the FEM is changed, so that one GPR ROM can predict the behavior of a wide range of systems and also quantify its predictive confidence. Through the advances presented in these two chapters, this dissertation opens up new possibilities for efficient and reliable model updating of geometrically nonlinear structures. In this respect, the third chapter explores an application of a data-driven ROM to FE model updating. The proposed updating method incorporates a GPR ROM into the model updating procedure, allowing one to use a single GPR ROM to efficiently update the FEM parameters to match actual field data. Lastly, a case study of model updating of a curved panel subjected to non-uniform thermal fields is demonstrated, which highlights how sensitive the structural response can be with respect to the localized thermal loads. This study suggests a need for a data-driven approach for updating the thermal model to account for highly uncertain and complicated thermal effects. This could be a fruitful avenue for future research.

Anomalous rheology is a material behavior that presents the fingerprint of power-laws, arising from anomalous diffusion in microstructures, and observed in a range of complex materials. Such microstructures often display a fractal nature with sub-diffusive dynamics, e.g., of entangled polymer chains, and defect interactions such as dislocation avalanches, cracks, and voids. The corresponding macroscopic non-exponential behavior makes integer-order models to lack a compact representation of the small-scale physics. Furthermore, classical linear viscoelastic models require arbitrary arrangements of Hookean/Newtonian elements, introducing a limited number of exponential relaxation modes that, at most, represent a truncated power-law approximation. While this may be satisfactory for short times at engineering accuracy, such models often yield high-dimensional parameter spaces and lack predictability for multiple time/length-scales. In this scenario, Fractional Calculus (FC) becomes an attractive modeling alternative since it naturally accounts for power-law kernels in its integro-differential operators. This allows accurate and predictive modeling of soft materials for multiple timescales, in which most standard models fail or become impractical.In this work, a data-driven framework for efficient, multi-scale fractional modeling and failure of anomalous materials is proposed. The overarching goal is to identify/construct efficient fractional rheological models, especially for soft materials, undergoing nonlinear response and failure. To this purpose, a fractional linear and nonlinear viscoelastic existence study is developed and employed for the first time to urinary bladder tissues undergoing large strains. The framework is extended to account for power-law viscoplastic behavior, and aiming for applications to larger systems, the resulting models are solved through a new approach called fractional return-mapping algorithm, that generalizes existing predictor-corrector schemes of classical elastoplasticity. Regarding the effects of fractional constitutive laws on structural dynamics, a few developed models are incorporated to beam and truss structures, where the effects of evolving constitutive laws on the anomalous dynamics of systems are analyzed. Although FC became an effective modeling tool in the last few decades, it requires careful considerations to satisfy basic thermodynamic conservation/dissipation laws. To this end, the thermodynamic consistency of the developed visco-elasto-plastic models with the addition of damage effects is proved. Furthermore, the associated energy release rate due to crack/void formation is consistent with the employed fractional rheological elements, which naturally introduces memory effects on damage evolution.Fractional differential equations (FDEs) inherently carry a functional nonlocal dependency and near-singular behaviors at bounded domains, which increases the computational complexity and degenerates the global accuracy of many existing numerical schemes. Therefore, two numerical contributions are proposed in the last part of the framework. The first one is a data-driven singularity-capturing approach that automatically addresses the low solution regularity and yields high accuracy for long time-integration. In the second contribution, fast implicit-explicit (IMEX) schemes are developed for stiff/nonlinear FDEs, which are shown to have larger stability regions than existing approaches.

Simulating the dynamics of large-scale complex, spatio-temporal systems requires prohibitively expensive computational resources. Moreover, the high-dimensional dynamics of such systems often lacks physical interpretability. However, the intrinsic dimensionality of the dynamics often remains quite low, meaning that the dynamics remains embedded in a low-dimensional attractor or manifold in a high-dimensional state-space. Leveraging this phenomenon, in model order reduction, reduced order models (ROMs) with low-dimensional states are derived that can approximate the high-dimensional dynamics of large-scale systems with reasonable accuracy. In this thesis, we study the model reduction of structural systems subjected to impact and nonsmooth boundary conditions, using proper Orthogonal Decomposition (POD), a data-driven projection-based dimension reduction technique. The dynamics of structural systems is typically characterized by partial differential equations (PDEs), which are often impossible to solve analytically. A direct attempt to numerically solve these PDEs to obtain approximate solutions leads to extremely high-dimensional systems of ordinary differential equations (ODEs). The larger the dimensionality of the system of ODEs, the greater is the accuracy of the approximate solution. As a result, often, the dimensionality of a problem is artificially inflated to achieve a more accurate solution, even though the intrinsic dimensionality of the original system is much lower, making the problem computationally intractable. However, data from such high-dimensional systems often exhibit certain dominant patterns, which are representative of the underlying low-dimensional dynamics. POD identifies these low-dimensional embedded patterns based on the dominant correlations present in the data and determines a subspace that contains the data to a desired level of accuracy. This subspace is spanned by a set of basis functions known as proper orthogonal modes (POMs). Mathematically, the POMs are constructed such that along those the variance of the data is maximized. A certain number of POMs are chosen to form a reduced subspace onto which the high dimensional model of the system is projected, yielding a reduced order model that can parsimoniously describe the dynamics of the high-dimensional system. A major part of my research addresses the question of how best to determine the number of POMs to be selected, which is also the dimension of the ROM. In standard implementations of POD, this is decided such that a predefined percentage of the total data variance is captured. However, a fundamental problem with variance-based mode selection is that it is difficult, a priori, to determine the percentage of total variance that will lead to an accurate ROM. Furthermore, the needed percentage of variance can differ widely from one system to the next, or even from one steady-state solution to another. There are two main reasons for this. First, POD is essentially a projection-based technique that ensures optimal reduction (in a mean-square statistical sense) of high-dimensional data. However, such projection optimality does not ensure the accuracy of a ROM. This is because, second, the variance of a data set, or any portion of it in a reduced subspace, has no direct connection with the dynamics of the system generating it. In particular, dynamically important modes that have small variance can still play a crucial role in transporting energy in and out of the system. The neglect of such small-variance degrees of freedom can result in a ROM with behavior that significantly deviates from the true system dynamics. A specific aim of our work was to go beyond merely statistical characterizations to gain a physics-based understanding of why, in specific cases, a given dimension of the reduced subspace is required for an accurate ROM. We were particularly interested in dynamical systems that are subjected to nonsmooth loading conditions, such as impacts, or that have nonsmooth constitutive behavior, such as piecewise linear springs. Such features typically result in numerous modes being excited in the system dynamics. While performing model reduction of such systems, it is essential to include all dynamically important modes. We studied the model reduction of an Euler-Bernoulli beam that was subjected to periodic impacts, using a semi-analytical approach. It was observed that using the conventional variance-based mode selection criterion yielded ROMs with substantial inaccuracies for impulsive loading conditions, with a maximum of 5% relative displacement error and 50% relative velocity error. However, selecting the number of POMs required to achieve energy balance on the corresponding reduced subspace (the span of the selected POMs) gave ROMs with errors that were smaller by approximately three orders of magnitude. These ROMs properly reflect the energetics of the full system, resulting in simulations that accurately represent the system's true behavior. With variance-based mode selection, in principle one may always formulate ROMs with any desired accuracy simply by increasing the reduced subspace dimension by trial and error. However, such an approach does not provide any insight as to why this needs to be done in specific cases. The energy closure method provides this physical insight. We further studied the general application of this energy closure criterion using discrete data, with and without measurement noise, as typically gathered in experiments or numerical simulations. We used the same model of the periodically kicked Euler-Bernoulli beam and formulated ROMs by applying POD to the steady-state discrete displacement field obtained from numerical simulations of the beam. An alternative approach to quantifying the degree of energy closure was derived. In this approach, the convergence of energy input to or dissipated from the system was obtained as a function of the subspace dimension, and the dimension capturing a predefined percentage of either energy is selected as the ROM-dimension. This was in agreement with our prior idea of selecting the ROM dimension by ensuring a balance between the energy dissipation and input on the subspace since the steady-state dynamics guarantees that an accurate estimate of either quantity will automatically lead to a balance between the two. This new metric for quantifying the degree of energy closure was, however, found to be more robust to data-discretization error and measurement noise while also being easier to interpret. The data processing necessary for implementing the new metric was discussed in detail. We showed that ROMs from the simulated data using our approach formulated accurately captured the dynamics of the beam for different sets of parameter values. Finally, we implemented this new metric to estimate energy-closure for the model order reduction of an experimental system consisting of a magnetically kicked nonlinear flexible oscillator. This was a piecewise linear, globally nonlinear system, and exhibited a wide range of dynamical behaviors: periodic, quasi-periodic, and chaotic. Furthermore, the nonsmooth nature of the forcing and the boundary conditions excited a large number of modes in the system. For high-fidelity simulations, we approximated the dynamics of the oscillator using linear models with 25 degrees of freedom. By applying POD on the discrete displacement data obtained from the simulations and using the energy-closure criterion, we were able to formulate a single ROM, with only 6 degrees of freedom, which accurately captured the different dynamical steady states shown by the original system. More importantly, it was observed that ROM was able to preserve the bifurcation structure of the system. We have thus shown, how a physics-informed understanding of estimating ROM-dimension can lead to accurate reduced order models in linear and nonlinear structural vibration problems.

J.-P. CALISTE, A. TRUYOL AND J. WESTBROOK The Series, "Data and Knowledge in a Changing World", exemplifies CODATA's primary purpose of collecting, from widely different fields, a wealth of information on efficient exploitation of data for progress in science and technology and making that information available to scientists and engineers. A separate and complementary CODATA Reference Series will present Directories of compiled and evaluated data and Glossaries of data-related terms. The present book "Thermodynamic Modeling and Materials Data Engineering" discusses thermodynamic, structural, systemic and heuristic approaches to the modeling of complex materials behavior in condensed phases, both fluids and solids, in order to evaluate their potential applications. Itwas inspired by the Symposium on "Materials and Structural Properties" held during the 14th International CODATA Conference in Chambery, France. The quality of the contributions to this Symposium motivated us to present" a coherent book of interest to the field. Updated contributions inspired by Symposium discussions and selections from other CODATA workshops concerning material properties data and Computer Aided Design combine to highlight the complexity of material data issues on experimental, theoretical and simulation levels Articles were selected for their pertinence in three areas. Complex data leading to interesting developments and tools such as: • new developments in state equations and their applications, • prediction and validation of physical and energy data by group correlations for pure compounds, • modeling and prediction of mixture properties.

Nonlinear Finite Elements for Continua and Structures p>Nonlinear Finite Elements for Continua and Structures This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXtended Finite Element Method (XFEM), multiresolution continuum theory for multiscale microstructures, and dislocation- density-based crystalline plasticity. Nonlinear Finite Elements for Continua and Structures, Second Edition focuses on the formulation and solution of discrete equations for various classes of problems that are of principal interest in applications to solid and structural mechanics. Topics covered include the discretization by finite elements of continua in one dimension and in multi-dimensions; the formulation of constitutive equations for nonlinear materials and large deformations; procedures for the solution of the discrete equations, including considerations of both numerical and multiscale physical instabilities; and the treatment of structural and contact-impact problems. Key features: Presents a detailed and rigorous treatment of nonlinear solid mechanics and how it can be implemented in finite element analysis Covers many of the material laws used in today’s software and research Introduces advanced topics in nonlinear finite element modelling of continua Introduction of multiresolution continuum theory and XFEM Accompanied by a website hosting a solution manual and MATLAB® and FORTRAN code Nonlinear Finite Elements for Continua and Structures, Second Edition is a must-have textbook for graduate students in mechanical engineering, civil engineering, applied mathematics, engineering mechanics, and materials science, and is also an excellent source of information for researchers and practitioners.