Download Free Physics Informed Model Reduction Of Dynamical Systems Subjected To Impacts And Discontinuity Book in PDF and EPUB Free Download. You can read online Physics Informed Model Reduction Of Dynamical Systems Subjected To Impacts And Discontinuity and write the review.

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.
Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications. Audience: anyone interested in model reduction, including graduate students and researchers in the fields of system and control theory, numerical analysis, and the theory of partial differential equations/computational fluid dynamics.
A textbook covering data-science and machine learning methods for modelling and control in engineering and science, with Python and MATLAB®.
Many physical, chemical, biomedical, and technical processes can be described by partial differential equations or dynamical systems. In spite of increasing computational capacities, many problems are of such high complexity that they are solvable only with severe simplifications, and the design of efficient numerical schemes remains a central research challenge. This book presents a tutorial introduction to recent developments in mathematical methods for model reduction and approximation of complex systems. Model Reduction and Approximation: Theory and Algorithms contains three parts that cover (I) sampling-based methods, such as the reduced basis method and proper orthogonal decomposition, (II) approximation of high-dimensional problems by low-rank tensor techniques, and (III) system-theoretic methods, such as balanced truncation, interpolatory methods, and the Loewner framework. It is tutorial in nature, giving an accessible introduction to state-of-the-art model reduction and approximation methods. It also covers a wide range of methods drawn from typically distinct communities (sampling based, tensor based, system-theoretic).?? This book is intended for researchers interested in model reduction and approximation, particularly graduate students and young researchers.
This book focuses on computational methods for large-scale statistical inverse problems and provides an introduction to statistical Bayesian and frequentist methodologies. Recent research advances for approximation methods are discussed, along with Kalman filtering methods and optimization-based approaches to solving inverse problems. The aim is to cross-fertilize the perspectives of researchers in the areas of data assimilation, statistics, large-scale optimization, applied and computational mathematics, high performance computing, and cutting-edge applications. The solution to large-scale inverse problems critically depends on methods to reduce computational cost. Recent research approaches tackle this challenge in a variety of different ways. Many of the computational frameworks highlighted in this book build upon state-of-the-art methods for simulation of the forward problem, such as, fast Partial Differential Equation (PDE) solvers, reduced-order models and emulators of the forward problem, stochastic spectral approximations, and ensemble-based approximations, as well as exploiting the machinery for large-scale deterministic optimization through adjoint and other sensitivity analysis methods. Key Features: Brings together the perspectives of researchers in areas of inverse problems and data assimilation. Assesses the current state-of-the-art and identify needs and opportunities for future research. Focuses on the computational methods used to analyze and simulate inverse problems. Written by leading experts of inverse problems and uncertainty quantification. Graduate students and researchers working in statistics, mathematics and engineering will benefit from this book.
This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.
A groundbreaking introduction to vectors, matrices, and least squares for engineering applications, offering a wealth of practical examples.
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.
The book focuses on the physical and mathematical foundations of model-based turbulence control: reduced-order modelling and control design in simulations and experiments. Leading experts provide elementary self-consistent descriptions of the main methods and outline the state of the art. Covered areas include optimization techniques, stability analysis, nonlinear reduced-order modelling, model-based control design as well as model-free and neural network approaches. The wake stabilization serves as unifying benchmark control problem.
Combining scientific computing methods and algorithms with modern data analysis techniques, including basic applications of compressive sensing and machine learning, this book develops techniques that allow for the integration of the dynamics of complex systems and big data. MATLAB is used throughout for mathematical solution strategies.