Book•
Certified Reduced Basis Methods for Parametrized Partial Differential Equations
01 Sep 2015-
TL;DR: In this article, the authors provide a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations, including model construction, error estimation and computational efficiency.
Abstract: This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.
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TL;DR: This work reviews the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.
Abstract: Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.
660 citations
Book•
28 Feb 2019TL;DR: In this paper, the authors bring together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science, and highlight many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy.
Abstract: Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.
563 citations
TL;DR: A computational morphogenesis tool, implemented on a supercomputer, that produces designs with giga-voxel resolution that provides insights into the optimal distribution of material within a structure that were hitherto unachievable owing to the challenges of scaling up existing modelling and optimization frameworks is reported.
Abstract: Giga-voxel-resolution computational morphogenesis is used to optimize the internal structure of a full-scale aeroplane wing, yielding light-weight designs with more similarities to animal bone structures than to current aeroplane wing designs. Computational morphogenesis is used to design the best possible shapes and material distributions for the desired structural properties, such as high strength at minimal weight. In plants and animals, morphogenesis occurs naturally through slow genetic evolution. In engineering, a much faster iterative approach for optimum material distribution has been adopted, called topology optimization. So far, it has been used to calculate only small or simple structures owing to limited resolution. Niels Aage et al. have developed a morphogenesis tool that can be run on a supercomputer and can calculate two orders of magnitude more voxels (the three-dimensional equivalents of pixels) than was previously attainable. This makes it possible to design structures with unprecedented detail, yielding new insights into optimal material distribution. The authors calculate an optimized full aircraft wing structure with remarkable structural detail at several length scales, which displays similarities to naturally occurring bone structures such as those seen in bird beaks. The new tool could inspire surprising design approaches for a range of structures, including wind turbine blades, tower masts and bridges. In the design of industrial products ranging from hearing aids to automobiles and aeroplanes, material is distributed so as to maximize the performance and minimize the cost. Historically, human intuition and insight have driven the evolution of mechanical design, recently assisted by computer-aided design approaches. The computer-aided approach known as topology optimization enables unrestricted design freedom and shows great promise with regard to weight savings, but its applicability has so far been limited to the design of single components or simple structures, owing to the resolution limits of current optimization methods1,2. Here we report a computational morphogenesis tool, implemented on a supercomputer, that produces designs with giga-voxel resolution—more than two orders of magnitude higher than previously reported. Such resolution provides insights into the optimal distribution of material within a structure that were hitherto unachievable owing to the challenges of scaling up existing modelling and optimization frameworks. As an example, we apply the tool to the design of the internal structure of a full-scale aeroplane wing. The optimized full-wing design has unprecedented structural detail at length scales ranging from tens of metres to millimetres and, intriguingly, shows remarkable similarity to naturally occurring bone structures in, for example, bird beaks. We estimate that our optimized design corresponds to a reduction in mass of 2–5 per cent compared to currently used aeroplane wing designs, which translates into a reduction in fuel consumption of about 40–200 tonnes per year per aeroplane. Our morphogenesis process is generally applicable, not only to mechanical design, but also to flow systems3, antennas4, nano-optics5 and micro-systems6,7.
460 citations
Cites background from "Certified Reduced Basis Methods for..."
...However, procedures for reducing the number of degrees of freedom by model reduction (17, 18), reducing the number of iterations, and increasing the granularity by deeper hierarchy (19), are being developed and are expected to result in significant reduction in computing requirements in the near future....
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TL;DR: The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs) to accurately approximate the coefficients of the reduced model.
315 citations
TL;DR: The case studies demonstrate the importance of embedding physical constraints within learned models, and highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.
247 citations
Cites background from "Certified Reduced Basis Methods for..."
...See for example [2, 3, 4, 5, 6, 7, 8] for summaries of state-of-the-art in projection-based parametric model reduction methods and applications....
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References
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5,359 citations
TL;DR: These six volumes as mentioned in this paper compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers.
Abstract: These six volumes - the result of a ten year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the Methoden der mathematischen Physik by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to caluclate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every fact of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. Volumes 5 and 6 cover problems of Transport and Evolution.
2,137 citations
01 Jan 1987
1,890 citations
TL;DR: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs).
Abstract: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.
1,695 citations
Book•
04 Jul 2013
TL;DR: In this paper, the authors discuss the algebraic aspects of saddle point problems in Hilbert spaces and approximate saddle point approximations in Finite Element Methods (FEM) in function spaces.
Abstract: Preface.- Variational Formulations and Finite Element Methods.- Function Spaces and Finite Element Approximations.- Algebraic Aspects of Saddle Point Problems.- Saddle Point Problems in Hilbert spaces.- Approximation of Saddle Point Problems.- Complements: Stabilisation Methods, Eigenvalue Problems.- Mixed Methods for Elliptic Problems.- Incompressible Materials and Flow Problems.- Complements on Elasticity Problems.- Complements on Plate Problems.- Mixed Finite Elements for Electromagnetic Problems.- Index.
1,484 citations