System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Computational time savings in multiscale fracture mechanics using model order reduction

Engineering problems are very often characterised by a large ratio between the scale of the structure and the scale at which the phenomena of interest need to be described. In fracture mechanics, the initiation and propagation of cracks is the result of localised microscopic phenomena. This local nature of fracture leads to large numerical models. Projection-based reduced order modelling is an increasingly popular technique for the fast solution of parametrised problems. However, traditional model order reduction methods are unable to reliably deal with either the initiation or the propagation of a crack or a local zone with high damage concentration. In this thesis, we look at the general problem of applying model order reduction to fracture/ damage mechanics, in the pursuit of rationalising the computational time involved in these kind of simulations. The first contribution of this thesis is the development of a reduced-order modelling for computational homogenisation, which is a general multiscale method used to take microscopic data into account when deriving an engineeringscale model. A specific strategy is used to reduce the cost of solving the representative element volume (RVE) boundary value problem traditionally formulated in this method. The second contribution was made by developing a partitioned reduced-order procedure for the case of parametrised nonlinear material deformations involving a local lack of correlation, which typically happens with fracture. The method allows to reduce the regions undergoing little non-linearities whilst computational work can be concentrated on regions of high non-linearity

Algebraic Multigrid for Meshfree Methods

This thesis deals with the development of a new Algebraic Multigrid method (AMG) for the solution of linear systems arising from Generalized Finite Difference Methods (GFDM). In particular, we consider the Finite Pointset Method, which is based on GFDM. Being a meshfree method, FPM does not rely on a mesh and can therefore deal with moving geometries and free surfaces is a natural way and it does not require the generation of a mesh before the actual simulation. In industrial use cases the size of the linear systems often becomes large, which means that classical linear solvers often become the bottleneck in terms of simulation run time, because their convergence rate depends on the discretization size. Multigrid methods have proven to be very efficient linear solvers in the domain of mesh-based methods. Their convergence is independent of the discretization size, yielding a run time that only scales linearly with the problem size. AMG methods are a natural candidate for the solution of the linear systems arising in the FPM, as this thesis will show. They need to be tuned to the specific characteristics of GFDM, though. The AMG methods that are developed in this thesis achieve a speed-up of up to 33x compared to the classical linear solvers and therefore allow much more accurate simulations in the future.Diese Dissertation beschäftigt sich mit der Entwicklung einer neuen Algebraischen Mehrgittermethode für die Lösung linearer Gleichungssysteme aus Generalisierten Finite Differenzen Methoden. Im Speziellen betrachten wir die sogenannte Finite Pointset Method, eine gitterfreie Lagrange Methode, welche auf Generalisierten Finite Differenzen Methoden basiert. Die Finite Pointset Method wurde insbesondere für Simulationen von Vorgängen mit freien Oberflächen und bewegten Geometrien entwickelt, bei denen der gitterfreie Charakter der Methode besonders große Vorteile liefert: An den freien Oberflächen und nahe der Geometrie muss zu keinem Zeitpunkt – auch nicht zu Beginn der Simulation – ein Gitter erstellt oder angepasst werden. Dies ist ein großer Vorteil gegenüber klassischen gitterbasierten Methoden. Wie in gitterbasierten Methoden entstehen auch in der Finite Pointset Method und anderen Generalisierten Finite Differenzen Methoden große, dünn besetze lineare Gleichungssysteme. Das Lösen dieser Gleichungssysteme wird bei fein aufgelösten Simulationen, wie sie in der Industrie oft nötig sind, schnell zum zeitlichen Flaschenhals der Gesamtsimulation. Ohne eine geeignete Methode zur Lösung dieser Gleichungssysteme dauern Simulationen oft sehr lange oder sind praktisch nicht durchführbar. Auch kann es vorkommen, dass klassische Lösungsverfahren divergieren und die Simulation damit unmöglich wird. Im Kontext von gitterbasierten Methoden sind Mehrgittermethoden ein etabliertes Werkzeug, um die entstehenden linearen Gleichungssysteme effizient und robust zu lösen. Besonders hervorzuheben ist dabei die lineare Skalierbarkeit dieser Methoden in der Größe der Matrix. Damit eignen sie sich besonders für fein aufgelöste Simulationen. Algebraische Mehrgittermethoden sind natürliche Kandidaten für die Lösung der Gleichungssysteme aus Generalisierten Finite Differenzen Methoden, wie diese Dissertation zeigen wird. Außerdem entwickeln wir eine neue Algebraische Mehrgittermethode, die auf den Einsatz in der Finite Pointset Method zugeschnitten ist und die Besonderheiten dieser Methode beachtet. Dazu zählen die Eigenschaften der einzelnen Matrizen, die wir ebenfalls analysieren werden, und auch die Veränderung der Matrizen über mehrere Zeitschritte hinweg, die im Vergleich mit gitterbasierten Verfahren eine größere Schwierigkeit darstellt. Wir evaluieren unsere neue Methode anhand von akademischen und realen Beispielen, sowohl mit nur einem Prozess als auch mit mehreren (MPI-)Prozessen. Die hier neu entwickelte Algebraische Mehrgittermethode ist um ein Vielfaches schneller als klassische Verfahren zur Lösung linearer Gleichungssysteme und erlaubt damit neue, genauere Simulationen mit gitterfreien Methoden

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Reduction of Multivariate Mixtures and Its Applications

We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation with fewer terms, up to user-selected accuracy. Our algorithm employs the well-known pivoted Cholesky decomposition of the Gram matrix constructed using terms of the mixture. Importantly, the multivariate mixtures do not have to be a separated representation of a function and complexity of the algorithm is independent of the number of variables (dimensions). The algorithm requires $\mathcal{O}\left(r^{2}N\right)$ operations, where $N$ is the initial number of terms in a multivariate mixture and $r$ is the number of selected terms. Due to the condition number of the Gram matrix, the resulting accuracy is limited to about $1/2$ digits of the used floating point arithmetic. We also consider two additional reduction algorithms for the same purpose. The first algorithm is based on orthogonalization of the multivariate mixture and have a similar performance as the approach based on Cholesky factorization. The second algorithm yields a better accuracy, but currently in high dimensions is only applicable to multivariate mixtures in a separated representation. We use the reduction algorithm to develop a new adaptive numerical method for solving differential and integral equations in quantum chemistry. We demonstrate the performance of this approach by solving the Hartree-Fock equations in two cases of small molecules. We also describe a number of initial applications of the reduction algorithm to solve partial differential and integral equations and to address several problems in data sciences. For data science applications in high dimensions we consider kernel density estimation (KDE) approach for constructing a probability density function (PDF) of a cloud of points, a far-field kernel summation method and the construction of equivalent sources for non-oscillatory kernels (used in both, computational physics and data science) and, finally, show how to use the reduction algorithm to produce seeds for subdividing a cloud of points into groups

Study of machine learning techniques for modeling large structural vibrations

The main purpose of the present work can be divided into two separate objectives. The first one and most important is to study the different computational techniques that are used in the structural vibrations modelling field in both the linear and nonlinear regimes, as well as their methodology of implementation. In particular, this thesis places special emphasis on a specific order-reducing model called hyperreduction. This approach makes use of Machine Learning techniques such as the Singular Value Decomposition to solve an optimization problem that selects the set of reduced elements to be integrated from the entirety of finite elements of the mesh. To do so, several simulations are launched an analysed throughout the study. These simulations intend to estimate the dynamic response that a simple 2D cantilever beam would suffer when subjected under different types of boundary conditions. Such a task is performed thanks to a Finite Element Method based on the Galerkin approximation. With the help of Matlab, the semi-discrete equation of motion that governs the dynamic response of the system can be computationally integrated making use of the Newmark-Bossak scheme and the Newton-Raphson algorithm. The results obtained from the Finite Element study will be compared with the analytical solution deduced from the modal analysis decomposition technique and also with the results obtained from the hyperreduction stage. On the other hand, the second goal of this thesis is to study the dynamic response of the above-mentioned structure when subjected to the action of real wind gusts. This study is included in the list of simulations to be launched in this project. In this sense, the Fast Fourier Transform (FFT) is also employed in order to obtain a continuous representation in the time domain of a discrete data set representing real measurements of wind speed. The main conclusion of this work is that the hyperreduction method is significantly more efficient than a standard Finite Element method since it requires less than 1% of the finite elements of the mesh to integrate the semi-discrete equation of motion, thus contributing to achieving reductions of up to 96% in computational time within an accuracy tolerance of the order of 10−3 . Therefore, this technique is a highly valuable tool in any engineering field that involves simulations with very refined meshes or with complex nonlinear conditions among others, since they require great computational efforts

Deep learning applied to computational mechanics: A comprehensive review, state of the art, and the classics

Three recent breakthroughs due to AI in arts and science serve as motivation: An award winning digital image, protein folding, fast matrix multiplication. Many recent developments in artificial neural networks, particularly deep learning (DL), applied and relevant to computational mechanics (solid, fluids, finite-element technology) are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. Hybrid methods combine traditional PDE discretizations with ML methods either (1) to help model complex nonlinear constitutive relations, (2) to nonlinearly reduce the model order for efficient simulation (turbulence), or (3) to accelerate the simulation by predicting certain components in the traditional integration methods. Here, methods (1) and (2) relied on Long-Short-Term Memory (LSTM) architecture, with method (3) relying on convolutional neural networks. Pure ML methods to solve (nonlinear) PDEs are represented by Physics-Informed Neural network (PINN) methods, which could be combined with attention mechanism to address discontinuous solutions. Both LSTM and attention architectures, together with modern and generalized classic optimizers to include stochasticity for DL networks, are extensively reviewed. Kernel machines, including Gaussian processes, are provided to sufficient depth for more advanced works such as shallow networks with infinite width. Not only addressing experts, readers are assumed familiar with computational mechanics, but not with DL, whose concepts and applications are built up from the basics, aiming at bringing first-time learners quickly to the forefront of research. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics, even in well-known references. Positioning and pointing control of a large-deformable beam is given as an example.Comment: 275 pages, 158 figures. Appeared online on 2023.03.01 at CMES-Computer Modeling in Engineering & Science

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described