A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

Three real-space discretization techniques in electronic structure calculations

A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the main approaches include finite-difference methods, various types of finite-elements and wavelets. This paper reports on the results of several code development projects that approach problems related to the electronic structure using these three different discretization methods. We review the ideas behind these methods, give examples of their applications, and discuss their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status solidi (b) - basic solid state physics" devoted to the CECAM workshop "State of the art developments and perspectives of real-space electronic structure techniques in condensed matter and molecular physics". v2: Minor stylistic and typographical changes, partly inspired by referee comment

Dense and sparse parallel linear algebra algorithms on graphics processing units

Una línea de desarrollo seguida en el campo de la supercomputación es el uso de procesadores de propósito específico para acelerar determinados tipos de cálculo. En esta tesis estudiamos el uso de tarjetas gráficas como aceleradores de la computación y lo aplicamos al ámbito del álgebra lineal. En particular trabajamos con la biblioteca SLEPc para resolver problemas de cálculo de autovalores en matrices de gran dimensión, y para aplicar funciones de matrices en los cálculos de aplicaciones científicas. SLEPc es una biblioteca paralela que se basa en el estándar MPI y está desarrollada con la premisa de ser escalable, esto es, de permitir resolver problemas más grandes al aumentar las unidades de procesado. El problema lineal de autovalores, Ax = lambda x en su forma estándar, lo abordamos con el uso de técnicas iterativas, en concreto con métodos de Krylov, con los que calculamos una pequeña porción del espectro de autovalores. Este tipo de algoritmos se basa en generar un subespacio de tamaño reducido (m) en el que proyectar el problema de gran dimensión (n), siendo m << n. Una vez se ha proyectado el problema, se resuelve este mediante métodos directos, que nos proporcionan aproximaciones a los autovalores del problema inicial que queríamos resolver. Las operaciones que se utilizan en la expansión del subespacio varían en función de si los autovalores deseados están en el exterior o en el interior del espectro. En caso de buscar autovalores en el exterior del espectro, la expansión se hace mediante multiplicaciones matriz-vector. Esta operación la realizamos en la GPU, bien mediante el uso de bibliotecas o mediante la creación de funciones que aprovechan la estructura de la matriz. En caso de autovalores en el interior del espectro, la expansión requiere resolver sistemas de ecuaciones lineales. En esta tesis implementamos varios algoritmos para la resolución de sistemas de ecuaciones lineales para el caso específico de matrices con estructura tridiagonal a bloques, que se ejecutan en GPU. En el cálculo de las funciones de matrices hemos de diferenciar entre la aplicación directa de una función sobre una matriz, f(A), y la aplicación de la acción de una función de matriz sobre un vector, f(A)b. El primer caso implica un cálculo denso que limita el tamaño del problema. El segundo permite trabajar con matrices dispersas grandes, y para resolverlo también hacemos uso de métodos de Krylov. La expansión del subespacio se hace mediante multiplicaciones matriz-vector, y hacemos uso de GPUs de la misma forma que al resolver autovalores. En este caso el problema proyectado comienza siendo de tamaño m, pero se incrementa en m en cada reinicio del método. La resolución del problema proyectado se hace aplicando una función de matriz de forma directa. Nosotros hemos implementado varios algoritmos para calcular las funciones de matrices raíz cuadrada y exponencial, en las que el uso de GPUs permite acelerar el cálculo.One line of development followed in the field of supercomputing is the use of specific purpose processors to speed up certain types of computations. In this thesis we study the use of graphics processing units as computer accelerators and apply it to the field of linear algebra. In particular, we work with the SLEPc library to solve large scale eigenvalue problems, and to apply matrix functions in scientific applications. SLEPc is a parallel library based on the MPI standard and is developed with the premise of being scalable, i.e. to allow solving larger problems by increasing the processing units. We address the linear eigenvalue problem, Ax = lambda x in its standard form, using iterative techniques, in particular with Krylov's methods, with which we calculate a small portion of the eigenvalue spectrum. This type of algorithms is based on generating a subspace of reduced size (m) in which to project the large dimension problem (n), being m << n. Once the problem has been projected, it is solved by direct methods, which provide us with approximations of the eigenvalues of the initial problem we wanted to solve. The operations used in the expansion of the subspace vary depending on whether the desired eigenvalues are from the exterior or from the interior of the spectrum. In the case of searching for exterior eigenvalues, the expansion is done by matrix-vector multiplications. We do this on the GPU, either by using libraries or by creating functions that take advantage of the structure of the matrix. In the case of eigenvalues from the interior of the spectrum, the expansion requires solving linear systems of equations. In this thesis we implemented several algorithms to solve linear systems of equations for the specific case of matrices with a block-tridiagonal structure, that are run on GPU. In the computation of matrix functions we have to distinguish between the direct application of a matrix function, f(A), and the action of a matrix function on a vector, f(A)b. The first case involves a dense computation that limits the size of the problem. The second allows us to work with large sparse matrices, and to solve it we also make use of Krylov's methods. The expansion of subspace is done by matrix-vector multiplication, and we use GPUs in the same way as when solving eigenvalues. In this case the projected problem starts being of size m, but it is increased by m on each restart of the method. The solution of the projected problem is done by directly applying a matrix function. We have implemented several algorithms to compute the square root and the exponential matrix functions, in which the use of GPUs allows us to speed up the computation.Una línia de desenvolupament seguida en el camp de la supercomputació és l'ús de processadors de propòsit específic per a accelerar determinats tipus de càlcul. En aquesta tesi estudiem l'ús de targetes gràfiques com a acceleradors de la computació i ho apliquem a l'àmbit de l'àlgebra lineal. En particular treballem amb la biblioteca SLEPc per a resoldre problemes de càlcul d'autovalors en matrius de gran dimensió, i per a aplicar funcions de matrius en els càlculs d'aplicacions científiques. SLEPc és una biblioteca paral·lela que es basa en l'estàndard MPI i està desenvolupada amb la premissa de ser escalable, açò és, de permetre resoldre problemes més grans en augmentar les unitats de processament. El problema lineal d'autovalors, Ax = lambda x en la seua forma estàndard, ho abordem amb l'ús de tècniques iteratives, en concret amb mètodes de Krylov, amb els quals calculem una xicoteta porció de l'espectre d'autovalors. Aquest tipus d'algorismes es basa a generar un subespai de grandària reduïda (m) en el qual projectar el problema de gran dimensió (n), sent m << n. Una vegada s'ha projectat el problema, es resol aquest mitjançant mètodes directes, que ens proporcionen aproximacions als autovalors del problema inicial que volíem resoldre. Les operacions que s'utilitzen en l'expansió del subespai varien en funció de si els autovalors desitjats estan en l'exterior o a l'interior de l'espectre. En cas de cercar autovalors en l'exterior de l'espectre, l'expansió es fa mitjançant multiplicacions matriu-vector. Aquesta operació la realitzem en la GPU, bé mitjançant l'ús de biblioteques o mitjançant la creació de funcions que aprofiten l'estructura de la matriu. En cas d'autovalors a l'interior de l'espectre, l'expansió requereix resoldre sistemes d'equacions lineals. En aquesta tesi implementem diversos algorismes per a la resolució de sistemes d'equacions lineals per al cas específic de matrius amb estructura tridiagonal a blocs, que s'executen en GPU. En el càlcul de les funcions de matrius hem de diferenciar entre l'aplicació directa d'una funció sobre una matriu, f(A), i l'aplicació de l'acció d'una funció de matriu sobre un vector, f(A)b. El primer cas implica un càlcul dens que limita la grandària del problema. El segon permet treballar amb matrius disperses grans, i per a resoldre-ho també fem ús de mètodes de Krylov. L'expansió del subespai es fa mitjançant multiplicacions matriu-vector, i fem ús de GPUs de la mateixa forma que en resoldre autovalors. En aquest cas el problema projectat comença sent de grandària m, però s'incrementa en m en cada reinici del mètode. La resolució del problema projectat es fa aplicant una funció de matriu de forma directa. Nosaltres hem implementat diversos algorismes per a calcular les funcions de matrius arrel quadrada i exponencial, en les quals l'ús de GPUs permet accelerar el càlcul.Lamas Daviña, A. (2018). Dense and sparse parallel linear algebra algorithms on graphics processing units [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/112425TESI

Multiscale Methods for Random Composite Materials

Simulation of material behaviour is not only a vital tool in accelerating product development and increasing design efficiency but also in advancing our fundamental understanding of materials. While homogeneous, isotropic materials are often simple to simulate, advanced, anisotropic materials pose a more sizeable challenge. In simulating entire composite components such as a 25m aircraft wing made by stacking several 0.25mm thick plies, finite element models typically exceed millions or even a billion unknowns. This problem is exacerbated by the inclusion of sub-millimeter manufacturing defects for two reasons. Firstly, a finer resolution is required which makes the problem larger. Secondly, defects introduce randomness. Traditionally, this randomness or uncertainty has been quantified heuristically since commercial codes are largely unsuccessful in solving problems of this size. This thesis develops a rigorous uncertainty quantification (UQ) framework permitted by a state of the art finite element package \texttt{dune-composites}, also developed here, designed for but not limited to composite applications. A key feature of this open-source package is a robust, parallel and scalable preconditioner \texttt{GenEO}, that guarantees constant iteration counts independent of problem size. It boasts near perfect scaling properties in both, a strong and a weak sense on over $15,000$ cores. It is numerically verified by solving industrially motivated problems containing upwards of 200 million unknowns. Equipped with the capability of solving expensive models, a novel stochastic framework is developed to quantify variability in part performance arising from localized out-of-plane defects. Theoretical part strength is determined for independent samples drawn from a distribution inferred from B-scans of wrinkles. Supported by literature, the results indicate a strong dependence between maximum misalignment angle and strength knockdown based on which an engineering model is presented to allow rapid estimation of residual strength bypassing expensive simulations. The engineering model itself is built from a large set of simulations of residual strength, each of which is computed using the following two step approach. First, a novel parametric representation of wrinkles is developed where the spread of parameters defines the wrinkle distribution. Second, expensive forward models are only solved for independent wrinkles using \texttt{dune-composites}. Besides scalability the other key feature of \texttt{dune-composites}, the \texttt{GenEO} coarse space, doubles as an excellent multiscale basis which is exploited to build high quality reduced order models that are orders of magnitude smaller. This is important because it enables multiple coarse solves for the cost of one fine solve. In an MCMC framework, where many solves are wasted in arriving at the next independent sample, this is a sought after quality because it greatly increases effective sample size for a fixed computational budget thus providing a route to high-fidelity UQ. This thesis exploits both, new solvers and multiscale methods developed here to design an efficient Bayesian framework to carry out previously intractable (large scale) simulations calibrated by experimental data. These new capabilities provide the basis for future work on modelling random heterogeneous materials while also offering the scope for building virtual test programs including nonlinear analyses, all of which can be implemented within a probabilistic setting

The Full Anisotropic Adaptive Fourier Modal Method and its Application to Periodic and Aperiodic Photonic Nanostructures

The thesis introduces the Fourier Modal Method as simulation tool for periodic photonic nanostructures, and extends the method towards the simulation of aperiodic structures using real and complex coordinate transformations. As exemplary cases, the method is applied for the characterization of a woodpile photonic crystal with the first complete photonic bandgap in the visible spectrum, and to the transmission properties of a long period fiber grating

Peridynamic-based multiscale frameworks for continuous and discontinuous material response

This PhD thesis aimed to develop two broad classes of multiscale frameworks for peridynamic theory to address two pressing needs: first is increased computational efficiency and the second is characterisation of heterogeneous media. To achieve these goals, two multiscale frameworks were proposed: model order reduction methodologies and homogenization frameworks. The model order reduction schemes were designed to improve computational efficiency, while the homogenization methodology aimed to provide frameworks for characterisation of heterogeneous materials within the peridynamic theory. Two specific model order reduction schemes were proposed, including a coarsening methodology and a model order reduction method based on static condensation. These schemes were applied to benchmark problems and shown to be effective in reducing the computational requirement of peridynamic models without compromising the fidelity of the simulation results. Additionally, a first-order nonlocal computational homogenization framework was proposed to characterise heterogeneous systems in the framework of peridynamics. This framework was utilised to characterise the behaviour of elastic and viscoelastic materials and materials with evolving microstructures. The results from these studies agreed with published results. The thesis achieved the goal of contributing to the development of efficient and accurate multiscale frameworks for peridynamic theory, which have potential applications in a wide range of fields, including materials science and engineering.This PhD thesis aimed to develop two broad classes of multiscale frameworks for peridynamic theory to address two pressing needs: first is increased computational efficiency and the second is characterisation of heterogeneous media. To achieve these goals, two multiscale frameworks were proposed: model order reduction methodologies and homogenization frameworks. The model order reduction schemes were designed to improve computational efficiency, while the homogenization methodology aimed to provide frameworks for characterisation of heterogeneous materials within the peridynamic theory. Two specific model order reduction schemes were proposed, including a coarsening methodology and a model order reduction method based on static condensation. These schemes were applied to benchmark problems and shown to be effective in reducing the computational requirement of peridynamic models without compromising the fidelity of the simulation results. Additionally, a first-order nonlocal computational homogenization framework was proposed to characterise heterogeneous systems in the framework of peridynamics. This framework was utilised to characterise the behaviour of elastic and viscoelastic materials and materials with evolving microstructures. The results from these studies agreed with published results. The thesis achieved the goal of contributing to the development of efficient and accurate multiscale frameworks for peridynamic theory, which have potential applications in a wide range of fields, including materials science and engineering

Computational Multiscale Methods

Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments

A Survey of High-Quality Computational Libraries and their Impactin Science and Engineering Applications

Recently, a number of important scientific and engineering problems have been successfully studied and solved by means of computational modeling and simulation. Many of these computational models and simulations benefited from the use of available software tools and libraries to achieve high performance and portability. In this article, we present a reference matrix of the performance of robust, reliable and widely used tools mapped to scientific and engineering applications that use them. We aim at regularly maintaining and disseminating this matrix to the computational science community. This matrix will contain information on state-of-the-art computational tools, their applications and their use