The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

Software concepts and algorithms for an efficient and scalable parallel finite element method

Software packages for the numerical solution of partial differential equations (PDEs) using the finite element method are important in different fields of research. The basic data structures and algorithms change in time, as the user\'s requirements are growing and the software must efficiently use the newest highly parallel computing systems. This is the central point of this work. To make efficiently use of parallel computing systems with growing number of independent basic computing units, i.e.~CPUs, we have to combine data structures and algorithms from different areas of mathematics and computer science. Two crucial parts are a distributed mesh and parallel solver for linear systems of equations. For both there exists multiple independent approaches. In this work we argue that it is necessary to combine both of them to allow for an efficient and scalable implementation of the finite element method. First, we present concepts, data structures and algorithms for distributed meshes, which allow for local refinement. The central point of our presentation is to provide arbitrary geometrical information of the mesh and its distribution to the linear solver. A large part of the overall computing time of the finite element method is spend by the linear solver. Thus, its parallelization is of major importance. Based on the presented concept for distributed meshes, we preset several different linear solver methods. Hereby we concentrate on general purpose linear solver, which makes only little assumptions about the systems to be solver. For this, a new FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) method is proposed. Those the standard FETI-DP method is quasi optimal from a mathematical point of view, its not possible to implement it efficiently for a large number of processors (> 10,000). The main reason is a relatively small but globally distributed coarse mesh problem. To circumvent this problem, we propose a new multilevel FETI-DP method which hierarchically decompose the coarse grid problem. This leads to a more local communication pattern for solver the coarse grid problem and makes it possible to scale for a large number of processors. Besides the parallelization of the finite element method, we discuss an approach to speed up serial computations of existing finite element packages. In many computations the PDE to be solved consists of more than one variable. This is especially the case in multi-physics modeling. Observation show that in many of these computation the solution structure of the variables is different. But in the standard finite element method, only one mesh is used for the discretization of all variables. We present a multi-mesh finite element method, which allows to discretize a system of PDEs with two independently refined meshes.Softwarepakete zur numerischen Lösung partieller Differentialgleichungen mit Hilfe der Finiten-Element-Methode sind in vielen Forschungsbereichen ein wichtiges Werkzeug. Die dahinter stehenden Datenstrukturen und Algorithmen unterliegen einer ständigen Neuentwicklung um den immer weiter steigenden Anforderungen der Nutzergemeinde gerecht zu werden und um neue, hochgradig parallel Rechnerarchitekturen effizient nutzen zu können. Dies ist auch der Kernpunkt dieser Arbeit. Um parallel Rechnerarchitekturen mit einer immer höher werdenden Anzahl an von einander unabhängigen Recheneinheiten, z.B.~Prozessoren, effizient Nutzen zu können, müssen Datenstrukturen und Algorithmen aus verschiedenen Teilgebieten der Mathematik und Informatik entwickelt und miteinander kombiniert werden. Im Kern sind dies zwei Bereiche: verteilte Gitter und parallele Löser für lineare Gleichungssysteme. Für jedes der beiden Teilgebiete existieren unabhängig voneinander zahlreiche Ansätze. In dieser Arbeit wird argumentiert, dass für hochskalierbare Anwendungen der Finiten-Elemente-Methode nur eine Kombination beider Teilgebiete und die Verknüpfung der darunter liegenden Datenstrukturen eine effiziente und skalierbare Implementierung ermöglicht. Zuerst stellen wir Konzepte vor, die parallele verteile Gitter mit entsprechenden Adaptionstrategien ermöglichen. Zentraler Punkt ist hier die Informationsaufbereitung für beliebige Löser linearer Gleichungssysteme. Beim Lösen partieller Differentialgleichung mit der Finiten Elemente Methode wird ein großer Teil der Rechenzeit für das Lösen der dabei anfallenden linearen Gleichungssysteme aufgebracht. Daher ist deren Parallelisierung von zentraler Bedeutung. Basierend auf dem vorgestelltem Konzept für verteilten Gitter, welches beliebige geometrische Informationen für die linearen Löser aufbereiten kann, präsentieren wir mehrere unterschiedliche Lösermethoden. Besonders Gewicht wird dabei auf allgemeine Löser gelegt, die möglichst wenig Annahmen über das zu lösende System machen. Hierfür wird die FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) Methode weiterentwickelt. Obwohl die FETI-DP Methode vom mathematischen Standpunkt her als quasi-optimal bezüglich der parallelen Skalierbarkeit gilt, kann sie für große Anzahl an Prozessoren (> 10.000) nicht mehr effizient implementiert werden. Dies liegt hauptsächlich an einem verhältnismäßig kleinem aber global verteilten Grobgitterproblem. Wir stellen eine Multilevel FETI-DP Methode vor, die dieses Problem durch eine hierarchische Komposition des Grobgitterproblems löst. Dadurch wird die Kommunikation entlang des Grobgitterproblems lokalisiert und die Skalierbarkeit der FETI-DP Methode auch für große Anzahl an Prozessoren sichergestellt. Neben der Parallelisierung der Finiten-Elemente-Methode beschäftigen wir uns in dieser Arbeit mit der Ausnutzung von bestimmten Voraussetzung um auch die sequentielle Effizienz bestehender Implementierung der Finiten-Elemente-Methode zu steigern. In vielen Fällen müssen partielle Differentialgleichungen mit mehreren Variablen gelöst werden. Sehr häufig ist dabei zu beobachten, insbesondere bei der Modellierung mehrere miteinander gekoppelter physikalischer Phänomene, dass die Lösungsstruktur der unterschiedlichen Variablen entweder schwach oder vollständig voneinander entkoppelt ist. In den meisten Implementierungen wird dabei nur ein Gitter zur Diskretisierung aller Variablen des Systems genutzt. Wir stellen eine Finite-Elemente-Methode vor, bei der zwei unabhängig voneinander verfeinerte Gitter genutzt werden können um ein System partieller Differentialgleichungen zu lösen

Accuracy, Scalability, and Efficiency of Mixed-Element USM3D for Benchmark Three-Dimensional Flows

The unstructured, mixed-element, cell-centered, finite-volume flow solver USM3D is enhanced with new capabilities including parallelization, line generation for general unstructured grids, improved discretization scheme, and optimized iterative solver. The paper reports on the new developments to the flow solver and assesses the accuracy, scalability, and efficiency. The USM3D assessments are conducted using a baseline method and the recent hierarchical adaptive nonlinear iteration method framework. Two benchmark turbulent flows, namely, a subsonic separated flow around a three-dimensional hemisphere-cylinder configuration and a transonic flow around the ONERA M6 wing are considered

Automatische Codegenerierung für Massiv Parallele Applikationen in der Numerischen Strömungsmechanik

Solving partial differential equations (PDEs) is a fundamental challenge in many application domains in industry and academia alike. With increasingly large problems, efficient and highly scalable implementations become more and more crucial. Today, facing this challenge is more difficult than ever due to the increasingly heterogeneous hardware landscape. One promising approach is developing domain‐specific languages (DSLs) for a set of applications. Using code generation techniques then allows targeting a range of hardware platforms while concurrently applying domain‐specific optimizations in an automated fashion. The present work aims to further the state of the art in this field. As domain, we choose PDE solvers and, in particular, those from the group of geometric multigrid methods. To avoid having a focus too broad, we restrict ourselves to methods working on structured and patch‐structured grids. We face the challenge of handling a domain as complex as ours, while providing different abstractions for diverse user groups, by splitting our external DSL ExaSlang into multiple layers, each specifying different aspects of the final application. Layer 1 is designed to resemble LaTeX and allows inputting continuous equations and functions. Their discretization is expressed on layer 2. It is complemented by algorithmic components which can be implemented in a Matlab‐like syntax on layer 3. All information provided to this point is summarized on layer 4, enriched with particulars about data structures and the employed parallelization. Additionally, we support automated progression between the different layers. All ExaSlang input is processed by our jointly developed Scala code generation framework to ultimately emit C++ code. We particularly focus on how to generate applications parallelized with, e.g., MPI and OpenMP that are able to run on workstations and large‐scale cluster alike. We showcase the applicability of our approach by implementing simple test problems, like Poisson’s equation, as well as relevant applications from the field of computational fluid dynamics (CFD). In particular, we implement scalable solvers for the Stokes, Navier‐Stokes and shallow water equations (SWE) discretized using finite differences (FD) and finite volumes (FV). For the case of Navier‐Stokes, we also extend our implementation towards non‐uniform grids, thereby enabling static mesh refinement, and advanced effects such as the simulated fluid being non‐Newtonian and non‐isothermal

New strategies for the aerodynamic design optimization of aeronautical configurations through soft-computing techniques

Premio Extraordinario de Doctorado de la UAH en 2013Lozano Rodríguez, Carlos, codir.This thesis deals with the improvement of the optimization process in the aerodynamic design of aeronautical configurations. Nowadays, this topic is of great importance in order to allow the European aeronautical industry to reduce their development and operational costs, decrease the time-to-market for new aircraft, improve the quality of their products and therefore maintain their competitiveness. Within this thesis, a study of the state-of-the-art of the aerodynamic optimization tools has been performed, and several contributions have been proposed at different levels: -One of the main drawbacks for an industrial application of aerodynamic optimization tools is the huge requirement of computational resources, in particular, for complex optimization problems, current methodological approaches would need more than a year to obtain an optimized aircraft. For this reason, one proposed contribution of this work is focused on reducing the computational cost by the use of different techniques as surrogate modelling, control theory, as well as other more software-related techniques as code optimization and proper domain parallelization, all with the goal of decreasing the cost of the aerodynamic design process. -Other contribution is related to the consideration of the design process as a global optimization problem, and, more specifically, the use of evolutionary algorithms (EAs) to perform a preliminary broad exploration of the design space, due to their ability to obtain global optima. Regarding this, EAs have been hybridized with metamodels (or surrogate models), in order to substitute expensive CFD simulations. In this thesis, an innovative approach for the global aerodynamic optimization of aeronautical configurations is proposed, consisting of an Evolutionary Programming algorithm hybridized with a Support Vector regression algorithm (SVMr) as a metamodel. Specific issues as precision, dataset training size, geometry parameterization sensitivity and techniques for design of experiments are discussed and the potential of the proposed approach to achieve innovative shapes that would not be achieved with traditional methods is assessed. -Then, after a broad exploration of the design space, the optimization process is continued with local gradient-based optimization techniques for a finer improvement of the geometry. Here, an automated optimization framework is presented to address aerodynamic shape design problems. Key aspects of this framework include the use of the adjoint methodology to make the computational requirements independent of the number of design variables, and Computer Aided Design (CAD)-based shape parameterization, which uses the flexibility of Non-Uniform Rational B-Splines (NURBS) to handle complex configurations. The mentioned approach is applied to the optimization of several test cases and the improvements of the proposed strategy and its ability to achieve efficient shapes will complete this study

New strategies for the aerodynamic design optimization of aeronautical configurations through soft-computing techniques

Premio Extraordinario de Doctorado de la UAH en 2013Lozano Rodríguez, Carlos, codir.This thesis deals with the improvement of the optimization process in the aerodynamic design of aeronautical configurations. Nowadays, this topic is of great importance in order to allow the European aeronautical industry to reduce their development and operational costs, decrease the time-to-market for new aircraft, improve the quality of their products and therefore maintain their competitiveness. Within this thesis, a study of the state-of-the-art of the aerodynamic optimization tools has been performed, and several contributions have been proposed at different levels: -One of the main drawbacks for an industrial application of aerodynamic optimization tools is the huge requirement of computational resources, in particular, for complex optimization problems, current methodological approaches would need more than a year to obtain an optimized aircraft. For this reason, one proposed contribution of this work is focused on reducing the computational cost by the use of different techniques as surrogate modelling, control theory, as well as other more software-related techniques as code optimization and proper domain parallelization, all with the goal of decreasing the cost of the aerodynamic design process. -Other contribution is related to the consideration of the design process as a global optimization problem, and, more specifically, the use of evolutionary algorithms (EAs) to perform a preliminary broad exploration of the design space, due to their ability to obtain global optima. Regarding this, EAs have been hybridized with metamodels (or surrogate models), in order to substitute expensive CFD simulations. In this thesis, an innovative approach for the global aerodynamic optimization of aeronautical configurations is proposed, consisting of an Evolutionary Programming algorithm hybridized with a Support Vector regression algorithm (SVMr) as a metamodel. Specific issues as precision, dataset training size, geometry parameterization sensitivity and techniques for design of experiments are discussed and the potential of the proposed approach to achieve innovative shapes that would not be achieved with traditional methods is assessed. -Then, after a broad exploration of the design space, the optimization process is continued with local gradient-based optimization techniques for a finer improvement of the geometry. Here, an automated optimization framework is presented to address aerodynamic shape design problems. Key aspects of this framework include the use of the adjoint methodology to make the computational requirements independent of the number of design variables, and Computer Aided Design (CAD)-based shape parameterization, which uses the flexibility of Non-Uniform Rational B-Splines (NURBS) to handle complex configurations. The mentioned approach is applied to the optimization of several test cases and the improvements of the proposed strategy and its ability to achieve efficient shapes will complete this study

A hierarchical parallel implementation model for algebra-based CFD simulations on hybrid supercomputers

(English) Continuous enhancement in hardware technologies enables scientific computing to advance incessantly and reach further aims. Since the start of the global race for exascale high-performance computing (HPC), massively-parallel devices of various architectures have been incorporated into the newest supercomputers, leading to an increasing hybridization of HPC systems. In this context of accelerated innovation, software portability and efficiency become crucial. Traditionally, scientific computing software development is based on calculations in iterative stencil loops (ISL) over a discretized geometry—the mesh. Despite being intuitive and versatile, the interdependency between algorithms and their computational implementations in stencil applications usually results in a large number of subroutines and introduces an inevitable complexity when it comes to portability and sustainability. An alternative is to break the interdependency between algorithm and implementation to cast the calculations into a minimalist set of kernels. The portable implementation model that is the object of this thesis is not restricted to a particular numerical method or problem. However, owing to the CTTC's long tradition in computational fluid dynamics (CFD) and without loss of generality, this work is targeted to solve transient CFD simulations. By casting discrete operators and mesh functions into (sparse) matrices and vectors, it is shown that all the calculations in a typical CFD algorithm boil down to the following basic linear algebra subroutines: the sparse matrix-vector product, the linear combination of vectors, and the dot product. The proposed formulation eases the deployment of scientific computing software in massively parallel hybrid computing systems and is demonstrated in the large-scale, direct numerical simulation of transient turbulent flows.(Català) La millora contínua en tecnologies de la informàtica possibilita a la comunitat de computació científica avançar incessantment i assolir ulteriors objectius. Des de l'inici de la cursa global per a la computació d'alt rendiment (HPC) d'exa-escala, s'han incorporat dispositius massivament paral·lels de diverses arquitectures als supercomputadors més nous, donant lloc a una creixent hibridació dels sistemes HPC. En aquest context d'innovació accelerada, la portabilitat i l'eficiència del programari esdevenen crucials. Tradicionalment, el desenvolupament de programari informàtic científic es basa en càlculs en bucles de patrons iteratius (ISL) sobre una geometria discretitzada: la malla. Tot i ser intuïtiva i versàtil, la interdependència entre algorismes i les seves implementacions computacionals en aplicacions de patrons sol donar lloc a un gran nombre de subrutines i introdueix una complexitat inevitable quan es tracta de portabilitat i sostenibilitat. Una alternativa és trencar la interdependència entre l'algorisme i la implementació per reduir els càlculs a un conjunt minimalista de subrutines. El model d'implementació portable objecte d'aquesta tesi no es limita a un mètode o problema numèric concret. No obstant això, i a causa de la llarga tradició del CTTC en dinàmica de fluids computacional (CFD) i sense pèrdua de generalitat, aquest treball està dirigit a resoldre simulacions CFD transitòries. Mitjançant la conversió d'operadors discrets i funcions de malla en matrius (disperses) i vectors, es demostra que tots els càlculs d'un algorisme CFD típic es redueixen a les següents subrutines bàsiques d'àlgebra lineal: el producte dispers matriu-vector, la combinació lineal de vectors, i el producte escalar. La formulació proposada facilita el desplegament de programari de computació científica en sistemes informàtics híbrids massivament paral·lels i es demostra el seu rendiment en la simulació numèrica directa de gran escala de fluxos turbulents transitoris.Enginyeria tèrmic

A hierarchical parallel implementation model for algebra-based CFD simulations on hybrid supercomputers

(English) Continuous enhancement in hardware technologies enables scientific computing to advance incessantly and reach further aims. Since the start of the global race for exascale high-performance computing (HPC), massively-parallel devices of various architectures have been incorporated into the newest supercomputers, leading to an increasing hybridization of HPC systems. In this context of accelerated innovation, software portability and efficiency become crucial. Traditionally, scientific computing software development is based on calculations in iterative stencil loops (ISL) over a discretized geometry—the mesh. Despite being intuitive and versatile, the interdependency between algorithms and their computational implementations in stencil applications usually results in a large number of subroutines and introduces an inevitable complexity when it comes to portability and sustainability. An alternative is to break the interdependency between algorithm and implementation to cast the calculations into a minimalist set of kernels. The portable implementation model that is the object of this thesis is not restricted to a particular numerical method or problem. However, owing to the CTTC's long tradition in computational fluid dynamics (CFD) and without loss of generality, this work is targeted to solve transient CFD simulations. By casting discrete operators and mesh functions into (sparse) matrices and vectors, it is shown that all the calculations in a typical CFD algorithm boil down to the following basic linear algebra subroutines: the sparse matrix-vector product, the linear combination of vectors, and the dot product. The proposed formulation eases the deployment of scientific computing software in massively parallel hybrid computing systems and is demonstrated in the large-scale, direct numerical simulation of transient turbulent flows.(Català) La millora contínua en tecnologies de la informàtica possibilita a la comunitat de computació científica avançar incessantment i assolir ulteriors objectius. Des de l'inici de la cursa global per a la computació d'alt rendiment (HPC) d'exa-escala, s'han incorporat dispositius massivament paral·lels de diverses arquitectures als supercomputadors més nous, donant lloc a una creixent hibridació dels sistemes HPC. En aquest context d'innovació accelerada, la portabilitat i l'eficiència del programari esdevenen crucials. Tradicionalment, el desenvolupament de programari informàtic científic es basa en càlculs en bucles de patrons iteratius (ISL) sobre una geometria discretitzada: la malla. Tot i ser intuïtiva i versàtil, la interdependència entre algorismes i les seves implementacions computacionals en aplicacions de patrons sol donar lloc a un gran nombre de subrutines i introdueix una complexitat inevitable quan es tracta de portabilitat i sostenibilitat. Una alternativa és trencar la interdependència entre l'algorisme i la implementació per reduir els càlculs a un conjunt minimalista de subrutines. El model d'implementació portable objecte d'aquesta tesi no es limita a un mètode o problema numèric concret. No obstant això, i a causa de la llarga tradició del CTTC en dinàmica de fluids computacional (CFD) i sense pèrdua de generalitat, aquest treball està dirigit a resoldre simulacions CFD transitòries. Mitjançant la conversió d'operadors discrets i funcions de malla en matrius (disperses) i vectors, es demostra que tots els càlculs d'un algorisme CFD típic es redueixen a les següents subrutines bàsiques d'àlgebra lineal: el producte dispers matriu-vector, la combinació lineal de vectors, i el producte escalar. La formulació proposada facilita el desplegament de programari de computació científica en sistemes informàtics híbrids massivament paral·lels i es demostra el seu rendiment en la simulació numèrica directa de gran escala de fluxos turbulents transitoris.Postprint (published version