Application of a Hierarchical Chromosome Based Genetic Algorithm to the Problem of Finding Optimal Initial Meshes for the Self-Adaptive hp-FEM

The paper presents an algorithm for finding the optimal initial mesh for the self-adaptive hp Finite Element Method (hp-FEM) calculations. We propose the application of the hierarchical chromosome based genetic algorithm for optimal selection of the initial mesh. The selection of the optimal initial mesh will optimize the convergence rate of the numerical error of the solution over the sequence of meshes generated by the self-adaptive hp-FEM. This is especially true in the case when material data are selected as a result of some stochastic algorithm and it is not possible to design optimal initial mesh by hand. The algorithm has been tested on the non-stationary mass transport problem modeling phase transition phenomenon

Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation

This paper presents an a-posteriori goal-oriented error analysis for a numerical approximation of the steady Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in position dependence. We derive computable error estimates and bounds for general target functionals of solutions of the steady Boltzmann equation based on the DGFE moment approximation. The a-posteriori error estimates and bounds are used to guide a model adaptive algorithm for optimal approximations of the goal functional in question. We present results for one-dimensional heat transfer and shock structure problems where the moment model order is refined locally in space for optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131

Petri Nets Modeling of Dead-End Refinement Problems in a 3D Anisotropic hp-Adaptive Finite Element Method

We consider two graph grammar based Petri nets models for anisotropic refinements of three dimensional hexahedral grids. The first one detects possible dead-end problems during the graph grammar based anisotropic refinements of the mesh. The second one employs an enhanced graph grammar model that is actually dead-end free. We apply the resulting algorithm to the simulation of resistivity logging measurements for estimating the location of underground oil and/or gas formations. The graph grammar based Petri net models allow to fix the self-adaptive mesh refinement algorithm and finish the adaptive computations with the required accuracy needed by the numerical solution

A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters

This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in PDE-based models and correspond to quantities such as density or pressure fields, elasto-plastic moduli and internal variables in solid mechanics, conductivity fields in heat diffusion problems, permeability fields in fluid flow through porous media etc. The proposed model has all the advantages of traditional Bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty. In contrast to existing approaches it utilizes a parsimonious, non-parametric formulation that favors sparse representations and whose complexity can be determined from the data. The proposed framework in non-intrusive and makes use of a sequence of forward solvers operating at various resolutions. As a result, inexpensive, coarse solvers are used to identify the most salient features of the unknown field(s) which are subsequently enriched by invoking solvers operating at finer resolutions. This leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy. It is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in Markov Chain Monte Carlo schemes

Finite element simulation of additive manufacturing with enhanced accuracy

Tesi en modalitat de compendi de publicacionsThis thesis develops numerical methods to improve the accuracy and computational efficiency of the part-scale simulation of Additive Manufacturing (AM) (or 3D printing) metal processes. AM is characterized by multiple scales in space and time, as well as multiple complex physics that occur in three-dimensional growing-in-time geometries, making its simulation a remarkable computational challenge. To this end, the computational framework is built by addressing four key topics: (1) a Finite Element technology with enhanced stress/strain accuracy including the incompressible limit; (2) an Adaptive Mesh Refinement (AMR) strategy accounting for geometric and solution accuracies; (3) a coarsening correction strategy to avoid loss of information in the coarsening AMR procedure, and (4) a GCodebased simulation tool that uses the exact geometric and process parameters data provided to the actual AM machinery. In this context, the mixed displacement/deviatoric-strain/pressure u/e/p FE formulation in (1) is adopted to solve incompressible problems resulting from the isochoric plastic flow in the Von Mises criterion typical of metals. The enhanced stress/strain accuracy of the u/e/p over the standard and u/p FE formulations is verified in a set of numerical benchmarks in iso-thermal and non-isothermal conditions. A multi-criteria AMR strategy in (2) is used to improve computational efficiency while keeping the number of FEs controlled and without the strictness of imposing the commonly adopted 2:1 balance scheme. Avoiding this enables to use high jumps on the refinement level between adjacent FEs; this improves the mesh resolution on the region of interest and keeps the mesh coarse elsewhere. Moving the FE solution from a fine mesh to a coarse mesh introduces loss of information. To prevent this, a coarsening correction strategy presented in (3) restores the fine solution in the coarse mesh, providing computational cost reduction and keeping the accuracy of the fine mesh solution accuracy. Lastly, design flexibility is one of the main advantages of AM over traditional manufacturing processes. This flexibility is observed in the design of complex components and the possibility to change the process parameters, i.e. power input, speed, waiting pauses, among others, throughout the process. In (4) a GCode-based simulation tool that replicates the exact path travelled and process parameters delivered to the AM machiney is developed. Furthermore, the GCode-based tool together with the AMR strategy allows to automatically generate an embedded fitted cartesian FE mesh for the evolving domain and removes the challenging task of mesh manipulation by the end-user. The FE framework is built on a high-performance computing environment. This framework enables to accelerate the process-to-performance understanding and to minimize the number of trial-and-error experiments, two key aspects to exploit the technology in the industrial environment.Esta tesis tiene como objetivo desarrollar métodos numéricos para mejorar la precisión y eficiencia computacionales en simulaciones de piezas fabricadas mediante Manufactura Aditiva (MA), también conocida como Impresión 3D. La manufactura aditiva es un problema complejo que involucra múltiples fenómenos físicos, que se desarolla en múltiples escalas, y cuya geometría evoluciona en el tiempo. Para tal fin, se plantean cuatro objetivos: (1) Desarrollo de una tecnología de elementos finitos para capturar con mayor precisión tanto tensiones como deformaciones en casos en el que el material tiene comportamiento isocórico; (2) Una estrategia de adaptividad de malla (AMR), que busca modificar la malla teniendo en cuenta la geometría y los errores en la solución numérica; (3) Una estrategia para minimizar la aproximación numérica durante el engrosamiento (coarsening) de la malla, crucial en la reducción de tiempos de cómputo en casos de piezas de grandes dimensiones; y (4) Un marco de simulación basado en la lectura de ficheros GCode, ampliamente usado por maquinaria de impresión en procesos de manufactura aditiva, un formato que no sólo proporciona los datos asociados a la geometría, sino también los parámetros de proceso. Con respecto a (1), esta tesis propone el uso de una formulación mixta en desplazamientos /deformación-desviadora / presión (u/e/p), para simular la deposición de materiales con deformación inelástica isocórica, como ocurre en los metales. En cuanto a la medición de la precisión en el cálculo de las tensiones y las deformaciones, en esta tesis se realiza un amplio número de experimentos tanto en condiciones isotérmicas como no isotérmicas para establecer una comparativa entre las dos formulaciones mixtas, u/e/p y u/p. Con respecto a (2), para mejorar la eficiencia computacional manteniendo acotado el número total de elementos finitos, se desarrolla una novedosa estrategia multicriterio de refinamiento adaptativo. Esta estrategia no se restringe a mallas con balance 2:1, permitiendo así tener saltos de nivel mayores entre elementos adyacentes. Por otra parte, para evitar la pérdida de información al proyectar la solución a mallas más gruesas, se plantee una corrección en (3), que tiene como objetivo recuperar la solución de la malla fina, garantizando así que la malla gruesa conserve la precisión obtenida en la malla fina. El proceso de manufactura aditiva se distingue por su gran flexibilidad comparándolo con otros métodos tradicionales de manufactura. Esta flexibilidad se observa en la posibilidad de construir piezas de gran complejidad geométrica, optimizando propiedades mecánicas durante el proceso de deposición. Por ese motivo, (4) se propone la lectura de ficheros en formato GCode que replica la ruta exacta del recorrido del láser que realiza la deposición del material. Los ingredientes lectura de comandos escritos en lenguaje Gcode, multicriterio de adaptividad de malla y el uso de mallas estructuradas basadas en octrees, permiten capturar con gran precisión el dominio discreto eliminando así la engorrosa tarea de generar un dominio discreto ad-hoc para la pieza a modelar. Los desarrollos de esta tesis se realizan en un entorno de computación de altas prestaciones (HPC) que permite acelerar el estudio de la ejecución del proceso de impresión y por ende reducir el número de experimentos destructivos, dos aspectos clave que permiten explorar y desarrollar nuevas técnicas en manufactura aditiva de piezas industriales.Postprint (published version

High-Performance Finite Elements with MFEM

The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability across leadership-class supercomputing facilities, including exascale supercomputers, as well as new capabilities and functionality, enabling a wider range of applications. Much of this work was undertaken as part of the Department of Energy’s Exascale Computing Project (ECP) in collaboration with the Center for Efficient Exascale Discretizations (CEED)