Detailed analysis of the lattice Boltzmann method on unstructured grids

The lattice Boltzmann method has become a standard for efficiently solving problems in fluid dynamics. While unstructured grids allow for a more efficient geometrical representation of complex boundaries, the lattice Boltzmann methods is often implemented using regular grids. Here we analyze two implementations of the lattice Boltzmann method on unstructured grids, the standard forward Euler method and the operator splitting method. We derive the evolution of the macroscopic variables by means of the Chapman-Enskog expansion, and we prove that it yields the Navier-Stokes equation and is first order accurate in terms of the temporal discretization and second order in terms of the spatial discretization. Relations between the kinetic viscosity and the integration time step are derived for both the Euler method and the operator splitting method. Finally we suggest an improved version of the bounce-back boundary condition. We test our implementations in both standard benchmark geometries and in the pore network of a real sample of a porous rock.Comment: 42 page

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the Finite Element Method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies available upon reques

Finite Element Modeling Driven by Health Care and Aerospace Applications

This thesis concerns the development, analysis, and computer implementation of mesh generation algorithms encountered in finite element modeling in health care and aerospace. The finite element method can reduce a continuous system to a discrete idealization that can be solved in the same manner as a discrete system, provided the continuum is discretized into a finite number of simple geometric shapes (e.g., triangles in two dimensions or tetrahedrons in three dimensions). In health care, namely anatomic modeling, a discretization of the biological object is essential to compute tissue deformation for physics-based simulations. This thesis proposes an efficient procedure to convert 3-dimensional imaging data into adaptive lattice-based discretizations of well-shaped tetrahedra or mixed elements (i.e., tetrahedra, pentahedra and hexahedra). This method operates directly on segmented images, thus skipping a surface reconstruction that is required by traditional Computer-Aided Design (CAD)-based meshing techniques and is convoluted, especially in complex anatomic geometries. Our approach utilizes proper mesh gradation and tissue-specific multi-resolution, without sacrificing the fidelity and while maintaining a smooth surface to reflect a certain degree of visual reality. Image-to-mesh conversion can facilitate accurate computational modeling for biomechanical registration of Magnetic Resonance Imaging (MRI) in image-guided neurosurgery. Neuronavigation with deformable registration of preoperative MRI to intraoperative MRI allows the surgeon to view the location of surgical tools relative to the preoperative anatomical (MRI) or functional data (DT-MRI, fMRI), thereby avoiding damage to eloquent areas during tumor resection. This thesis presents a deformable registration framework that utilizes multi-tissue mesh adaptation to map preoperative MRI to intraoperative MRI of patients who have undergone a brain tumor resection. Our enhancements with mesh adaptation improve the accuracy of the registration by more than 5 times compared to rigid and traditional physics-based non-rigid registration, and by more than 4 times compared to publicly available B-Spline interpolation methods. The adaptive framework is parallelized for shared memory multiprocessor architectures. Performance analysis shows that this method could be applied, on average, in less than two minutes, achieving desirable speed for use in a clinical setting. The last part of this thesis focuses on finite element modeling of CAD data. This is an integral part of the design and optimization of components and assemblies in industry. We propose a new parallel mesh generator for efficient tetrahedralization of piecewise linear complex domains in aerospace. CAD-based meshing algorithms typically improve the shape of the elements in a post-processing step due to high complexity and cost of the operations involved. On the contrary, our method optimizes the shape of the elements throughout the generation process to obtain a maximum quality and utilizes high performance computing to reduce the overheads and improve end-user productivity. The proposed mesh generation technique is a combination of Advancing Front type point placement, direct point insertion, and parallel multi-threaded connectivity optimization schemes. The mesh optimization is based on a speculative (optimistic) approach that has been proven to perform well on hardware-shared memory. The experimental evaluation indicates that the high quality and performance attributes of this method see substantial improvement over existing state-of-the-art unstructured grid technology currently incorporated in several commercial systems. The proposed mesh generator will be part of an Extreme-Scale Anisotropic Mesh Generation Environment to meet industries expectations and NASA\u27s CFD visio

Modelo computacional de remodelamiento óseo mediante estructuras discretas

Libro de tesis, algunas figuras en color, mayoría en blanco y negro.ilustraciones, tablasIn-silico models applied to bone remodeling are widely used to investigate bone mechanics, bone diseases, bone-implant interactions, and also the effect of treatments in bone pathologies. This work proposes a new methodology to solve the bone remodeling problem using one-dimensional (1D) elements to discretize trabecular structures more efficiently. First a concept review on the bone remodelling process and mathematical approaches, such as homogenization for its modelling are revised along with famous previous works on this field, later, in chapter two, the discrete modelling approach is validated by comparing FE simulations with experimental results for a cellular like material created using additive manufacturing and following a tessellation algorithm, and later, applying an optimization scheme based on maximum stiffness for a given porosity. In chapter three, an Euler integration scheme for a bone remodelling problem is coupled with the momentum equations to obtain the evolution of material density at each step. For the simulations, the equations were solved by using the finite element method and a direct formulation, and two benchmark tests were solved varying mesh parameters in two dimensions, an additional three-dimensional benchmark was addressed with the same methodology. Proximal femur and calcaneus bone were selected as study cases given the vast research available on the topology of these bones, and compared with the anatomical features of trabecular bone reported in the literature, the study cases were examined mainly in two dimensions, but the main trabecular groups for the femur were also obtained in three dimensions. The presented methodology has proven to be efficient in optimizing topologies of lattice structures; It can predict the trend in formation patterns of the main trabecular groups from two different cancellous bones (femur and calcaneus) using domains set up by discrete elements as a starting point. Preliminary results confirm that the proposed approach is suitable and useful in bone remodeling problems in 2D and 3D leading to a considerable computational cost reduction. Characteristics similar to those encountered in topological optimization algorithms were identified in the benchmark tests as well, showing the viability of the proposed approach in other applications such as bio-inspired design. Finally, in the last part of this work, the discrete approach developed in chapter two and three is coupled with two classic bone remodelling models, forming a new model that takes into account a variety of biological parameters such as paracrine and autocrine regulators and is able to predict different periodical responses in the bone remodelling process within a 2D domain with mechanical field variables. (Text taken from source)Los modelos in-silico aplicados a la remodelación ósea son ampliamente utilizados para investigar la mecánica del hueso, las enfermedades óseas, las interacciones hueso-implante y también el efecto de los tratamientos en las patologías óseas. Este trabajo propone una nueva metodología para resolver el problema de la remodelación ósea utilizando elementos unidimensionales (1D) para discretizar las estructuras trabeculares de forma más eficiente. En primer lugar se revisa una revisión conceptual sobre el proceso de remodelación ósea y las aproximaciones matemáticas, como el método de homogeneización para su modelización, junto con famosos trabajos previos en este campo, posteriormente, en el capítulo dos, se valida la modelación discreta comparando las simulaciones de FE (elementos finitos) con los resultados experimentales para un material similar al celular creado mediante fabricación aditiva y siguiendo un algoritmo de teselación, y posteriormente, aplicando un esquema de optimización basado en la máxima rigidez para una determinada porosidad. En el capítulo tres, se acopla un esquema de integración de Euler para un problema de remodelación ósea con las ecuaciones de momento para obtener la evolución de la densidad del material en cada paso de tiempo. Para las simulaciones, las ecuaciones se resolvieron utilizando el método de los elementos finitos y una formulación directa, y se resolvieron dos pruebas de referencia variando los parámetros de la malla en dos dimensiones, adicionalmente, se abordó una prueba de referencia tridimensional adicional con la misma metodología. Se seleccionaron el fémur proximal y el hueso calcáneo como casos de estudio, dada la amplia investigación disponible sobre la topología de estos huesos, y se compararon con las características anatómicas del hueso trabecular reportadas en la literatura, los casos de estudio se examinaron principalmente en dos dimensiones, pero los principales grupos trabeculares para el fémur también se obtuvieron en tres dimensiones. La metodología presentada ha demostrado ser eficaz en la optimización de las topologías de estructuras reticulares; puede predecir la tendencia de los patrones de formación de los principales grupos trabeculares de dos huesos esponjosos diferentes (fémur y calcáneo) utilizando dominios establecidos por elementos discretos como punto de partida. Los resultados preliminares confirmaron que el enfoque propuesto es adecuado y útil en problemas de remodelación ósea en 2D y 3D, lo que conlleva una considerable reducción del coste computacional. En las pruebas de referencia también se identificaron características similares a las encontradas en los algoritmos de optimización topológica, lo que demuestra la viabilidad del enfoque propuesto en otras aplicaciones como el diseño bioinspirado. Finalmente, en la última parte de este trabajo, el enfoque discreto desarrollado en los capítulos dos y tres se acopla con dos modelos clásicos de remodelación ósea, formando un nuevo modelo que tiene en cuenta una variedad de parámetros biológicos como los reguladores paracrinos y autocrinos, y es capaz de predecir diferentes respuestas periódicas en el proceso de remodelación ósea dentro de un dominio 2D con variables de campo mecánico. (Texto tomado de la fuente)MaestríaMagíster en Ingeniería BiomédicaMecánica computaciona

Level set-fitted polytopal meshes with application to structural topology optimization

We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method

TNL: NUMERICAL LIBRARY FOR MODERN PARALLEL ARCHITECTURES

We present Template Numerical Library (TNL, www.tnl-project.org) with native support of modern parallel architectures like multi–core CPUs and GPUs. The library offers an abstract layer for accessing these architectures via unified interface tailored for easy and fast development of high-performance algorithms and numerical solvers. The library is written in C++ and it benefits from template meta–programming techniques. In this paper, we present the most important data structures and algorithms in TNL together with scalability on multi–core CPUs and speed–up on GPUs supporting CUDA