Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

Parametric reciprocal structures: workshop of design and fabrication

[Excerpt] Foreword: One of the central challenges that needs to be maintained throughout new structural and constructional design pedagogy is how to impart knowledge about structural and constructive concepts in a manner that enhances the capacity to understand and apply them in design. Promoted under the auspices of the International Conference on Structures and Architecture — ICSA2016, the workshop “Parametric Reciprocal Structure: workshop of design and fabrication” had it genesis in proposals developed by students of the course of Special Structures of the Master in Architecture of the School of Architecture of the University of Minho (EAUM). The solutions designed by the students focused on the design of a reciprocal structure to be built at the Design Institute of Guimarães (former Tanning Factory of Ramada). The reciprocal structures workshop was organized as part of the special structures course. The workshop involved students and staff to implement constructive solutions, in the manufacturing and in the assembling of the structure. The initiative aimed to explore architectural and structural design concepts, embracing the research of: methods and processes of designing thinking; simulation and processing tools; and manufacturing concepts and materials. The computational model Reciprocalizer, developed by Prof. Dario Parigi from the University of Aalborg, was used for the morphological design exploration. This model allows the generation of three-dimensional reciprocal grids, characterized by a high degree of freedom and formal experimentation. The proposed combination of creative aspects in the conception and construction of structures, advanced technologies and complex architectural and structural applications represents a valuable learning experience of collaborative work. [...]This book and workshop had financial support of the Project Lab2PT — Landscapes, Heritage and Territory laboratory - AUR/04509 with the financial support from FCT/MCTES through national funds (PIDDAC) and co-financing from the European Regional Development Fund (FEDER) POCI-01-0145-FEDER-007528, in the aim of the new partnership agreement PT2020 throught COMPETE 2020 — Competitiveness and Internationalization Operational Program (POCI)

Combinatorial meshing for mechanical FEM

Diese Dissertation führt die Forschung zur Erzeugung von FEM Netzen für mechanische Simulationen fort. Zur zielgerichteten Steuerung der weiteren Forschung in diesem Feld wurde eine Umfrage zur Identifikation der Kerninteressen der Anwender durchgef¨uhrt. Das vorgestellte Verfahren des Combinatorial Meshing ist ein neuartiges Konzept im Bereich Grid Based Meshing. Im Gegensatz zu den kartesischen Gittern, die im Grid Based Meshing Anwendung finden wird ein an das Problem angepasstes Gitter genutzt. Dieses Precursor Mesh wird durch Analyse des CAD Strukturbaums der Geometrie gewählt. Die Zellen des Precursor Mesh werden mit vorberechneten Netzsegmenten – sogenannten Superelementen gefüllt. Die Wahl passender Superelemente wird als combinatorisches Optimierungsproblem modelliert. Dieses wird mit Hilfe von Answer Set Programming (ASP) und einem alternativen heuristischen Ansatz gelöst. Beide Verfahren werden in Hinblick auf Zeitkomplexität und Ergebnisqualität verglichen. Das resultierende Netz ist eine Grobe Näherung der Zielgeometrie, die an geometrische Elemente angebunden werden muss. Für diesen Prozess wird ein neuer Algorithmus vorgestellt, der automatisch identifizieren kann, an welche Geometrieelemente Oberflächenknoten des Netzes gebunden werden müssen um die Zielgeometrie möglichst exakt abzubilden. Für die Erzeugung der Superelemente wird ein neues Verfahren auf Basis von ASP entwickelt. Um die Generierung von FEM Netzen mit ASP zu ermöglichen, wird das Problem der Netzgenerierung als graphentheoretisches Problem modelliert. Dieses ist die Wahl eines optimalen Subgraphen aus einem Primärgraph. Dieses Problem wird mit einem ASP Solver für verschiedene Optimierungsziele gelöst. Die Graphenformulierung ist zudem ein Fortschritt im theoretischen Verständnis der Komplexität der Netzgenerierung.his dissertation advances the research of mesh generation for Finite Element Method simulation for mechanical applications. In order to target further research at user needs, a survey is conducted to identify the most pressing issues in FEM software. The concept of Combinatorial Meshing is proposed as a novel approach to grid based meshing. While conventional grid based meshing works on trivial Cartesian grids, the use of a Precursor Mesh instead of a grid is proposed. Appropriate Precursor Meshes are selected by analyzing the internal feature structure of the provided CAD data. The cells of this Precursor Mesh are then filled with precomputed mesh templates – called Super Elements. The selection of appropriate Super Elements is modeled as a combinatorial optimization problem. To solve this problem, Answer Set programming (ASP) and a heuristic approach are compared with respect to their time complexity and result quality. The resulting mesh is a rough approximation of the target geometry which then has to be fitted to the geometric entities. For this process a novel algorithm is presented which is able to automatically identify the geometric entities on which the surface nodes of the mesh have to be drawn in order to generate high quality meshes and correctly approximate the desired geometry. For the generation of Super Element Meshes, a novel approach based on ASP is developed. In order to enable meshing with ASP, a graph representation of a mesh is developed and the meshing process is formulated as a graph selection problem. It is then solved with an ASP solver for multiple optimization goals. The graph formulation will also aid the theoretical understanding of meshing complexity

A New Approach to Automatic Generation of an all Pentagonal Finite Element Mesh for Numerical Computations over Convex Polygonal Domains

A new method is presented for subdividing a large class of solid objects into topologically simple subregionssuitablefor automatic finite element meshing withpentagonalelements. It is known that one can improve the accuracy of the finite element solutionby uniformly refining a triangulation or uniformly refining a quadrangulation.Recently a refinement scheme of pentagonal partition was introduced in [31,32,33]. It is demonstrated that the numerical solutionbased on the pentagonal refinement scheme outperforms the solutions based on the traditional triangulation refinement scheme as well as quadrangulation refinement scheme. It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes with a hope to offer even further improvement. It is shown in literature that one cannot refine a hexagon using hexagons of smaller size. In general, one can only refine an n-gon by n-gons of smaller size if n = 5. Furthermore, we introduce a refinement scheme of a generalpolygon based on the pentagon scheme. This paper first presents a pentagonalization (or pentagonal conversion) scheme that can create a pentagonal mesh from any arbitrary mesh structure. We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh

Robin Schwarz algorithm for the NICEM Method: the Pq finite element case

In Gander et al. [2004] we proposed a new non-conforming domain decomposition paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on Schwarz type methods that allows for the use of Robin interface conditions on non-conforming grids. The error analysis was done for P1 finite elements, in 2D and 3D. In this paper, we provide new numerical analysis results that allow to extend this error analysis in 2D for piecewise polynomials of higher order and also prove the convergence of the iterative algorithm in all these cases.Comment: arXiv admin note: substantial text overlap with arXiv:0705.028

The multiscale hybrid mixed method in general polygonal meshes

This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and k+1, k 0, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree k+1 posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order k +1 and k +2 in the broken H1 and L2 norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides

Stability Aware Delaunay Refinement

Good quality meshes are extensively used for finding approximate solutions for partial differential equations for fluid flow in two dimensional surfaces. We present an overview of existing algorithms for refinement and generation of triangular meshes. We introduce the concept of node stability in the refinement of Delaunay triangulation. We present two algorithms for generating stable refinement of Delaunay triangulation. We also present an experimental investigation of a triangulation refinement algorithm based on the location of the center of gravity and the location of the center of circumcircle. The results show that the center of gravity based refinement is more effective in refining interior nodes for a given distribution of nodes in two dimensions

The H2-wavelet method

the present paper, we introduce the H2-wavelet method for the fast solution of nonlocal operator equations on unstructured meshes. On the given mesh, we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates is compressed to O(N log N) relevant matrix coefficients, where N denotes the number of boundary elements. The compressed system matrix is computed with nearly linear complexity by using the fast H2-matrix approach. Numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet Galerkin scheme on unstructured triangular or quadrangular meshes

The H2-wavelet method

the present paper, we introduce the H2-wavelet method for the fast solution of nonlocal operator equations on unstructured meshes. On the given mesh, we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates is compressed to O(N log N) relevant matrix coefficients, where N denotes the number of boundary elements. The compressed system matrix is computed with nearly linear complexity by using the fast H2-matrix approach. Numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet Galerkin scheme on unstructured triangular or quadrangular meshes