Frame Fields for Hexahedral Mesh Generation

As a discretized representation of the volumetric domain, hexahedral meshes have been a popular choice in computational engineering science and serve as one of the main mesh types in leading industrial software of relevance. The generation of high quality hexahedral meshes is extremely challenging because it is essentially an optimization problem involving multiple (conflicting) objectives, such as fidelity, element quality, and structural regularity. Various hexahedral meshing methods have been proposed in past decades, attempting to solve the problem from different perspectives. Unfortunately, algorithmic hexahedral meshing with guarantees of robustness and quality remains unsolved. The frame field based hexahedral meshing method is the most promising approach that is capable of automatically generating hexahedral meshes of high quality, but unfortunately, it suffers from several robustness issues. Field based hexahedral meshing follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a feature-aligned frame field and (2) generation of an integer-grid map that best aligns with the frame field. The main robustness issue stems from the fact that smooth frame fields frequently exhibit singularity graphs that are inappropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The thesis aims at analyzing the gap between the topologies of frame fields and hexahedral meshes and developing algorithms to realize a more robust field based hexahedral mesh generation. The first contribution of this work is an enumeration of all local configurations that exist in hexahedral meshes with bounded edge valence and a generalization of the Hopf-Poincaré formula to octahedral (orthonormal frame) fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large non-linear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field. In the collaboration work with colleagues [BRK+22], the dataset HexMe consisting of practically relevant models with feature tags is set up, allowing a fair evaluation for practical hexahedral mesh generation algorithms. The extendable and mutable dataset remains valuable as hexahedral meshing algorithms develop. The results of the standard field based hexahedral meshing algorithms on the HexMesh dataset expose the fragility of the automatic pipeline. The major contribution of this thesis improves the robustness of the automatic field based hexahedral meshing by guaranteeing local meshability of general feature aligned smooth frame fields. We derive conditions on the meshability of frame fields when feature constraints are considered, and describe an algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite the fact that local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 57%, when compared on the challenging HexMe dataset

Interface theory and percolation

This thesis is mainly concerned with percolation on general infinite graphs, as well as the approximation of conformal maps by square tilings, which are defined using electrical networks. The first chapter is concerned with the smoothness of the percolation density on various graphs. In particular, we prove that for Bernoulli percolation on Z d , d ≥ 2, the percolation density is an analytic function of the parameter in the supercritical interval (pc(Z d ), 1]. This answers a question of Kesten [1981]. The analogous result is also proved for the Boolean model of continuum percolation in R 2 , answering a question of Last et al. [2017]. In order to prove these results, we introduce the notion of interfaces, which is studied extensively in the current thesis. For dimensions d ≥ 3, we use renormalisation tecnhiques. Furthermore, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that pc < 1/2 for bond percolation on certain families of triangulations for which Benjamini & Schramm conjectured that pc ≤ 1/2 for site percolation. For the latter result, we use the well-known circle packing theorem of He and Schramm [1995], a discrete analogue of the Riemann mapping theorem. In Chapter 2, we continue the study of interfaces, and in particular, we consider the exponential growth rate br of the number of interfaces of a given size as a function of their surface-to-volume ratio r. We prove that the values of the percolation parameter p for which the interface size distribution has an exponential tail are uniquely determined by br by comparison with a dimension-independent function f(r) := (1+r) 1+r r r . We also point out a formula for translating any upper bound on the percolation threshold of a lattice G into a lower bound on the exponential growth rate of lattice animals a(G) and vice-versa. We exploit this in both directions. We obtain the rigorous lower bound pc(Z 3 ) > 0.2522 for 3-dimensional site percolation. We also improve on the best known asymptotic lower and upper bounds on a(Z d ) as d → ∞. We also prove that the rate of the exponential decay of the cluster size distribution, defined as c(p) := limn→∞ (Pp(|Co| = n))1/n, is a continuous function of p. The proof makes use of the Arzel`a-Ascoli theorem but otherwise boils down to elementary calculations. The analogous statement is also proved for the interface size distribution. For this we first establish that the rate of exponential decay is well-defined. In Chapter 3, we use interfaces to obtain upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. The results of this chapter are inspired by well-known conjectures of Benjamini and Schramm [1996b] for percolation on general graphs. We prove a conjecture by Benjamini and Schramm [1996b] stating that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2. We also make progress on a conjecture of Angel et al. [2018] that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. In the process, we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of [Lyons and Peres, 2016, Question 6.20]. Another topic of this thesis is the discrete approximation of conformal maps using another discrete analogue of the Riemann mapping theorem, namely the square tilings of Brooks et al. [1940]. This result is analogous to a well-known the orem of Rodin & Sullivan, previously conjectured by Thurston, which states that the circle packing of the intersection of a lattice with a simply connected planar domain Ω into the unit disc D converges to a Riemann map from Ω to D when the mesh size converges to 0. As a result, we obtain a new algorithm that allows us to numerically compute the Riemann map from any Jordan domain onto a square