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Program
The program at a glance:
Thursday 24/08:
- 12:30-14:00 Lunch
- 14:00-15:30 Talks
- 15:30-16:00 Coffee break
- 16:00-17:30 Talks
- 19:00 Dinner at Le grand café Foy
Friday 25/08:
- 9:30-10:30 Talks
- 10:30-11:00 Coffee break
- 11:00-12:30 Talks
- 12:30 Lunch and conference end
Thursday 24/08:
(14:00-14:30) Guillaume Coiffier (University of Lorraine): Towards grid-preserving surface parametrization in a single optimization
The frame-field based approach for quadmeshing is an algorithmic pipeline divided in the computation of a smooth frame field, of a seamless parametrization and a grid-preserving (quantized) parametrization from which quads can be extracted. While each subproblem is challenging on its own, failure cases may also appear due to suboptimal decisions made in previous steps.
In an effort to improve robustness and user-control, recent works propose a joint optimization of a smooth frame field and its associated seamless parametrization. Single-optimization methods for computing a grid-preserving parametrization directly from a frame field have also been explored in the past by the _periodic global parametrization_ (PGP) algorithm and related works.
In this talk, our goal is to integrate ideas from the two to design a variant of the PGP algorithm that computes a grid-preserving parametrization in a single continuous (non-linear) optimization. By defining a different periodic function over uv-coordinates, we derive a set of equations which optimization would allow PGP to keep full control over the singularity distribution as well as over the parametrization's distortion, allowing it to freely place new cones instead of T-junctions.
(14:30-15:00) Sofiane Benzait (CEA): Hybrid Mesh Generation using metric and frame fields
Today, fully tetrahedral mesh generation is a solved problem, while hexahedral mesh generation is not. A benefit of using hexahedral meshes is to help to study numerical solutions in certain given directions in a simpler way than with tetrahedral meshes. Given the difficulty of generating a whole hexahedral mesh, an is to combine hexahedra and tetrahedra to generate hybrid meshes. This solution makes it possible to put hexahedra in the areas of interest for the numerical simulation and to use tetrahedra everywhere else.
In this presentation, I propose to generate such hybrid meshes using both a frame field and a metric field as constraints. We choose to develop a frontal approach for creating the vertices that will form 3D cells and we use the cavity operator for performing all the mesh modification operations. Starting from an initial tetrahedral mesh T, we apply an advancing-front algorithm to generate vertices along geometrical curves, then on the surfaces, and eventually in the volume. Generated vertices, noted G, are potentially “connected” between layers and inside a layer. Once the vertices of G are generated, we iteratively insert them in T and remove some “useless” vertices from T. Edges and faces are also forced in the mesh using the connection between the vertices of G and the cavity operator. The last step consists in reconstructing the hexahedra that could not be created by frontal generation by combining the tetrahedra to generate hexahedra. This last step is controlled by a pre-imposed quality factor.
(15:00-15:30) David Bommes (University of Bern) : Singularity Optimization for High-Quality Integer-Grid Maps
Integer-Grid Maps based approaches enable high-quality quadrilateral meshes and have recently been integrated into several commercial applications. Equipped with similar strengths, their volumetric counterparts are a promising research direction for high-quality hexahedral mesh generation. However, the direct construction of integer-grid maps corresponds to a difficult non-linear and non-convex mixed-integer problem. Consequently, all available algorithms decompose the overall difficult task into several subsequent easier steps, typically first optimizing a frame field to obtain suitable singularities and then generating a locally injective map with fixed singularities. Unfortunately, such a decomposition frequently has a negative impact on the quality of the resulting integer-grid map since singularities cannot be altered anymore when minimizing the distortion of the locally injective map.
In this ongoing project, we explore a novel approach for the optimization of locally injective seamless and integer-grid maps, where the singularities are part of the optimization. Different to prior methods, singularities can move continuously, effectively avoiding low-quality local minima often observed in schemes that restrict the singularity positions to vertices of an input triangle mesh. Moreover, a novel strategy is proposed to identify discrete locations where additional pairs of singularities are most beneficial. The resulting optimization framework is general in the sense that it can be used for any distortion objective, including the setting of meshing w.r.t. a given metric field.
(15:30-16:00) Coffee break
(16:00-16:30) Valentin Nigolian (University of Bern): Expansion Cones: A Progressive Volumetric Mapping Framework
Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with guarantees regarding map bijectivity. Through this work, we aim at opening the door to a new family of methods by providing a novel framework based on the concept of progressive expansion. Starting from an initial map of a tetrahedral mesh whose image may contain degeneracies but no inversions, we incrementally adjust vertex images to expand degenerate elements. By restricting movement to so-called expansion cones, it is done in such a way that the number of degenerate elements decreases in a strictly monotonic manner, without ever introducing any inversion. Adaptive local refinement of the mesh is performed to facilitate this process. We describe a prototype algorithm in the realm of this framework for the computation of maps from ball-topology tetrahedral meshes to convex or star-shaped domains. This algorithm is evaluated and compared to state-of-the-art methods, demonstrating its benefits in terms of bijectivity. We also discuss the associated cost in terms of sometimes significant mesh refinement to obtain the necessary degrees of freedom required for establishing a valid mapping. Our conclusions include that while this algorithm is only of limited immediate practical utility due to efficiency concerns, the general framework has the potential to inspire a range of novel methods improving on the efficiency aspect.
(16:30-17:00) Heng Liu (University of Bern): Locally Meshable Frame Fields
The main robustness issue of state-of-the-art frame field based hexahedral mesh generation algorithms originates from non-meshable topological configurations, which do not admit the construction of an integer-grid map but frequently occur in smooth frame fields. In this article, we investigate the topology of frame fields and derive conditions on their meshability, which are the basis for a novel algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite 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 58%, when compared on the challenging HexMe dataset.
(17:00-17:30) Hendrik Brückler and Marcel Campen (Osnabrück University): The Integer DoFs in Frame Field based Meshing
Rightfully, the „meshability gap“ between frame fields on the one hand and global parametrizations on the other hand is receiving significant attention in recent years; the structural degrees of freedom of a hexahedral mesh (or a global parametrization) in terms of its singularity structure are hard to characterize and thus hard to take into account in the field design process. But there also is a second gap in the frame field based meshing pipeline, the gap between a continuous global parametrization (obtainable by integrating a meshable frame field) and a parametrization that actually implies a proper hexahedral mesh — a so-called integer-grid map. Bridging this gap hinges on dealing with the inherent integer degrees of freedom. This problem has received somewhat less attention, possibly because a simple rounding-based solution is sufficient if one sets the target mesh sizing just fine enough. This, however, besides requiring trial-and-error, can significantly limit flexibility and practical utility. Especially for the generation of intentionally coarse block-structured meshes, a reliable way of dealing with the integer degrees of freedom is crucial. In this talk we report recent advances on this aspect, involving 3D motorcycle complexes and differential quantization and remapping strategies.
19:00 Dinner at Le grand café Foy (1 Pl. Stanislas, 54000 Nancy)
Friday 25/08:
(9:30-10:00) Mattéo Couplet (UC Louvain): Integrable Frame Fields for Flexible Quadrilateral Meshing
A popular approach to quad meshing is to first generate a frame field, i.e., a continuous assignment of a set of 2 orthogonal directions (a frame) representing the local orientation of the mesh elements. Most existing frame field-driven meshing approaches generate a non-scaled frame field, and compute a parametrization that follows the frame field as closely as possible. In many cases however, the parametrization strongly deviates from the prescribed frame field, which prevents the user from flexibly applying desired orientation and size constraints. Instead of treating the frame field as a guide as in previous methods, we tackle this issue by imposing a priori that the frame field is integrable, i.e., locally the 2 branches form the gradients of a parametrization.
The key insight of our method is a new energy that expresses integrability of frame fields, which we represent using the well-known Odeco tensor that is invariant to the frame symmetries.
Our method is able to create the correct singularities and scaling required to respect the prescribed constraints. The induced parametrization closely follows the computed frame field and allows us to generate the desired mesh.
This contribution paves the way for a robust quad mesh generator suitable for the geometries typically found in industrial contexts.
(10:00-10:30) Yuri Nesterenko (Siemens): Octupoles for octahedral symmetry
Spherical harmonics of degree 4 are widely used in volumetric frame fields design due to their ability to reproduce octahedral symmetry. We will show how to use harmonics of degree 3 (octupoles) for the same purpose, thereby reducing number of parameters and computational complexity. The key ingredients of the presented approach:
- implicit equations for the manifold of octupoles possessing octahedral symmetry up to multiplication by -1,
- corresponding rotationally invariant measure of octupole’s deviation from the specified symmetry,
- smoothing penalty term compensating the lack of octupoles’ symmetries during a field optimization.
It will also be shown that this approach is particularly effective in combination with immersed boundary conditions technique.
- implicit equations for the manifold of octupoles possessing octahedral symmetry up to multiplication by -1,
- corresponding rotationally invariant measure of octupole’s deviation from the specified symmetry,
- smoothing penalty term compensating the lack of octupoles’ symmetries during a field optimization.
It will also be shown that this approach is particularly effective in combination with immersed boundary conditions technique.
(10:30-11:00) Coffee break
(11:00-11:30) Sébastien Mestrallet (CEA): Towards validity-first polycube-based hex-meshing
Polycubes (axis-aligned frame fields) have been a fruitful approach for hex-meshing, greatly improving robustness at the cost of lower mesh quality. A convenient representation of polycubes involves a supporting tetrahedral mesh, and consists in associating boundary triangles to the six principal axes ±{X,Y,Z}. However, existing labeling algorithms often have two issues : they try to optimize validity and quality in an undifferentiated way, and their validity criteria are neither necessary nor sufficient. We will present ongoing work on labeling validity criteria and validity-first polycube generation.
(11:30-12:00) David Lopez (Tessael): 2.5D Hexahedral meshing for reservoir simulations
We present a new method to generate pure hexahedral meshes for reservoir simulations. The grid is obtained by extruding a quadrangular mesh, using ideas from the latest advances in computational geometry and more specifically the generation of semi-structured quadrangular meshes based on global parameterization.
Hexahedral elements are automatically constructed to smoothly honor input features' geometry (domain boundaries, faults and horizons), thus, making it possible to be used for multiple types of physical simulations on the same mesh.
Main contributions are: the introduction of a new semi-structured hexahedral meshing workflow producing high quality meshes for a wide range of fault systems, and the study and definition of weak verticality on triangulated surface meshes, allowing us to design better and more robust algorithms during the extrusion phase along non-vertical faults.
We demonstrate 1) the simplicity of using such hexahedral meshes generated using the proposed method to do coupled flow-geomechanics simulations using state-of-the-art simulators for reservoir studies and 2) the possibilty of using such semi-structured hexahedral meshes in commercial structured flow simulators, offering a gridding alternative to treat a wider family of fault networks without the recourse to the stair-step fault approximation.
(12:00-12:30) Paul Viville (University of Strasbourg): Skeleton based hexahedral mesh generation
Pure hexahedral meshes are notoriously challenging to generate because of the inherent difficulty in computing a proper connectivity for domains of arbitrary genus and shape. For a subset of domains, those that can be efficiently represented by a 1D or 2D skeleton, we offer a solution that produces high quality hexahedral meshes efficiently. A skeleton is a simplified descriptor of the geometry that accurately represents the topology of the object, its analysis is much simpler than that of a 3D volume.
By leveraging this representation, we build a scaffold to encode the local connectivity of the domain, in essence an implicit segmentation of the domain. The segmentation is then filled with predefined construction blocks that are joined together through the scaffold.This process is based on mostly local decisions, making it very easy to parallelize.
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