2023Activity reportProject-TeamPIXEL

2.1 Scientific grounds

Mesh generation is a notoriously difficult task. A quick search on the NSF grant web page with “mesh generation AND finite element” keywords returns more than 30 currently active grants for a total of $8 million. NASA indicates mesh generation as one of the major challenges for 2030  38, and estimates that it costs 80% of time and effort in numerical simulation. This is due to the need for constructing supports that match both the geometry and the physics of the system to be modeled. In our team we pay a particular attention to scientific computing, because we believe it has a world changing impact.

It is very unsatisfactory that meshing, i.e. just “preparing the data” for the simulation, eats up the major part of the time and effort. Our goal is to change the situation, by studying the influence of shapes and discretizations, and inventing new algorithms to automatically generate meshes that can be directly used in scientific computing. This goal is a result of our progressive shift from pure graphics (“Geometry and Lighting”) to real-world problems (“Shape Fidelity”).

Meshing is central in geometric modeling because it provides a way to represent functions on the objects being studied (texture coordinates, temperature, pressure, speed, etc.). There are numerous ways to represent functions, but if we suppose that the functions are piecewise smooth, the most versatile way is to discretize the domain of interest. Ways to discretize a domain range from point clouds to hexahedral meshes; let us list a few of them sorted by the amount of structure each representation has to offer (refer to Figure 1).

• At one end of the spectrum there are point clouds: they exhibit no structure at all (white noise point samples) or very little (blue noise point samples). Recent explosive development of acquisition techniques (e.g. scanning or photogrammetry) provides an easy way to build 3D models of real-world objects that range from figurines and cultural heritage objects to geological outcrops and entire city scans. These technologies produce massive, unstructured data (billions of 3D points per scene) that can be directly used for visualization purposes, but this data is not suitable for high-level geometry processing algorithms and numerical simulations that usually expect meshes. Therefore, at the very beginning of the acquisition-modeling-simulation-analysis pipeline, powerful scan-to-mesh algorithms are required.

  During the last decade, many solutions have already been proposed  34, 18, 30, 29, 21, but the problem of building a good mesh from scattered 3D points is far from being solved. Beside the fact that the data is unusually large, the existing algorithms are challenged also by the extreme variation of data quality. Raw point clouds have many defects, they are often corrupted with noise, redundant, incomplete (due to occlusions): they all are uncertain.

• Triangulated surfaces are ubiquitous, they are the most widely used representation for 3D objects. Some applications like 3D printing do not impose heavy requirements on the surface: typically it has to be watertight, but triangles can have an arbitrary shape. Other applications like texturing require very regular meshes, because they suffer from elongated triangles with large angles.

  While being a common solution for many problems, triangle mesh generation is still an active topic of research. The diversity of representations (meshes, NURBS, ...) and file formats often results in a “Babel” problem when one has to exchange data. The only common representation is often the mesh used for visualization, that has in most cases many defects, such as overlaps, gaps or skinny triangles. Re-injecting this solution into the modeling-analysis loop is non-trivial, since again this representation is not well adapted to analysis.

• Tetrahedral meshes are the volumic equivalent of triangle meshes, they are very common in the scientific computing community. Tetrahedral meshing is now a mature technology. It is remarkable that still today all the existing software used in the industry is built on top of a handful of kernels, all written by a small number of individuals 23, 36, 42, 25, 35, 37, 24, 46.

  Meshing requires a long-term, focused, dedicated research effort that combines deep theoretical studies with advanced software development. We have the ambition to bring this kind of maturity to a different type of mesh (structured, with hexahedra), which is highly desirable for some simulations, and for which, unlike tetrahedra, no satisfying automatic solution exists. In the light of recent contributions, we believe that the domain is ready to overcome the principal difficulties.

• Finally, at the most structured end of the spectrum there are hexahedral meshes composed of deformed cubes (hexahedra). They are preferred for certain physics simulations (deformation mechanics, fluid dynamics ...) because they can significantly improve both speed and accuracy. This is because (1) they contain a smaller number of elements (5-6 tetrahedra for a single hexahedron), (2) the associated tri-linear function basis has cubic terms that can better capture higher-order variations, (3) they avoid the locking phenomena encountered with tetrahedra 16, (4) hexahedral meshes exploit inherent tensor product structure and (5) hexahedral meshes are superior in direction dominated physical simulations (boundary layer, shock waves, etc). Being extremely regular, hexahedral meshes are often claimed to be The Holy Grail for many finite element methods  17, outperforming tetrahedral meshes both in terms of computational speed and accuracy.

  Despite 30 years of research efforts and important advances, mainly by the Lawrence Livermore National Labs in the U.S. 41, 40, hexahedral meshing still requires considerable manual intervention in most cases (days, weeks and even months for the most complicated domains). Some automatic methods exist 28, 44, that constrain the boundary into a regular grid, but they are not fully satisfactory either, since the grid is not aligned with the boundary. The advancing front method 15 does not have this problem, but generates irregular elements on the medial axis, where the fronts collide. Thus, there is no fully automatic algorithm that results in satisfactory boundary alignment.

3.1 Point clouds

Currently, transforming the raw point cloud into a triangular mesh is a long pipeline involving disparate geometry processing algorithms:

• Point pre-processing: colorization, filtering to remove unwanted background, first noise reduction along acquisition viewpoint;
• Registration: cloud-to-cloud alignment, filtering of remaining noise, registration refinement;
• Mesh generation: triangular mesh from the complete point cloud, re-meshing, smoothing.

The output of this pipeline is a locally-structured model which is used in downstream mesh analysis methods such as feature extraction, segmentation in meaningful parts or building Computer-Aided Design (CAD) models.

It is well known that point cloud data contains measurement errors due to factors related to the external environment and to the measurement system itself  39, 33, 19. These errors propagate through all processing steps: pre-processing, registration and mesh generation. Even worse, the heterogeneous nature of different processing steps makes it extremely difficult to know how these errors propagate through the pipeline. To give an example, for cloud-to-cloud alignment it is necessary to estimate normals. However, the normals are forgotten in the point cloud produced by the registration stage. Later on, when triangulating the cloud, the normals are re-estimated on the modified data, thus introducing uncontrollable errors.

We plan to develop new reconstruction, meshing and re-meshing algorithms, with a specific focus on the accuracy and resistance to all defects present in the input raw data. We think that pervasive treatment of uncertainty is the missing ingredient to achieve this goal. We plan to rethink the pipeline with the position uncertainty maintained during the whole process. Input points can be considered either as error ellipsoids  43 or as probability measures  27. In a nutshell, our idea is to start by computing an error ellipsoid  45, 31 for each point of the raw data, and then to cumulate the errors (approximations) made at each step of the processing pipeline while building the mesh. In this way, the final users will be able to take the knowledge of the uncertainty into account and rely on this confidence measure for further analysis and simulations. Quantifying uncertainties for reconstruction algorithms, and propagating them from input data to high-level geometry processing algorithms has never been considered before, possibly due to the very different methodologies of the approaches involved. At the very beginning we will re-implement the entire pipeline, and then attack the weak links through all three reconstruction stages.

3.2 Parameterizations

One of the favorite tools we use in our team are parameterizations, and we have major contributions to the field: we have solved a fundamental problem formulated more than 60 years ago 2. Parameterizations provide a very powerful way to reveal structures on objects. The most omnipresent application of parameterizations is texture mapping: texture maps provide a way to represent in 2D (on the map) information related to a surface. Once the surface is equipped with a map, we can do much more than a mere coloring of the surface: we can approximate geodesics, edit the mesh directly in 2D or transfer information from one mesh to another.

Parameterizations constitute a family of methods that involve optimizing an objective function, subject to a set of constraints (equality, inequality, being integer, etc.). Computing the exact solution to such problems is beyond any hope, therefore approximations are the only resort. This raises a number of problems, such as the minimization of highly nonlinear functions and the definition of direction fields topology, without forgetting the robustness of the software that puts all this into practice.

 

We are particularly interested in a specific instance of parameterization: hexahedral meshing. The idea 6, 4 is to build a transformation $f$ from the domain to a parametric space, where the distorted domain can be meshed by a regular grid. The inverse transformation $f^{-1}$ applied to this grid produces the hexahedral mesh of the domain, aligned with the boundary of the object. The strength of this approach is that the transformation may admit some discontinuities. Let us show an example: we start from a tetrahedral mesh (Figure 2, left) and we want to deform it in a way that its boundary is aligned with the integer grid. To allow for a singular edge in the output (the valency 3 edge, Figure 2, right), the input mesh is cut open along the highlighted faces and the central edge is mapped onto an integer grid line (Figure 2, middle). The regular integer grid then induces the hexahedral mesh with the desired topology.

Current global parameterizations allow grids to be positioned inside geometrically simple objects whose internal structure (the singularity graph) can be relatively basic. We wish to be able to handle more configurations by improving three aspects of current methods:

• Local grid orientation is usually prescribed by minimizing the curvature of a 3D steering field. Unfortunately, this heuristic does not always provide singularity curves that can be integrated by the parameterization. We plan to explore how to embed integrability constraints in the generation of the direction fields. To address the problem, we already identified necessary validity criteria: for example, the permutation of axes along elementary cycles that go around a singularity must preserve one of the axes (the one tangent to the singularity). The first step to enforce this (necessary) condition will be to split the frame field generation into two parts: first we will define a locally stable vector field, followed by the definition of the other two axes by a 2.5D directional field (2D advected by the stable vector field).
• The grid combinatorial information is characterized by a set of integer coefficients whose values are currently determined through numerical optimization of a geometric criterion: the shape of the hexahedra must be as close as possible to the steering direction field. Thus, the number of layers of hexahedra between two surfaces is determined solely by the size of the hexahedra that one wishes to generate. In these settings, degenerate configurations arise easily, and we want to avoid them. In practice, mixed integer solvers often choose to allocate a negative or zero number of layers of hexahedra between two constrained sheets (boundaries of the object, internal constraints or singularities). We will study how to inject strict positivity constraints into these cases, which is a very complex problem because of the subtle interplay between different degrees of freedom of the system. Our first results for quad-meshing of surfaces give promising leads, notably thanks to motorcycle graphs  20, a notion we wish to extend to volumes.
• Optimization for the geometric criterion makes it possible to control the average size of the hexahedra, but it does not ensure the bijectivity (even locally) of the resulting parameterizations. Considering other criteria, as we did in 2D  26, would probably improve the robustness of the process. Our idea is to keep the geometry criterion to find the global topology, but try other criteria to improve the geometry.

3.3 Hexahedral-dominant meshing

All global parameterization approaches are decomposed into three steps: frame field generation, field integration to get a global parameterization, and final mesh extraction. Getting a full hexahedral mesh from a global parameterization means that it has positive Jacobian everywhere except on the frame field singularity graph. To our knowledge, there is no solution to ensure this property, but some efforts are done to limit the proportion of failure cases. An alternative is to produce hexahedral dominant meshes. Our position is in between those two points of view:

1. We want to produce full hexahedral meshes;
2. We consider as pragmatic to keep hexahedral dominant meshes as a fallback solution.

The global parameterization approach yields impressive results on some geometric objects, which is encouraging, but not yet sufficient for numerical analysis. Note that while we attack the remeshing with our parameterizations toolset, the wish to improve the tool itself (as described above) is orthogonal to the effort we put into making the results usable by the industry. To go further, our idea (as opposed to  32, 22) is that the global parameterization should not handle all the remeshing, but merely act as a guide to fill a large proportion of the domain with a simple structure; it must cooperate with other remeshing bricks, especially if we want to take final application constraints into account.

For each application we will take as an input domains, sets of constraints and, eventually, fields (e.g. the magnetic field in a tokamak). Having established the criteria of mesh quality (per application!) we will incorporate this input into the mesh generation process, and then validate the mesh by a numerical simulation software.

4.1 Geometric Tools for Simulating Physics with a Computer

Numerical simulation is the main targeted application domain for the geometry processing tools that we develop. Our mesh generation tools will be tested and evaluated within the context of our cooperation with Hutchinson, experts in vibration control, fluid management and sealing system technologies. We think that the hex-dominant meshes that we generate have geometrical properties that make them suitable for some finite element analyses, especially for simulations with large deformations.

We also have a tight collaboration with a geophysical modeling specialists via the RING consortium. In particular, we produce hexahedral-dominant meshes for geomechanical simulations of gas and oil reservoirs. From a scientific point of view, this use case introduces new types of constraints (alignments with faults and horizons), and allows certain types of nonconformities that we did not consider until now.

Our cooperation with RhinoTerrain pursues the same goal: reconstruction of buildings from point cloud scans allows to perform 3D analysis and studies on insolation, floods and wave propagation, wind and noise simulations necessary for urban planification.

5.1 Awards

• Étienne Corman has received the Young Investigator Award from the Shape Modeling International society.
• Our software Geogram received the Symposium on Geometry Processing Software Award.
• François Protais received the second prize for his PhD from GdR IG-RV

5.2 Programming contests

• Yoann Coudert–Osmont finished second in the final of Tech Challenger, Tech Challenger 2023, a French algorithmic contest that attracted over 3'000 contestants. After four qualification rounds and a semi-final, 100 challengers gathered for a final in Paris. At each round, the participants have one hour to solve three algorithmic problems in as little time as possible.
• Yoann Coudert–Osmont received the Genius prize for the best student for the Master Dev de France 2023, a contest similar to Tech Challenger. The contest takes place on a single day with the format of five qualification round each one allowing 40 more participants to reach the final. The competition is attended by about 2000 participants.
• Yoann Coudert–Osmont won the first prize for the Meritis, Code on Time 2023. This contest lasts for 2 weeks. 332 French participants competed for the highest score by solving three exact problems and maximizing their scores with heuristics on a final NP-hard problem.
• Yoann Coudert–Osmont ranked second in the ICPC 2023 Online Spring Challenge powered by Huawei:.Few years ago, the ICPC foundation started to organize some regular challenges with their partner Huawei. The ICPC foundation organizes the International Collegiate Programming Contest (ICPC formerly ACM ICPC), the most prestigious algorithmic contest in the world. The 2023 Online Spring Challenge is the fifth one. The challenge lasted two weeks and gathered 1344 participants around the world. The goal of this edition was to design a buffer sharing algorithm for a multi-tenant database environment.
• ICPC Challenge Championship powered by Huawei: In August 2023, the ICPC foundation and Huawei invited 58 of the best contestants from the five previous ICPC challenges to a new onsite challenge in the Ox Horn campus, Dongguan, China. Participants came from 25 different countries. This new challenge, strongly inspired by the previous online challenge, lasted 5 hours and Yoann Coudert–Osmont came third out of the 58 contestants and about 100 participants from a training camp for the upcoming ICPC world finals which were taking place in the same time in Dongguan.

6.1 New software

7.1 Integer seamless maps

 

A large family of quadmeshing methods is based on a parametrization of the input surface. Intuitively, a quadmesh can be computed by first laying an input triangle mesh onto the plane using a parametrization technique, overlaying the uv-coordinates with an orthogonal grid of the plane and lifting the result back onto the surface. In other words, using a parametrization for quadmeshing boils down to applying a grid texture onto a surface and extracting the resulting connectivity, refer to Figure 3. However, one needs to be careful of what happens at the seams and the boundary of the input mesh in order to avoid discontinuities and retrieve complete and valid quads. Computing a global seamless parametrization is a challenging problem in geometry processing, mostly due to the fact that any attempt at solving it needs to determine two very different sets of variables. On the one hand, a seamless parametrization is defined by a discrete set of singular points (or cones) that concentrate all the curvature in multiples of $\pi /2$. On the other hand, the actual mapping needs to be optimized to account for specific constraints depending on the application, such as alignment with feature edges or creases, local sizing of elements or controlled distortion. Yet, the mapping's topology is determined by the cone distribution, which can drastically alter its quality. In particular, finding cone positions minimizing the mapping's distortion is challenging and often leads to sophisticated optimization problems.

To overcome this apparent complexity a common approach is to decouple the problem into two independent subproblems: first computing a rotationally seamless parameterization, and then compute a truly seamless (integer seamless) map, refer to Figure 4 for an illustration. This year we contributed 7, 8 to both parts of the pipeline.

 

7.1.2 Quad Mesh Quantization Without a T‐Mesh

As we have mentioned earlier, truly seamless maps can be obtained by solving a mixed integer optimization problem: real variables define the geometry of the charts and integer variables define the combinatorial structure of the decomposition. To make this optimization problem tractable, a common strategy is to ignore integer constraints at first by computing a rotationally seamless map, then to enforce them in a so-called quantization step.

Actual quantization algorithms exploit the geometric interpretation of integer variables to solve an equivalent problem: they consider that the final quadmesh is a subdivision of a T-mesh embedded in the surface, and optimize the number of subdivisions for each edge of this T-mesh (Fig. 5). We propose 8 to operate on a decimated version of the original surface instead of the T-mesh. One motivation for this work is the easiness of implementation adaptation to constraints such as free boundaries, complex feature curves network, etc. In addition to that, this approach is also very promising for its possible extensions to the 3D (hex meshing) case.

 

    

Contour plots of our method with 1-Lipschitz network on a toy example.

 

 

 

7.2 Non mesh-related results

This year we have also contributed to few topics not directly related to the core topic of the team.

8.1 Bilateral contracts with industry

Company: TotalEnergies

Duration: 01/10/2020 – 30/03/2024

Participants: Dmitry Sokolov, Nicolas Ray and David Desobry

Abstract: The goal of this project is to improve the accuracy of rubber behavior simulations for certain parts produced by TotalEnergies, notably gaskets. To do this, both parties need to develop meshing methods adapted to the simulation of large deformations in non-linear mechanics. The Pixel team has a strong expertise in hex-dominant meshing, TotalEnergies has a strong background in numerical simulation within an industrial context. This collaborative project aims to take advantage of both expertises.

David defended his PhD 12 on August 23, 2023.

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

9.3 Popularization

10.1 Major publications

• 1 articleJ.Justine Basselin, L.Laurent Alonso, N.Nicolas Ray, D.Dmitry Sokolov, S.Sylvain Lefebvre and B.Bruno Lévy. Restricted Power Diagrams on the GPU.Computer Graphics Forum402June 2021HALDOI
• 2 articleV.Vladimir Garanzha, I.Igor Kaporin, L.Liudmila Kudryavtseva, F.François Protais, N.Nicolas Ray and D.Dmitry Sokolov. Foldover-free maps in 50 lines of code.ACM Transactions on GraphicsVolume 40issue 4July 2021, Article No.102, pp 1–16HALDOIback to text
• 3 articleN.Nicolas Ray, D.Dmitry Sokolov, S.Sylvain Lefebvre and B.Bruno Lévy. Meshless Voronoi on the GPU.ACM Trans. Graph.376December 2018, 265:1--265:12URL: http://doi.acm.org/10.1145/3272127.3275092DOI
• 4 articleN.Nicolas Ray, D.Dmitry Sokolov and B.Bruno Lévy. Practical 3D Frame Field Generation.ACM Trans. Graph.356November 2016, 233:1--233:9URL: http://doi.acm.org/10.1145/2980179.2982408DOIback to text
• 5 articleN.Nicolas Ray and D.Dmitry Sokolov. Robust Polylines Tracing for N-Symmetry Direction Field on Triangulated Surfaces.ACM Trans. Graph.333June 2014, 30:1--30:11URL: http://doi.acm.org/10.1145/2602145DOI
• 6 articleD.Dmitry Sokolov, N.Nicolas Ray, L.Lionel Untereiner and B.Bruno Lévy. Hexahedral-Dominant Meshing.ACM Transactions on Graphics3552016, 1 - 23HALDOIback to text

10.2 Publications of the year

10.3 Cited publications

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• 17 inproceedingsT.Ted Blacker. Meeting the Challenge for Automated Conformal Hexahedral Meshing.9th International Meshing Roundtable2000, 11--20back to text
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• 43 inproceedingsM.Markus Ylimäki, J.Juho Kannala and J.Janne Heikkilä. Robust and Practical Depth Map Fusion for Time-of-Flight Cameras.Image AnalysisChamSpringer International Publishing2017, 122--134back to text
• 44 inproceedingsY.Yongjie Zhang and C.Chandrajit Bajaj. Adaptive and Quality Quadrilateral/Hexahedral Meshing from Volumetric Data.International Meshing Roundtable conf. proc.2004back to text
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• 46 miscCgal, Computational Geometry Algorithms Library.http://www.cgal.orgback to text