Convex polyhedral meshing for robust solid modeling

Abstract

We introduce a new technique to create a mesh of convex polyhedra representing the interior volume of a triangulated input surface. Our approach is particularly tolerant to defects in the input, which is allowed to self-intersect, to be non-manifold, disconnected, and to contain surface holes and gaps. We guarantee that the input surface is exactly represented as the union of polygonal facets of the output volume mesh. Thanks to our algorithm, traditionally difficult solid modeling operations such as mesh booleans and Minkowski sums become surprisingly robust and easy to implement, even if the input has defects. Our technique leverages on the recent concept of indirect geometric predicate to provide an unprecedented combination of guaranteed robustness and speed, thus enabling the practical implementation of robust though flexible solid modeling systems. We have extensively tested our method on all the 10000 models of the Thingi10k dataset, and concluded that no existing method provides comparable robustness, precision and performances.

Interactive and Robust Mesh Booleans

“…Its faster version [Hu et al 2020] uses floating point calculations to create a conforming tetrahedral mesh, though with no formal guarantees. The creation of a volumetric mesh which conforms to the input was also used in [Diazzi and Attene 2021], where exact arithmetic was replaced by indirect predicates and Boolean operations could be extracted at a much higher speed while maintaining the correctness guarantees. The problem of partitioning the space based on input facets was also tackled in [Paoluzzi et al 2020[Paoluzzi et al , 2019 based on floating point computation, using the language of geometric and algebraic topology.…”

“…Robust methods for inside/outside partitioning exist, but they are either computationally inefficient [Jacobson et al 2013] or are efficient at query time but are approximate and require initialization [Barill et al 2018]. The alternative is to eschew the use of floating point arithmetic and represent point coordinates with rational numbers or implicitly [Diazzi and Attene 2021]. These techniques ensure a topologically correct result at the cost of 1 to 2 orders of magnitude slow down w.r.t.…”

Hex-Mesh Generation and Processing: a Survey

“…Most of the techniques discussed in this section make assumptions on the topology and geometry of the input mesh and are not able to operate on meshes containing topological (e.g., open boundaries, holes, or non-manifold elements) or geometric (e.g., intersecting or degenerate elements) defects. Methods that operate on a supporting tetrahedral mesh may leverage robust tetrahedralization techniques such as [Diazzi and Attene 2021;Hu et al 2020. Methods that operate on surface meshes can sanitize their inputs with known robust surface processing algorithms, such as [Attene 2010;Attene et al 2013;Cherchi et al 2020;.…”

“…Considering the important information encoded in the connectivity of these meshes, when simulating complex body dynamics that involve multiple organs it becomes important to create composite simulation domains that preserve as much as possible the connectivity of each original mesh. Blending multiple meshes into a single one is a widely studied problem in the literature, especially for the case of unstructured meshes composed of triangles or tetrahedra [Cherchi et al 2020;Diazzi and Attene 2021;. For structured meshes made of quads or hexahedra, the problem is more complex because the necessary changes of the local connectivity have a global footprint.…”

“…Tuchinsky and Clark observed that since a hexahedral mesh can cover the same volume of a tetrahedral mesh with roughly one-quarter of the elements, there is a 75% saving in terms of computational cost [Tuchinsky and Clark 1997]. This estimate is based on the assumption that "analysis setup and pre-processing requires the same time for hex-and tet-based work", which does not reflect the current state of mesh generation because creating and processing tetrahedral meshes is significantly easier and more robust than creating hexahedral ones [Diazzi and Attene 2021;Hu et al 2020. Regarding accuracy, it seems to be well understood and established that linear tetrahedra are to be avoided because they introduce artificial stiffness in the problem (i.e., they "lock") [Wang et al 2004], whereas linear hexahedral elements do not introduce such artifact.…”