Tetrahedral mesh deformation with positional constraints
Graphical abstract
Introduction
Tetrahedral meshes are a common representation of 3D objects in science, engineering and medicine. They provide a compact representation for both the boundary surface and the interior volume, which benefits many applications such as computer aided design, finite element analysis and computational simulation. The tetrahedral mesh representation also facilitates the rendering of the interior region and structure of objects. Extensive research has been conducted on generating quality tetrahedral meshes from various inputs and some excellent algorithms have been developed (Si (2006); Goksel and Salcudean (2011); Si (2015)).
This paper considers the problem of deforming a tetrahedral mesh under positional constraints that some vertices of the mesh are moved to the required new locations. The problem is motivated by applications in computer animation (Zhang et al. (2015)) and design optimization (Sieger et al. (2013)), where two or more topologically compatible tetrahedral meshes are involved. For example, example-based animation requires examples (tetrahedral meshes) to have the same topology as the animated model and the examples are usually created from an initial model by displacing some vertices. Computer aided design frequently changes the shape of the model according to a set of parameters driven by the optimization process and the change is often achieved by updating the existing mesh to conform to the user specified constraints, which simplifies the simulation (Staten et al. (2012)). In general, generating a deformed tetrahedral mesh with good quality involves considerable effort and manual adjustment is often needed.
Especially when the positional difference between the driving vertices on the input tetrahedral mesh and the target is large, mesh deformation may cause some tetrahedra to invert. A tetrahedral mesh with inverted elements is not allowed for numerical analysis since it yields a physically invalid solution and/or unstable simulation. A good method should produce inversion-free deformation. Also, different from conventional mesh generation, mesh deformation creates a new mesh that is often required to have the topology compatible with the existing one. However, some remeshing strategies used to eliminate inverted elements or other artefacts (Dai and Schmidt (2005)) likely result in topology incompatibility. Moreover, similar to conventional mesh generation, the output mesh is required to have good quality especially when the input mesh has good quality. All these requirements actually impose challenges on constrained tetrahedral mesh deformation.
We present a solution that achieves the constrained deformation through an iterative warping process. The two underlying technical components of the process are radial basis function (RBF) interpolation and adaptive refinement. The approach is an extension of our previous work presented at the conference on Computer-Aided Design and Computer Graphics (Zhang et al. (2013)), where the deformation of a tetrahedral mesh is driven only by the boundary surface. Inspired by (Ma et al. (2015)), we propose to use RBF-interpolation to achieve the required positional constraints and meanwhile the global smooth triharmonic RBF delivers a robust and quality deformation. The combination of the selection of safe warping stepsizes for iterative RBF interpolation and the local refinement delivers an inversion-free and effective warping without too many iterations or over-refinement (Shontz and Vavasis (2010)). Unlike previous work that uses remeshing, we perform refinement based on the edge bisection that allows us to refine the warped mesh and the original mesh concurrently to ensure consistent topology while keeping the shape of the original mesh unchanged. Consequently our method can output an inversion-free and topology compatible deformation mesh that satisfies the positional constraints. Fig. 1 shows one example where simply deforming the mesh (top-left) introduces inversion (bottom-left) and by iterative warping and local refinement, our method can generate an inversion-free deformed mesh (bottom-right) with the same topology as the refined input mesh (top-right).
The rest of the paper is organized as follows. Section 2 briefly reviews related work. Then Section 3 presents the proposed deformation algorithm, followed by theoretical analysis of the algorithm in Section 4. Section 5 reports experimental results and provides some discussions. Finally Section 6 concludes the paper.
Section snippets
Related work
Transforming an existing tetrahedral mesh to conform to a modified geometry (particularly the boundary surface) has been studied in different scenarios. Various techniques have been developed, such as mesh morphing (Vurputoor et al. (2008)), mesh warping (Shontz and Vavasis (2010)) and mesh moving (Tezduyar and Benney (2003)). Several popular techniques are reviewed in (Staten et al. (2012)). The techniques can roughly be classified into mesh-based approaches or mesh-less approaches according
Deformation through warping
A tetrahedral mesh is called conforming if the intersection of two distinct tetrahedra is either a common face, a common edge, a common vertex, or empty. The input to our algorithm includes: (1) a conforming tetrahedral mesh with n vertices and m tetrahedra where X is the vertex set of the mesh and T represents the connectivity of the vertices to form the mesh; and (2) two vertex sets and where vertex is moved to . We also assume that vertices
Theoretical analysis
This section analyzes some properties of the proposed algorithm.
We first prove that Arnold et al.'s edge bisection refinement (Arnold et al. (2000)) can generate sufficiently small tetrahedra.
Lemma 4.1 Given a non-degenerated tetrahedron , for an arbitrary positive number , there exists a natural number such that any tetrahedron generated from by Arnold et al.'s edge bisection refinement after n times edge bisection has the maximal edge length less than β. Proof We prove the lemma by
Experiments and discussions
This section presents some experimental results. The examples used in our experiments are selected from two fields: computer-aided design and computer graphics. The bore and pipe models are the same as the models used in (Sieger et al. (2013)). The other three are from computer graphics. All the tetrahedral meshes are generated with Tetgen (Si (2006)). The statistics of these models are summarized in Table 3.
CAD models. A good mesh deformation algorithm should allow large shape changes (Sieger
Conclusions
We have described an iterative RBF-based tetrahedral mesh deformation algorithm that warps a tetrahedral mesh to conform to positional constraints. The algorithm warps the mesh iteratively with a safe stepsize, assuring no inversion. Locally refinement is introduced to allow larger displacement for the constrained vertices and meanwhile avoid over-refinement. An edge bisection process is employed, which is applied to the original and warped meshes to ensure the topology compatibility. The
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was conducted in collaboration with HP Inc. and partially supported by the Singapore Government through the Industry Alignment Fund Industry Collaboration Projects Grant. It was also partially supported by the Ministry of Education, Singapore, under its MOE Tier-2 Grant (2017-T2-1-076).
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