Cutting and pasting to combine different elements into a common structure are widely used operations that have been successfully adapted to many media types. Surface design could also benefit from the availability of a general, robust, and efficient cut-andpaste tool, especially during the initial stages of design when a large space of alternatives needs to be explored. Techniques to support cut-and-paste operations for surfaces have been proposed in the past, but have been of limited usefulness due to constraints on the type of shapes supported and the lack of real-time interaction. In this paper, we describe a set of algorithms based on multiresolution subdivision surfaces that perform at interactive rates and enable intuitive cut-and-paste operations.

In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to free-form solids bounded by multiresolution subdivision surfaces.

We present an effective and easy-to-implement angle-based smoothing scheme for triangular, quadrilateral and tri-quad mixed meshes. For each mesh node our algorithm compares all the pairs of adjacent angles incident to the node and adjusts these angles so that they become equal in the case of a triangular mesh and a quadrilateral mesh, or they form the ideal ratio in the case of a tri-quad mixed mesh. The size and shape quality of the mesh after this smoothing algorithm is much better than that after Laplacian smoothing. The proposed method is superior to Laplacian smoothing by reducing the risk of generating inverted elements and increasing the uniformity of element sizes. The computational cost of our smoothing method is yet much lower than optimization-based smoothing. To prove the effectiveness of this algorithm, we compared errors in approximating a given analytical surface by a set of bi-linear patches corresponding to a mesh with Laplacian smoothing and a mesh with the proposed smoothing method. The experiments show that a mesh smoothed with our method has roughly 20 % less approximation error.

A key step in the nite element method is to generate well-shaped meshes in 3D. A mesh is well-shaped if every tetrahedron element has a small aspect ratio. It is an old outstanding problem to generate well-shaped Delaunay meshes in three or more dimensions. Existing algorithms do not completely solve this problem, primarily because they can not eliminate all slivers. A sliver is a tetrahedron whose vertices are almost coplanar and whose circumradius is not much larger than its shortest edge length. We present two new algorithms to generate sliver-free Delaunay meshes. The rst algorithm locally moves the vertices of an almost-good mesh, whose tetrahedra have small circumradius to shortest edge length ratio. We show that the Delaunay triangulation of the perturbed mesh vertices is still almost good. Furthermore, most slivers disappear after a mild perturbation of the mesh vertices. The remaining slivers migrate to the boundary where they can be peeled o or can be treated with boundary enforcement heuristics. The second algorithm adds points to generate well-shaped meshes. It is based on the following observations. Any tetrahedron will disappear from the Delaunay triangulation if a point is added inside the circumsphere of the tetrahedron. Among the tetrahedra created by

The quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. The objective function is derived from some quality measure of the local submesh, that is, the set of tetrahedra connected to the adjustable or free node. Although these objective functions are suitable to improve the quality of a mesh in which there are non-inverted elements, they are not when the mesh is tangled. This is due to the fact that usual objective functions are not defined on all R 3 and they present several discontinuities and local minima that prevent the use of conventional optimisation procedures. Otherwise, when the mesh is tangled, there are local submeshes for which the free node is out of the feasible region, or this does not exist. In this paper we propose the substitution of objective functions having barriers by modified versions that are defined and regular on all R 3. With these modifications, the optimisation process is also directly applicable to meshes with inverted elements, making a previous untangling procedure unnecessary. This simultaneous procedure allows to reduce the number of iterations for reaching a prescribed quality. To illustrate the effectiveness of our approach, we present several applications where it can be seen that our results clearly improve those obtained by other authors.

We develop and implement new algorithms for smoothing triangular and tetrahedral unstructured meshes. Our approach is based on a variation of the force-directed method used in graph drawing. This method assumes that on each vertex a certain force is applied that moves the vertex relative to its neighbors so that the shapes of its incident elements are improved. The final stable configuration often corresponds to a graph with good global properties. In this paper we show that this method can be successfully applied to mesh smoothing and describe some details of our implementation and test results.

CCR-9706572, and ACI-0086093. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the NSF or the US government. Keywords: mesh generation, computational geometry, Bézier curves, Bézier triangles, We present in this report a new framework for maintaining good quality of two dimensional triangular moving meshes. The use of curved elements is the key idea that allows us to avoid excessive refinement and still obtain good quality meshes consisting of a low number of well shaped elements. We use B-splines curves to model object boundaries and objects are meshed with second order Bézier triangles. As the mesh evolves according to a nonuniform flow velocity field, we keep track of object boundaries and, if needed, carefully modify the mesh to keep it well shaped by applying a combination of vertex insertion and deletion, edge flipping, and curve smoothing operations at each time step. Our algorithms for these tasks are extensions of known algorithms for meshes build of straight–sided elements and are designed for any fixed-order Bézier elements and B-splines. We discuss a calculus of geometric primitives for Bézier curves and triangles that we employ to implement such

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We study the design and implementation of algorithms...

: A novel method is presented for automatically generating quadrilateral meshes on arbitrary two-dimensional domains. Global minimization of a potential function governs mesh formation and characteristics. Comprised of several terms, the potential function distributes the elements throughout the domain and aligns the edges of the elements to form valid connectivities. If there are any remaining unlinked element edges, the local connectivity is examined and a "hole elimination" algorithm is applied that successively finds alternative connectivities. Unlinked edges, representing holes in the mesh, are moved to either coalesce, or to a boundary. The components of the potential, the minimization procedure, and the connectivity refinement algorithm are presented. The method shows promise for extension to automatic three-dimensional hexahedral meshing. Initial conditions required to ensure mesh closure include an even number of elements on the boundary and a closed boundary. The desired mesh...