Centroidal Voronoi diagrams for isotropic surface remeshing

This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.     

Polygonal surface remeshing with Delaunay refinement

Polygonal meshes are used to model smooth surfaces in many applications. Often these meshes need to be remeshed for improving the quality, density or gradedness. We apply the Delaunay refinement paradigm to design a provable algorithm for isotropic remeshing of a polygonal mesh that approximates a smooth surface. The proofs provide new insights and our experimental results corroborate the theory.     

High quality surface remeshing with equilateral triangle grid

This study proposes a robust and efficient 3D surface remeshing algorithm for mesh quality optimization. Instead of the global mesh relaxation method proposed in the previous study conducted on remeshing, this study proposes an equilateral triangle grid-resampling scheme for achieving mesh optimization more efficiently. In order to improve the feasibility of resampling by directly using an equilateral triangle grid, the surface structure of the original model is correctly extracted by an automatic surface segmentation technique before the resampling step is executed. Results of this study show that the proposed remeshing algorithm can automatically and substantially improve the quality of triangulation, as well as automatically preserve shape features under an acceptable level of measurement error in the shape approximation, which is suitable for a mesh with a specific topology.     

Matrix representations for toric parametrizations

In this paper we show that a surface in parametrized over a 2-dimensional toric variety can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over established in [Busé, L., Jouanolou, J.-P., 2003. J. Algebra 265 (1), 312–357] and [Busé, L., Chardin, M.J., 2005. Symbolic Comput. 40 (4–5), 1150–1168]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case in detail and give numerous examples.     

Vertex location optimisation for improved remeshing

Remeshing aims to produce a more regular mesh from a given input mesh, while representing the original geometry as accurately as possible. Many existing remeshing methods focus on where to place new mesh vertices; these samples are placed exactly on the input mesh. However, considering the output mesh as a piecewise linear approximation of some geometry, this simple scheme leads to significant systematic error in non-planar regions. Here, we use parameterised meshes and the recent mathematical development of orthogonal approximation using Sobolev-type inner products to develop a novel sampling scheme which allows vertices to lie in space near the input surface, rather than exactly on it. The algorithm requires little extra computational effort and can be readily incorporated into many remeshing approaches. Experimental results show that on average, approximation error can be reduced by 40% with the same number of vertices.     

Adaptive tetrahedral remeshing for modified solid models

During the engineering design process, a designed engineering component is usually repeatedly modified and analyzed to reach final design objectives, and completely mesh regeneration for each design change is very time consuming. An efficient remeshing approach for modified solid models using existing prior existing meshes is proposed in this paper to resolve this issue. It is mainly achieved via three main steps: different face (between the input model and its modification) identification, local destruction zone generation, and local mesh regeneration. In this approach, by carefully selecting a destruction zone to be removed from the original mesh model, a final geometric adaptive mesh model of the modified model is generated, which is very important for downstream analysis accuracy control. Additionally, the complex boundary of the variation region that needs to be remeshed is set up by merging the boundaries of modified region and the destruction zone without using complex intersection operations, which ensure the approach’s robustness. Experimental results on practical engineering parts are also shown to demonstrate the method’s performance.     

Optimizing polycube domain construction for hexahedral remeshing

Polycube mapping can provide regular and global parametric representations for general solid models. Automatically constructing effective polycube domains, however, is challenging. We present an algorithm for polycube construction and volumetric parameterization. The algorithm has three steps: pre-deformation, polycube construction and optimization, and mapping computation. Compared with existing polycube mapping methods, our algorithm can robustly generate desirable polycube domain shape and low-distortion volumetric parameterization. It can be used for automatic high-quality hexahedral mesh generation.     

On parametrizations for the minimal partial realization problem

The paper considers parametrizations for minimal partial realizations of a finite sequence of Markov parameters. First it is shown that all minimal partial realizations have the same set of input and output Kronecker indices. A parametrization is obtained formulated in terms of a minimal set of extension entries. Then an I/O canonical form is used as a parametrization for the set of minimal partial realizations. It is shown that this allows to define the invariants of the problem. Moreover, the I/O canonical form is shown to contain a set of parameters that can be chosen such that any minimal partial realization can be represented.     

语义特征模型网格重划的研究*

本文提出一个新的方法,基于原模型的网格来有效构造一个新网格。该技术利用语义特征模型具有操作局部化的特点,通过细胞元变化模型得出 (修改前/后)两个特征模型的相关部分,将原模型的复制网格节点直接关联到修正模型中,并结合对自由节点扩大自由集来进行优化过程。利用这种方法,可以在语义特征模型的几何属性中找到其固有的一致性,并且与修改前模型的网格相关联,最终在保持现有有限元的质量条件下快速产生完整质量网格。     

Automatic construction of boundary parametrizations for geometric multigrid solvers

We present an algorithm that constructs parametrizations of boundary and interface surfaces automatically. Starting with high-resolution triangulated surfaces describing the computational domains, we iteratively simplify the surfaces yielding a coarse approximation of the boundaries with the same topological type. While simplifying we construct a function that is defined on the coarse surface and whose image is the original surface. This function allows access to the correct shape and surface normals of the original surface as well as to any kind of data defined on it. Such information can be used by geometric multigrid solvers doing adaptive mesh refinement. Our algorithm runs stable on all types of input surfaces, including those that describe domains consisting of several materials. We have used our method with success in different fields and we discuss examples from structural mechanics and biomechanics. Editorial responsibility: P. Deuflhard     

System identification using balanced parametrizations

Some general issues in the “black-box” identification of multivariable systems are first discussed. It is then suggested that balanced parametrizations can be used to give identifiable forms. A particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models, including stable minimal models, minimum-phase models, positive-real models, and normalized coprime factor models. Before optimizing the parameters of balanced parametrizations, an initial model must be found. We use realization-based methods and so-called “subspace” methods for this purpose. These methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes. The paper ends with two simulation examples, which compare the use of balanced parametrizations with more traditional ones, and three “real” examples based on practical problems: a distillation column, an industrial dryer, and the (irrational) spectrum or sea waves     

Minimal rational parametrizations of canal surfaces

It is well known that canal surfaces defined by a rational spine curve and a rational radius function are rational. The aim of the present paper is to construct a rational parameterization of low degree. The author uses the generalized stereographic projection in order to transform the problem to a parameterization problem for ruled surfaces. Two problems are discussed: parameterization with boundary conditions (design of canal surfaces with two curves on it, as is the case for rolling ball blends) and parameterization without boundary conditions.     

Surface remeshing in arbitrary codimensions

We present a method for remeshing surfaces that is both general and efficient. Existing efficient methods are restrictive in the type of remeshings they produce, while methods that are able to produce general types of remeshings are generally based on iteration, which prevents them from producing remeshes at interactive rates. In our method, the input surface is directly mapped to an arbitrary (possibly high-dimensional) range space, and uniformly remeshed in this space. Because the mesh is uniform in the range space, all the quantities encoded in the mapping are bounded, resulting in a mesh that is simultaneously adapted to all criteria encoded in the map, and thus we can obtain remeshings of arbitrary characteristics. Because the core operation is a uniform remeshing of a surface embedded in range space, and this operation is direct and local, this remeshing is efficient and can run at interactive rates.     

Minimal parametrizations of single-input linear dynamical systems

The parametrization of the set of all controllable single-input time-invariant linear dynamical systems is defined as a map π from a parameter set M_π onto a given set of canonical forms 𝒸_π. A parametrization is called ‘minimal’ if it induces a continuous canonical map and the number of parameters is minimal. Some of its general properties, necessary and sufficient conditions, and several new concrete forms suitable for identification are also given.     

Manifold-based approach to semi-regular remeshing

This paper describes a method for semi-regular remeshing of arbitrary shapes. The proposed approach is based on building a parameterization map which is smooth with respect to a differential structure built on the base domain. A global parametric energy functional is introduced and optimized in order to establish a globally smooth parameterization. The proposed approach avoids using meta-mesh construction during the parameterization and resampling stages which allows for an easier implementation. A simple extension of the method is proposed to improve the approximation properties of the resulting remesh.     

Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems

Multivariable systems can be represented, in a uniquely identifiable way, either by canonical forms or by so-called overlapping forms. The advantage of the latter is that they do not require the a priori estimation of a set of structural invariants (e.g. Kronecker invariants). We show here how to define uniquely identifiable overlapping parametrizations for state-space and ARMA models. We show that these parametrizations are all related to a set of intrinsic invariants, which are obtained from the Markov parameters of the system. Different forms of overlapping ARMA parametrizations are derived and their properties discussed.     

Surface remeshing with robust high-order reconstruction

Remeshing is an important problem in variety of applications, such as finite element methods and geometry processing. Surface remeshing poses some unique challenges, as it must deliver not only good mesh quality but also good geometric accuracy. For applications such as finite elements with high-order elements (quadratic or cubic elements), the geometry must be preserved to high-order (third-order or higher) accuracy, since low-order accuracy may undermine the convergence of numerical computations. The problem is particularly challenging if the CAD model is not available for the underlying geometry, and is even more so if the surface meshes contain some inverted elements. We describe remeshing strategies that can simultaneously produce high-quality triangular meshes, untangling mildly folded triangles and preserve the geometry to high-order of accuracy. Our approach extends our earlier works on high-order surface reconstruction and mesh optimization by enhancing its robustness with a geometric limiter for under-resolved geometries. We also integrate high-order surface reconstruction with surface mesh adaptation techniques, which alter the number of triangles and nodes. We demonstrate the utilization of our method to meshes for high-order finite elements, biomedical image-based surface meshes, and complex interface meshes in fluid simulations.     

Out-of-Core remeshing of large polygonal meshes

We propose an out-of-core method for creating semi-regular surface representations from large input surface meshes. Our approach is based on a streaming implementation of the MAPS remesher of Lee et al. [18]. Our remeshing procedure consists of two stages. First, a simplification process is used to obtain the base domain. During simplification, we maintain the mapping information between the input and the simplified meshes. The second stage of remeshing uses the mapping information to produce samples of the output semi-regular mesh. The out-of-core operation of our method is enabled by the synchronous streaming of a simplified mesh and the mapping information stored at the original vertices. The synchronicity of two streaming buffers is maintained using a specially designed write strategy for each buffer. Experimental results demonstrate the remeshing performance of the proposed method, as well as other applications that use the created mapping between the simplified and the original surface representations.     

Efficient tetrahedral remeshing of feature models for finite element analysis

Finite element analysis is nowadays widely used for product testing. At various moments during the design phase, aspects of the physical behaviour of the product are simulated by performing an analysis of the model. For each analysis, a mesh needs to be created that represents the geometry of the model at that point. In particular during the later stages of the development cycle, often only minor modifications are made to a model between design iterations. In that case it can be beneficial to reuse part of the previous mesh, especially if it was costly to construct. A new method is presented that efficiently constructs a tetrahedral mesh based on a tetrahedral mesh of a feature model at an earlier point of the design cycle. This is done by analysing the difference of the two feature models from the point of view of the individual features. By this means we can find a natural correspondence between the geometries of the feature models, and relate this to the mesh of the earlier model. We discuss the algorithm, gained improvements, quality of the results, and conditions for this method to be effective.     

Subdivision connectivity remeshing and its applications

This paper presents a subdivision connectivity remeshing approach for closed genus 0 meshes. It is based on spherical parameterization and umbrella‐operator smoothing. Our main contribution lies in adopting a low‐distortion spherical parameterization approach to generate high‐quality subdivision connectivity meshes. Besides, a simple and efficient point location method on the sphere based on the uniform partition of the rectangle is presented, which is used to find the containing triangle in the spherical mesh for each point on the sphere rapidly. Our method can generate high‐quality subdivision connectivity meshes fast, which can be applied to level of detail and progressive transmission. All the application examples demonstrate that our remeshing procedure is robust and efficient. Copyright © 2011 John Wiley & Sons, Ltd.