Centroidal Voronoi tessellation‐based finite element superconvergence

Huang, Yunqing; Qin, Hengfeng; Wang, Desheng

doi:10.1002/nme.2374

Cited by 25 publications

(15 citation statements)

References 36 publications

(65 reference statements)

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“…Given the superior quality of the CVT/CVDT mesh, it is not a coincidence that the resulting finite volume methods exhibit superconvergent properties as observed in [43]. Similar studies on the superconvergence of finite element solutions based on the CVT/CVDT meshes were presented in [66].…”

Section: Cvt-based Finite Volume and Finite Element Methodssupporting

confidence: 54%

“…A numerical example from [70] is given in Figure 10. Superconvergence based a posterior error estimator was further developed in [67] for CVT/CVDT based meshes, utilizing their superconvergence properties as shown in [66]. This leads to another CVT/CVDT based adaptive finite element algorithm for numerical PDEs.…”

Section: Cvt-based Adaptive Algorithms For Numerical Pdesmentioning

confidence: 99%

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Advances in Studies and Applications of Centroidal Voronoi Tessellations

Du¹,

Gunzburger²,

Ju³

2010

NMTMA

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Abstract. Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concept and a few of its generalizations and well-known properties. We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs. Whenever possible, we point out some outstanding issues that still need investigating.

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Section: Cvt-based Finite Volume and Finite Element Methodssupporting

confidence: 54%

Section: Cvt-based Adaptive Algorithms For Numerical Pdesmentioning

confidence: 99%

Advances in Studies and Applications of Centroidal Voronoi Tessellations

Du¹,

Gunzburger²,

Ju³

2010

NMTMA

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“…Successful applications of CVT/CVDT-based adaptive algorithms have been conducted in numerical analysis of partial differential equations (PDEs) [48,54]. In [55], adaptive finite element methods based on the superconvergence properties [56,57] of CVT/CVDT are further developed. It is worth mentioning that the computations of CVT/CVDT density functions are very similar to the calculations of sizing functions in the context of adaptive meshing.…”

Section: Centroidal Voronoi Tessellationmentioning

confidence: 99%

Adaptive meshing for finite element analysis of heterogeneous materials

You

Kou

Tan

2015

Computer-Aided Design

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Adaptive meshing of 2D planar regions, curved surfaces as well as 3D volumes has been extensively studied in Finite Element Analysis (FEA) in the past few decades. Despite the maturity of these adaptive meshing approaches, most of the existing schemes assume the domain or sub-domain of interest is homogeneous. In the context of FEA of heterogeneous objects, traditional adaptive mesh generation strategies become inadequate as they fail to take into account the material heterogeneities. This paper is motivated to tackle such problems and propose an adaptive mesh generation scheme for FEA of versatile heterogeneous materials. The proposed approach takes full advantages of the material heterogeneity information, and the mesh density is formulated with a specific function of the material variations. Dual triangulation of centroidal Voronoi tessellation is then constructed and necessary mesh subdivision is applied with respect to a predefined material threshold. Experiments show that the proposed approach distributes the material composition variation over mesh elements as equally as possible and thus minimizes the number of elements in terms of the given material threshold. FEA results show that the proposed method can significantly decrease the mesh complexities as well as computational resources in FEA of heterogeneous objects and compared with existing approaches, significant mesh reduction can be achieved without sacrifice in FEA qualities.

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“…The centroidal voronoi tessellation (CVT)-based methods have been applied to high quality mesh generation [9,11]. Triangles generated are almost equilateral.…”

Section: Examplementioning

confidence: 99%

High order compact schemes for gradient approximation

Huang

Qin

2010

Sci. China Math.

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In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results.

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Centroidal Voronoi tessellation‐based finite element superconvergence

Cited by 25 publications

References 36 publications

Advances in Studies and Applications of Centroidal Voronoi Tessellations

Advances in Studies and Applications of Centroidal Voronoi Tessellations

Adaptive meshing for finite element analysis of heterogeneous materials

High order compact schemes for gradient approximation

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