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Structural and Multidisciplinary Optimization

Erschienen in:

01.03.2012 | Educational Article

PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab

verfasst von: Cameron Talischi, Glaucio H. Paulino, Anderson Pereira, Ivan F. M. Menezes

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 3/2012

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Abstract

We present a simple and robust Matlab code for polygonal mesh generation that relies on an implicit description of the domain geometry. The mesh generator can provide, among other things, the input needed for finite element and optimization codes that use linear convex polygons. In topology optimization, polygonal discretizations have been shown not to be susceptible to numerical instabilities such as checkerboard patterns in contrast to lower order triangular and quadrilaterial meshes. Also, the use of polygonal elements makes possible meshing of complicated geometries with a self-contained Matlab code. The main ingredients of the present mesh generator are the implicit description of the domain and the centroidal Voronoi diagrams used for its discretization. The signed distance function provides all the essential information about the domain geometry and offers great flexibility to construct a large class of domains via algebraic expressions. Examples are provided to illustrate the capabilities of the code, which is compact and has fewer than 135 lines.

Nächster Artikel PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes

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For example, consider \(\Omega_{1}=\left\{ \left(x_{1},x_{2}\right)\in\mathbb{R}^{2}:x_{1}<0\right\} \) and \(\Omega_{2}=\left\{ \left(x_{1},x_{2}\right)\in\mathbb{R}^{2}:x_{2}<0\right\} \). The formula \(d_{\Omega_{1}\cup\Omega_{2}}(\mathbf{x})=\min\left(d_{\Omega_{1}}(\mathbf{x}),d_{\Omega_{2}}(\mathbf{x})\right)\) has incorrect distance “value” in the third quadrant, i.e., for x ₁ < 0, x ₂ < 0. In this region, the closest boundary point is the new corner x = (0, 0) formed by the union operation.

A tessellation or tiling of Δ is a collection of open sets S _i such that \(\cup_{i}\overline{S}_{i}=\overline{\Delta}\) and S _i ∩ S _j = ∅ if i ≠ j.

The cell structure allows for storing vectors of different size and is therefore suitable for connectivity of polygonal elements with different number of nodes.

This small overhead can be removed after a few iterations once a good estimate value is obtained.

Since the gradation in mesh is often dictated by the geometry of the domain, it is natural that both μ and h be defined based on the distance function d _Ω.

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Titel: PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab
verfasst von: Cameron Talischi
Glaucio H. Paulino
Anderson Pereira
Ivan F. M. Menezes
Publikationsdatum: 01.03.2012
Verlag: Springer-Verlag
Erschienen in: Structural and Multidisciplinary Optimization / Ausgabe 3/2012
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-011-0706-z

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