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Structural and Multidisciplinary Optimization

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01.03.2012 | Educational Article

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

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 an efficient Matlab code for structural topology optimization that includes a general finite element routine based on isoparametric polygonal elements which can be viewed as the extension of linear triangles and bilinear quads. The code also features a modular structure in which the analysis routine and the optimization algorithm are separated from the specific choice of topology optimization formulation. Within this framework, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We address issues pertaining to the use of unstructured meshes and arbitrary design domains in topology optimization that have received little attention in the literature. Also, as part of our examination of the topology optimization problem, we review the various steps taken in casting the optimal shape problem as a sizing optimization problem. This endeavor allows us to isolate the finite element and geometric analysis parameters and how they are related to the design variables of the discrete optimization problem. The Matlab code is explained in detail and numerical examples are presented to illustrate the capabilities of the code.

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

Nächster Artikel Dynamical systems, SIMP, bone remodeling and time dependent loads

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Polygonal discretizations have been used in computational solid mechanics, see for example Ghosh (2010). In topology optimization, they have been shown to outperform linear triangles and quads since they eliminate numerical instabilities such as checkerboarding (Langelaar 2007; Saxena 2008; Talischi et al. 2009, 2010).

This is in contrast to the relaxation setting which begins with \(\mathcal{O}\) defined as the set of all measurable subsets of Ω and addresses ill-posedness of the problem by further enlarging the space.

Here L ^∞ Ω;K) denotes the space of measurable functions defined on Ω that take values in \(K\subseteq\mathbb{R}\). For example, \(L^{\infty}(\Omega;\left\{ 0,1\right\} )\) and \(L^{\infty}(\Omega;\left[0,1\right])\) denote the space of measurable functions that take values in \(\left\{ 0,1\right\} \) and interval \(\left[0,1\right]\), respectively.

This function takes value of 1 at point x ∈ Ω if ρ(x) ≥ 0.5 and is zero otherwise – this is a simple choice of post-processing.

Variants of density methods involve different material interpolation functions, but are similar in spirit (Stolpe and Svanberg 2001b; Bruns 2005).

The distinction between the η and the admissible function \(\mathcal{P}_{s}(\eta)\) should be more apparent here: η is defined on half of the domain while \(\mathcal{P}_{s}(\eta)\) is defined over all of Ω.

m _E essentially determines the dependence of the state equation on the design.

Filtering, symmetry, pattern repetition, and extrusion constraints can all be implemented by means of such linear maps

Alternatively we can view \(\mathcal{P}_{h}=\mathcal{I}_{h}\circ\mathcal{P}\) where \(\mathcal{I}_{h}\) maps any ρ _h to \(\tilde{\rho}_{h}\), that is, \(\mathcal{I}_{h}(\rho)=\sum_{\ell=1}^{N}\rho(\mathbf{x}_{\ell}^{*})\chi_{\Omega_{\ell}}\)

In the remainder of the paper, we understand m _E(y) and m _V(y) as vectors with entries m _E(y _ℓ) and m _V(y _ℓ).

We caution that the PolyMesher files should be added to the Matlab path in order for this call to be successful. This mesh generator can be of course replaced by any other mesh generator (e.g. distMesh (Persson and Strang 2004)) as long as the node list, element connectivity cell and load and support vectors follow the same format.

Again we understand m _E′(y) and m _V′(y) as vectors with elements m _E′(y _ℓ) and m _V′(y _ℓ). Essentially m _E′(y) and m _V′(y) are the diagonal entries of Jacobian matrices \(J_{m_{E}}(\mathbf{y)}\) and \(J_{m_{V}}(\mathbf{y})\).

The local stiffness matrices are obtained by calling the function LocalK on either line 68 or line 70. If the mesh is known to be uniform, by setting the fem.Reg tag equal to 1 in the initialization of the fem structure, only one such call is made (on line 68) and thus there is no overhead for repeated calculation of the same element stiffness matrices.

These matrices are expected to have the following format: Supp must have three columns, the first holding the node number, the second and third columns giving support conditions for that node in the x- and y-direction, respectively. Value of 0 indicates that the node is free, and value of 1 specifies a fixed node. The nodal load vector Load is structured in a similar way, except for the values in the second and third columns, which represent the magnitude of the x- and y-components of the force.

Regarding the connection between (25) and the well-known expression \(\int_{\Omega_{\ell}}\mathbf{B}_{I}^{T}\mathbf{D}\mathbf{B}_{J}d\mathbf{x}\) for the element stiffness matrix, we refer the reader to Section 2.8 of Hughes (2000).

By convention, we set p _n + 1 = p ₁ in this expression.

The Voronoi mesh may be different from the one used in our results due to the random placement of seeds in PolyMesher.

The larger the radius of filtering, the longer it takes to compute the filtering matrix and also the larger the amount of memory needed to store it.

Note, however, that in PolyTop the volume function is normalized by the volume of the entire domain and elements can have different areas (so 1 is replaced by A in PolyTop). Also, the constraint function is defined as the difference between the normalized volume and specified volume fraction \(\overline{v}\).

The significance of approximating response dependent cost functions in “reciprocal” variables, i.e., when a = − 1, are discussed in Groenwold and Etman (2008) and references therein. For a = 1, one recovers the usual Taylor linearization.

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Titel: PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes
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/s00158-011-0696-x

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