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.
► Tikhonov regularization is used to address ill-posedness of topology optimization. ► Semi-implicit discretization of gradient flow produces a form of Helmholtz filtering. ► For a certain projection ...map, we recover the forward-backward splitting algorithm.
In this work, we investigate a Tikhonov-type regularization scheme to address the ill-posedness of the classical compliance minimization problem. We observe that a semi-implicit discretization of the gradient descent flow for minimization of the regularized objective function leads to a convolution of the original gradient descent step with the Green’s function associated with the modified Helmholtz equation. The appearance of “filtering” in this update scheme is different from the current density and sensitivity filtering techniques in the literature. The next iterate is defined as the projection of this provisional density onto the space of admissible density functions. For a particular choice of projection mapping, we show that the algorithm is identical to the well-known forward–backward splitting algorithm, an insight that can be further explored for topology optimization. Also of interest is that with an appropriate choice of the projection parameter, nearly all intermediate densities are eliminated in the optimal solution using the common density material models. We show examples of near binary solutions even for large values of the regularization parameter.
We explore the recently-proposed Virtual Element Method (VEM) for the numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the linear ...elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.
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.
Polygonal finite elements for finite elasticity Chi, Heng; Talischi, Cameron; Lopez-Pamies, Oscar ...
International journal for numerical methods in engineering,
27 January 2015, Letnik:
101, Številka:
4
Journal Article
Recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations. On the one hand, these elements are well suited ...to model complex microstructures (e.g. particulate microstructures and microstructures involving different length scales) and incorporating periodic boundary conditions. On the other hand, polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements, and to produce more accurate results in bending and shear. With mixed formulations, lower order mixed polygonal elements are also shown to be numerically stable on Voronoi-type meshes without any additional stabilization treatment. However, polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes. As a result, non-convergent finite element results typically occur, which severely limit their applications. In this work, a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field. With the correction scheme, the recovery of optimal convergence for solutions of displacement-based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity. In addition, for mixed polygonal elements, the various choices of the pressure field approximations are discussed, and their performance on stability and accuracy are numerically investigated. We present applications of those elements in physically-based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers.
We present a Matlab implementation of topology optimization for fluid flow problems in the educational computer code
PolyTop
(Talischi et al.
2012b
). The underlying formulation is the ...well-established porosity approach of Borrvall and Petersson (
2003
), wherein a dissipative term is introduced to impede the flow in the solid (non-fluid) regions. Polygonal finite elements are used to obtain a stable low-order discretization of the governing Stokes equations for incompressible viscous flow. As a result, the same mesh represents the design field as well as the velocity and pressure fields that characterize its response. Owing to the modular structure of
PolyTop
, incorporating new physics, in this case modeling fluid flow, involves changes that are limited mainly to the analysis routine. We provide several numerical examples to illustrate the capabilities and use of the code. To illustrate the modularity of the present approach, we extend the implementation to accommodate alternative formulations and cost functions. These include topology optimization formulations where both viscosity and inverse permeability are functions of the design; and flow control where the velocity at a certain location in the domain is maximized in a prescribed direction.
Polygonal finite elements for incompressible fluid flow Talischi, Cameron; Pereira, Anderson; Paulino, Glaucio H. ...
International journal for numerical methods in fluids,
20 January 2014, Letnik:
74, Številka:
2
Journal Article
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