A penalty method for topology optimization subject to a pointwise state constraint

Lat. Am. j. solids struct.

2015

The topological derivative measures the sensitivity of a shape functional with respect to an infinitesimal singular domain perturbation, such as the insertion of holes, inclusions or source-terms. The topological derivative has been successfully applied in obtaining the optimal topology for a large class of physics and engineering problems. In this paper the topological derivative is applied in the context of topology optimization of structures subject to multiple load-cases. In particular, the structural compliance under plane stress or plane strain assumptions is minimized under volume constraint. For the sake of completeness, the topological asymptotic analysis of the total potential energy with respect to the nucleation of a small circular inclusion is developed in all details. Since we are dealing with multiple load-cases, a multi-objective optimization problem is proposed and the topological sensitivity is obtained as a sum of the topological derivatives associated with each load-case. The volume constraint is imposed through the Augmented Lagrangian Method. The obtained result is used to devise a topology optimization algorithm based on the topological derivative together with a level-set domain representation method. Finally, several finite element-based examples of structural optimization are presented. Keywords

Section: Introductionmentioning

confidence: 99%

Topological derivative-based topology optimization of structures subject to multiple load-cases

Lopes

Santos

Lat. Am. j. solids struct.

2015

“…We assume that the derivatives Φ ′ and Φ ′′ are bounded. Then, the penalty functional is defined as

G_{Ω} : = \int_{Ω^{⋆}} normalΦ false(S_{M}^{2} false) .

In particular, we shall adopt a function Φ of the following functional form:

normalΦ false(t false) \equiv Φ_{p} false(t false),

where p ≥1 is a given real parameter and

Φ_{p} : R_{+} \to R_{+}

is defined as

Φ_{p} false(t false) = {([], 1 + t^{p})}^{1 false/ p} - 1 .

Therefore, the constrained optimization problem with pointwise constraints can be approximated by the following penalized optimization problem:

\underset{normalΩ \subset scriptD}{Minimize} {scriptJ}_{normalΩ}^{α} : = J_{Ω} + α G_{Ω},

with the scalar α>0 used to denote a given penalty coefficient. Following the original ideas presented in the work of Amstutz and Novotny, the local stress constraints in are replaced by the penalty functional .…”

Section: Structural Topology Optimization Problemmentioning

confidence: 99%

“…In practice, we chose α to be as large as possible according to each problem under consideration, and p is set as 32, which produces the desired sharp variation of

{normalΦ}_{p}^{'}

around t =1 (see the work of Amstutz for more details).…”

Section: Structural Topology Optimization Problemmentioning

confidence: 99%

Reliability‐based topology optimization of structures under stress constraints

Santos

Torii

Numerical Meth Engineering

2018

Summary In this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.

“…This relatively new concept was introduced in the fundamental paper [56] and has been successfully applied to many relevant fields such as shape and topology optimization [1,8,11,12,15,17,29,38,40,48,49,50,59], inverse problems [10,19,20,21,23,30,32,34,36,42], imaging processing [13,14,31,33,39], multiscale material design [9,26,27,28,52] and mechanical modeling including damage [2] and fracture [60] evolution phenomena. Regarding the theoretical development of the topological asymptotic analysis, see for instance [6,7,22,24,25,35,37,41,43,44,45,46,47,…”

Section: Introductionmentioning

confidence: 99%

Topological Derivative Method for Electrical Impedance Tomography Problems

Ferreira

Sokołowski

2016

IAPGOS

In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data. ZASTOSOWANIE METODY POCHODNEJ TOPOLOGICZNEJ W ELEKTRYCZNEJ TOMOGRAFII IMPEDANCYJNEJStreszczenie. W dziedzinie optymalizacji kształtu i topologii zaproponowano nową koncepcję pochodnej topologicznej danego funkcjonału kształtu. Zastosowano asymptotyczną analizę w celu określenia pochodnej topologicznej funkcjonału kształtu dla zagadnień eliptycznych. Pochodna Topologiczna -PT (ang. the topological derivative -TD) jest miarą wpływu wtrącenia w postaci małego defektu na funkcjonał kształtu w badanym obszarze dla eliptycznego zagadnienia brzegowego. Obszar z małym defektem traktowany jest jako obszar zaburzony przez zmiany topologii. Oznacza to, że dana pochodna topologiczna stanowi aproksymację pierwszego rzędu ze względu na mały parametr, który określa objętość defektu dla obliczanego funkcjonału kształtu w zaburzonym obszarze. PT jest funkcją zdefiniowaną w obszarze niezaburzonym, który może być wyznaczony na podstawie znajomości rozwiązania zagadnienia brzegowego w tym (niezaburzonym) obszarze. Oznacza to że PT może być w...