Antonio Andre Novotny, Sebastian Miguel Giusti and Samuel Amstutz
Àlex Ferrer and Sebastián Miguel Giusti
The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.
Abstract
Purpose
The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.
Design/methodology/approach
The optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.
Findings
The optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.
Originality/value
The proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.
Details
Keywords
Diego Esteves Campeão, Sebastian Miguel Giusti and Andre Antonio Novotny
– The purpose of this paper is to compare between two methods of volume control in the context of topological derivative-based structural optimization of Kirchhoff plates.
Abstract
Purpose
The purpose of this paper is to compare between two methods of volume control in the context of topological derivative-based structural optimization of Kirchhoff plates.
Design/methodology/approach
The compliance topology optimization of Kirchhoff plates subjected to volume constraint is considered. In order to impose the volume constraint, two methods are presented. The first one is done by means of a linear penalization method. In this case, the penalty parameter is the coefficient of a linear term used to control the amount of material to be removed. The second approach is based on the Augmented Lagrangian method which has both, linear and quadratic terms. The coefficient of the quadratic part controls the Lagrange multiplier update of the linear part. The associated topological sensitivity is used to devise a structural design algorithm based on the topological derivative and a level-set domain representation method. Finally, some numerical experiments are presented allowing for a comparative analysis between the two methods of volume control from a qualitative point of view.
Findings
The linear penalization method does not provide direct control over the required volume fraction. In contrast, through the Augmented Lagrangian method it is possible to specify the final amount of material in the optimized structure.
Originality/value
A strictly simple topology design algorithm is devised and used in the context of compliance structural optimization of Kirchhoff plates under volume constraint. The proposed computational framework is quite general and can be applied in different engineering problems.
Details
Keywords
Marcel Xavier and Nicolas Van Goethem
In the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.
Abstract
Purpose
In the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.
Design/methodology/approach
The Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.
Findings
Numerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.
Originality/value
This is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.