scholarly journals A Shape Optimization Problem on Planar Sets with Prescribed Topology

2021 ◽  
Author(s):  
Luca Briani ◽  
Giuseppe Buttazzo ◽  
Francesca Prinari
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Torsional Rigidity ◽  
Connected Components ◽  
Topological Constraints ◽  
Ill Posed ◽  
Complementary Set ◽  
Cost Functionals ◽  
Well Posed

AbstractWe consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $$P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}$$ P ( Ω ) T q ( Ω ) | Ω | - 2 q - 1 / 2 , and the class of admissible domains consists of two-dimensional open sets $$\Omega $$ Ω satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when $$q<1/2$$ q < 1 / 2 an optimal relaxed domain exists. When $$q>1/2$$ q > 1 / 2 , the problem is ill-posed, and for $$q=1/2$$ q = 1 / 2 , the explicit value of the infimum is provided in the cases $$k=0$$ k = 0 and $$k=1$$ k = 1 .

scholarly journals Shape Optimization by Pursuing Diffeomorphisms

2015 ◽  
Vol 15 (3) ◽  
pp. 291-305 ◽  
Author(s):  
Ralf Hiptmair ◽  
Alberto Paganini
Keyword(s):  
Finite Element ◽  
Shape Optimization ◽  
Optimization Problems ◽  
Approximation Properties ◽  
The Finite Element Method ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
B Spline ◽  
Ill Posed ◽  
Well Posed

AbstractWe consider PDE constrained shape optimization in the framework of finite element discretization of the underlying boundary value problem. We present an algorithm tailored to preserve and exploit the approximation properties of the finite element method, and that allows for arbitrarily high resolution of shapes. It employs (i) B-spline based representations of the deformation diffeomorphism, and (ii) superconvergent domain integral expressions for the shape gradient. We provide numerical evidence of the performance of this method both on prototypical well-posed and ill-posed shape optimization problems.

scholarly journals A class of shape optimization problems for some nonlocal operators

2018 ◽  
Vol 11 (4) ◽  
pp. 373-386 ◽  
Author(s):  
Julián Fernández Bonder ◽  
Antonella Ritorto ◽  
Ariel Martin Salort
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
State Equation ◽  
The State ◽  
Local State ◽  
Nonlocal Operator ◽  
State Equations ◽  
Nonlocal Operators ◽  
Shape Optimization Problem

AbstractIn this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

Optimal Cantilever Design by Topology and Shape Optimization Methods

2015 ◽  
Vol 789-790 ◽  
pp. 306-310
Author(s):  
Jin Woo Lee
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Mode Frequency ◽  
Torsional Mode ◽  
Initial Shape ◽  
Topology Optimization Problem ◽  
Torsion Mode ◽  
Topology And Shape Optimization

This work presents the framework to optimally design a cantilever for torsion mode frequency maximization. A cantilever design problem is formulated by topology and shape optimization methods. The torsion mode frequency is selected as an objective function, and the volume of the cantilever and the first bending mode frequency are constrained. Two optimization problems are defined and sequentially solved for the specific values. A new idea in this work is using a final topology obtained in the topology optimization problem as an initial shape in the shape optimization problem. The torsional mode frequency of the optimized cantilever is well improved in comparison with a nominal cantilever.

scholarly journals Symmetry results for variational energies on convex polygons

2020 ◽  
Author(s):  
Dorin Bucur ◽  
Ilaria Fragala
Keyword(s):  
Shape Optimization ◽  
Optimization Problems ◽  
Torsional Rigidity ◽  
Logarithmic Capacity ◽  
Convex Polygons ◽  
Principal Frequency

We prove that, in the class of convex polygons with a given number of sides,  the regular $n$-gon is optimal for some shape optimization problems involving the torsional rigidity, the principal frequency of the Laplacian, or the logarithmic capacity.

Shape Optimization of Flexible Robotic Manipulators

2005 ◽  
Vol 128 (3) ◽  
pp. 559-565 ◽  
Author(s):  
U. S. Dixit ◽  
R. Kumar ◽  
S. K. Dwivedy
Keyword(s):  
Shape Optimization ◽  
Cross Sections ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Robotic Manipulators ◽  
Flexible Manipulator ◽  
The Finite Element Method ◽  
Cross Sectional ◽  
Euler Bernoulli Beam ◽  
Cartesian Manipulator

In this work, the problem of shape optimization of flexible robotic manipulators of circular cross sections is studied. Two different manipulators are considered—a manipulator with revolute joint and a roller supported Cartesian manipulator. The finite element method is used to find the natural frequency and dynamic response of a flexible manipulator by treating it as an Euler-Bernoulli beam. The cross-sectional diameter is varied along the length keeping the constraint on the mass of the manipulator and static tip deflection in order to maximize the fundamental frequency of the beam. This optimization problem is compared with other optimization problems (with different objective functions and constraints). It is observed that the proposed optimization problem is superior to other optimization problems.

scholarly journals Application of Nontraditional Optimization Techniques for Airfoil Shape Optimization

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
R. Mukesh ◽  
K. Lingadurai ◽  
U. Selvakumar
Keyword(s):  
Genetic Algorithm ◽  
Simulated Annealing ◽  
Shape Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Optimization Techniques ◽  
Aerodynamic Shape Optimization ◽  
Shape Optimization Problem ◽  
Aerodynamic Shape

The method of optimization algorithms is one of the most important parameters which will strongly influence the fidelity of the solution during an aerodynamic shape optimization problem. Nowadays, various optimization methods, such as genetic algorithm (GA), simulated annealing (SA), and particle swarm optimization (PSO), are more widely employed to solve the aerodynamic shape optimization problems. In addition to the optimization method, the geometry parameterization becomes an important factor to be considered during the aerodynamic shape optimization process. The objective of this work is to introduce the knowledge of describing general airfoil geometry using twelve parameters by representing its shape as a polynomial function and coupling this approach with flow solution and optimization algorithms. An aerodynamic shape optimization problem is formulated for NACA 0012 airfoil and solved using the methods of simulated annealing and genetic algorithm for 5.0 deg angle of attack. The results show that the simulated annealing optimization scheme is more effective in finding the optimum solution among the various possible solutions. It is also found that the SA shows more exploitation characteristics as compared to the GA which is considered to be more effective explorer.

scholarly journals Regularized Deblurring using Directional Prior with Sparse Representation

2019 ◽  
Vol 9 (1S3) ◽  
pp. 206-210
Keyword(s):  
Optimization Problem ◽  
Blind Deconvolution ◽  
Augmented Lagrangian Method ◽  
Real Data ◽  
Performance Comparison ◽  
Simultaneous Estimation ◽  
Deconvolution Problem ◽  
Ill Posed ◽  
Regularized Optimization ◽  
Well Posed

Blind deconvolution defined as simultaneous estimation and removal of blur is an ill-posed problem that can be solved with well-posed priors. In this paper we focus on directional edge prior based on orientation of gradients. Then the deconvolution problem is modeled as L2-regularized optimization problem which seeks a solution through constraint optimization. The constrained optimization problem is done in frequency domain with an Augmented Lagrangian Method (ALM). The proposed algorithm is tested on various synthetic as well as real data taken from various sources and the performance comparison is carried out with other state of the art existing methods.

On a characteristic property of conditionally well-posed problems

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Inverse Problems ◽  
Input Data ◽  
Characteristic Property ◽  
Optimization Problems ◽  
Data Driven ◽  
Error Models ◽  
Regularizing Algorithms ◽  
Stochastic Error ◽  
Ill Posed ◽  
Well Posed

AbstractThe aim of this paper is to discuss and illustrate the fact that conditionally well-posed problems stand out among all ill-posed problems as being regularizable via an operator independent of the level of errors in input data. We give examples of corresponding purely data driven regularizing algorithms for various classes of conditionally well-posed inverse problems and optimization problems in the context of deterministic and stochastic error models.

scholarly journals A free boundary approach to shape optimization problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140273 ◽  
Author(s):  
D. Bucur ◽  
B. Velichkov
Keyword(s):  
Free Boundary ◽  
Shape Optimization ◽  
Optimization Problem ◽  
Free Boundary Problems ◽  
Optimization Problems ◽  
General Shape ◽  
Dirichlet Laplacian ◽  
Boundary Problems ◽  
Survey Article ◽  
The Laplace Operator

The analysis of shape optimization problems involving the spectrum of the Laplace operator, such as isoperimetric inequalities, has known in recent years a series of interesting developments essentially as a consequence of the infusion of free boundary techniques. The main focus of this paper is to show how the analysis of a general shape optimization problem of spectral type can be reduced to the analysis of particular free boundary problems. In this survey article, we give an overview of some very recent technical tools, the so-called shape sub- and supersolutions, and show how to use them for the minimization of spectral functionals involving the eigenvalues of the Dirichlet Laplacian, under a volume constraint.

scholarly journals Well-Posedness and Porosity for Symmetric Optimization Problems

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1253
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Metric Space ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Complete Metric Space ◽  
Well Posedness ◽  
Porous Set ◽  
Minimization Problems ◽  
Symmetric Optimization ◽  
Lower Semi ◽  
Well Posed

In the present work, we investigate a collection of symmetric minimization problems, which is identified with a complete metric space of lower semi-continuous and bounded from below functions. In our recent paper, we showed that for a generic objective function, the corresponding symmetric optimization problem possesses two solutions. In this paper, we strengthen this result using a porosity notion. We investigate the collection of all functions such that the corresponding optimization problem is well-posed and prove that its complement is a σ-porous set.

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