Applications of Primal-dual Interior Methods in Structural Optimization

2003 ◽  
Vol 3 (1) ◽  
pp. 159-176 ◽  
Author(s):  
Ronald H. W. Hoppe ◽  
Svetozara I. Petrova
Keyword(s):  
Structural Optimization ◽  
Optimization Problems ◽  
Null Space ◽  
Inequality Constraints ◽  
Composite Ceramic ◽  
Ceramic Materials ◽  
Design Parameters ◽  
State Variables ◽  
Primal Dual ◽  
Search Approach

AbstractWe are concerned with structural optimization problems for technological processes in material science that are described by partial differential equations. In particular, we consider the topology optimization of conductive media in high-power electronic devices described by Maxwell equations and the optimal design of composite ceramic materials by homogenization modeling. All these tasks lead to constrained nonconvex minimization problems with both equality and inequality constraints on the state variables and design parameters. After discretization by finite elements, we solve the discretized optimization problems by a primal-dual Newton interior-point method. Within a line-search approach, transforming iterations are applied with respect to the null space decomposition of the condensed primal-dual system to find the search direction. Some numerical experiments for the two applications are presented.

DISTRIBUTED PROXIMAL-GRADIENT METHOD FOR CONVEX OPTIMIZATION WITH INEQUALITY CONSTRAINTS

2014 ◽  
Vol 56 (2) ◽  
pp. 160-178 ◽  
Author(s):  
JUEYOU LI ◽  
CHANGZHI WU ◽  
ZHIYOU WU ◽  
QIANG LONG ◽  
XIANGYU WANG
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Distributed Optimization ◽  
Inequality Constraints ◽  
Gradient Algorithm ◽  
Subgradient Methods ◽  
Penalty Function Method ◽  
Proximal Gradient Method ◽  
Primal Dual

AbstractWe consider a distributed optimization problem over a multi-agent network, in which the sum of several local convex objective functions is minimized subject to global convex inequality constraints. We first transform the constrained optimization problem to an unconstrained one, using the exact penalty function method. Our transformed problem has a smaller number of variables and a simpler structure than the existing distributed primal–dual subgradient methods for constrained distributed optimization problems. Using the special structure of this problem, we then propose a distributed proximal-gradient algorithm over a time-changing connectivity network, and establish a convergence rate depending on the number of iterations, the network topology and the number of agents. Although the transformed problem is nonsmooth by nature, our method can still achieve a convergence rate, ${\mathcal{O}}(1/k)$, after $k$ iterations, which is faster than the rate, ${\mathcal{O}}(1/\sqrt{k})$, of existing distributed subgradient-based methods. Simulation experiments on a distributed state estimation problem illustrate the excellent performance of our proposed method.

scholarly journals Null space gradient flows for constrained optimization with applications to shape optimization

2020 ◽  
Vol 26 ◽  
pp. 90 ◽  
Author(s):  
F. Feppon ◽  
G. Allaire ◽  
C. Dapogny
Keyword(s):  
Shape Optimization ◽  
Constrained Optimization ◽  
Objective Function ◽  
Optimization Problems ◽  
Null Space ◽  
Gradient Flow ◽  
Inequality Constraints ◽  
Variation Method ◽  
Constrained Problems ◽  
Space Step

The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita (Math. Program. 18 (1980) 155–168) for equality constrained problems: the search direction is a combination of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints. The solution to this problem allows to identify the inequality constraints to which the optimization trajectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of – infinite-dimensional – Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard’s boundary variation method. The cornerstone of this formulation is the classical operation of extension and regularization of shape derivatives. The numerical efficiency and ease of implementation of our algorithm are demonstrated on realistic shape optimization problems.

scholarly journals Relaxed gradient projection algorithm for constrained node-based shape optimization

2021 ◽  
Author(s):  
Ihar Antonau ◽  
Majid Hojjat ◽  
Kai-Uwe Bletzinger
Keyword(s):  
Shape Optimization ◽  
Optimization Problems ◽  
Gradient Projection ◽  
Projection Algorithm ◽  
Design Parameters ◽  
Active Set ◽  
Physical Constraints ◽  
Original Algorithm ◽  
Gradient Projection Algorithm ◽  
Non Linear

AbstractIn node-based shape optimization, there are a vast amount of design parameters, and the objectives, as well as the physical constraints, are non-linear in state and design. Robust optimization algorithms are required. The methods of feasible directions are widely used in practical optimization problems and know to be quite robust. A subclass of these methods is the gradient projection method. It is an active-set method, it can be used with equality and non-equality constraints, and it has gained significant popularity for its intuitive implementation. One significant issue around efficiency is that the algorithm may suffer from zigzagging behavior while it follows non-linear design boundaries. In this work, we propose a modification to Rosen’s gradient projection algorithm. It includes the efficient techniques to damp the zigzagging behavior of the original algorithm while following the non-linear design boundaries, thus improving the performance of the method.

Catalytic pyrolysis of propane-butane hydrocarbon mixture on the composite ceramic materials in an open system

2010 ◽  
Vol 80 (4) ◽  
pp. 742-748 ◽  
Author(s):  
Y. A. Aleksandrov ◽  
E. I. Tsyganova ◽  
V. M. Shekunova ◽  
I. I. Didenkulova ◽  
I. A. Pishchurova ◽  
...  
Keyword(s):  
Open System ◽  
Catalytic Pyrolysis ◽  
Composite Ceramic ◽  
Ceramic Materials ◽  
Hydrocarbon Mixture ◽  
Composite Ceramic Materials ◽  
Propane Butane

Analysis, Design, and Optimization of High Speed Vehicle Suspensions Using State Variable Techniques

1974 ◽  
Vol 96 (2) ◽  
pp. 193-203 ◽  
Author(s):  
J. K. Hedrick ◽  
G. F. Billington ◽  
D. A. Dreesbach
Keyword(s):  
High Speed ◽  
Optimization Problems ◽  
Vertical Plane ◽  
Computer Application ◽  
Vehicle Suspension ◽  
State Variables ◽  
State Variable ◽  
Suspension Parameters ◽  
Suspension Model ◽  
High Speed Vehicle

This article applies state variable techniques to high speed vehicle suspension design. When a reasonably complex suspension model is treated, the greater adaptability of state variable techniques to digital computer application makes it more attractive than the commonly used integral transform method. A vehicle suspension model is developed, state variable techniques are applied, numerical methods are presented, and, finally, an optimization algorithm is chosen to select suspension parameters. A fairly complete bibliography is included in each of these areas. The state variable technique is illustrated in the solution of two suspension optimization problems. First, the vertical plane suspension of a high speed vehicle subject to guideway and aerodynamic inputs will be analyzed. The vehicle model, including primary and secondary suspension systems, and subject to both heave and pitch motions, has thirteen state variables. Second, the horizontal plane suspension of a high speed vehicle subject to guideway and lateral aerodynamic inputs is analyzed. This model also has thirteen state variables. The suspension parameters of both these models are optimized. Numerical results are presented for a representative vehicle, showing time response, mean square values, optimized suspension parameters, system eigenvalues, and acceleration spectral densities.

Multi-objective structural optimization of a wind turbine blade using NSGA-II algorithm and FSI

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ramazan Özkan ◽  
Mustafa Serdar Genç
Keyword(s):  
Structural Optimization ◽  
Wind Turbine ◽  
Turbine Blade ◽  
Wind Turbine Blade ◽  
Design Parameters ◽  
Aerodynamic Optimization ◽  
Nsga Ii ◽  
Content Type ◽  
Multi Objective ◽  
Optimization Study

Purpose Wind turbines are one of the best candidates to solve the problem of increasing energy demand in the world. The aim of this paper is to apply a multi-objective structural optimization study to a Phase II wind turbine blade produced by the National Renewable Energy Laboratory to obtain a more efficient small-scale wind turbine. Design/methodology/approach To solve this structural optimization problem, a new Non-Dominated Sorting Genetic Algorithm (NSGA-II) was performed. In the optimization study, the objective function was on minimization of mass and cost of the blade, and design parameters were composite material type and spar cap layer number. Design constraints were deformation, strain, stress, natural frequency and failure criteria. ANSYS Composite PrepPost (ACP) module was used to model the composite materials of the blade. Moreover, fluid–structure interaction (FSI) model in ANSYS was used to carry out flow and structural analysis on the blade. Findings As a result, a new original blade was designed using the multi-objective structural optimization study which has been adapted for aerodynamic optimization, the NSGA-II algorithm and FSI. The mass of three selected optimized blades using carbon composite decreased as much as 6.6%, 11.9% and 14.3%, respectively, while their costs increased by 23.1%, 29.9% and 38.3%. This multi-objective structural optimization-based study indicates that the composite configuration of the blade could be altered to reach the desired weight and cost for production. Originality/value ACP module is a novel and advanced composite modeling technique. This study is a novel study to present the NSGA-II algorithm, which has been adapted for aerodynamic optimization, together with the FSI. Unlike other studies, complex composite layup, fiber directions and layer orientations were defined by using the ACP module, and the composite blade analyzed both aerodynamic pressure and structural design using ACP and FSI modules together.

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