Accelerated fixed-point formulation of topology optimization: Application to compliance minimization problems

2020 ◽  
Vol 103 ◽  
pp. 103469 ◽  
Author(s):  
Weichen Li ◽  
Phanish Suryanarayana ◽  
Glaucio H. Paulino
Keyword(s):  
Fixed Point ◽  
Topology Optimization ◽  
Compliance Minimization ◽  
Minimization Problems

Numerical Experiments in Using Voronoi Cell Finite Elements for Topology Optimization

2009 ◽  
Author(s):  
James M. Gibert ◽  
Georges M. Fadel
Keyword(s):  
Topology Optimization ◽  
Isotropic Material ◽  
Numerical Experiments ◽  
Voronoi Cell ◽  
Material Model ◽  
Topological Optimization ◽  
Advantages And Disadvantages ◽  
Compliance Minimization ◽  
Design Variables ◽  
Standard Linear

This paper provides two separate methodologies for implementing the Voronoi Cell Finite Element Method (VCFEM) in topological optimization. Both exploit two characteristics of VCFEM. The first approach utilizes the property that a hole or inclusion can be placed in the element: the design variables for the topology optimization are sizes of the hole. In the second approach, we note that VCFEM may mesh the design domain as n sided polygons. We restrict our attention to hexagonal meshes of the domain while applying Solid Isotropic Material Penalization (SIMP) material model. Researchers have shown that hexagonal meshes are not subject to the checker boarding problem commonly associated with standard linear quad and triangle elements. We present several examples to illustrate the efficacy of the methods in compliance minimization as well as discuss the advantages and disadvantages of each method.

Approximating the minimum-norm fixed point of pseudocontractive mappings

2015 ◽  
Vol 08 (02) ◽  
pp. 1550036
Author(s):  
H. Zegeye ◽  
O. A. Daman
Keyword(s):  
Fixed Point ◽  
Iterative Process ◽  
Convergence Result ◽  
Monotone Mappings ◽  
Minimum Norm ◽  
Nonlinear Operators ◽  
Linear Inverse Problems ◽  
Minimization Problems ◽  
Pseudocontractive Mappings ◽  
Convex Minimization Problems

We introduce an iterative process which converges strongly to the minimum-norm fixed point of Lipschitzian pseudocontractive mapping. As a consequence, convergence result to the minimum-norm zero of monotone mappings is proved. In addition, applications to convexly constrained linear inverse problems and convex minimization problems are included. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

scholarly journals A New Modified Hybrid Steepest-Descent by Using a Viscosity Approximation Method with a Weakly Contractive Mapping for a System of Equilibrium Problems and Fixed Point Problems with Minimization Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Uamporn Witthayarat ◽  
Thanyarat Jitpeera ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Approximation Method ◽  
Contractive Mapping ◽  
Steepest Descent ◽  
Common Element ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Weakly Contractive Mapping ◽  
Minimization Problems ◽  
Weakly Contractive

The purpose of this paper is to consider a modified hybrid steepest-descent method by using a viscosity approximation method with a weakly contractive mapping for finding the common element of the set of a common fixed point for an infinite family of nonexpansive mappings and the set of solutions of a system of an equilibrium problem. The sequence is generated from an arbitrary initial point which converges in norm to the unique solution of the variational inequality under some suitable conditions in a real Hilbert space. The results presented in this paper generalize and improve the results of Moudafi (2000), Marino and Xu (2006), Tian (2010), Saeidi (2010), and some others. Finally, we give an application to minimization problems and a numerical example which support our main theorem in the last part.

Tracing Pareto-Optimal Frontiers in Topology Optimization

2010 ◽  
Author(s):  
Krishnan Suresh
Keyword(s):  
Fixed Point ◽  
Topology Optimization ◽  
Iteration Scheme ◽  
Stochastic Method ◽  
Pareto Optimal ◽  
Topological Sensitivity ◽  
Fixed Point Iteration ◽  
Numerical Examples ◽  
Multi Objective ◽  
Sensitivity Field

In multi-objective topology optimization, a design is defined to be “pareto-optimal” if no other design exists that is better with respect to one objective, and as good with respect to others. This unfortunately suggests that unless other ‘better’ designs are found, one cannot declare a particular topology to be pareto-optimal. In this paper, we first show that a topology can be guaranteed to be (locally) pareto-optimal if certain inherent properties associated with the topological sensitivity field are satisfied, i.e., no further comparison is necessary. This, in turn, leads to a deterministic, i.e., non-stochastic, method for directly tracing pareto-optimal frontiers using the classic fixed-point iteration scheme. The proposed method can generate the full set of pareto-optimal topologies in a single-run, and is therefore both efficient and predictable, as illustrated through numerical examples.

scholarly journals Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Hong-Kun Xu ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Viscosity Approximation Method ◽  
Convex Minimization Problem ◽  
Viscosity Approximation ◽  
Minimization Problems ◽  
Strict Pseudocontraction ◽  
Constrained Minimization Problems ◽  
Viscosity Approximation Methods

We introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex minimization problem and a suitable equilibrium problem.

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