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.

Tracing the Envelope of the Objective-Space in Multi-Objective Topology Optimization

2011 ◽  
Author(s):  
Inna Turevsky ◽  
Krishnan Suresh
Keyword(s):  
Topology Optimization ◽  
Eigenvalue Problems ◽  
Optimization Problems ◽  
Pareto Optimal ◽  
Pareto Optimal Solutions ◽  
Optimal Solutions ◽  
Topological Sensitivity ◽  
Numerical Examples ◽  
Multi Objective ◽  
Objective Space

In multi-objective problems, one is often interested in generating the envelope of the objective-space, where the envelope is, in general, a superset of pareto-optimal solutions. In this paper, we propose a method for tracing the envelope of multi-objective topology optimization problems, and generating the corresponding topologies. The proposed method exploits the concept of topological sensitivity, and is applied to bi-objective optimization, namely eigenvalue-volume, eigenvalue-eigenvalue and compliance-eigenvalue problems. The robustness and efficiency of the method is illustrated through numerical examples.

Predicting the Benefits of Topology Optimization

2015 ◽  
Author(s):  
Shiguang Deng ◽  
Krishnan Suresh
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Shape Optimization ◽  
Systematic Method ◽  
Topological Sensitivity ◽  
Numerical Examples ◽  
Initial Design

Topology optimization is a systematic method of generating designs that maximize specific objectives. While it offers significant benefits over traditional shape optimization, topology optimization can be computationally demanding and laborious. Even a simple 3D compliance optimization can take several hours. Further, the optimized topology must typically be manually interpreted and translated into a CAD-friendly and manufacturing friendly design. This poses a predicament: given an initial design, should one optimize its topology? In this paper, we propose a simple metric for predicting the benefits of topology optimization. The metric is derived by exploiting the concept of topological sensitivity, and is computed via a finite element swapping method. The efficacy of the metric is illustrated through numerical examples.

scholarly journals Geometric Mean Method to Solve Multi-Objective Transportation Problem Under Fuzzy Environment

2020 ◽  
Vol 9 (5) ◽  
pp. 1739-1744
Keyword(s):  
Transportation Problem ◽  
Geometric Mean ◽  
Method Comparison ◽  
Optimal Solution ◽  
Pareto Optimal Solution ◽  
Pareto Optimal ◽  
Numerical Examples ◽  
Multi Objective ◽  
Fuzzy Environment ◽  
Single Objective

Transportation problem is a very common problem for a businessman. Every businessman wants to reduce cost, time and distance of transportation. There are several methods available to solve the transportation problem with single objective but transportation problems are not always with single objective. To solve transportation problem with more than one objective is a typical task. In this paper we explored a new method to solve multi criteria transportation problem named as Geometric mean method to Solve Multi-objective Transportation Problem Under Fuzzy Environment. We took a problem of transportation with three objectives cost, time and distance. We converted objectives into membership values by using a membership function and then geometric mean of membership values is taken. We also used a procedure to find a pareto optimal solution. Our method gives the better values of objectives than other methods. Two numerical examples are given to illustrate the method comparison with some existing methods is also made.

A Class of Boundary Value Problems Arising in Mathematical Physics: A Green’s Function Fixed-point Iteration Method

2015 ◽  
Vol 70 (5) ◽  
pp. 343-350 ◽  
Author(s):  
Suheil Khuri ◽  
Ali Sayfy
Keyword(s):  
Fixed Point ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Mathematical Physics ◽  
High Accuracy ◽  
Iteration Scheme ◽  
Fixed Point Iteration

AbstractThis paper presents a method based on embedding Green’s function into a well-known fixed-point iteration scheme for the numerical solution of a class of boundary value problems arising in mathematical physics and geometry, in particular the Yamabe equation on a sphere. Convergence of the numerical method is exhibited and is proved via application of the contraction principle. A selected number of cases for the parameters that appear in the equation are discussed to demonstrate and confirm the applicability, efficiency, and high accuracy of the proposed strategy. The numerical outcomes show the superiority of our scheme when compared with existing numerical solutions.

scholarly journals Inverting deformation fields using a fixed point iteration scheme

2010 ◽  
Author(s):  
Marcel Luethi
Keyword(s):  
Image Analysis ◽  
Fixed Point ◽  
Medical Image ◽  
Medical Image Analysis ◽  
Iteration Scheme ◽  
Deformation Field ◽  
Fixed Point Iteration ◽  
Fixed Point Algorithm

Being able to quickly compute the inverse of a deformation field is often useful in the context of medical image analysis. While ITK supports this functionality, the current algorithms are slow and do not always yield accurate results. In this paper we describe an ITK implementation of a fixed point algorithm for the approximate inversion of deformation fields that was recently proposed by M. Chen and co-workers. The algorithm has been shown to be both faster and more accurate than those currently implemented in ITK.

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