A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Survey of multi-objective optimization methods for engineering

A level set method for structural topology optimization

The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http://www.topopt.dtu.dk.

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A 99 line topology optimization code written in Matlab

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization approaches: A comparative review

Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: A review.

A path-dependent level set topology optimization with fracture criterion

This paper presents a simple evolutionary method for optimization of plates subject to constant weight, where design variable thicknesses are discrete. Sensitivity numbers for sizing elements are derived using optimality criteria methods. An optimal design with minimum displacement or minimum strain energy is obtained by gradually shifting material from elements to the others according to their sensitivity numbers. A simple smoothing technique is additionally employed to suppress formation of checkerboard patterns. It is shown that the proposed method can directly deal with discrete design variables. Examples are provided to show the capacity of the proposed evolutionary method for structural optimization with discrete design variables.

An evolutionary method for optimal design of plates with discrete variable thicknesses subject to constant weight

In previous work by the authors, a Genetic Algorithm (GA) based shape optimization technique was introduced. The method was shown to be capable of producing high-fidelity optimal shapes. However, the process was computationally expensive and required constant re-meshing due to distorted boundary elements resulting from large boundary movements. This paper combines the Fixed Grid (FG) method of Finite Element Analysis (FEA) and the GA shape optimization module to create a hybrid that effectively addresses these problems. The FG solver is found to be significantly faster than conventional FEA, and the fixed FE mesh frees boundary movements from meshing constraints. The Fixed-Grid Genetic-Algorithm (FGGA) shape optimization method is detailed in this paper, and the key algorithms used in the FG and the GA components are explained. The method is also applied to a number of shape optimization problems, and the results are presented and discussed.

On improving the GA step-wise shape optimization method through the application of the Fixed Grid FEA paradigm

The performance of countless engineering systems usually depends on various physical phenomena belonging to different disciplines, such as solid mechanics, fluid mechanics, and heat transfer. Such ...

Grant P. Steven

Papers

A path-dependent level set topology optimization with fracture criterion

An evolutionary method for optimal design of plates with discrete variable thicknesses subject to constant weight

On improving the GA step-wise shape optimization method through the application of the Fixed Grid FEA paradigm

Topology Optimization Applied to Transpiration Cooling

Machine learning based topology optimization of fiber orientation for variable stiffness composite structures