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.

In computing eigenvalues for a large finite element system it has been observed that the eigenvalue extractors produce eigenvectors that are in some sense more accurate than their corresponding eigenvalues. From this observation the paper uses a patch type technique based on the eigenvector for one mesh quality to provide an eigenvalue error indicator. Tests show this indicator to be both accurate and reliable.

Natural frequency error estimation for 3D brick elements

An algorithm for the generation of general shape functions is presented here. The algorithm is coded using the Macsyma program which has symbolic/algebraic manipulation capabilities. The current technique is based on the direct approach of calculating shape functions which is seldom used due to manual algebraic tediousness. Listing of the code is included in this paper as well as examples showing how it correctly generates some of the common shape functions with ease. Two examples of new shape functions were created for C2 elements. Hence, the more useful purpose of this algorithm is that it can be used to investigate the existence of new types of shape functions by making use of the power of symbolic computational software. Copyright © 2000 John Wiley & Sons, Ltd.

Shape functions generation using Macsyma

Periodic topology optimization has been suggested as an effective means to design efficient structures which address a range of practical constraints, such as manufacturability, transportability, replaceability and ease of assembly. This study proposes a new approach for design of finite periodic structures by allowing variable orientation states of individual unit-cells. In some design instances of periodic structures, the unit-cell may exhibit certain geometric features allowing multiple possible assembly orientations (e.g. facing up or down). For the first time, this work incorporates such assembly flexibility within the periodic topology optimization, which enables to greatly expand the conventional periodic design space and take more advantage of structural periodicity. Given its broad applications, a methodology for the design of more efficient periodic structures while maintaining the same degree of periodic constraint may be of significant benefit to engineering practice. In this study, several numerical examples are presented to demonstrate the effectiveness of this new approach for both static and vibratory criteria. Brute force analysis is also utilized to compare all possible assembly configurations for several periodic structures with a small number of unit-cells. A heuristic approach is suggested for selecting more beneficially oriented configurations in periodic structures with a large number of unit-cells for which an exhaustive search may be computationally infeasible. It is found that in all the presented cases, the oriented periodic structures outperform the conventional non-oriented (or namely translational) periodic counterparts. Finally, an educational MATLAB code is provided for replication of the design results in this paper.

Finite periodic topology optimization with oriented unit-cells

A simple evolutionary procedure for structural optimisation was modified to allow for the shape optimisation of metallic inserts into composite materials. The aim of the shape optimisation of inserts is to reduce the high stress concentration factor present in mechanically fastened composite joints to increase the bearing strength of pin loaded and bolted joints. Both optimised shape aluminium inserts obtained from the evolutionary structural optimisation package achieved a reduction in compressive stress concentration of approximately 46% when compared to a 2 mm circular insert, and a reduction of 67% when compared to a laminate without metallic insert. A comparison of the reduction of compressive stress at the interface per unit mass of test specimen shows that the optimum shapes developed by means of this process are approximately 40% more efficient than a 2 mm circular aluminium insert. A testing program is under way to prove the concept.

Shape Optimisation of Metallic Inserts in Composite Bolted Joints

In computing eigenvalues for a large finite element system, it has been observed that the eigenvalue extractors produce eigenvectors that are in some sense more accurate than their corresponding eigenvalues. From this observation, uses a patch‐type technique based on the eigenvector for one mesh quality to provide an eigenvalue error indicator for the complex problem of out‐of‐plane plate vibration analysis. Tests show this indicator to be both accurate and reliable.

Grant P. Steven

Papers

Natural frequency error estimation for 3D brick elements

Shape functions generation using Macsyma

Finite periodic topology optimization with oriented unit-cells

Shape Optimisation of Metallic Inserts in Composite Bolted Joints

Error estimation for natural frequency analysis using plate elements