In engineering, design is the most sophisticated step as it involves redefining ground conditions in intrinsic computer programs. With the advent of material distribution methods, topology optimization of continuum structures has become a popular field of research and development. To this end, binary variable algorithms diversifying their application to almost all fields of engineering and design have been reported. The ultimate goal of such topology optimization algorithms has been to determine the best locations and geometries of cavities in a given design domain. Literature has it that the homogenization and solid isotropic material with penalization (SIMP) are most popular. Unfortunately, they suffer a drawback that limit their full adoption. Specifically, they lead to optimal solutions that lack defined structural boundaries. Fortunately, the bidirectional evolutionary structural optimization (BESO) approach has been of late reported to solve the discrete problem directly. However, this also has drawbacks in that a predefined amount of material has to be specified, thus, it requires evolutionary ratios to determine a target volume for the next iteration and does not use mathematical optimization. This means that each iteration in the BESO method is not guaranteed to be an optimal step.
In a recent publication, Dr. David Munk formerly at The University of Sydney and currently at Aerospace Division at Defence Science and Technology developed a bidirectional evolutionary structural optimization algorithm that employed integer linear programming to compute optimal solutions to topology optimization problems with the objective of mass minimization. In other words, he developed a BESO algorithm that employed mathematical programming, specifically integer linear programming (ILP), to update the design variables at every iteration. His work is currently published in International Journal for Numerical Methods in Engineering.
In brief, his approach entailed linearization of the objective and constraint functions using the Taylor’s first-order approximation, thereby allowing the method to handle all types of constraints without using Lagrange multipliers or sensitivity thresholds. Afterwards, a relaxation of the constraint targets was performed such that only small changes in topology were allowed during a single update, thus ensuring the existence of feasible solutions. A number of single and multiple constraint problems were then solved using the proposed method. A neighborhood constraint was later employed, ensuring that the approximation of the first-order derivatives was accurate at each step.
Dr. Munk highlighted that, for the real-world problem of designing the internal structure of a transport aircraft wing-box for minimum mass with stress and buckling constraints, the algorithm was able to produce a final design with a 65% reduction in mass and maintain the minimum buckling load factor and maximum stress above/below the limit values. Overall, he reported that the main design driver was buckling of the skin panels and indicated that the conventional method of stringers would possibly allow even more mass to be removed because they would reinforce the skin panels and hence reduce the dominance of buckling on the design.
In summary, Dr. Munk presented a new BESO algorithm that employed mathematical programming methods to solve a series of integer linear programs to update the design, instead of using sensitivity threshold methods. In the developed approach, a series of subproblems were solved, over a small neighborhood, converging to a final optimum topology for a minimum mass objective. Remarkably, he demonstrated the novel approach on a series of nontrivial structural constraints, including compliance, stress, displacement, frequency, and buckling. Altogether, he was able to find excellent solutions to large scale discrete topology optimization problems.
David J. Munk. A bidirectional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints. International Journal for Numerical Methods in Engineering: 2019; volume 118: page 93–120Go To International Journal for Numerical Methods in Engineering