Evolution strategies for solving discrete optimization problems

A method to solve discrete optimization problems using evolution strategies (ESs) is described. The ESs imitate biological evolution in nature and have two characteristics that differ from other conventional optimization algorithms: (a) ESs use randomized operators instead of the usual deterministic ones; (b) instead of a single design point, the ESs work simultaneously with a population of design points in the space of variables. The important operators of ESs are mutation, selection and recombination. The ESs are commonly applied for continuous optimization problems. For the application to discrete problems, several modifications on the operators mutation and recombination are suggested here. Several examples from the literature are solved with this modified ES and the results compared. The examples show that the modified ES is robust and suitable for discrete optimization problems.     

Efficient strategies for reliability-based design optimization of variable stiffness composite structures

This study investigates efficient design optimization frameworks for composite structures with uncertainties related to material properties and loading. The integration of two decoupled reliability-based design optimization methodologies with a decoupled discrete material optimization is proposed to determine material and fiber orientation for three-dimensional composite structures. First, a deterministic and decoupled discrete material optimization is used for baseline comparison. The objective is to minimize the cost of composite structures with the design variables comprising of the piecewise patch orientations and material properties of the fiber reinforced composites. The reliability-based design optimization includes a hybrid method, and also the sequential optimization and reliability assessment method. In the sequential optimization and reliability assessment method, the inverse reliability analysis is evaluated using a stochastic response surface method and a first order reliability approach. Comparing the methods based on the optimal material and fiber orientations, the uncertainties in loads and material properties lead to different optimal layouts compared to the deterministic solutions. The numerical results also reveal that the hybrid method applied in reliability based designs results in negligible additional computational cost.     

Methods based on discrete optimization for finding road network rehabilitation strategies

We study a dynamic optimization problem arising in the (long-term) planning of road rehabilitation activities. In this area one seeks a pavement resurfacing plan for a road network under budget constraints. Our main approach is to model this as an integer programming problem with underlying dynamic programming structure. We investigate properties of this model and propose a solution method based on Lagrangian relaxation where one gets subproblems that are shortest path problems. Some computational experiences based on realistic data are reported.     

Material interpolation schemes for unified topology and multi-material optimization

This paper presents two multi-material interpolation schemes as direct generalizations of the well-known SIMP and RAMP material interpolation schemes originally developed for isotropic mixtures of two isotropic material phases. The new interpolation schemes provide generally applicable interpolation schemes between an arbitrary number of pre-defined materials with given (anisotropic) properties. The method relies on a large number of sparse linear constraints to enforce the selection of at most one material in each design subdomain. Topology and multi-material optimization is formulated within a unified parametrization.     

PolyMat: an efficient Matlab code for multi-material topology optimization

Structural and Multidisciplinary Optimization - We present a Matlab implementation of topology optimization for compliance minimization on unstructured polygonal finite element meshes that...     

Modified element stacking method for multi-material topology optimization with anisotropic materials

Structural and Multidisciplinary Optimization - This article presents a modified element stacking method for anisotropic multi-material topology optimization. This method can transform standard...     

A B-spline multi-parameterization method for multi-material topology optimization of thermoelastic structures

A B-spline multi-parameterization method (MPM) is presented in this paper for topology optimization of thermoelastic structures. As thermoelastic topology     

Nonprobabilistic reliability oriented topological optimization for multi-material heat-transfer structures with interval uncertainties

Structural and Multidisciplinary Optimization - This study presents a nonprobabilistic reliability-based topology optimization (NRBTO) framework that combines a multi-material interpolation model...     

Optimization design of multi-material micropump using finite element method

This paper presents a micropump fabricated from low cost materials with specific goal of cost reduction. The micropump does not require any valve flap and comprises one plastic pump polyether–ether–ketone (PEEK) body, one metal diaphragm, and three piezoelectric ceramics to form piezoelectrically actuated diaphragm valves. The valve actuation simplifies micropump structural designs and assembly processes to make the pump attractive for low cost bio-medical drug delivery applications. A detailed optimization design of geometric parameters of the piezoelectrically actuated diaphragm is undertaken by use of 3D finite element method (FEM) to maximize piezoelectric actuation capability and ensure actuation reliability. An optimized geometric dimensional design: the ratio of thicknesses between the piezoelectric ceramics and the metal diaphragm, and the lateral dimension of the piezoelectric ceramic, is obtained through simulations. Based on the optimized design, a good agreement has been reached between simulated and measured strokes of the micropumps. The tested results show that the micropump has a high pump flow rate for air, up to 39 ml/min, and for water, up to 1.8 ml/min, and is capable of ensuring diaphragm’s maximum stress and strain is within material strength for reliable work.     

Hybrid remap for multi-material ALE

Remapping is one of the essential parts of most arbitrary Lagrangian–Eulerian (ALE) methods. In this short paper we focus on multi-material fluid flows. We present a hybrid remapping method combining the swept remapping algorithm in pure regions with the intersection-based remapping algorithm close to material interfaces. We describe the hybrid remapping method in two formulations, as a one-step and a two-step procedure and compare behaviour of both approaches with the standard intersection-based algorithm using several numerical examples.     

A discrete post-processing method for structural optimization

This paper introduces a discrete variable post-processing method for structural design optimization. The motivation behind the method is to find a good discrete solution at manageable cost while the traditional discrete optimization algorithms are regarded as impractical for large-scale structural design problems. In this paper, the Design of Experiments (DOE) and Conservative Discrete Design (CDD) approaches have been proposed to deal with discrete variables with limited computational cost. Both methods work on the explicit approximate discreteproblem to explore the discrete design. These two approaches, together with engineering rounded-off methods, can be used to process discrete variables at any specified continuous design optimization cycle for structural design problems. Brief background and a theoretical discussion about these approaches are given in this paper. Finally, the methods that have been implemented in MSC.Nastran are demonstrated by academic and real engineering examples.     

Direct zigzag search for discrete multi-objective optimization

Multiple objective optimization (MOO) models and solution methods are commonly used for multi-criteria decision making in real-life engineering and management applications. Much research has been conducted for continuous MOO problems, but MOO problems with discrete or mixed integer variables and black-box objective functions arise frequently in practice. For example, in energy industry, optimal development problems of oil gas fields, shale gas hydraulic fracturing, and carbon dioxide geologic storage and enhanced oil recovery, may consider integer variables (number of wells, well drilling blocks), continuous variables (e.g. bottom hole pressures, production rates), and the field performance is typically evaluated by black-box reservoir simulation. These discrete or mixed integer MOO (DMOO) problems with black-box objective functions are more challenging and require new MOO solution techniques. We develop a direct zigzag (DZZ) search method by effectively integrating gradient-free direct search and zigzag search for such DMOO problems. Based on three numerical example problems including a mixed integer MOO problem associated with the optimal development of a carbon dioxide capture and storage (CCS) project, DZZ is demonstrated to be computationally efficient. The numerical results also suggest that DZZ significantly outperforms NSGA-II, a widely used genetic algorithms (GA) method.     

Design of thermoelastic multi-material structures with graded interfaces using topology optimization

A level-set based shape and topology optimization framework is used to investigate the influence of graded interfaces in the optimization of micro-architectured multi-materials. In contrast to other studies found in the literature, interfaces are considered as smooth and graded transitions between constitutive phases instead of sharp delimitations. Case studies for extreme thermoelastic properties of 2D isotropic composites are analyzed and optimal designs are presented. It is shown that explicitly accounting for interfaces can influence the design of heterogeneous materials in composite microstructures.     

3D multi-material and multi-joint topology optimization with tooling accessibility constraints

Structural and Multidisciplinary Optimization - This paper proposes a method for performing both multi-material topology optimization and multi-joint topology optimization. The algorithm can...     

Quantum-inspired firefly algorithm with particle swarm optimization for discrete optimization problems

The firefly algorithm is a recent meta-heuristic inspired from nature. It is based on swarm intelligence of fireflies and generally used for solving continuous optimization problems. This paper proposes a new algorithm called “Quantum-inspired Firefly Algorithm with Particle Swarm Optimization (QIFAPSO)” that among other things, adapts the firefly approach to solve discrete optimization problems. The proposed algorithm uses the basic concepts of quantum computing such as superposition states of Q-bit and quantum measure to ensure a better control of the solutions diversity. Moreover, we use a discrete representation for fireflies and we propose a variant of the well-known Hamming distance to compute the attractiveness between them. Finally, we combine two strategies that cooperate in exploring the search space: the first one is the move of less bright fireflies towards the brighter ones and the second strategy is the PSO movement in which a firefly moves by taking into account its best position as well as the best position of its neighborhood. Of course, these two strategies of fireflies’ movement are adapted to the quantum representation used in the algorithm for potential solutions. In order to validate our idea and show the efficiency of the proposed algorithm, we have used the multidimensional knapsack problem which is known as an NP-Complete problem and we have conducted various tests of our algorithm on different instances of this problem. The experimental results of our algorithm are competitive and in most cases are better than that of existing methods.     

POD-assisted strategies for structural topology optimization

We propose a new numerical tool for structural optimization design. To cut down the computational burden typical of the Solid Isotropic Material with Penalization (SIMP) method, we apply Proper Orthogonal Decomposition on SIMP snapshots computed on a fixed grid to construct a rough structure (predictor) which becomes the input of a SIMP procedure performed on an anisotropic adapted mesh (corrector). The benefit of the proposed design tool is to deliver smooth and sharp layouts which require a contained computational effort before moving to the 3D printing production phase.     

Steady-state genetic algorithms for discrete optimization of trusses

This paper presents the applications of steady-state genetic algorithms to discrete optimization of trusses. It is mathematically formulated as a constrained nonlinear optimization problem with discrete design variables. Discrete design variables are treated by a two-stage mapping process which is constructed by the mapping relationships between unsigned decimal integers and discrete values. With small generation gap and careful modification, steady-state genetic algorithms can significantly reduce the computational effort and promote the computational efficiency. The effectiveness, robustness and fast convergence of steady-state genetic algorithms are demonstrated through several examples. The performance of four crossover operators is also compared.     

A relative difference quotient algorithm for discrete optimization

According to the characteristics of discrete optimization, the concept of a relative difference quotient is proposed, and a highly accurate heuristic algorithm, a relative difference quotient algorithm, is developed for a class of discrete optimization problems with monotonic objective functions and constraint functions. The algorithm starts from the minimum point of the objective function outside the feasible region and advances along the direction of minimum increment of the objective function and maximum decrement of constraint functions to find a better approximate optimum solution. In order to evaluate the performance of the algorithm, a stochastic numerical test and a statistical analysis for the test results are also completed. The algorithm has been successfully applied to the discrete optimization of structures.