Discrete Sizing Research Articles

An efficient k-NN-based rao optimization method for optimal discrete sizing of truss structures

This study proposes a new method called k-nearest neighbor comparison (k-NNC) to address the computational cost issue of truss optimization with discrete variables using metaheuristic algorithms. The k-NNC judges a new design candidate is worth evaluating by comparing its k available closest designs (k-nearest neighbors) with another design in the population. The new design will be eliminated without evaluating it if the majority of the k nearest neighboring designs are inferior to the one being compared. The k-NNC is combined with Rao algorithms along with Deb’s constraint handling rules and the rounding technique to be suitable for constrained optimization problems with discrete variables. The new optimization Rao algorithms based on k-NNC are used in five truss optimization examples, including both planar trusses and spatial trusses, to evaluate their effectiveness. The numerical results demonstrate that the proposed k-NNC-based Rao algorithms outperform the original Rao algorithms in terms of computational costs. Moreover, the overall performance of k-NNC-based Rao algorithms is similar to or better than that of some state-of-the-art metaheuristic algorithms conducted on the same examples.

Discrete optimal designs for distributed energy systems with nonconvex multiphase optimal power flow

The optimal selection, sizing, and location of small-scale technologies within a grid-connected distributed energy system (DES) can contribute to reducing carbon emissions, consumer costs, and network imbalances. There is a significant lack of studies on how DES designs, especially those with electrified heating systems, impact unbalanced low-voltage distribution networks to which most DES are connected. This is the first study to present an optimisation framework for obtaining discrete technology sizing and selection for grid-connected DES design, while simultaneously considering multiphase optimal power flow (MOPF) constraints to accurately represent unbalanced low-voltage distribution networks. An algorithm is developed to solve the resulting Mixed-Integer Nonlinear Programming (MINLP) formulation. It employs a decomposition based on Mixed-Integer Linear Programming (MILP) and Nonlinear Programming (NLP) and uses integer cuts and complementarity reformulations to obtain discrete designs that are also feasible with respect to the network constraints. A heuristic modification to the original algorithm is also proposed to improve computational speed. Improved formulations for selecting feasible combinations of air source heat pumps (ASHPs) and hot water storage tanks are also presented. Two networks of varying size are used to test the optimisation methods. Designs with electrified heating (ASHPs and tanks) are compared to those with conventional gas boilers. The algorithms outperform one of the existing state-of-the-art commercial deterministic MINLP solvers, which fails to find any solutions in two instances within specified time limits. While feasible solutions were obtained for all cases, convergence was not achieved for all, especially for those involving the larger network. Where converged, the algorithm with the heuristic modification has achieved results up to 70% faster than the original algorithm. Results for case studies suggest that including ASHPs can support up to 16% higher renewable generation capacity compared to gas boilers, albeit with higher ASHP investment costs, as local generation and consumption minimises network violations associated with excess power export. The results also show the importance of including nonlinear power flow constraints in DES design problems. The optimisation framework and results can be used to inform stakeholders such as policymakers and network operators, to increase renewable energy capacity and aid the decarbonisation of domestic heating systems.

Discrete sizing optimization method based on dividing rectangles algorithm and local response surface for steel frame structures

It has always been the goal of structural engineers to construct safe and stable buildings using the least amount of materials. Utilizing a quick and effective way to optimize the cross-section size is crucial because conventional building design methods are impacted by the designers' knowledge, and it is challenging to prevent material waste. Due to its implicit optimization function, discrete design variables, and expensive individual evaluation, sizing optimization of high-rise buildings is very challenging to achieve. In order to effectively handle this optimization problem, a two-stage discrete sizing optimization method for high-rise buildings based on the DIviding RECTangles (DIRECT) algorithm and local response surface is proposed in this paper. The optimization method suggested in this research consists of two key stages: the global search stage and the local search stage. In the global search stage, the entire design domain is divided using a modified DIRECT algorithm to swiftly identify potentially optimal subregions that may contain the best points. In the local search stage, the local response surface model is constructed to approximate the objective and constraint functions using the sampling points from the previous stage, and the discrete optimal solution is rapidly found through mathematical iterative solving. A sizing optimization calculation program for high-rise buildings was developed in Microsoft Visual Studio 2015 on the basis of C++ to achieve automatic optimization. The new method was applied to optimize three high-rise steel frame buildings with different heights and plan shapes. The results showed that the material cost could be successfully saved compared with the conventional design, and the over-limit constraints could be adjusted automatically, which demonstrated the viability and efficacy of the two-stage optimization method.

MILP-based discrete sizing and topology optimization of truss structures: new formulation and benchmarking

Discrete sizing and topology optimization of truss structures subject to stress and displacement constraints has been formulated as a Mixed-Integer Linear Programming (MILP) problem. The computation time to solve a MILP problem to global optimality via a branch-and-cut solver highly depends on the problem size, the choice of design variables, and the quality of optimization constraint formulations. This paper presents a new formulation for discrete sizing and topology optimization of truss structures, which is benchmarked against two well-known existing formulations. Benchmarking is carried out through case studies to evaluate the influence of the number of structural members, candidate cross sections, load cases, and design constraints (e.g., stress and displacement limits) on computational performance. Results show that one of the existing formulations performs significantly worse than all other formulations. In most cases, the new formulation proposed in this work performs best to obtain near-optimal solutions and verify global optimality in the shortest computation time.

The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses

The paper presents the Mixed-Integer Non-linear Programming (MINLP) approach to the synthesis of trusses. The solution of continuous/discrete non-convex and non-linear optimization problems is discussed with respect to the simultaneous topology, shape and discrete sizing optimization of trusses. A truss MINLP superstructure of different topology and design alternatives has been generated, and a special MINLP model formulation for trusses has been developed. In the optimization model, a mass objective function of the structure has been defined and subjected to design, load and dimensioning constraints. The MINLP problems are solved using the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm. Multi-level MINLP strategies are introduced to accelerate the convergence of the algorithm. The Modified Two-Phase and the Sequential Two-Phase MINLP strategies are proposed in order to solve highly combinatorial topology, shape and discrete sizing optimization problems. The importance of local buckling constraints on topology optimization is also discussed. Some simple numerical examples are shown at the end of the paper to demonstrate the suitability and efficiency of the proposed method.

Discrete sizing, cross-sectional shape, topology optimization, and material selection of a framed automotive body

This study proposed a discrete structural optimization method for a framed automotive body. Up to four types of discrete design variables are considered simultaneously, that is, the sizing, cross-sectional shape, topology, and material variables. Firstly, to solve the nonconvex and nonlinear optimization problem, the original non-dominated sorting genetic algorithm, the third version (NSGA-III), is adapted. An improved extreme points identification scheme and a new mutation operator are proposed to stabilize the normalization of the population and accommodate the manufacturing constraints, respectively. Two test problems demonstrate that the modified NSGA-III can handle continuous and discontinuous multiple objective optimization. Subsequently, the classical 10-bar truss is used to illustrate the proposed method. A weight reduction of 4.5 kg is achieved as compared to previous optimal designs in the literature. Finally, a framed automotive body is optimized for maximizing the first order natural frequency and minimizing the total mass, the maximum stresses and the maximum displacements in different load cases and the manufacturing cost. The results obtained by different optimization procedures are presented and discussed. The results demonstrate the feasibility and effectiveness of the proposed method. A weight reduction of 17.59% is achieved while other structural performances satisfy the design requirements.

Incremental Lagrangian Relaxation Based Discrete Gate Sizing and Threshold Voltage Assignment

Timing closure remains one of the most critical challenges of a physical synthesis flow, especially when the design operates under multiple operating conditions. Even if timing is almost closed at the end of the flow, last-mile placement and routing congestion optimizations may introduce new timing violations. Correcting such violations needs minimally disruptive techniques such as threshold voltage reassignment and gate sizing that affect only marginally the placement and routing of the almost finalized design. To this end, we transform a powerful Lagrangian-relaxation-based optimizer, used for global timing optimization early in the design flow, into a practical incremental timing optimizer that corrects small timing violations with fast runtime and without increasing the area/power of the design. The proposed approach was applied to already optimized designs of the ISPD 2013 benchmarks assuming that they experience new timing violations due to local wire rerouting. Experimental results show that in single corner designs, timing is improved by more than 36% on average, using 45% less runtime. Correspondingly, in a multicorner context, timing is improved by 39% when compared to the fully-fledged version of the timing optimizer.

Discrete sizing design of steel truss bridges through teaching-learning-based and biogeography-based optimization algorithms involving dynamic constraints

In this paper, Teaching-Learning Based Optimization (TLBO) and Biogeography-Based Optimization (BBO) algorithms are presented to examine the optimum discrete sizing design of steel truss steel bridges for minimizing the structural weights. Both proposed nature-inspired metaheuristic optimization algorithms are encoded in MATLAB with integration of a structural analysis program (SAP2000) via open application programming interface (OAPI). At the end, optimal steel profiles are selected from available discrete section lists by satisfying the structural restrictions, such as stress and displacement, involved by American Institute of Steel Construction-Allowable Stress Design (AISC-ASD). Additional to these, optimum discrete sizing design process is performed for the cases with and without dynamic constraints, which are adopted from natural periods of the bridge structures with respect to the mode shapes. The algorithmic performance of the proposed algorithms outperforms on both planar and spatial steel truss bridges. To prove this obtained optimal solutions are compared with previously reported optimum designs attaining via different metaheuristics. The final optimum discrete sizing designs of the steel truss bridges reveal that the proposed TLBO and BBO algorithms can easily be applied to discrete nonlinear programming problems.

Enhanced Imperialist Competitive Algorithm for Optimal Structural Design

Solution of complex engineering problems by meta-heuristics, requires powerful operators to maintain sufficient diversification as well as proper intensification during the search. Standard Imperialist Competitive Algorithm, ICA, delays the search intensification by propagating it via a number of artificial empires that compete each other until one concurs the others. An Enhanced Imperialist Competitive Algorithm is developed here by adding an evolutionary operator to the standard ICA followed by greedy replacement; in order to improve its effectiveness. The new operator introduces a walk step directed from the less fit to the fitter individual in each pair of the search agents together with a random scaling and pick up scheme. EICA performance is then compared with ICA as well as GA, PSO, DE, CBO, TLBO, SOS; first in a set of fifteen test functions. Second, a variety of continuous and discrete engineering benchmarks and structural sizing problems are solved to evaluate EICA in constrained optimization. In this regard, a diversity index is traced as well as the other convergence metrics. The results exhibit considerable improvement of the algorithm by the proposed features of EICA and its competitive performance with respect to the other treated methods.

Simultaneous sizing, shape, and layout optimization and automatic member grouping of dome structures

This paper presents both discrete and continuous sizing, shape, and layout structural optimization problems concerning the weight minimization of dome structures and considering displacements, stresses, natural frequencies of vibration, and global stability as constraints. The layout optimization searches for the best structural configuration through the maintenance of the bars’ grouping by introducing a new design variable. This variable refers to the number of standard modules used to generate the structural configuration of optimized domes. It can be attractive to use a reduced number of distinct cross-sectional areas, minimizing the costs of fabrication, transportation, storing, checking, and welding, providing labor saving measures. To obtain an automatic member grouping of the bars of the trusses analyzed in this paper, a specific encoding using cardinality constraints is considered. As result, trade-off curves are provided, showing the optimized weights in comparison with the number of distinct cross-sectional areas used in the solutions. Differential Evolution is the search algorithm adopted in this paper. The structural optimization problems showed better solutions when compared with those presented in the literature.

Multi-objective truss structural optimization considering natural frequencies of vibration and global stability

Conflicting objectives such as minimizing weight and minimizing the maximum nodal displacement, with constraints on normal stresses in the bars, is a common multi-objective structural optimization problem widely found in the literature. This paper proposes multi-objective structural optimization problems with the combination of new conflicting objectives functions and constraints, such as the natural frequencies of vibration and the load factors concerning the global stability of the structure. The solution for these problems may be of great interest in the field of structural engineering, not yet discussed in the literature. The problems analyzed in this paper deal with both discrete and continuous sizing, shape, and layout design variables. The search algorithm adopted here is a modified version of the Differential Evolution called the Third Evolution Step Differential Evolution (GDE3). Several experiments are analyzed with their Pareto-fronts showing the non-dominated solutions. The solutions are defined after obtaining the Pareto curve, which is one of the most important steps and a task that may not be trivial for the Decision Maker. This paper involves a strategy that establishes criteria defining weights (importance) for each objective function and, through these values, enables comparison scenarios. The numerical experiments include plane and spatial benchmark trusses.

Discrete Sizing Design of Truss Structure Using an Approximate Model and Post-Processing

Discrete Sizing Design of Truss Structure Using an Approximate Model and Post-Processing

Discrete sizing optimization of stepped cylindrical silo using PSO method and implicit dynamic FE analysis

Considering the size discreteness of commercially available metal plates and the intrinsic buckling strength vulnerability of slender silo, this paper proposed a methodology, which integrated the nonlinear implicit dynamic Finite Element Method (FEM), Particle Swarm Optimization (PSO) algorithm and MATLAB programming, to optimize the wall-thickness layout for stepped thin-walled cylindrical silo, with the objective of minimizing silo mass while ensuring its structural stability. Taking into account both practicality and reliability, silo discharge loads were amplified 1.6 times to try to reflect the comprehensive effectiveness of negative and positive factors on slender silo buckling strength. When evaluating the fitness of PSO method, nonlinear implicit dynamic FEM results, such as kinetic energy history data plots, total energy history data plots, etc., were used to intuitively determine whether silo buckled or not. In essence, the optimal wall-thickness layout problem of a stepped silo is an NP-hard combinational optimization problem. The discrete thicknesses of rolled metal plates set an unavoidable constraint on stepped silo size optimization, which implies that there are only a few specific thickness values could be selected. In addition, the data of plate width are also discrete and one width value might correspond to several thickness data. For reasons for saving the potential cutting costs, the heights of most silo segments should be an integral multiple of the corresponding plate width value as far as possible, while the overall height of the silo should be kept still. To realize this goal, numerical processing techniques, such as generating a random number from a uniformly distributed set of discrete positive integers, linear normalization and linear interpolation, etc., were applied in this study.

Discrete sizing/layout/topology optimization of truss structures with an advanced Jaya algorithm

Discrete optimization of truss structures is a hard computing problem with many local minima. Metaheuristic algorithms are naturally suited for discrete optimization problems as they do not require gradient information. A recently developed method called Jaya algorithm (JA) has proven itself very efficient in continuous engineering problems. Remarkably, JA has a very simple formulation and does not utilize algorithm-specific parameters. This study presents a novel JA formulation for discrete optimization of truss structures under stress and displacement constraints. The new algorithm, denoted as discrete advanced JA (DAJA), implements efficient search mechanisms for generating new trial designs including discrete sizing, layout and topology optimization variables. Besides the JA’s basic concept of moving towards the best design of the population and moving away from the worst design, DAJA tries to form a set of descent directions in the neighborhood of each candidate designs thus generating high quality trial designs that are very likely to improve current population. Results collected in seven benchmark problems clearly demonstrate the superiority of DAJA over other state-of-the-art metaheuristic algorithms and multi-stage continuous–discrete optimization formulations.

Robust optimal discrete arc sizing for tree-shaped potential networks

We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing’s optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.

Enhancing sensitivity-based power reduction for an industry IC design context

For many years, discrete gate sizing has been widely used for timing and power optimization in VLSI designs. The importance of gate sizing optimization has been emphasized by academia for many years, especially since the 2012/2013 ISPD gate sizing contests [1, 2]. These contests have provided practical impetus to academic sizers through the use of realistic constraints and benchmark formats. At the same time, due to simplified delay/power Liberty models and timing constraints, the contests fail to address real-world criteria for gate sizing that are highly challenging in practice. We observe that lack of consideration of practical issues such as electrical and multi-corner constraints – along with limited sets of benchmarks – can misguide the development of contest-focused academic sizers. Thus, we study implications of the “gap” between academic sizers and product design use cases. In this paper, we note important constraints of modern industrial designs that are generally not comprehended by academic sizers. We also point out that various optimization techniques used in academic sizers can fail to offer benefits in product design contexts due to differences in the underlying optimization formulation and constraints. To address this gap, we develop a new robust academic sizer, Sizer, from a fresh implementation of Trident [3]. Experimental results show that Sizer is able to achieve up to 10% leakage power and 4% total power reductions compared to leading commercial tools on designs implemented with foundry technologies, and 7% leakage power reduction on a modern industrial design in the multi-corner multi-mode (MCMM) context.

Optimized Continuous Multicolumn Chromatography Enables Increased Productivities and Cost Savings by Employing More Columns.

The advantages of continuous chromatography with respect to increased capacity are well established. However, the impact of different loading scenarios and total number of columns on the process economics has not been addressed. Here four different continuous multicolumn chromatography (MCC) loading scenarios are evaluated for process performance and economics in the context of a Protein A mAb capture step. To do so, a computational chromatography model is validated experimentally. The model is then used to predict process performance for each of the loading methods. A wide range of feed concentrations and residence times are considered, and the responses of operating binding capacity, specific productivity, and the number of process columns are calculated. Processes that are able to add more columns proved to be up to 65% more productive, especially at feed concentrations above 5 g L-1 . An investigation of the operating costs shows that discrete column sizing and process performance metrics do not always correlate and that the most productive process is not necessarily the most cost effective. However, adding more columns for the non-load steps at higher feed concentrations allows for overall cost savings of up to 32%.

Hybrid Improved Dolphin Echolocation and Ant Colony Optimization for Optimal Discrete Sizing of Truss Structures

This paper presents a robust hybrid improved dolphin echolocation and ant colony optimization algorithm (IDEACO) for optimizing the truss structures with discrete sizing variables. The dolphin echolocation (DE) is inspired by the navigation and hunting behavior of dolphins. An improved version of dolphin echolocation (IDE), as the main engine, is proposed and uses the positive attributes of ant colony optimization (ACO) to increase the efficiency of the IDE. Here, ACO is employed to improve the precision of the global optimization solution. In the proposed hybrid optimization method, the balance between exploration and exploitation process was the main factor to control the performance of the algorithm. IDEACO algorithm performance is tested on several problems of benchmarks discrete truss structure optimization. The results indicate the excellent performance of the proposed algorithm in optimum design and rate of convergence in comparison with other metaheuristic optimization methods, so IDEACO offers a good degree of competitiveness against other existing metaheuristic methods.

Seeding the initial population with feasible solutions in metaheuristic optimization of steel trusses

In spite of considerable research work on the development of efficient algorithms for discrete sizing optimization of steel truss structures, only a few studies have addressed non-algorithmic issues affecting the general performance of algorithms. For instance, an important question is whether starting the design optimization from a feasible solution is fruitful or not. This study is an attempt to investigate the effect of seeding the initial population with feasible solutions on the general performance of metaheuristic techniques. To this end, the sensitivity of recently proposed metaheuristic algorithms to the feasibility of initial candidate designs is evaluated through practical discrete sizing of real-size steel truss structures. The numerical experiments indicate that seeding the initial population with feasible solutions can improve the computational efficiency of metaheuristic structural optimization algorithms, especially in the early stages of the optimization. This paves the way for efficient metaheuristic optimization of large-scale structural systems.

Observer-Teacher-Learner-Based Optimization: An enhanced meta-heuristic for structural sizing design

Structural sizing is a rewarding task due to its non-convex constrained nature in the design space. In order to provide both global exploration and proper search refinement, a hybrid method is developed here based on outstanding features of Evolutionary Computing and Teaching-Learning-Based Optimization. The new method introduces an observer phase for memory exploitation in addition to vector-sum movements in the original teacher and learner phases. Proper integer coding is suited and applied for structural size optimization together with a fly-to-boundary technique and an elitism strategy. Performance of the proposed method is further evaluated treating a number of truss examples compared with teaching-learning-based optimization. The results show enhanced capability of the method in efficient and stable convergence toward the optimum and effective capturing of high quality solutions in discrete structural sizing problems.