Sizing Design Variables Research Articles

Multiscale topology optimization for the design of spatially-varying three-dimensional lattice structure

This paper introduces three dimensional (3D) topology optimization specifically tailored for designing spatially-varying primitive-cubic (CP) type lattice structures. The developed design process consists of three steps: pre-processing, main processing, and post-processing. In the pre-processing step, a surrogate model between lattice geometry variables and effective elasticity tensor is constructed using an artificial neural network. This surrogate model is essential because it enables the quick calculation of the effective elasticity tensor of lattice microstructure. In the subsequent main processing step, multiscale topology optimization is performed. During this step, lattice microstructure size and rotation design variables are optimized together with the macrostructure density design variables. Macrostructures are designed using a well-established three-field density scheme with a SIMP formulation. Formulations for spatially-varying 3D lattice microstructure are newly developed based on a unit quaternion rotation representation scheme. In the final post-processing step, a spatially-varying microstructure is restored using a de-homogenization scheme, newly developed particularly for the primitive-cubic (CP) type 3D lattice structures. The effectiveness and robustness of the proposed formulations are validated through three design examples for the compliance minimization problem. Moreover, a designed lattice structure is fabricated using an additive manufacturing machine.

Automatic Minimization of the Drift Performance of RC 3D Irregular Buildings Using Genetic Algorithm

This study introduces the application of genetic algorithms for the optimal design of the seismic torsional drift performance of three-dimensional reinforced concrete buildings. Attempts have been made to achieve an optimal automatic design of the torsional drift of the storeys of reinforced concrete buildings with plan irregularities to build torsional balanced structures. The storey torsional drift response generated by static and dynamic loads can be clearly expressed in terms of vertical structure elements’ sizing design variables. Two examples are provided to demonstrate the efficiency and practicability of the proposed optimum design approach. The performance of the structures was assessed as per the procedure prescribed in modern seismic code languages. Mathematical and finite element modelling were used to perform seismic analysis on buildings. MATLAB® programming was used as a solution to the sizing optimization problem. The results confirmed the proposed genetic algorithm’s ability to find efficient optimum solutions to three-dimensional reinforced concrete structures through the problem of size optimization.

Optimum design of steel frames with sizing and orientation design variables using a reformulated evolution strategy

Sizing optimization of steel frame structures has been traditionally conducted based on fixed orientations of structural members. However, it seems more expedient to search for an optimum design where orientations of column members are considered as design variables together with sizing of all members for the minimum weight design of a steel frame. In order to solve the resulting optimization problem effectively, a reformulated evolution strategy (ES) algorithm is proposed and employed in the present study. The numerical investigations are carried out using three test problems regarding the optimum design of steel space frames. In each test problem, a frame is first designed for the minimum weight considering sizing design variables only, where initial orientations of the column members are kept unchanged. Next, sizing and orientation design variables are considered simultaneously for minimizing the weight of the same frame. The results obtained in the foregoing two cases are compared and the associated optimal layouts for the column orientations are depicted. The study reveals that the optimal selection of column orientations has a significant influence on the minimum weight design of steel frames.

Sizing optimization and experimental characterization of a variable stiffness shape memory polymer filled honeycomb composite

Variable stiffness structures and materials have been considered for many applications, including active vibration control and shape morphing. With regards to shape morphing, variable stiffness materials and composites have been considered for reconfigurable skin materials in aerospace vehicles. Of the many concepts that have been developed for such applications, shape memory polymers (SMPs) are one such promising materials for shape morphing. SMPs exhibit both high modulus ratios and recoverable strains but suffer from a low overall modulus and often require reinforcements, such as honeycomb. This work investigates the design space of such honeycomb reinforced SMPs as variable stiffness materials. Unit cell finite element models are developed for the material, and parametric studies are completed for varying honeycomb cell geometries. A multiobjective, constrained Pareto front optimization is completed for two honeycomb material models and in two loading directions using selected sizing design variables. Pareto fronts are established, and cell geometries are selected and fabricated to experimentally verify the optimized model predictions. The results both predict and demonstrate the advantages of using honeycomb reinforcements for SMPs. Effective in-plane moduli as high as 45 GPa are predicted while achieving a change in modulus of 450X. Compared to existing reinforcement strategies for shape memory polymers, these composites exhibit favorable combinations of both high stiffness and high changes in stiffness with a high degree of tailorability through the honeycomb cell geometry and predicted performances that meet and exceed the state of the art.

Success history based adaptive multi-objective differential evolution variants with an interval scheme for solving simultaneous topology, shape and sizing truss reliability optimisation

A general approach based on the most probable point (MPP) method for solving reliability truss optimisation with simultaneous topology, shape and sizing (TSS) design variables is developed. The design problems are solved using double loop optimisation where the inner loop is for reliability index and the probability of failure approximation is solved by Harris Hawk Optimisation (HHO). The outer loop, the main TSS truss optimisation loop, is solved by two newly developed algorithms, namely interval success history based adaptive multi-objective differential evolution (iSHAMODE) and its hybrid variant with the whale optimisation algorithm (iSHAMODE-WO). Six TSS truss optimisation problems are evaluated. The results from the proposed method for reliability approximation and First Order Second Moment (FOSM) are compared and validated with Monte Carlo Simulation (MCS). The proposed method shows more consistent and accurate results compared to FOSM. Furthermore, the efficiency of the proposed optimisation algorithms (iSHAMODE and iSHAMODE-WO) is proved; they can outperform their predecessors and several state-of-the-art algorithms including multi-objective multi-verse optimisation algorithm (MOMVO), multi-objective grasshopper optimisation algorithm (MOGOA), multi-objective dragonfly optimisation (MODA), multi-objective salp swarm algorithm (MSSA), hybridisation of real-code population-based incremental learning and differential evolution (RPBILDE), and multi-objective meta-heuristic with iterative parameter distribution estimation (MMIPDE).

Topology and size optimization for X-bracing system of nonlinear inelastic space steel frames

In this article, a Python-programmed advanced design paradigm is firstly introduced to topology and size optimization of the X-bracing system of nonlinear inelastic space steel frames. For that purpose, an advanced analysis method considering both geometric and material nonlinearities is utilized as an effective finite element analysis (FEA) solver. In which, X-bracing members are modeled by truss elements, while the beam and column members are simulated by beam-column ones. The bracing members’ cross-sectional area and their position are respectively treated as discrete size and topology design variables. The problem aims to minimize the weight of X-bracing system so that the constraints on the strength, inter-story drift and maximum displacement are satisfied. An adaptive hybrid evolutionary firefly algorithm (AHEFA) is employed as an optimizer. Numerical examples are exhibited to illustrate the powerful ability of the present methodology.

Aero-structural optimization-based tailoring of bridge deck geometry

The deck cross-section is usually identified in engineering practice as the most important design variable in the wind-resistant design of long-span bridges. This is certainly true in most cases since it controls the aerodynamic and mechanical contribution of the deck to the global bridge performance. However, the effectiveness of deck shape modifications to handle the aeroelastic constraints is highly influenced by the aeroelastic performance requirements. This paper seeks to delve into the aero-structural design optimization of long-span bridges by analyzing possible design scenarios depending on the ability of the deck shape to meet the imposed aeroelastic requirements. A long-span cable-stayed bridge is optimized focusing on the buffeting response and considering different sets of limit values for the design constraints. According to the effectiveness of the deck shape design variables and other size design variables to manage the aeroelastic design constraints, three types of aero-structural optimization problems are identified: type I, aeroelastic constraints are not active and structural constraints drive the design; type II, aeroelastic constraints are active and effectively controlled by deck shape modifications; and type III, aeroelastic constraints are active and demand both shape and size modifications. The engineering significance for practical design is discussed.

Optimum design of truss structures with frequency constraints by an enhanced particle swarm optimization method with gradient directions based on emigration philosophy

In this article, the combined method of particle swarm optimization (PSO) with gradient directions (PSOG) is further extended to increase the capabilities of exploration and exploitation. Four different enhancements are added to the PSOG, based on emigration ideas. To investigate the power of the presented methods, 29 complicated functions are optimized with several local optima. Then, the suggested approaches are used in structural design problems with multiple frequency constraints containing sizing and shape design variables for two- and three-dimensional truss structures. The results are compared with some of the methods in the literature and the superiority of the suggested techniques is demonstrated.

ANFIS-Based Optimum Design of Real Power Transmission Towers with Size, Shape and Panel Design Variables using BBO Algorithm

The present study aims to provide a suitable approach to optimize transmission line towers with size, shape and panel design variables. MSTOWER software was used for modeling, analysis and design of the transmission tower. The design requirements applied to the structure were in accordance with ASCE10-97 standard. Transmission line towers are optimized in two ways. The first method combines Biogeography based Optimization (BBO) algorithm with MSTOWER software called BBO-MSTOWER and the second method, called BBO-ANFIS, uses an adaptive fuzzy neural inference system (ANFIS). To analyze the structure in the optimization process with the aim of reducing the computations and time of the optimization process. In order to evaluate the proposed method, two full-scale transmission towers were optimized as numerical models using the above two methods. Finally, the optimal design and time required in the two optimization methods were compared with each other as well as with the initial design of the tower and the results reported in previous studies. The results showed that with acceptable accuracy, ANFIS resulted in a significant reduction in the calculation time of the entire optimization process.

Parameter free Jaya algorithm for truss sizing-layout optimization under natural frequency constraints

In this study, the parameter free Jaya algorithm (PFJA) is developed for sizing and layout optimization of truss structures subject to natural frequency constraints. The distinctive feature of PFJA is that it uses neither algorithm-specific parameters nor common parameters in the search process. Besides using an elitist strategy where new structural analyses are performed only if strictly necessary, PFJA adaptively changes population size in the optimization process. The validity of proposed PFJA is demonstrated by solving eight classical truss weight minimization problems including up to 59 sizing and layout design variables. The results obtained by the PFJA are compared with those of standard JA, modified Jaya algorithm (MJA) and other state-of-art metaheuristic algorithms in terms of optimized weight, convergence speed and several statistical parameters. Optimization results prove the superiority of PFJA over standard JA, MJA and other metaheuristic optimizers available 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.

Size and Shape Design Optimization of Truss Structures Using the Jaya Algorithm

The metaheuristic algorithm is proposed to solve the weight minimization problem of truss structures, considering the shape and sizing design variables. Design variables are discrete and/or continuous. The design of truss structures is optimized by an efficient optimization algorithm called Jaya. The main feature of Jaya is that it does not require setting algorithm-specific parameters. The algorithm has a very simple formulation in which the basic idea is to approach the best solution and escape from the worst solution [6]. Analyses of structures are performed by a finite element code in MATLAB. The effectiveness of the Jaya algorithm is demonstrated using two benchmark examples: planar truss 18-bar and spatial truss 39-bar, and compared with results in references.

Weight Minimization of Spatial Trusses with Genetic Algorithm

Abstract A genetic algorithm is proposed to solve the weight minimization problem of spatial truss structures considering size and shape design variables. A very recently developed metaheuristic method called JAYA algorithm (JA) is implemented in this study for optimization of truss structures. The main feature of JA is that it does not require setting algorithm specific parameters. The algorithm has a very simple formulation where the basic idea is to approach the best solution and escape from the worst solution. Analyses of structures are performed by a finite element code in MATLAB. The effectiveness of JA algorithm is demonstrated through benchmark spatial truss 39-bar, and compare with results in references.

Optimal dome design considering member-related design constraints

This study proposes to optimize the design of geometrically nonlinear dome structures. A new Multi-objective Optimization Algorithm named Pareto Archived Genetic Algorithm (PAGA), which has an ability of integrating the nonlinear structural analysis with the provisions of American Petroleum Institute specification is employed to optimize the design of ellipse and sphere-shaped dome configurations. Thus, it is possible to investigate how the qualities of optimal designations vary considering the shape, size, and topology-related design variables. Furthermore, the computing efficiency of PAGA is evaluated considering six multi-objective optimization algorithms and eight quality measuring indicators. It is shown that PAGA has a capability of both exploring an increased number of pareto solutions and predicting a pareto front with a higher convergence degree. Moreover, the inclusion of shape-related design variables leads to a decrease in both the weights of dome structures and their load-carrying capacities. However, the designer easily determines the most requested optimal design through the archiving feature of PAGA. Thus, it is also demonstrated that the proposed optimal design procedure increases the correctness degree in the evaluation of optimal dome designs through the tradeoff analysis. Consequently, PAGA is recommended as an optimization tool for the design optimization of geometrically nonlinear dome structures.

A Unified Optimal Design Approach for Geometrically Nonlinear Skeletal Dome Structures

In this study, a unified optimal design approach is proposed for the design of skeletal dome structure (SDS). Thus, this study has three objectivities, i) presenting the emergence of proposed design integrity, ii) applying the proposed optimal design approach for the design optimization geometrically nonlinear SDS with both ellipse and sphere-shaped forms considering both the shape, size and topology-related design variables, iii) determining the dominant design criteria in the design of SDS. In this framework, the design of SDS is optimized thereby minimizing its entire weight and joint displacements and maximizing its member forces at the same time. The design constraints are borrowed from the provisions of American Petroleum Institute (API RP2A-LRFD) specification. A multi-objective optimization algorithm (MOA) named Pareto Archived Genetic Algorithm (PAGA), as an optimization tool is integrated by an automatic dome generating tool. Therefore, the novelty of this study comes from being the first attempt to obtain the optimal design in a way of integrating both member and joint-related design constraints by the geometrically nonlinear structural analysis. Consequently, it is displayed that that the proposed optimal design approach facilitates to determine an appropriate optimal design through a tradeoff analysis for designers depending on their preferences. The design concepts concerned with buckling, axial stress, combination of axial & bending, and yielding have the higher dominant effects in the optimal design of SDS. Furthermore, it is also demonstrated that the inclusion of diagonal members into the design of SDS provides a reduction in the violation of dominant design constraints.

Bilevel Programming Weight Minimization of Composite Flying-Wing Aircraft with Curvilinear Spars and Ribs

This paper studies weight minimization of a composite flying-wing aircraft using bioinspired arbitrarily shaped spars and ribs, known as SpaRibs, for the internal structural layout design. A generalized mathematical model is developed to parameterize the shape of SpaRibs using a four-node nonuniform rational basis spline curve. Additionally, for reducing weight, composite structures and lift redistribution using multiple control surfaces are considered. A stacking sequence table is used to parameterize the wing laminate configuration with ply drops for flexible wing design. For the optimization problem including topology as well as discrete and size design variables, a bilevel programming optimization is employed by using a previously developed parallel particle swarm optimization and a gradient-based optimization, respectively, for the upper- and lower-level optimization problems. The upper-level optimization problem is employed to satisfy such design requirements as the flutter constraints for achieving a specific body freedom flutter mode. The lower-level optimization is used to optimize the control surface rotations to minimize the wing root bending moment, which is considered to be a surrogate for wing weight. Different internal structural layouts and flutter constraints are studied for comparison. Optimization studies show that using SpaRibs for aircraft wing box internal structural layout design can reduce the wing weight by 21.8% as compared to the base model and by 7.3% as compared to the wing box with traditional straight spars and ribs. The additional specific flutter result constraints lead to a 4.2% increase in the wing weight for aircraft design with SpaRibs as compared to the one obtained only with the lower bound on the flutter speed.

Shape and size optimization of trusses with dynamic constraints using a metaheuristic algorithm

Metaheuristic algorithm is used to solve the weight minimization problem of truss structures considering shape, and sizing design variables. The cross-sectional areas of the line element in trusses are the design variables for size optimization and the changeable joint coordinates are the shape optimization used in this study. The design of plane and spatial truss structures are optimized by metaheuristic technique named Teaching-Learning-Based Optimization (TLBO). Finite element analyses of structures and optimization process are carried out by the computer program visually developed by the authors coded in MATLAB. The four benchmark problems (trusses 2D ten-bar, 3D thirty-seven-bar, 3D seventy-two-bar and 2D two-hundred-bar) taken from literature are optimized and the optimal solution compared the results given by previous studies.

An engineering method for complex structural optimization involving both size and topology design variables

SummaryThis work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second‐level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second‐level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.

An isogeometric multimesh design approach for size and shape optimization of multidirectional functionally graded plates

This article firstly presents a novel numerical methodology to concurrently optimize material distribution (size) and thickness variation (shape) of multidirectional functionally graded (MFG) plates under free vibration within the isogeometric analysis (IGA) framework. An isogeometric multimesh design (IMD) approach is proposed to generate two distinct non-uniform rational B-spline (NURBS) surfaces via the k-refinement strategy. A finer analysis one relied upon a combination of the IGA and a generalized shear deformation theory (GSDT) is utilized for the unknown solution approximation in finite element analyses (FEAs). Whilst the other coarser design one is employed for the exact geometry representation as well as the optimal material and thickness depiction. Size and shape design variables are in turn the ceramic volume fraction and z-axis coordinate of the top side of the MFG plate coincidentally assigned to each of control points on this surface. Flexibly utilizing such two surfaces helps diminish a large number of design variables and considerably save the computational cost in optimization problems, yet still appropriately manifesting optimal material and thickness profiles. Additionally, this definition accurately simulates mechanical behavior of MFG plates in analysis ones as well. A recently developed derivative-free adaptive hybrid evolutionary firefly algorithm (AHEFA) is used to solve constrained frequency maximization problems. Several numerical examples are executed to verify the effectiveness and robustness of the present paradigm.

Simultaneous topology, shape, and size optimization of trusses, taking account of uncertainties using multi-objective evolutionary algorithms

This paper proposes the design of trusses using simultaneous topology, shape, and size design variables and reliability optimization. Objective functions consist of structural mass and reliability, while the probability of failure is set as a design constraint. Design variables are treated to simultaneously determine structural topology, shape, and sizes. Six test problems are posed and solved by a number of multi-objective evolutionary algorithms, and it is found that Hybridized Real-Code Population-Based Incremental Learning and Differential Evolution is the best performer. This work is considered an initial study for the combination of reliability optimization and simultaneous topology, shape, and sizing optimization of trusses.