Discrete Truss Research Articles

Sizing and Shape Optimization of Discrete Truss Employing a Target-Oriented Krill Herd Algorithm

Abstract The krill herd (KH) algorithm is widely used for optimizing truss structures as no gradient information is necessary, and only a few parameters require adjustment. However, when the truss structure becomes discrete and complex, KH tends to fall into a local optimum. Therefore, a novel target-oriented KH (TOKH) algorithm is proposed in this study to optimize the design of discrete truss structures. Initially, a crossover operator is established between the “best krill” and “suboptimal krill” to generate a robust “cross krill” for global exploration. Additionally, an improved local mutation and crossover (ILMC) operator is introduced to fine-tune the “center of food” and candidate solutions for local exploitation. The proposed method and other optimization approaches are experimentally compared considering 15 benchmark functions. Then, the performance of the TOKH algorithm is evaluated based on four discrete truss structure optimization problems under multiple loading conditions. The obtained optimization results indicate that the proposed method presents competitive solutions in terms of accuracy, unlike other algorithms in the literature, and avoids falling into a local minimum.

Weight Optimization of Discrete Truss Structures Using Quantum-Based HS Algorithm

Recently, a new field that combines metaheuristic algorithms and quantum computing has been created and is being applied to optimization problems in various fields. However, the application of quantum computing-based metaheuristic algorithms to the optimization of structural engineering is insufficient. Therefore, in this paper, we tried to optimize the weight of the truss structure using the QbHS (quantum-based harmony search) algorithm, which combines quantum computing and conventional HS (harmony search) algorithms. First, the convergence performance according to the parameter change of the QbHS algorithm was compared. The parameters selected for the comparison of convergence performance are QHMS, QHMCR, QPAR, ϵ, and θr. The selected parameters were compared using six benchmark functions, and the range for deriving the optimal convergence performance was found. In addition, weight optimization was performed by applying it to a truss structure with a discrete cross-sectional area. The QbHS algorithm derived a lower weight than the QEA (quantum-inspired evolutionary algorithm) and confirmed that the convergence performance was better. A new algorithm that combines quantum computing and metaheuristic algorithms is required for application to various engineering problems, and this effort is essential for the expansion of future algorithm development.

A Symbolic Approach to Discrete Structural Optimization Using Quantum Annealing

With the advent of novel quantum computing technologies and the new possibilities thereby offered, a prime opportunity has presented itself to investigate the practical application of quantum computing. This work investigates the feasibility of using quantum annealing for structural optimization. The target problem is the discrete truss sizing problem—the goal is to select the best size for each truss member so as to minimize a stress-based objective function. To make the problem compatible with quantum annealing devices, the objective function must be translated into a quadratic unconstrained binary optimization (QUBO) form. This work focuses on exploring the feasibility of making this translation. The practicality of using a quantum annealer for such optimization problems is also assessed. A method is eventually established to translate the objective function into a QUBO form and have it solved by a quantum annealer. However, scaling the method to larger problems faces some challenges that would require further research to address.

Implementation of Modified Compression Field Theory to Simulate the Behavior of Fiber-Reinforced Polymer Shear-Strengthened Reinforced Concrete Beams under Monotonic Loading

The numerical modeling of structures is a widely preferable approach to investigate the structural behavior of RC beams since it delivers inexpensive predictions for confirming the required goals concurrently with reducing casting, testing time, and effort. Shear-strengthening of reinforced concrete (RC) beams using externally bonded (EB) fiber-reinforced polymers (FRPs) has attracted much attention due to the fact that the FRP strengthening technique has the ability to alter the distribution of stresses between the structural elements and increase the load-carrying capacity. A significant number of experimental studies have been carried out to test the monotonic behavior of FRP shear-strengthened RC beams. Conversely, limited numerical research has been performed to investigate such performance. The VecTor2 software is developed based on the modified compression field theory (MCFT) and is directed to examine the monotonic behavior of retrofitted specimens using fiber-reinforced polymer (FRP) composites. To the authors’ knowledge, the behavior of FRP shear-strengthened beams has not been explored in the literature using the MCFT modeling approach. The main objective of this study is to detect the software’s capability of predicting the experimental outcomes of FRP shear-strengthened RC beams. This research study is carried out in two stages. Initially, the numerical study involves the development of an accurate finite element model to simulate the control specimens. The quality of this model is assessed by comparing the numerical results with the experimental outcomes. In the second phase of the numerical study, the control beam model is modified to accommodate the presence of external FRP composites. The accuracy of this model is again measured by comparing its predictions with the experimental measurements. The goal of these phases is to ensure that the numerical model captures the actual behavior of the tested beams. Additionally, two distinctive modeling approaches are investigated to represent the behavior of FRP composites. The accuracy of the numerical models is verified through comparisons of numerical predictions to experimental results in terms of ultimate loading capacity, load–deflection relationships, and failure modes. It can be stated that the validated numerical model provides alternate means for evaluating the monotonic behavior of both strengthened and non-strengthened RC beams. The predicted results compare very well with the test results of the control specimens when discrete truss elements are employed for the FRP composites. Furthermore, the numerical model provides useful information on the crack patterns and failure modes.

Chaotic enhanced teaching-based differential evolution algorithm applied to discrete truss optimization

In this paper, we develop a discrete chaotic enhanced hybrid teaching-based differential evolution algorithm (CETDE) that can deal with realistic yet challenging constrained discrete truss design problems more efficiently. CETDE adaptively executes multiple mutation operators to quickly search for the solution, but at the same time, premature convergence, a common issue in discrete optimization, is also prevented by embedding the chaotic logistic rule into the mutations. An improved logical strategy that only allows potential mutants to go through structural analysis is introduced to reduce computation. An enhanced chaotic local search strategy and a one-component change technique are also developed for further efficiency. Besides, CETDE simultaneously keeps continuous and discrete population throughout the evolution, which is distinguishable from existing discrete optimizers, to preserve the stochastic nature. Compared to other state-of-the-art performers in the literature, CETDE illustrates both higher efficiency and solution optimality.

Topology Optimization of Graded Truss Lattices Based on On-the-Fly Homogenization

Abstract We introduce a computational framework for the topology optimization of cellular structures with spatially varying architecture, which is applied to functionally graded truss lattices under quasistatic loading. We make use of a first-order homogenization approach, which replaces the discrete truss by an effective continuum description to be treated by finite elements in a macroscale boundary value problem. By defining the local truss architecture through a set of Bravais vectors, we formulate the optimization problem with regards to the spatially varying basis vectors and demonstrate its feasibility and performance through a series of benchmark problems in 2D (though the method is sufficiently general to also apply in 3D, as discussed). Both the displacement field and the topology are continuously varying unknown fields on the macroscale, and a regularization is included for well posedness. We argue that prior solutions obtained from aligning trusses along the directions of principal stresses are included as a special case. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).

Development of a coupled numerical framework of steel fiber reinforced self-compacting concrete

A coupled numerical framework comprised of a computational fluid dynamics (CFD) based method and a finite element method is presented in this paper to model the steel fiber reinforced self-compacting concrete (SFRSCC). The computational fluid dynamics based method is applied to simulate the casting process of fresh SFRSCC to determine the coordinates of the steel fibers, which are used as input to the finite element model of hardened SFRSCC specimens. In the finite element model, SFRSCC is assumed to be a heterogeneous medium consisting of a homogeneous phase (concrete matrix) and a discrete phase (steel fibers). The nonlinear behavior of the concrete matrix is modelled by the continuum damage plasticity model, while the steel fibers are explicitly modeled as discrete truss elements. In order to take into account the fiber pull-out response, the equivalent stress–strain laws derived from an analytical load-slip curve are assigned to the steel fibers depending on their orientation angles. To verify the numerical framework, the casting process and four-point bending tests of the SFRSCC beam specimens were numerically simulated. The good agreement with the simulated and experimental results proves the feasibility of the proposed numerical framework.

On transmissible load formulations in topology optimization

Transmissible loads are external loads defined by their line of action, with actual points of load application chosen as part of the topology optimization process. Although for problems where the optimal structure is a funicular, transmissible loads can be viewed as surface loads, in other cases such loads are free to be applied to internal parts of the structure. There are two main transmissible load formulations described in the literature: a rigid bar (constrained displacement) formulation or, less commonly, a migrating load (equilibrium) formulation. Here, we employ a simple Mohr’s circle analysis to show that the rigid bar formulation will only produce correct structural forms in certain specific circumstances. Numerical examples are used to demonstrate (and explain) the incorrect topologies produced when the rigid bar formulation is applied in other situations. A new analytical solution is also presented for a uniformly loaded cantilever structure. Finally, we invoke duality principles to elucidate the source of the discrepancy between the two formulations, considering both discrete truss and continuum topology optimization formulations.

Discrete Size and Shape Optimization of Truss Structures Based on Job Search Inspired Strategy and Genetic Operations

A meta-heuristic algorithm for discrete size and shape optimization of trusses via a job search inspired strategy together with genetic operators of mutation, selection, and crossover is proposed. The alternation of movements with respect to objective function and load bearing capacity of constructive decisions is provided. Being introduced is an intermediate search goal connected in terms of posed limitations with heightened suitability levels of individuals meeting the current requirements for the initial objective function. As soon as these conditions allow achieving a structure type which meets task limitations, requirements for the function value are redefined. This technique does not demand penalty functions that provide strict control of limitations in any algorithm usage, greater stability of the results received, and finding better solutions. The efficiency of this approach in terms of solution accuracy is demonstrated through five benchmark design examples, in comparison with other methods of discrete truss structure optimization.

A new Newton metaheuristic algorithm for discrete performance-based design optimization of steel moment frames

In this paper a new and efficient metaheuristic algorithm is proposed for discrete performance-based seismic design optimization of steel moment frames. The proposed metaheuristic uses the Newton gradient-based method as its updating scheme in a population-based framework and therefore it is termed as Newton Metaheuristic Algorithm (NMA). In order to enable the NMA to effectively explore the discrete design space, a term containing the best solution found is added to the basic updating rule of the algorithm. In addition, a simple and efficient method is proposed in order to establish a balance between local and global search abilities of the proposed algorithm. The efficiency of the NMA is illustrated by presenting two benchmark discrete truss optimization problems. Moreover, three steel moment frames are optimized in the framework of performance-based design by the NMA and the results are compared with those of some recent metaheuristics. The performance of the algorithms is analyzed using statistical parametric and non-parametric tests indicating that NMA outperforms the other algorithms in literature.

Continuum models for stretching- and bending-dominated periodic trusses undergoing finite deformations

Advances in additive manufacturing across scales have enabled the creation of random, periodic, or hierarchical truss networks containing millions and more of individual truss members. In order to significantly reduce computational costs while accurately capturing the dominant deformation mechanisms, we introduce a simple yet powerful homogenized continuum description of truss lattices, which is based on applying a multi-lattice Cauchy–Born rule to a representative unit cell (RUC). Beam theory applied at the level of the RUC introduces rotational degrees of freedom and leads to a generalized continuum model that depends on the effective deformation gradients, rotation, and curvature on the macroscale. While affinely deforming the RUC is shown to produce excellent results for simple Bravais lattices, a multi-lattice extension is required for general and especially bending-dominated lattices, which cannot be described by a pure Taylor expansion of the RUC deformation; the importance of the multi-lattice concept is demonstrated through analytical examples. The resulting method is a beneficial compromise between inefficient FE2 techniques and micropolar theories with limited applicability. By implementing the model within a finite element framework, we solve and report several benchmark tests in 2D to illustrate the accuracy and efficiency of the model, which comes with only a small fraction of the computational costs associated with the full, discrete truss calculation. By using a corotational beam description, we also capture finite beam rotations. We further demonstrate that a second-gradient homogenization formulation is beneficial in examples involving localization, providing higher local accuracy at the RUC level than a first-gradient scheme, while affecting the global response only marginally.

A quasicontinuum theory for the nonlinear mechanical response of general periodic truss lattices

We present a framework for the efficient, yet accurate description of general periodic truss networks based on concepts of the quasicontinuum (QC) method. Previous research in coarse-grained truss models has focused either on simple bar trusses or on two-dimensional beam lattices undergoing small deformations. Here, we extend the truss QC methodology to nonlinear deformations, general periodic beam lattices, and three dimensions. We introduce geometric nonlinearity into the model by using a corotational beam description at the level of individual truss members. Coarse-graining is achieved by the introduction of representative unit cells and an affine interpolation analogous to traditional QC. General periodic lattices defined by the periodic assembly of a single unit cell are modeled by retaining all unique degrees of freedom of the unit cell (identified by a lattice decomposition into simple Bravais lattices) at each macroscopic point in the simulation, and interpolating each degree of freedom individually. We show that this interpolation scheme accurately captures the homogenized properties of periodic truss lattices for uniform deformations. In order to showcase the efficiency and accuracy of the method, we perform simulations to predict the brittle fracture toughness of multiple lattice architectures and compare them to results obtained from significantly more expensive discrete truss simulations. Finally, we demonstrate the applicability of the method for nonlinear elastic truss lattices undergoing finite deformations. Overall, the new technique shows convincing agreement with exact, discrete results for most lattice architectures, and offers opportunities to reduce computational expenses in structural lattice simulations and thus to efficiently extract the effective mechanical performance of discrete networks.

A novel approach to discrete truss design problems using mixed integer neighborhood search

Discrete truss sizing problems are very challenging to solve due to their combinatorial, nonlinear, non-convex nature. Consequently, truss sizing problems become unsolvable as the size of the truss grows. To address this issue, we consider various mathematical formulations for the truss design problem with the objective of minimizing weight, while the cross-sectional areas of the bars take only discrete values. Euler buckling constraints, Hooke’s law, and bounds for stress and displacements are also considered. We propose mixed integer linear optimization (MILO) reformulations of the non-convex mixed integer models. The resulting MILO models are not solvable with existing MILO solvers as the size of the problem grows. Our novel methodology provides high-quality solutions for large-scale real truss sizing problems, as demonstrated through extensive numerical experiments.

A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures

A new optimization method called Electromagnetism-like Firefly Algorithm (EFA), which is a novel hybrid between the Electromagnetism-like Algorithm (EM) and the Firefly Algorithm (FA) for discrete structural optimization is proposed. The EFA inherits the advantages of both the FA and the EM. This proposed optimization algorithm is then presented to improve both solution accuracy and convergence speed, as well as to treat constrained optimization problems with discrete design variables. In EFA, modified formulas of interactive forces are used to increase the diversification of the population, and the constraint violations are embedded into the charges of all electromagnetic fireflies to avoid becoming trapped in unfeasible domains. A mechanism called “current-to-best” electromagnetic movement is incorporated with traditional interactive movements to balance the exploration and the exploitation abilities of the EFA. In the local search phase, a newly bi-directional searching procedure is performed on the best firefly to intensify effectively its local optimum. In order to guarantee the convergence capability of the EFA, a harmonized selection mechanism combined with the traditional and elitist selections is proposed and applied if the algorithm cannot find a better optimal solution during a number of predetermined optimization loops. In addition, in this study, the FA and EM with some improvements in several phases are also proposed to solve the constrained optimization problems. Finally, a rounding technique is applied to the proposed EFA, FA and EM for solving discrete optimization problems. The improved performance of the EFA in comparison with the FA, EM as well as other optimization algorithms in the literature is demonstrated by six popular truss optimization problems with discrete variables.

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.

Layout and sizing optimization of discrete truss based on continuum

A new method combined continuum topology with discrete truss is proposed for truss optimization. The optimized material distribution is obtained by means of continuum topology optimization and then transformed into initial truss. Based on the element principal stress and 8-neighborhood elements of continuum, force flow path extraction method is presented to obtain the initial truss layout. For section size optimization, criterion is deduced from a mathematical model by Lagrange multiplier method. Leveraging the free-form nature of continuum, this method can avoid the restriction of traditional ground structure. And it can achieve a better solution with less mass, smaller deformation and more uniform stress.

Local and nonlocal continuum modeling of inelastic periodic networks applied to stretching-dominated trusses

We present a nonlocal continuum model and its numerical implementation to describe the macroscale response of periodic discrete networks via second-order homogenization. The scale-bridging technique is applied to the specific example of stretching-dominated elastic and inelastic periodic truss networks. Experiments on small-scale truss structures have highlighted the importance of nodal connections on the effective stiffness and strength. Therefore, we describe the mechanics of trusses by accounting for the stretching of truss members and the deformation of nodes. For the representative 2D examples of lattices having square and triangular architectures and for example bar and nodal constitutive laws, we show that a simple continuum model based on affinely deforming a representative unit cell is sufficient to reproduce the nonlinear elastic behavior of discrete trusses. By contrast, localization that arises, e.g., from inelastic deformation requires a refined model. This is where the presented nonlocal continuum model is capable of accurately capturing details of localized deformation. We illustrate the performance of the model by comparing the results of example finite element simulations using the continuum constitutive model to discrete lattice calculations with elastic–plastic bars. Optimal performance is achieved when the representative unit cell of the continuum model agrees with the actual size of the discrete truss unit cell, which accounts for size effects even in regimes where a separation of scales between finite element size and unit cell size does not strictly apply.

Discrete Planar Truss Optimization by Node Position Variation Using Grammatical Evolution

The majority of existing discrete truss optimization methods focus primarily on optimizing global truss topology using a ground structure approach, in which all possible node and beam locations are specified a priori . The ground structure discrete optimization method has been shown to be restrictive as it limits derivable solutions to what is explicitly defined. Greater representational freedom can improve performance. In this paper, grammatical evolution is applied. It can represent a variable number of nodes and their locations on a continuum. A novel method of connecting evolved nodes using a Delaunay triangulation algorithm shows that fully triangulated, kinematically stable structures can be generated. Discrete beam-truss structures can be optimized without the need for any information about the desired form of the solution other than the design envelope. Our technique is compared to existing discrete optimization techniques, and notable savings in structure self-weight are demonstrated. In particular, our new method can produce results superior to those reported in the literature in cases in which the problem is ill-defined and the structure of the solution is not known a priori .

A Hybrid Harmony Search algorithm for discrete sizing optimization of truss structure

This paper presents a new variant of the Harmony Search (HS) algorithm. This Hybrid Harmony Search (HHS) algorithm follows a new approach to improvisation: while retaining HS algorithm Harmony Memory and pitch adjustment functions, it replaces the HS algorithm randomization function with Global-best Particle Swarm Optimization (PSO) search and neighbourhood search. HHS algorithm performance is tested on six discrete truss structure optimization problems under multiple loading conditions. Optimization results demonstrate the excellent performance of the HHS algorithm in terms of both optimum solution and the convergence behaviour in comparison with various alternative optimization methods.

Wave propagation in equivalent continuums representing truss lattice materials

Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation methods. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.