Topology optimization of cellular materials with periodic microstructure under stress constraints

Material design is a critical development area for industries dealing with lightweight construction. Trying to respond to these industrial needs topology optimization has been extended from structural optimization to the design of material microstructures to improve overall structural performance. Traditional formulations based on compliance and volume control result in stiffness-oriented optimal designs. However, strength-oriented designs are crucial in engineering practice. Topology optimization with stress control has been applied mainly to (macro) structures, but here it is applied to material microstructure design. Here, in the context of density-based topology optimization, well-established techniques and analyses are used to address known difficulties of stress control in optimization problems. A convergence analysis is performed and a density filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element meshes associated with highly non-linear stress behavior. A stress-constraint relaxation technique (qp-approach) is applied to overcome the singularity phenomenon. Parallel computing is used to minimize the impact of the local nature of the stress constraints and the finite difference design sensitivities on the overall computational cost of the problem. Finally, several examples test the developed model showing its inherent difficulties.

     

Improving machine dynamics via geometry optimization

The central thesis of this paper is that the dynamic performance of machinery can be improved dramatically in certain cases through a systematic and meticulous evolutionary algorithm search through the space of all structural geometries permitted by manufacturing, cost and functional constraints. This is a cheap and elegant approach in scenarios where employing active control elements is impractical for reasons of cost and complexity. From an optimization perspective the challenge lies in the efficient, yet thorough global exploration of the multi-dimensional and multi-modal design spaces often yielded by such problems. Moreover, the designs are often defined by a mixture of continuous and discrete variables—a task that evolutionary algorithms appear to be ideally suited for. In this article we discuss the specific case of the optimization of crop spraying machinery for improved uniformity of spray deposition, subject to structural weight and manufacturing constraints. Using a mixed variable evolutionary algorithm allowed us to optimize both shape and topology. Through this process we have managed to reduce the maximum roll angle of the sprayer by an order of magnitude, whilst allowing only relatively inexpensive changes to the baseline design. Further (though less dramatic) improvements were shown to be possible when we relaxed the cost constraint. We applied the same approach to the inverse problem of reducing the mass while maintaining an acceptable roll angle—a 2% improvement proved possible in this case.     

A new look at ESO and BESO optimization methods

The “hard-kill” optimization methods such as evolutionary structural optimization (ESO) and bidirectional evolutionary structural optimization (BESO) may result in a nonoptimal design (Zhou and Rozvany in Struct Multidisc Optim 21:80–83, 2001) when these methods are implemented and used inadequately. This note further examines this important problem and shows that failure of ESO may occur when a prescribed boundary support is broken for a statically indeterminate structure. When a boundary support is broken, the structural system could be completely changed from the one originally defined in the initial design and even BESO would not be able to rectify the nonoptimal design. To avoid this problem, it is imperative that the prescribed boundary conditions for the structure be checked and maintained at each iteration during the optimization process. Several simple procedures for solving this problem are suggested. The benchmark problem proposed by Zhou and Rozvany (Struct Multidisc Optim 21:80–83, 2001) is revisited, and it is shown that the highly nonoptimal design can be easily avoided.     

An optimality criteria based method for discrete design optimization taking into account buildability constraints

This paper presents a heuristic design optimization method specifically developed for practicing structural engineers. Practical design optimization problems are often governed by buildability constraints. The majority of optimization methods that have recently been proposed for design optimization under buildability constraints are based on evolutionary computing. While these methods are generally easy to implement, they require a large number of function evaluations (finite element analyses), and they involve algorithmic parameters that require careful tuning. As a consequence, both the computation time and the engineering time are high. The discrete design optimization algorithm presented in this paper is based on the optimality criteria method for continuous optimization. It is faster than an evolutionary algorithm and it is free of tuning parameters. The algorithm is successfully applied to two classical benchmark problems (the design of a ten-bar truss and an eight-story frame) and to a practical truss design optimization problem.     

Uniform crashworthiness optimization of car body for high-speed trains

The paper deals with a new uniform crashworthiness concept of car bodies optimization of high-speed trains. The design optimization was done from the point of view of structural protection of occupants’ survival space. For the reason that it is impossible to find a highly probable scenario for the derailment, the authors decided to find the solution in the form of rigid frame structure (survival cells), which will provide safety space for the passengers. In the optimization example a typical passenger car body was divided into cells of approximately equal dimensions. The optimization problem was to minimize the mass of the structure with stress constraints. The survival cell was subjected to a sequence of high value loads. The loads are acting in an asynchronous way in three load directions what gives the optimized structure uniform crashworthiness. The optimization strategy consists of three stages. In the first step, the constant criterion surface algorithm (CCSA) of topology optimization is applied to find a preliminary solutions. For improving the manufacture properties of this solution, a new concept of design space constraints was proposed. The sizing optimization with evolutionary algorithms was used to define a thin-walled structure in the second step. For evolutionary optimization a standard procedure was employed. Finally, CCSA optimization algorithm was applied again to remove excessive material from a car body structure. As the optimization result a new design proposition of a car body with multiple survival cells of high uniform stiffness was obtained. By maintaining passengers’ survival space, the passive safety of a high-speed car body was significantly increased.     

Truss structure optimization using adaptive multi-population differential evolution

This paper applies multi-population differential evolution (MPDE) with a penalty-based, self-adaptive strategy—the adaptive multi-population differential evolution (AMPDE)—to solve truss optimization problems with design constraints. The self-adaptive strategy developed in this study is a new adaptive approach that adjusts the control parameters of MPDE by monitoring the number of infeasible solutions generated during the evolution process. Multiple different minimum weight optimization problems of the truss structure subjected to allowable stress, deflection, and kinematic stability constraints are used to demonstrate that the proposed algorithm is an efficient approach to finding the best solution for truss optimization problems. The optimum designs obtained by AMPDE are better than those found in the current literature for problems that do not violate the design constraints. We also show that self-adaptive strategy can improve the performance of MPDE in constrained truss optimization problems, especially in the case of simultaneous optimization of the size, topology, and shape of truss structures.     

Constraint-based modelling and optimization to support the design of complex multi-domain engineering problems

We present a constraint-based methodology which is successfully applied to a variety of engineering problems from a wide range of disciplines. Initially conceived from investigations of the engineering design process, the methodology has helped design engineers to identify and understand the initial limitations placed upon a system. Written as a set of algebraic expressions, the design objectives and design constraints can be formulated and minima found using numerical optimization techniques. These solutions provide initial configurations for the system, corresponding to how “true” all of the constraints are. A bespoke constraint-based modelling environment has been created which embodies the methodology. This is able to resolve large systems, comprising over 100 degrees-of-freedom, using an assortment of optimization routines—direct, gradient and evolutionary algorithms. These algorithms are appropriate for a number of problem types and their inclusion increase the scope of applicability of the methodology which is demonstrated using case studies from a number of engineering domains. Machines and mechanisms; human modelling; force and flow; structural geology and discrete disassembly processes are all studied using constraint-based formulations. The contribution of the paper lies in thus proving that complex (heterogeneous) systems-of-systems can be solved if the connectivity between the systems is expressed using constraint-rules.     

Metamorphic development: a new topology optimization method for continuum structures

An effective optimization procedure for finding structural shapes and topologies that minimize structural compliance and weight subject to stress and deflection constraints is presented. This new approach, called “Metamorphic Development” (MD), can allow a structure to grow and degenerate towards an optimum topological layout. In this method, the optimization can start from the simplest possible geometry (layout) or any degree of development of the structure rather than from a complex ground mesh. The structure is then developed metamorphically using rectangular and triangular elements that can be of any specified sizes. Examples demonstrate the potential of the MD optimization procedure to generate innovative solutions to structural design problems. Results are given and the growth and degeneration histories during optimization are illustrated. Received August 20, 1999     

Stress isolation through topology optimization

This paper introduces a problem of stress isolation in structural design and presents an approach to the problem through topology optimization. We model the stress isolation problem as a topology optimization problem with multiple stress constraints in different regions. The shape equilibrium constraint approach is employed to effectively control the local stress constraints. The level set based structural optimization is implemented with the extended finite element method (X-FEM) for providing an adequately accurate stress analysis. Numerical examples of stress isolation design in two dimensions are investigated as a benchmark test of the proposed method. The results, from the force transmittance point of view, suggest that the guard “grooves” obtained can change the force path to successfully realize the stress isolation in the structure.     

Adaptive topology optimization of elastoplastic structures

Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.     

Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition

This paper describes a phase field method for the optimization of multimaterial structural topology with a generalized Cahn–Hilliard model. Similar to the well-known simple isotropic material with penalization method, the mass concentration of each material phase is considered as design variable. However, a variational approach is taken with the Cahn–Hilliard theory to define a thermodynamic model, taking into account of the bulk energy and interface energy of the phases and the elastic strain energy of the structure. As a result, the structural optimization problem is transformed into a phase transition problem defined by a set of nonlinear parabolic partial differential equations. The generalized Cahn–Hilliard model regularizes the original ill-posed topology optimization problem and provides flexibility of topology changes with interface coalescence and break-up due to phase separation and coarsening. We employ a powerful multigrid algorithm and extend it to include four material phases for numerical solution of the Cahn–Hilliard equations. We demonstrate our approach through several 2-D and 3-D examples to minimize mean compliance of the multimaterial structures.     

A New Approach for Solving Nonlinear Equations Systems

This paper proposes a new perspective for solving systems of complex nonlinear equations by simply viewing them as a multiobjective optimization problem. Every equation in the system represents an objective function whose goal is to minimize the difference between the right and left terms of the corresponding equation. An evolutionary computation technique is applied to solve the problem obtained by transforming the system into a multiobjective optimization problem. The results obtained are compared with a very new technique that is considered as efficient and is also compared with some of the standard techniques that are used for solving nonlinear equations systems. Several well-known and difficult applications (such as interval arithmetic benchmark, kinematic application, neuropsychology application, combustion application, and chemical equilibrium application) are considered for testing the performance of the new approach. Empirical results reveal that the proposed approach is able to deal with high-dimensional equations systems.     

Transit network design by Bee Colony Optimization

The transit network design problem is one of the most significant problems faced by transit operators and city authorities in the world. This transportation planning problem belongs to the class of difficult combinatorial optimization problem, whose optimal solution is difficult to discover. The paper develops a Swarm Intelligence (SI) based model for the transit network design problem. When designing the transit network, we try to maximize the number of satisfied passengers, to minimize the total number of transfers, and to minimize the total travel time of all served passengers. Our approach to the transit network design problem is based on the Bee Colony Optimization (BCO) metaheuristics. The BCO algorithm is a stochastic, random-search technique that belongs to the class of population-based algorithms. This technique uses a similarity among the way in which bees in nature look for food, and the way in which optimization algorithms search for an optimum of a combinatorial optimization problem. The numerical experiments are performed on known benchmark problems. We clearly show that our approach, based on the BCO algorithm, is competitive with other approaches in the literature, and it can generate high-quality solutions.     

Optimization of reinforced concrete frames subjected to historical time-history loadings using DMPSO algorithm

In this paper, the seismic design of reinforced concrete (RC) frames subjected to time-history loadings was formulated as an optimization problem. Because finding the optimum design is relatively difficult and time-consuming for structural dynamics problems, an innovative algorithm combining multi-criterion decision-making (DM) and Particle Swarm Optimization (PSO), called DMPSO, was presented for accelerating convergence toward the optimum solution. The effectiveness of the proposed algorithm was illustrated in some benchmark reinforced concrete optimization problems. The main goal was to minimize the cost or weight of structures subjected to time-history loadings while satisfying all design requirements imposed by building design codes. The results confirmed the ability of the proposed algorithm to find the optimal solutions for structural optimization problems subjected to time-history loadings.     

Development of an Enhanced Ant Lion Optimization Algorithm and its Application in Antenna Array Synthesis

In order to design a highly effective communication system, antenna plays a vital role and antenna array adds to the performances. And to achieve such a goal, the crucial challenge is to determine the optimum spacing between the elements and their excitations. In order to address this issue a novel optimization technique named as enhanced ant lion optimization (e-ALO) algorithm has been developed by modifying the basic Ant lion optimization algorithm. Further, to validate the efficacy of the proposed algorithm, few benchmark functions have been successfully tested and significant improvement is obtained in comparison to other reported optimization approaches. The proposed scheme is applied to antenna array synthesis problem to optimize the inter-element spacing and excitation of the elements for different antenna geometries, with an objective to minimize the sidelobe levels while keeping other constraints within boundary limits. The encouraging results obtained from the study have emphatically placed the proposed e-ALO algorithm in the optimization arena as a dominant player.     

Comparison of evolutionary-based optimization algorithms for structural design optimization

In this paper, a comparison of evolutionary-based optimization techniques for structural design optimization problems is presented. Furthermore, a hybrid optimization technique based on differential evolution algorithm is introduced for structural design optimization problems. In order to evaluate the proposed optimization approach a welded beam design problem taken from the literature is solved. The proposed approach is applied to a welded beam design problem and the optimal design of a vehicle component to illustrate how the present approach can be applied for solving structural design optimization problems. A comparative study of six population-based optimization algorithms for optimal design of the structures is presented. The volume reduction of the vehicle component is 28.4% using the proposed hybrid approach. The results show that the proposed approach gives better solutions compared to genetic algorithm, particle swarm, immune algorithm, artificial bee colony algorithm and differential evolution algorithm that are representative of the state-of-the-art in the evolutionary optimization literature.     

Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator

This paper deals with the dynamic modeling and design optimization of a three Degree-of-Freedom spherical parallel manipulator. Using the method of Lagrange multipliers, the equations of motion of the manipulator are derived by considering its motion characteristics, namely, all the components rotating about the center of rotation. Using the derived dynamic model, a multiobjective optimization problem is formulated to optimize the structural and geometric parameters of the spherical parallel manipulator. The proposed approach is illustrated with the design optimization of an unlimited-roll spherical parallel manipulator with a main objective to minimize the mechanism mass in order to enhance both kinematic and dynamic performances.     

School-based optimization for performance-based optimum seismic design of steel frames

The performance-based optimum seismic design of steel frames is one of the most complicated and computationally demanding structural optimization problems. Metaheuristic optimization methods have been successfully used for solving engineering design problems over the last three decades. A very recently developed metaheuristic method called school-based optimization (SBO) will be utilized in the performance-based optimum seismic design of steel frames for the first time in this study. The SBO actually is an improved/enhanced version of teaching–learning-based optimization (TLBO), which mimics the teaching and learning process in a class where learners interact with the teacher and between themselves. Ad hoc strategies are adopted in order to minimize the computational cost of SBO results. The objective of the optimization problem is to minimize the weight of steel frames under interstory drift and strength constraints. Three steel frames previously designed by different metaheuristic methods including particle swarm optimization, improved quantum particle swarm optimization, firefly and modified firefly algorithms, teaching–learning-based optimization, and JAYA algorithm are used as benchmark optimization examples to verify the efficiency and robustness of the present SBO algorithm. Optimization results are compared with those of other state-of-the-art metaheuristic algorithms in terms of minimum structural weight, convergence speed, and several statistical parameters. Remarkably, in all test problems, SBO finds lighter designs with less computational effort than the TLBO and other methods available in metaheuristic optimization literature.

     

Shape optimization for 2-D mixed-mode fracture using Extended FEM (XFEM) and Level Set Method (LSM)

Weight and service life are often the two most important considerations in design of structural components. This research incorporates a novel crack propagation analysis technique into shape optimization framework to support design of 2-D structural components under mixed-mode fracture for: (1) maximum service life, subject to an upper limit on volume, and (2) minimum weight subject to specified minimum service life. In both cases, structural performance measures are selected as constraints and CAD dimensions are employed as shape design variables. Fracture parameters, such as crack growth rate and crack growth direction are computed using extended finite element method (XFEM) and level set method (LSM). XFEM employs special enrichment functions to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM utilizes level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor re-mesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. An accurate and efficient semi-analytical design sensitivity analysis (DSA) method is developed for calculating gradients of fracture parameters. Two different approaches—a batch-mode, gradient-based, nonlinear algorithm and an interactive what-if analysis—are used for optimization. An engine connecting rod example is used to demonstrate the feasibility of the proposed method.     

Performance enhancement of multiband antennas through a two‐stage optimization technique

This paper presents an optimization technique that aims at multiband antenna design. The proposed method is based on the framework of multi‐objective evolutionary algorithms, and a two‐stage mechanism that balances the degree of optimizing impedance matching and the degree of providing a wide impedance bandwidth is incorporated. Conventionally, the design optimization of multiband antennas relies on minimizing the maximum reflection coefficient or maximizing the area under the return‐loss curve over targeted frequency bands. However, these widely used methods direct an optimization algorithm to improper solution sub‐domains in the multiband design problem. To overcome the limitation of these conventional methods, the general rule of objective functions is thoroughly investigated in this paper. Furthermore, a two‐stage optimizer is designed based on what the multiband optimization problem needs. With the use of the proposed method, two multiband antennas for mobile communication systems covering 824–960 MHz and 1710–2170 MHz are successfully developed. Simulated and measured results show that the proposed technique outperforms conventional optimization approaches significantly.