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Usually, an optimal topology is obtained by optimizing the material distribution within a prescribed domain; for example, a rectangular domain with a specified length and width for a plane problem. However, the dimensions (i.e. aspect ratio) of a rectangular design domain have significant influence on the resultant optimal topology. In this paper, a minimum Averaged Compliance Density (ACD) based method for topology optimization of structures is proposed. Unlike the conventional topology optimization method, the ACD is taken as the objective function, and the topology and domain dimensions of the structure are optimized simultaneously. As an example, the topology of a cantilever beam with large aspect ratio will be optimized, which is often difficult for traditional topology optimization algorithms. Through optimizing the topology and the dimensions of the design domain, a base structure is obtained, which is repeated to yield the whole, assembled beam. The influence of the relative values of shear force and moment is analyzed numerically. Results show that as the value of the bending moment increases relative to the shear force, the optimal topology changes from a truss‐like structure to a vertically stiffened box‐like structure.

This research presents a design method for designing greenhouse structures based on topology optimization. Moreover, the structural design of a gothic greenhouse is proposed in which its structural strength has been improved by using this proposed method. In this method, the design of the structure is done mathematically; therefore, in the design process, more attention can be focused on the constraint space and boundary conditions. It was also shown how the static reliability and fatigue coefficients will change as a result of the design of the greenhouse structure with this method. Another purpose of this study is to find the weakest part of the greenhouse structure against lateral winds and other general loads on the greenhouse structure.

In the proposed method, the outer surface and the allowable volume as a constraint domain were considered. The desired loads can be located on the constraint domain. The topology optimization was used to minimize the mass and structural compliance as the objective function. The obtained volume was modified for simplifying the construction. The changes in the shape of the greenhouse structure were investigated by choosing three different penalty numbers for the topology optimization algorithm. The final design of the proposed structure was performed based on the total simultaneous critical loads on the structure. The results of the proposed method were compared in the order of different volume fractions. This showed that the volume fraction approach can significantly reduce the weight of the structure while maintaining its strength and stability.

Topology optimization results showed different strut and chords composition because of the changes in maximum mass limit and volume fraction. The results showed that the fatigue was more hazardous, and it decreased the strength of structure nearly three times more than a static analysis. Further, it was noticed that how the penalty numbers can affect topology optimization results. An optimal design based on topology optimization results was presented to improve the proposed greenhouse design against destruction and demolition. Furthermore, this study shows the most sensitive part of the greenhouse against the standard loads of wind, snow, and crop.

The obtained designs were compared with a conventional arch greenhouse, and then the structural performances were shown based on standard loads. The results showed that in designing the proposed structure, the optimized changes increased the structure strength against the standard loads compared to a simple arch greenhouse. Moreover, the stress safety factor and fatigue safety factor because of different designs of this structure were also compared with each other.

The purpose of this paper is to develop a topology optimization algorithm considering natural frequencies.

To incorporate natural frequency as design criteria of shell-infill structures, two types of design models are formulated: (1) type I model: frequency objective with mass constraint; (2) type II model: mass objective with frequency constraint. The interpolation functions are constructed by the two-step density filtering approach to describe the fundamental topology of shell-infill structure. Sensitivities of natural frequencies and mass with respect to the original element densities are derived, which will be used for both type I model and type II model. The method of moving asymptotes is used to solve both models in combination with derived sensitivities.

Mode switching is one of the challenges faced in eigenfrequency optimization problems, which can be overcome by the modal-assurance-criterion-based mode-tracking strategy. Furthermore, a shifting-frequency-constraint strategy is recommended for type II model to deal with the unsatisfactory topology obtained under direct frequency constraint. Numerical examples are systematically investigated to demonstrate the effectiveness of the proposed method.

In this paper, a topology optimization method considering natural frequencies is proposed by the author, which is useful for the design of shell-infill structures to avoid the occurrence of resonance in dynamic conditions.

The structural design problem can be viewed as an iterative design loop with each iteration involving two stages for topology and shape designs with genetic algorithm (GA) as the optimization tool for both.

The topology optimization problem, which is ill posed, is regularized using a constraint on perimeter and solved using GA. The problem is formulated as one of compliance minimization subject to volume constraint for the single loading case. A dual formulation of this has been used for the multiple loading cases resulting in as many behavioral constraints as there are loading cases. The tentative topology given by the topology optimization module is taken and the domain boundary is approximated using straight lines, B‐splines and cubic spline curves and design variables are selected among the boundary defining points. Optimum boundary shape of the problem has been obtained using GA in two different ways: without stress constraints; and with stress constraints.

The proposed two stage strategy has been tested on benchmark structural optimization problems and its performance is found to be extremely good.

The strategy appears to be eminently suitable for implementation in a general purpose FE software as an add‐on module for structural design optimization.

It has been observed that the integrated topology and shape design method is robust and easy to implement in comparison with other techniques. The computing time requirements for the GA does not appear daunting in the present scenario of high performance parallel computing and improved GA techniques.

This paper proposes a bar‐system graph representation for structural topology optimization using a genetic algorithm (GA).

Based on graph theory, a graph is first used to represent a skeletal structure consisting of joining paths in the design domain, each of which is represented by a chain subgraph with finite number of vertices. Based on the edges of this graph, a bar‐system representation is proposed to define all the bars and the resulting topology is obtained by mapping each bar with its corresponding thickness to the design domain which is discretized into a regular mesh. The design variables are thus reduced to the spatial distribution of the vertices and the thickness of each bar. This method combines the advantages of both continuum and ground structure optimization methods.

The overall procedure is applied to classical structural topology optimization problems and its good performance is illustrated in the numerical examples.

It is suggested that the present representation method is both physically meaningful and computationally effective in the framework of topological optimum design using GAs.

Application of the engineering design optimization methods and tools to the design of automotive body‐in‐white (BIW) structural components made of polymer metal hybrid (PMH) materials is considered. Specifically, the use of topology optimization in identifying the optimal initial designs and the use of size and shape optimization techniques in defining the final designs is discussed. The optimization analyses employed were required to account for the fact that the BIW structural PMH component in question may be subjected to different in‐service loads be designed for stiffness, strength or buckling resistance and that it must be manufacturable using conventional injection over‐molding. The paper demonstrates the use of various engineering tools, i.e. a CAD program to create the solid model of the PMH component, a meshing program to ensure mesh matching across the polymer/metal interfaces, a linear‐static analysis based topology optimization tool to generate an initial design, a nonlinear statics‐based size and shape optimization program to obtained the final design and a mold‐filling simulation tool to validate manufacturability of the PMH component.

Considering the randomness of physical parameters of structural material, dynamic characteristic topology optimization mathematical model based on reliability of planar continuum structures is built in this paper. In which topology information variables of the structure are taken as design variables, minimizing the mean value of total structural weight as objective function and satisfying the reliability requirement of structural dynamic characteristic as constraints. In the process of optimization, the ESO method based on probability is adopted as solution strategy. At the same time, distribution function method is utilized to convert the reliability constraints into conventional constraints formally. A square thin plate with four sides fixed is used as an example to demonstrate the rationality and validity of the presented model.

The purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.

The new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.

Two-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.

This paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

The purpose of this paper is to develop a heuristic method for topology optimization of a continuum with bi-modulus material which is frequently occurred in practical engineering.

The essentials of this model are as follows: First, the original bi-modulus is replaced with two isotropic materials to simplify structural analysis. Second, the stress filed is adopted to calculate the effective strain energy densities (SED) of elements. Third, a floating reference interval of SED is defined and updated by active constraint. Fourth, the elastic modulus of an element is updated according to its principal stresses. Final, the design variables are updated by comparing the local effective SEDs and the current reference interval of SED.

Numerical examples show that the ratio between the tension modulus and the compression modulus of the bi-modulus material in a structure has a significant effect on the final topology design, which is different from that in the same structure with isotropic material. In the optimal structure, it can be found that the material points with the higher modulus are reserved as much as possible. When the ratio is far more than unity, the material can be considered as tension-only material. If the ratio is far less than unity, the material can be considered as compression-only material. As a result, the topology optimization of continuum structures with tension-only or compression-only materials can also be solved by the proposed method.

The value of this paper is twofold: the bi-modulus material layout optimization in a continuum can be solved by the method proposed in this paper, and the layout difference between the structure with bi-modulus material and the same structure but with isotropic material shows that traditional topology optimization result could not be suitable for a real bi-modulus layout design project.

The purpose of this paper is to explore the possibility of combining additive manufacturing (AM) with topology optimization to generate support structures for addressing the challenging overhang problem. The overhang problem is considered as a constraint, and a novel algorithm based on continuum topology optimization is proposed.

A mathematical model is formulated, and the overhang constraint is embedded implicitly through a Heaviside function projection. The algorithm is based on the Solid Isotropic Material Penalization (SIMP) method, and the optimization problem is solved through sensitivity analysis.

The overhang problem of the support structures is fixed. The optimal topology of the support structures is developed from a mechanical perspective and remains stable as the material volume of support structures changes, which allows engineers to adjust the material volume to save cost and printing time and meanwhile ensure sufficient stiffness of the support structures. Three types of load conditions for practical application are considered. By discussing the uniform distributive load condition, a compromise result is achieved. By discussing the point load condition, the removal work of support structures after printing is alleviated. By discussing the most unfavorable load condition, the worst collapse situation of the printing model during printing process is sufficiently considered. Numerical examples show feasibility and effectiveness of the algorithm.

The proposed algorithm involves time-consuming finite element analysis and iterative solution, which increase the computation burden. Only the overhang constraint and the minimum compliance problem are discussed, while other constraints and objective functions may be of interest.

Compared with most of the existing heuristic or geometry-based support-generating algorithms, the proposed algorithm develops support structures for AM from a mechanical perspective, which is necessary for support structures particularly used in AM for mega-scale construction such as architectures and sculptures to ensure printing success and accuracy of the printed model.

With the rapid development of AM, complicated structures result from topology optimization are available for fabrication. The present paper demonstrates a combination of AM and topology optimization, which is the trend of fabricating manner in the future.

This paper remarks the first of attempts to use continuum topology optimization method to generate support structures for AM. The methodology used in this work is theoretically meaningful and conclusions drawn in this paper can be of important instruction value and practical significance.