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This paper aims to present a multiscale fuzzy optimization (FO) method to optimize both the density distribution and macrotopology of a uniform octet-truss lattice structure.

The design formulae for the strut radii are presented based on the effective mechanical properties obtained from the representative volume element. The proposed basic lattice material is applied in a normalization process to determine the material model with penalization. The solid isotropic material with penalization (SIMP) method is extended to solve the minimum compliance problem using the optimality criteria. The evolutionary deletion process is proposed to delete elements corresponding to thin-strut unit cells and to obtain the optimal macrotopology.

Both numerical cases indicate that the FO results significantly improved in structural performance compared with the results of the conventional SIMP. The deleting threshold controls the macrotopology of the graded-density lattice structures with negligible effects on the mechanical properties.

This paper presents one of the first multiscale optimization methods to optimize both the relative density and macrotopology of uniform octet-truss lattices. The material model and corresponding optimality criteria of octet-truss lattices are proposed and implemented in the optimization.

In this work, the authors deal with topology optimization in electromagnetism using solid isotropic material with penalization (SIMP) method associated with a gradient-based algorithm. The purpose of this study is to propose and investigate the impact of new generalized material interpolation scheme (MIS) used in SIMP approaches.

The variable domains of this kind of electromagnetism design problem are decomposed into small squares which represent a material point (iron here) or void. A least square function where the magnetic field in a target zone has to be as close as possible to a fixed one is minimized. Then the binary optimization problem is relaxed to a continuous one. By using the adjoint variable method, the gradient is provided. By penalizing the objective function using MIS, gradient-based algorithms can then be directly applied to provide efficient solutions close to the binary ones.

In this work, new general MISs are proposed. It is shown on numerous numerical instances that the so-obtained design solutions are more precise to define the zones with or without materials.

Only the linearity of the materials is addressed because the associated adjoint method needs this assumption. However, the new penalization approaches are not dependent directly on this assumption.

The new MISs are efficiently applied to design of a hall effect thruster (HET) magnetic circuits. Furthermore, these schemes are generic and can then be applied to other topology optimization applications in electromagnetism as well as and in mechanism.

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.

The presented study aims to minimize the energy consumed by a Hall effect thruster (HET) under a constraint which makes it possible to generate a specified magnetic field in a target region of the thruster.

Herein topology optimization (TO) is used to reduce the energy consumption of an HET while keeping its performance unchanged. The design variables are the current densities in the coils and the distribution of materials in the polar pieces of the thruster. Intermediate values of material distribution are penalized using the solid isotropic material with penalization method to favor binary solutions. By means of the adjoint method, this paper provides the derivatives of the objective and constraint functions with respect to material distribution and current density variables.

The TO-based design methodology is developed and validated on a design example involving 2,051 variables. The approach shows its interest and its effectiveness of on a large scale two-criteria problem.

In this paper, TO is presented as a tool that has allowed to explore new and innovative designs. However, although the design presented is original, its fabrication is not feasible. Despite this, the designs found give a good idea of the starting points for shape and parametric optimization tools.

Through the HET design problem, TO shows the ability to explore more original design possibilities of a complex magnetostatic design problem and to discover designs that make a HET more efficient with respect to several criteria at the same time.

A new way to reduce the energy consumption of a HET is presented. To achieve this, an adjoint-based TO method is developed and then implemented in a simple way. This approach shows that, for efficiency purposes, TO is a key tool for extending the state of the art of HET designs.

Three-dimensional (3D) printing provides more possibilities for composite manufacturing. Composites can no longer just be layered or disorderly mixed as before. This paper aims to introduce a new algorithm for dual-material 3D printing design.

A novel topology design method: solid isotropic material with penalization (SIMP) for hybrid lattice structure is introduced in this paper. This algorithm extends the traditional SIMP topology optimization, transforming the original 0–1 optimization into A–B optimization. It can be used to optimize the spatial distribution of bi-material composite structures.

A novel hybrid structure with high damping and strength efficiency is studied as an example in this work. By using the topology method, a hybrid Kagome structure is designed. The 3D Kagome truss with face sheet was manufactured by selective laser melting technology, and the thermosetting polyurethane was chosen as filling material. The introduced SIMP method for hybrid lattice structures can be considered an effective way to improve lattice structures’ stiffness and vibration characteristics.

The fabricated hybrid lattice has good stiffness and damping characteristics and can be applied to aerospace components.

This is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. The authors trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods.

Starting from the global optimum of the compliance minimization problem, the authors employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum and how the initial guess has some weight in determining the final optimum.

The non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application.

In this article, the authors present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. The authors put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.

Structural topology optimization is computationally expensive due to the involvement of high-resolution mesh and repetitive use of finite element analysis (FEA) for computing the structural response. Since FEA consumes most of the computational time in each optimization iteration, a novel GPU-based parallel strategy for FEA is presented and applied to the large-scale structural topology optimization of 3D continuum structures.

A matrix-free solver based on preconditioned conjugate gradient (PCG) method is proposed to minimize the computational time associated with solution of linear system of equations in FEA. The proposed solver uses an innovative strategy to utilize only symmetric half of elemental stiffness matrices for implementation of the element-by-element matrix-free solver on GPU.

Using solid isotropic material with penalization (SIMP) method, the proposed matrix-free solver is tested over three 3D structural optimization problems that are discretized using all hexahedral structured and unstructured meshes. Results show that the proposed strategy demonstrates 3.1× –3.3× speedup for the FEA solver stage and overall speedup of 2.9× –3.3× over the standard element-by-element strategy on the GPU. Moreover, the proposed strategy requires almost 1.8× less GPU memory than the standard element-by-element strategy.

The proposed GPU-based matrix-free element-by-element solver takes a more general approach to the symmetry concept than previous works. It stores only symmetric half of the elemental matrices in memory and performs matrix-free sparse matrix-vector multiplication (SpMV) without any inter-thread communication. A customized data storage format is also proposed to store and access only symmetric half of elemental stiffness matrices for coalesced read and write operations on GPU over the unstructured mesh.

Topology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity of interest (QoI) subject to constraints. The purpose of this study is to develop a problem-agnostic automatic differentiation (AD) framework to compute sensitivities of the QoI required for density distribution-based topology optimization in an unstructured co-located cell-centered finite volume framework. Using this AD framework, the authors develop and demonstrate the topology optimization procedure for multi-dimensional steady-state heat conduction problems.

Topology optimization is performed using the well-established solid isotropic material with penalization approach. The method of moving asymptotes, a gradient-based optimization algorithm, is used to perform the optimization. The sensitivities of the QoI with respect to design variables, required for optimization algorithm, are computed using a discrete adjoint method with a novel AD library named residual automatic partial differentiator (Rapid).

Topologies that maximize or minimize relevant quantities of interest in heat conduction applications are presented. The efficacy of the technique is demonstrated using a variety of realistic heat transfer applications in both two and three dimensions, in conjugate heat transfer problems with finite conductivity ratios and in non-rectangular/non-cuboidal domains.

In contrast to most published work which has either used finite element methods or Cartesian finite volume methods for transport applications, the topology optimization procedure is developed in a general unstructured finite volume framework. This permits topology optimization for flow and heat transfer applications in complex design domains such as those encountered in industry. In addition, the Rapid library is designed to provide a problem-agnostic pathway to automatically compute all required derivatives to machine accuracy. This obviates the necessity to write new code for finding sensitivities when new physics are added or new cost functions are considered and permits general-purpose implementations of topology optimization for complex industrial applications.