Topology optimization of multiphase material structures under design dependent pressure loads

When a structure under design-dependent pressure loads is being topologically optimized, the magnitude and/or direction of these loads may change as the structure’s topology is being updated simultaneously. Therefore, two critical aspects in the optimization to be addressed are the identification of changing topological boundaries and the correct application of pressure loads to newly formed pressure surfaces. In this study, we present a simple boundary identification and load evolution (BILE) model to resolve these aspects. As the name suggests, a boundary identification step utilizes a threshold volume fraction to define topological boundaries and a load evolution step that occurs for a number of iterations between two successive boundary identification iterations. This model is particularly attractive because of its ease of application and computational cost-effectiveness as most numerical results for 2D problems using 5000 to 13,000 isoparametric quad elements were obtained under 90 s and 100 iterations using the optimality criteria method (OCM). Based on the modified SIMP method, this model is compared with existing methodologies by presenting several numerical examples for validation.

In this paper, topology optimization (TO) problems of structures subjected to design-dependent loads are formulated in the context of a special density-based TO approach wherein a Non Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor. Unlike classical density-based TO approaches, the NURBS-density-based method allows representing the pseudo-density field through a purely geometric computer-aided design compatible entity. In this context, the problem is formulated in the most general case of inhomogeneous Neumann-Dirichlet boundary conditions. Moreover, a thorough study of the penalty function of the design-dependent loads is carried out to investigate its effect on the optimized topologies and overcome the singularity effect related to the zones characterized by low values of the pseudo-density field. Finally, a wide campaign of sensitivity analyses is conducted to investigate the influence of the integer parameters of the NURBS entity, of the combination of design-depedent loads and inhomogeneous Neumann-Dirichlet boundary conditions, and of the concentrated load on the optimized topology. The effectiveness of the approach is tested on 2D and 3D benchmark problems.

A new fuzzy strategy for size and topology optimization of truss structures

Robust topology optimization of multi-material structures under load uncertainty using the alternating active-phase method

The center of mass and the inertia tensor are important parameters characterizing the kinetic properties of many structural systems, e.g., the rotor, spacecraft, and aircraft. In these occasions, designs not consistent with the requirements on the center of mass and inertia tensor cannot be accepted. In this paper, mathematical models are established to carry out the topology optimization of continua considering the mass and inertia characteristics. Two numerical examples, one for minimizing the static compliance considering locations of the center of mass and the other for maximizing the fundamental eigenfrequency considering both locations of the center of mass and values of the inertia products, are presented to demonstrate that the requirements on mass and inertia characteristics could have a profound influence on the final optimized layouts in structural topology optimization problems. On the other hand, the mass and inertia characteristics can also be tuned by topology optimization methods. In the second numerical problem, the smooth minimum operator is used to approximate the fundamental eigenfrequency with the first 4 eigenfrequencies so that the possible non-differentiability of the repeated eigenfrequencies can be avoided. Due to its extreme importance in engineering practice, the work of this paper can be useful to carry out designs in accordance with the requirements on the mass and inertia characteristics under the framework of traditional density-based structural topology optimization methods.

Concurrent topology and anisotropy optimisation of variable-stiffness composite structures under design-dependent loads

Topology optimization of laminated composite structures with design-dependent loads

Isogeometric analysis (IGA) connects CAE with standard CAD design tools and has been utilized for a decade in structural optimization. FEA has a number of drawbacks, including a lack of linkage between analytic and geometric models, a shortage of continuity between components, and a lack of accuracy for high-order elements. Isogeometric analysis effectively overcomes these obstacles. This paper proposes the isogeometric topological optimization mathematical model depending on the asymmetric constitutive relationship of tension and compression microscopic materials. We provide a new density-based topology optimization formulation in which the design domain is limited to the B-spline space. A rectangular domain with a tensor-product is embedded in a design domain that is arbitrarily shaped. The density field is represented using B-splines. The intrinsic filter provided by the B-spline representation for topology optimization is governed by the degrees of B-spline and the knot spans. This B-spline effectively removes numerical artifacts and controls the minimum feature-length in; when the B-spline basis functions span many analysis elements. Benchmark problems in structural topology optimization are computed through Evolutionary Structural Optimization (ESO) algorithm. Comparisons of the optimal structure topologies using finite element analysis and isogeometric collocation methods are analyzed. The influence of different tension-compression stiffness ratios on the final structure topologies is listed and analyzed. The numerical results show the reliability and performance of the proposed mathematical formulation and asymmetry of natural structure.

Topology optimization of continuum structures is a recent field in structural optimization. However, an increasing research activity in this area has been developed since the statement of the very first formulations. These formulations try to obtain the most adequate material distribution that satisfies the imposed structural limitations. The existence or absence of material in each part of the domain is usually defined by using a continuum variable (the relative density) in order to avoid dealing with a discrete optimization problem. This continuum approach of the material properties present important advantages since conventionaloptimization algorithms can be used. However, numerical models must be considered in order to develop the structural analysis for intermediate values of the relative densities. In this paper, we present some improvements in a minimum weight approach of the structural topology optimization problem. The main goal of this paper is to present an improved formulation that tries to reach binary 0-1 material distributions by using a continuum approach of the design variables. Furthermore, a perimeter penalization is included in the objective function to simplify the solutions obtained. In addition, some computational aspects are considered in order to reduce the computational effort. Finally, we compare the solutions obtained by using these formulations in two application examples.

This paper will give a short survey about topology optimization of structures. It is particularly focused on topological shape optimization of structures using level-set methods, including the level-set based standard methods and the level-set based alternative methods. The former often directly solve the Hamilton-Jacobi partial differential equation (H-J PDE) to obtain the boundary velocity field using Finite Differential Methods (FDM), and the later commonly employ parametric or equivalent methods to evaluate the velocity field without directly solving the H-J PDE. The unique characteristics of the level-set based topology optimization methods are discussed, and a future perspective and prospects in this research area is also included. A benchmark numerical example is used to showcase the effectiveness of the level-set based methods.

The increasing structural flexibility of large aircraft leads to significant aeroelastic effects. More efficient topology optimization techniques are required for the design to further take advantage of aeroelasticity and obtain lightweight structures. This paper proposes a moving boundary meshfree topology optimization that combines the Galerkin method of weighted residuals and non-uniform rational B-splines (NURBS). The solution domain is described by the control points of NURBS and its property is calculated adaptively with an integration subtraction technique. The minimal compliance is searched for using the globally convergent method of moving asymptotes (GCMMA) by designing the locations of control points as subject to volume and flux constraints. The method is first applied to a typical two-dimensional design example with symmetric boundary conditions. The results show that the shape constraints can be conveniently applied, and smoother boundaries are obtained with fewer parameters. Then, a three-dimensional wing structure with asymmetric boundary conditions is optimized. A three-dimensional flight load that combines the high-order-panel and meshfree methods is employed to calculate the elastic loads and update asymmetric external loads during the optimization process. The designed wing satisfies engineering requirements and the presented method can solve the practical topology optimization problems of three-dimensional structures.

A multi-mesh strategy is employed to reduce the computational costs associated with incorporating correlated uncertainties into structural topology optimization problems. Uncertainty quantification is achieved through an existing perturbation-based method. This method transforms uncertainties in stiffness, such as geometric and material property randomness, into an augmented deterministic topology optimization problem. Although the approach has proven quite efficient in general, the uncertainty sources are tied to the finite element analysis mesh. Computational costs then rise significantly when refining the finite element mesh in the presence of correlated random variables. We therefore propose separating the analysis and uncertainty mesh. The technique is evaluated on problems of expected stiffness with uncertainties in elastic modulus and compared to solutions found using the standard perturbation and Monte Carlo topology optimization algorithms.

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size

The goal of this article is to develop a new topology optimization framework that can secure the manufacturability of the synchronous reluctance motors (SynRMs) with multiple iron webs and bridges while satisfying the required electromagnetic and mechanical performances. In the SynRM, multiple iron webs and bridges are crucial to achieve a higher torque and a structural safety. A minimum thickness of the web and bridge should also be guaranteed to achieve manufacturability. To acquire the optimal SynRM that can satisfy the above characteristics, a two-stage topology optimization framework is proposed with an individual filtering-and-penalization scheme. This proposed scheme can individually control the number and thickness of multiple iron webs and bridges in the rotor to obtain the manufacturable and structurally safe SynRMs. For the abovementioned purpose, topology optimization is formulated to minimize a torque ripple while satisfying the desired average torque, structural compliance, and a rotational moment of inertia. Average torque and structural compliance are evaluated by performing the electromagnetic and structural finite element analyses, respectively, at each iteration of topology optimization. Here, design-dependent loads are considered in the radial and circumferential directions to reflect a high-speed rotation. Through investigating the optimized SynRM designs with various filter radii, the relationship among the torque, structural pattern, and stress distribution is examined. Finally, the experimental results with the manufactured prototypes validate the feasibility and potential of the proposed design method.

Bi-directional evolutionary topology optimization of continuum structures subjected to inertial loads