Topology optimization of frame structures with flexible joints

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

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

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

A bit-array representation method for structural topology optimization using the GA is proposed. The importance of design connectivity is further emphasized and a hierarchical violation penalty method is proposed to penalize the violated constraint functions so that the problem of representation degeneracy can be overcome and the GA search can be driven towards the combination of better structural performance, less unusable material and fewer connected objects in the design domain. An identical initialization method is also proposed to test the performance of the GA operators. With the appropriately selected GA operators, the bit-array representation GA is applied to the structural topology optimization problems of minimum weight. Numerical results demonstrate that the present GA can achieve better accuracy with less computational cost and suggest that the GA performance can be significantly improved by handling the design connectivity properly.

Discrete ground structure topology optimization design methods have to date received considerable attention in structural engineering. An alternative class of structural topology optimization methods, which have not yet received much attention in structural engineering, but which have undergone considerable development in the past decade, are the so-called continuum formulations. In this work, a continuum structural topology optimization formulation is presented and applied to the concept design optimization of structural bracing systems that are needed to stiffen tall structures against sidesway under lateral-wind and seismic-type loading. Although demonstrated here in the context of these specific design examples, continuum structural topology optimization methods are believed to hold potential as a design tool for a wide range of civil engineering type structures. A variety of continuum topology design formulations, including static compliance minimization and eigenvalue optimization, are explored, and solution parameters are varied to show that a number of design possibilities can be realized as solutions.

Bone is a dynamic living tissue that undergoes continuous adaptation of its mass and structure in response to mechanical and biological environment demands. In this paper, we firstly propose a mathematical model based on cross-type reaction diffusion equations of bone adaptation during a remodeling cycle due to mechanical stimulus. The model captures qualitatively very well the bone adaptation and cell interactions during the bone remodeling. Secondly assuming the bone structure to be a self-optimizing biological material which maximizes its own structural stiffness, bone remodeling model coupled with finite element method by using the add and remove element a new topology optimization of continuum structure is presented. Two Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the others well known methods for structural topology optimization in open literatures.

Topology optimization of multi-material structures considering anisotropic yield strengths

Structural topology optimization is a crucial approach for achieving lightweight design. An effective topology optimization algorithm must strike a balance between the objective functions, constraints, and design variables, which essentially reflects the symmetry and tradeoff between the objective and constraints. In this study, a topology optimization method grounded in load path theory is proposed. Element bearing capacity is quantified using the element birth and death method, with an explicit formulation derived via finite element theory. The effectiveness in evaluating structural performance is assessed through comparisons with stress distributions and topology optimization density maps. In addition, a novel evaluation index for element bearing capacity is proposed as the objective function in the topology optimization model, which is validated through thin plate optimization. Subsequently, sensitivity redistribution mitigates checkerboard patterns, while mesh filtering suppresses multi-branch structures and prevents local optima. The method is applied for the lightweight design of a triangular arm, with results benchmarked against the variable density method, demonstrating the feasibility and effectiveness of the proposed method. The element bearing capacity seeks to homogenize the load distribution of each element; the technique in this study can be extended to the optimization of symmetric structures.

Hybrid electric vehicles (HEVs) are equipped with multiple power sources and involve complex mode-switching processes, which lead to intricate torsional vibration characteristics in their powertrains. This paper presents a novel clutch torsional damper with arc springs (CTD-AS) that combines the advantages of traditional clutch torsional dampers (CTDs) and dual-mass flywheels (DMFs). The structure and working principle of the CTD-AS are introduced in the paper, and a torsional damper parameter design method is discussed. During the study, a 13-degree-of-freedom concentrated mass model was established for an HEV powertrain, and optimization schemes based on multiple sets of operating conditions were obtained for a CTD, a DMF, and the CTD-AS. The optimized simulation results demonstrated that the average torsional vibration transmissibility value of the CTD-AS under the three working conditions was approximately 13.8% and 29.7% lower than that of the DMF and the CTD, respectively, and the overall performance is the best. The novel structural topology and parameter optimization method for the torsional damper of a parallel HEV proposed in this paper can effectively enhance driving smoothness under a variety of working conditions.

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.

SummaryIn this work, an asymptotically concentrated topology optimization method based on the solid isotropic material with logistic function interpolation is proposed. The asymptotically concentrated method is introduced into the process of optimization cycle after updating the design variables. At the same time, with the use of the solid isotropic material with logistic function interpolation, all the candidate densities are reasonably polarized, relying on the characteristic of the interpolation curve itself. The asymptotically concentrated method can effectively suppress the generation of intermediate density and speed up the process of updating the design variables, hence improving the optimization efficiency. Moreover, the above polarization can weaken the influence of low‐related‐density elements and enhance the influence of high‐related‐density elements. For the single‐material topology optimization problem, gray‐scale elements can be effectively eliminated, and clear boundary and smaller compliance can be obtained by this method. For the multimaterial topology optimization problem, minimum compliance with high efficiency can be achieved by this method. The proposed method mainly includes the following advantages: concentrated density variables, reasonable interpolation, high computational efficiency, and good topological results.

A wing that can be deformed the geometrical shape seamlessly and continuously to improve aerodynamic performance of wing is called a morphing wing. A compliant mechanism has been proposed as the internal structure of the wing to achieve the desirable deformation. Most previous studies have adopted the ground structure approach to obtain the internal structural configuration of the morphing devices. We consider to adopt the level set-based topology optimization method to improve the structural feature of the morphing wing. Then, we begin with the problem formulation to minimize the difference between the ideal given deformed shape and the deformation of the obtained compliant configuration. This manuscript gives current status of the study by illustrating some simple numerical examples.

A new bionic topology optimization method by combining reaction-diffusion equations describing bone adaptation process with finite element analysis is presented in this study. The major idea of the present approach is to consider the structure to be optimized as a piece of bone that obeys bone adaptation and the process of finding the optimum topology of a structure is equivalent to the bone remodeling process. Two widely used numerical examples demonstrate that the proposed approach greatly improves numerical efficiency compared with the othert well known methods for structural topology optimization in open literature. The results show that the optimal designs from the present bionic topology optimization method without use mathematical programming and numerical instability control techniques. The proposed method results in a better and faster convergence.

A robust and stable phase field method for structural topology optimization

A PID-optimality criteria method for structural topology optimization