Robust topology optimization of trusses with automatic grouping of cross-sections

Topology optimization of truss lattices, using the ground structure method, is a practical engineering tool that allows for improved structural designs. However, in general, the final topology consists of a large number of undesirable thin bars that may add artificial stiffness and degenerate the condition of the system of equations, sometimes even leading to an invalid structural system. Moreover, most work in this field has been restricted to linear material behavior, yet real materials generally display nonlinear behavior. To address these issues, we present an efficient filtering scheme, with reduced-order modeling, and demonstrate its application to two- and three-dimensional topology optimization of truss networks considering multiple load cases and nonlinear constitutive behavior. The proposed scheme accounts for proper load levels during the optimization process, yielding the displacement field without artificial stiffness by simply using the truss members that actually exist in the structure (spurious members are removed), and improving convergence performance. The nonlinear solution scheme is based on a Newton-Raphson approach with line search, which is essential for convergence. In addition, the use of reduced-order information significantly reduces the size of the structural and optimization problems within a few iterations, leading to drastically improved computational performance. For instance, the application of our method to a problem with approximately 1 million design variables shows that the proposed filter algorithm, while offering almost the same optimized structure, is more than 40 times faster than the standard ground structure method.

In this paper, a topology optimization method is constructed for thermal problems with generic heat transfer boundaries in a fixed design domain that include design-dependent effects. First, the topology optimization method for thermal problems is briefly explained using a homogenization method for the relaxation of the design domain, where a continuous material distribution is assumed, to suppress numerical instabilities and checkerboards. Next, a method is developed for handling heat transfer boundaries between material and void regions that appear in the fixed design domain and move during the optimization process, using the Heaviside function as a function of node-based material density to extract the boundary of the target structure being optimized so that the heat transfer boundary conditions can be set. Shape dependencies concerning heat transfer coefficients are also considered in the topology optimization scheme. The optimization problem is formulated using the concept of total potential energy and an optimization algorithm is constructed using the Finite Element Method and Sequential Linear Programming. Finally, several numerical examples are presented in order to confirm the usefulness of the proposed method.

In this paper, a topology optimization method is constructed for thermal problems with generic heat transfer boundaries in a fixed design domain that includes design-dependent effects. First, the topology optimization method for thermal problems is briefly explained using a homogenization method for the relaxation of the design domain, where a continuous material distribution is assumed, to suppress numerical instabilities and checkerboards. Next, a method is developed for handling heat transfer boundaries between material and void regions that appear in the fixed design domain and move during the optimization process, using the Heaviside function as a function of node-based material density to extract the boundaries of the target structure being optimized so that the heat transfer boundary conditions can be set. Shape dependencies concerning heat transfer coefficients are also considered in the topology optimization scheme. The optimization problem is formulated using the concept of total potential energy and an optimization algorithm is constructed using the Finite Element Method and Sequential Linear Programming. Finally, several numerical examples are presented to confirm the usefulness of the proposed method.

Combination of topology optimization and optimal control method

The purpose of this study is to develop a topology optimization scheme based on big bang–big crunch (BB–BC) algorithm, inspired from the evolution of the universe called big bang and big crunch theory. In order to apply the BB–BC algorithm to topology optimization for static and dynamic stiffness problems, the parameters of the algorithm were transformed to those of topology optimization scheme. In addition, some parameters such as big bang (BB) range, BB search, population and non-exchange limit were newly introduced to topology optimization scheme. Also, a parametric study for the parameters involved in the topology optimization scheme was performed to reduce the number of parameters, and find the appropriate ranges for topology optimization. Some examples were provided to examine the effectiveness of the developed topology optimization scheme for both static and dynamic stiffness problems throughout comparing with other metaheuristic topology optimization algorithms and the BESO (bi-directional evolutionary structural optimization) method. It was verified that the suggested algorithm shows superior to the compared typical metaheuristic topology optimization algorithms in the viewpoints of stability, robustness, accuracy and the convergence rate.

By adopting topology optimization based on numerical solvers, the geometries of the piezoelectric sensors can be optimized to produce higher electrical output in a certain loading directions. 2D topology optimization and simulation studies are carried out with ANSYS using Piezo and MEMS extensions for coupled systems. Topology optimization is based on Solid isotropic material with the penalization method, where the design variables are the pseudo densities that control material distribution at each finite element. The optimization problem is solved using Sequential Convex Programming. The approximation of the objective function happens through a uniform convex function. The objective is to reduce the fundamental frequency of the piezoelectric sensor for given constraints and boundary conditions (maximum sensor size of 30 × 30 mm and reduction of sensor mass by 50%). Two optimized shapes are chosen for further analysis. All sensors are made of 10 vol% PZT piezoelectric ceramic in High-Temperature V2 photopolymer resin. Sensors are manufactured using a simulated Direct Light Processing (DLP) type 3D printing process by tape-casting them on glass and exposing them to UV light. The performance of sensors is measured on a 4-point bending setup. Experiments show the enhanced performance of the optimized sensors even when their mass is reduced by 50%.

Transient sensitivity analysis and topology optimization of particle suspended in transient laminar fluid

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.

In the design optimization process design variables are selected in the deterministic way though those have uncertainties in nature. To consider variances in design variables reliability-based design optimization problem is formulated by introducing the probability distribution function. The concept of reliability has been applied to the topology optimization based on a reliability index approach or a performance measure approach. Since these approaches, called double-loop singlevariable approach, requires the nested optimization problem to obtain the most probable point in the probabilistic design domain, the time for the entire process makes the practical use infeasible. In this work, new reliability-based topology optimization method is proposed by utilizing single-loop singlevariable approach, which approximates searching the most probable point analytically, to reduce the time cost and dealing with several constraints to handle practical design requirements. The density method in topology optimization including SLP (Sequential Linear Programming) algorithm is implemented with object-oriented programming. To examine uncertainties in the topology design of a structure, the modulus of elasticity of the material and applied loadings are considered as probabilistic design variables. The results of a design example show that the proposed method provides efficiency curtailing the time for the optimization process and accuracy satisfying the specified reliability.

Multi-material topology optimization considering interface behavior via XFEM and level set method

In order to generate a reliable design the nonlinear structural response, e.g. buckling or plasticity, has to be considered in topology and shape optimization. In the present study material topology optimization determining the basic layout is extended to elastoplasticity. Afterwards the shape of the boundaries is optimized by shape optimization also considering the nonlinear material behavior. An elastoplastic von Mises material model with linear, isotropic hardening/softening for small strains is used. The objective of the design problem is to maximize the structural ductility defined by the strain energy over a given range of a prescribed displacement. With respect to the specific features of topology and shape optimization, e.g. the number of optimization variables or local-global influence of optimization variables on the structural response, different numerical methods are applied to solve the respective optimization problem. In topology optimization the gradient of the ductility is determined by the variational adjoint approach. In shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach. Topology optimization problems are solved by optimality criteria (OC) methods, shape optimization problems by mathematical programming (MP) methods, i.e. SQP-algorithm. In topology optimization a geometrically adaptive optimization procedure is additionally applied in order to increase the numerical efficiency and to avoid artificial stress singularities. The numerical procedures are verified by a 2D design problem under plane stress conditions.

We propose a new topological shape optimization scheme based on the artificial bee colony algorithm (ABCA). Since the level set method (LSM) and phase field method (PFM) in topological shape optimization have been developed, one of any algorithms in this field has not yet been proposed. To perform the topological shape optimization based on the ABCA, a variable called the “Boundary Element Indicator (BEI),” is introduced, which serves to define the boundary elements whenever a temporary candidate solution is found in the employed and onlooker bee phases. Numerical examples are provided to verify the performance of the suggested ABCA compared with the discrete LSM and the ABCA for topology optimization. The numerical examples showed that holes in the structure are naturally created in the ABCA for topological shape optimization. Moreover, the objective function of the suggested ABCA is lower than that of the ABCA for topology optimization, and is similar to that of the discrete LSM. The convergence rate of the suggested ABCA is the fastest among the comparison methods. Therefore, it can be verified that the suggested topological shape optimization scheme, based on the ABCA, is the most effective among the comparison methods.

To solve generalized shape optimization problems, material topology optimization is applied to determine the basic layout, i.e., to insert and eliminate holes in the design space. In addition, the inner and outer contours are optimized in detail by shape optimization. To improve the quality of the results and to increase the numerical efe ciency, the design model is adapted during the optimization process. Two aspects of design adaptivity are considered. On the one hand, the parametrization of the design space is varied during the material topology optimization process and, on the other hand, the kind of design model is adapted during the overall optimization process switching between material topology and shape optimization. The adaptive techniques are discussed, and adaptive design modeling is verie ed for topology optimization problems of shell structures.

To solve generalized shape optimization problems, material topology optimization is applied to determine the basic layout, i.e., to insert and eliminate holes in the design space. In addition, the inner and outer contours are optimized in detail by shape optimization. To improve the quality of the results and to increase the numerical efe ciency, the design model is adapted during the optimization process. Two aspects of design adaptivity are considered. On the one hand, the parametrization of the design space is varied during the material topology optimization process and, on the other hand, the kind of design model is adapted during the overall optimization process switching between material topology and shape optimization. The adaptive techniques are discussed, and adaptive design modeling is verie ed for topology optimization problems of shell structures.

The paper presents a gradient‐based topology optimization formulation that allows to solve acoustic–structure (vibro‐acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface‐coupling integrals, however, such a formulation does not allow for free material re‐distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p‐formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two‐dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.