A robust topological derivative-based multi-material optimization approach: Optimality condition and computational algorithm

An approach for the concurrent homogenization-based microstructure type and topology optimization problem

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

With the rapid advancement of computing technology, mathematical modeling and computer simulations have become common and standard methods for design of complex systems. To exercise these models intelligently and eliminate the burden of manual iteration, manipulating inputs and reviewing outputs, optimization strategies are applied in a simulation-based design environment. In reality, the knowledge that engineers have about the design problem is imperfect and incomplete, uncertainty always exists in simulation-based design. The conventional deterministic optimization techniques do not consider the impact of such uncertainties. If the system responses driving the design decisions are very sensitive to these uncertainties, serious problems might confront the designer. Performance of the actual system may differ greatly from expected optimum performance. Again, if the design solution is near one or more constraint boundaries, then even slight uncertainties or changes in the operating environment could produce failed, unsafe designs, and/or result in substantial performance degradation. In recent years, the concept of reliability and robust design has become very popular. Probabilistic design analysis and statistical modeling allow the identification of designs that qualify as not only feasible, but as consistently feasible in the face of uncertainty. Reliability based topology optimization (RBTO) extends the reliability notion to the area of structural topology optimization. Although reliability based design optimization (RBDO) is a well established research area, very few work has been presented in the area of RBTO. In this study, we give an overview of the various works done in the field of RBTO. Ever since Bendsoe and Kikuchi introduced the topology optimization using a homogenization method, many topology optimization methods have been developed for the both linear and nonlinear structures. Traditional topology optimization methods drive the topology of a structure to an optimum design based on the constraint on mass. Kharmanda et al. introduced reliability constraints in deterministic topology optimization problem. They proposed a heuristic strategy that aims to reduce mass while improving the reliability level of the structure without greatly increasing its weight. But the limit state function used by them was not based on failure criteria for the structure. Their formulation considered uncertainty with respect to ∗Design Automation Laboratory †Advisor geometrical dimension and applied load only. Also their reliability analysis seems to be independent of the boundary and loading condition, so their results showed similar values for the uncertain variables for different structures. Tovar et al. have developed the Hybrid Cellular Automaton (HCA) method for structural synthesis of continuum material where the state of each cell is defined by both density and strain energy. In Agarwal, a decoupled RBDO approach is employed such that the topology optimization is separate from the reliability analysis. Patel et al. showed the use of RBTO using the gradient free Hybrid Cellular Automata (HCA) method. Their formulation incorporates uncertainty with respect to material property also. They considered limit state function based on failure modes on the output displacements. This study aims at giving a comparative study on these methods. It points out the advantages and disadvantages of the methods, and their significance in the field of RBTO. As large numbers of design variables are associated with continuum topology optimization problems, RBTO methods are inherently computationally expensive because of additional required system analysis associate with RBDO. Based on the observations, this study presents an efficient method for incorporating reliability analysis in topology optimization.

Variational approach to relaxed topological optimization: Closed form solutions for structural problems in a sequential pseudo-time framework

A globally convergent method to accelerate topology optimization using on-the-fly model reduction

Topology optimization is among the most effective tools for innovative and lightweight structural designs. Multi-material design is considered to achieve better structural performance than single-material design. To significantly reduce the design space dimensionality and facilitate the optimization of multi-material structural design problems, this study proposes an effective topological representation and dimensionality reduction approach based on the material-field series expansion (MFSE) model. In the proposed method, a specified number of material phases is described within a single material field with a piecewise Heaviside projection function. The topology optimization problem is solved by determining the optimal MFSE coefficients. Owing to the single material-field topological description and series expansion, the number of design variables is independent of the finite element mesh and the number of material phases. In terms of dimensionality reduction, the proposed method outperformed all reported state-of-the-art algorithms for multi-material topology optimization. The validity and universality of the proposed method are illustrated in two- and three-dimensional numerical examples.

This paper presents an efficient and comprehensive MATLAB code to solve two-dimensional structural topology optimization problems, including minimum mean compliance, compliant mechanism synthesis and multi-load compliance problems. The Unsmooth Variational Topology Optimization (UNVARTOP) method, developed by the authors in a previous work, is used in the topology optimization code, based on the finite element method (FEM), to compute the sensitivity and update the topology. The paper also includes instructions to improve the bisection algorithm, modify the computation of the Lagrangian multiplier by using an Augmented Lagrangian to impose the constraint, implement heat conduction problems and extend the code to three-dimensional topology optimization problems. The code, intended for students and newcomers in topology optimization, is included as an appendix (Appendix A) and it can be downloaded from https://github.com/DanielYago together with supplementary material.

In this research, a reliability based topology optimization (RBTO) for structural design methodology using the Hybrid Cellular Automata (HCA) method is proposed. More speciflcally, a decoupled reliability based design optimization (RBDO) approach is utilized, so that the topology optimization is separate from the reliability analysis. In this paper, a maximum allowable displacement failure mode is considered. In this methodology, starting from a continuum design space of uniform material distribution and initial uncertain variable values, a deterministic topology optimization is followed by a reliability assessment of the resulting structure to determine the most probable point of failure (MPP) for the current structure. The MPP is determined with respect to the maximum allowable de∞ection of the structure when loaded. This is generally a computationally expensive process using traditional techniques due to the large number of design variables associated with topology optimization problem. However, combining the e‐cient methods of the non-gradient HCA algorithm with the decoupled approach for RBDO aims to reduce this burden. The topology optimization was without constraint in previous applications of the HCA method. To accommodate RTBO, a mechanism for a global constraint for maximum allowable displacement is developed. This paper details the methodology for the six-sigma design of structures using topology optimization.

In this paper we consider the classical minimum compliance topology optimization in linear elasticity. In our approach, the objective is to minimize a cost functional that comprises the work of applied body and traction forces obtaining simultaneously the optimal topology and the optimal localization for proportional actuators on the concerning structural domain. A proportional actuator is considered as a spring and one of the objectives is to obtain the optimal local to couple it on the structure. The problem is treated as a topology optimization problem where a given cost functional must be minimized under suitable constraints for external forces, control energy, volume restriction and a set of project variables. The topology optimization procedure here performed uses the solid isotropic material with penalization approach, based on the concept of optimizing the material distribution. The solution of the optimization problem is obtained by using a sequential linear programming and the finite element method is used to discretize the design domain. Several numerical examples are presented showing the effectiveness of the proposed approach.

SummaryThis article proposes an efficient approach for solving three‐dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)‐based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.

This work presents the framework to optimally design a cantilever for torsion mode frequency maximization. A cantilever design problem is formulated by topology and shape optimization methods. The torsion mode frequency is selected as an objective function, and the volume of the cantilever and the first bending mode frequency are constrained. Two optimization problems are defined and sequentially solved for the specific values. A new idea in this work is using a final topology obtained in the topology optimization problem as an initial shape in the shape optimization problem. The torsional mode frequency of the optimized cantilever is well improved in comparison with a nominal cantilever.

In this study, we present a framework based on the concept of multifidelity design optimization with the purpose of indirectly solving complex—computationally heavy and/or unstable—topology optimization problems. Our primary idea is to divide an original topology optimization problem into two types of subproblems for low-fidelity optimization and high-fidelity evaluation. To realize this idea, artificial design parameters, which we refer to as seeding parameters, are incorporated into the low-fidelity optimization problem for generating various patterns of topology-optimized candidates. The low-fidelity optimization problem is deliberately formulated as an easily solvable one by decreasing the nonlinearity of the original physical phenomena. Notably, selecting valid seeding parameters in the low-fidelity optimization problem is essential for employing the proposed framework. The aim of high-fidelity evaluation is to obtain a satisfactory solution using a high-fidelity analysis model, which considers the nonlinearity of the original physical phenomena, from the data set of the topology-optimized candidates, via the design of experiments. We apply the proposed framework to two case studies of isothermal and thermal turbulent flow problems, and discuss its efficacy as an alternative strategy for solving complex topology optimization problems.

Topology optimization for minimum compliance using a control strategy

A classical problem in topology optimization concerns the minimization of the compliance of a static structure, subject to a volume constraint upon the available material. Assuming that the structure is under small displacements and it is composed of a linear elastic material, the evaluation of the objective function demands the solution of a linear system. Hence, within the computational optimization process of addressing topology optimization problems, the cost of evaluating the objective function may be an issue, especially as the discretized mesh is refined. This work pursues the approximate reanalysis technique in combination with the Sequential Piecewise Linear Programming method for obtaining optimized structures. Numerical evidences are presented to corroborate the usage of this blend in a study composed by three distinct strategies in three benchmark test problems. A further analysis has been performed concerning the impact of the computation of the gradient vector of the objective function, pointing out room for additional savings.