Revisiting topology optimization with buckling constraints

Structural and Multidisciplinary Optimization - We review some features of topology optimization with a lower bound on the critical load factor, as computed by linearized buckling analysis. The...     

On topology optimization of linear and nonlinear plate problems

In this paper we propose a new restriction method based on employing C ⁰-continuous fields of density defined on a set of meshes different from the one used for the finite element analysis. The optimization procedure starts with using a coarse density-mesh compared to the finite element one. Once the convergence is obtained in the optimization steps, a finer density-mesh is nominated for the further steps. Linear and nonlinear plate behaviors are considered and formulated by Kirchhoff or Mindlin–Reissner hypothesis. Comparison is made with element/nodal based approaches using filter. The results show excellent and robust performance of the proposed method.     

Aerothermoelastic topology optimization with flutter and buckling metrics

This work develops a framework for SIMP-based topology optimization of a metallic panel structure subjected to design-dependent aerodynamic, inertial, elastic, and thermal loads. Multi-physics eigenvalue-based design metrics such as thermal buckling and dynamic flutter are derived, along with their adjoint-based design derivatives. Locating the flutter point (Hopf-bifurcation) in a precise and efficient manner is a particular challenge, as is outfitting the optimization problem with sufficient constraints such that the critical flutter mode does not switch during the design process. Results are presented for flutter-optimal topologies of an unheated panel, thermal buckling-optimal topologies, and flutter-optimality of a heated panel (where the latter case presents a topological compromise between the former two). The effect of various constraint boundaries, temperature gradients, and (for the flutter of the heated panel) thermal load magnitude are assessed. Off-design flutter and thermal buckling boundaries are given as well.     

On concave constraint functions and duality in predominantly black-and-white topology optimization

We study the ‘classical’ discrete, solid-void or black-and-white topology optimization problem, in which minimum compliance is sought, subject to constraints on the available material resource. We assume that this problem is solved using methods that relax the discreteness requirements during intermediate steps, and that the associated programming problems are solved using sequential approximate optimization (SAO) algorithms based on duality. More specifically, we assume that the advantages of the well-known Falk dual are exploited. Such algorithms represent the state-of-the-art in (large-scale) topology optimization when multiple constraints are present; an important example being the method of moving asymptotes (MMA).We depart by noting that the aforementioned SAO algorithms are invariably formulated using strictly convex subproblems. We then numerically illustrate that strictly concave constraint functions, like those present in volumetric penalization, as recently proposed by Bruns and co-workers, may increase the difficulty of the topology optimization problem when strictly convex approximations are used in the SAO algorithm. In turn, volumetric penalization methods are of notable importance, since they seem to hold much promise for generating predominantly solid-void or discrete designs.We then argue that the nonconvex problems we study may in some instances efficiently be solved using dual SAO methods based on nonconvex (strictly concave) approximations which exhibit monotonicity with respect to the design variables.Indeed, for the topology problem resulting from SIMP-like volumetric penalization, we show explicitly that convex approximations are not necessary. Even though the volumetric penalization constraint is strictly concave, the maximum of the resulting dual subproblem still corresponds to the optimum of the original primal approximate subproblem.     

Solving optimization problems with linear-fractional objective functions and additional constraints on arrangements

The paper proposes an exact method to solve an optimization problem on arrangements with a linear-fractional objective function and additional linear constraints. The efficiency of the solution algorithm is analyzed by means of numerical experiments. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 79–85, September–October 2006.     

On some fundamental properties of structural topology optimization problems

We study some fundamental mathematical properties of discretized structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0–1. We show, by examples which can be solved by hand calculations, that the optimal solutions to the problems in general are not unique and that the discrete problems possibly have inactive volume or compliance constraints. These observations have immediate consequences on the theoretical convergence properties of penalization approaches based on material interpolation models. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis. The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.     

Difficulties in truss topology optimization with stress and local buckling constraints

The aim of this note is to discuss problems associated with local buckling constraints in the context of topology optimization. It is shown that serious difficulties are encountered unless additional measures are introduced.     

某SUV车架多目标拓扑优化设计

为得到同时满足刚度和动态要求的SUV车架,基于SIMP材料插值方法,分别以刚度最大和低阶模态固有频率最大作为优化目标建立拓扑优化模型,利用折中规划法建立多工况下刚度和低阶固有频率多目标优化模型,通过拓扑优化迭代得到新的SUV车架;对新车架进行仿真分析,得到其位移和应力分布及前4阶固有频率.其静态特性满足材料要求且有很大提高,第1阶固有频率提高到30.3 Hz,新车架质量减轻到193.3 kg.计算结果表明该方法能够很好地解决多目标下的结构优化问题。     

Shape and topology optimization for periodic problems

The present paper deals with the implementation of an optimization algorithm for periodic problems which alternates shape and topology optimization; the theoretical background about shape and topological derivatives was developed in Part I (Barbarosie and Toader, Struct Multidiscipl Optim, 2009). The proposed numerical code relies on a special implementation of the periodicity conditions based on differential geometry concepts: periodic functions are viewed as functions defined on a torus. Moreover the notion of periodicity is extended and cases where the periodicity cell is a general parallelogram are admissible. This approach can be adapted to other frameworks (e.g. Bloch waves or fluid dynamics). The numerical method was tested for the design of periodic microstructures. Several examples of optimal microstructures are given for bulk modulus maximization, maximization of rigidity for shear response, maximization of rigidity in a prescribed direction, minimization of the Poisson coefficient.     

An alternative interpolation scheme for minimum compliance topology optimization

We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material used. We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q. This increases the probability to obtain a zero-one solution of the relaxed problem. Received July 20, 2000     

Solving stress constrained problems in topology and material optimization

This article is a continuation of the paper Ko?vara and Stingl (Struct Multidisc Optim 33(4?C5):323?C335, 2007). The aim is to describe numerical techniques for the solution of topology and material optimization problems with local stress constraints. In particular, we consider the topology optimization (variable thickness sheet or ??free sizing??) and the free material optimization problems. We will present an efficient algorithm for solving large scale instances of these problems. Examples will demonstrate the efficiency of the algorithm and the importance of the local stress constraints. In particular, we will argue that in certain topology optimization problems, the addition of stress constraints must necessarily lead not only to the change of optimal topology but also optimal geometry. Contrary to that, in material optimization problems the stress singularity is treated by the change in the optimal material properties.     

Revisiting density-based topology optimization for fluid-structure-interaction problems

This study revisits the application of density-based topology optimization to fluid-structure-interaction problems. The Navier-Cauchy and Navier-Stokes equations are discretized using the finite element method and solved in a unified formulation. The physical modeling is limited to two dimensions, steady state, the influence of the structural deformations on the fluid flow is assumed negligible, and the structural and fluid properties are assumed constant. The optimization is based on adjoint sensitivity analysis and a robust formulation ensuring length-scale control and 0/1 designs. It is shown, that non-physical free-floating islands of solid elements can be removed by combining different objective functions in a weighted multi-objective formulation. The framework is tested for low and moderate Reynolds numbers on problems similar to previous works in the literature and two new flow mechanism problems. The optimized designs are consistent with respect to benchmark examples and the coupling between the fluid flow, the elastic structure and the optimization problem is clearly captured and illustrated in the optimized designs. The study reveals new features of topology optimization of FSI problems and may provide guidance for future research within the field.     

On nonunique solutions in topology optimization

This paper deals with the problem of non-unique solutions in topology optimization. Depending on the optimization path, the solutions, in other words the topologies of a structure, are different. The nonuniqueness problem in topology optimization is presented in connection with the testing of different lower material mass value bounding functions and the use of different material properties updating functions and different threshold functions. The structure strain energy minimum criterion is applied to find the optimum topology. A comparison of the topologies obtained from the energy criterion point of view is made.     

Improved proportional topology optimization algorithm for solving minimum compliance problem

Structural and Multidisciplinary Optimization - This paper presents some practical formulations for heat conduction topology optimization problems. In post-optimization analysis, temperature...     

Difficulties in truss topology optimization with stress,local buckling and system stability constraints

A serlous difficulty in topology optimization with only stress andlocal buckling constraints was pointed out recently by Zhou (1996a). Possibilities for avoiding this pitfall are (i) inclusion of system stability constraints and (ii) application of imperfections in the ground structure. However, it is shown in this study that the above modified procedures may also lead to erroneous solutions which cannot be avoided without changing the ground structure.     

Analysis of an algorithm for solution of conditional optimization problems with linear-fractional objective functions over permutations

A method is considered to solve a conditional optimization problem with a linear-fractional objective function over permutations. The performance of sub algorithms to solve this problem is evaluated. The practical efficiency of the algorithm is analyzed by conducting numerical experiments. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 133–146, July–August 2007.     

A level set topology optimization method for the buckling of shell structures

Structural and Multidisciplinary Optimization - Shell structures are some of the most widely used in engineering applications. Flat plates, stiffened panels, and wing ribs are each examples of...     

Stability of vector optimization problems with fuzzy weights in the objective functions and fuzzy matrix parameters in the constraints

The aim of this paper is to investigate the stability of multiobjective nonlinear programming problems with fuzzy weights in the objective functions and fuzzy matrix parameters in the constraints and represent, in addition, the related dual problems for which the set of feasible parameters and the solvability set are studied. These fuzzy weights and fuzzy matrix parameters are characterized by fuzzy numbers. The existing results concerning the basic notions parametric space in convex programs are redefined and analyzed qualitatively under the concept of α-Pareto optimality. An illustrative example is given to clarify the obtained results.