Topology optimization under topologically dependent material uncertainties

Structural and Multidisciplinary Optimization - A methodology allowing for the algorithmic integration of topologically dependent random fields of material parameters in topology optimization...     

Topology optimization with worst-case handling of material uncertainties

Structural and Multidisciplinary Optimization - In this article, a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case...     

Computational methods in optimization considering uncertainties - An overview

This article presents a brief survey on some of the most relevant developments in the field of optimization under uncertainty. In particular, the scope and the relevance of the papers included in this Special Issue are analyzed. The importance of uncertainty quantification and optimization techniques for producing improved models and designs is thoroughly discussed. The focus of the discussion is in three specific research areas, namely reliability-based optimization, robust design optimization and model updating. The arguments presented indicate that optimization under uncertainty should become customary in engineering design in the foreseeable future. Computational aspects play a key role in analyzing and modeling realistic systems and structures.     

Robust topology optimization of frame structures under geometric or material properties uncertainties

A robust topology optimization algorithm is proposed for frame structures in the presence of geometric or material properties uncertainties. While geometric uncertainties were modeled with uncorrelated random variables expressing the node locations of the structure, material properties uncertainties were modeled with a correlated random field of the material Young’s modulus with an exponentially decaying correlation structure throughout the domain. The proposed algorithm uses stochastic perturbation method for propagating these uncertainties to the structural response level, measured in terms of compliance, and optimizes the expected value plus multiple factors of the standard deviation of the response. A comparison between the resulting robust designs and deterministic designs is made, and changes to the final topologies are discussed. Moreover, using Monte Carlo simulation, it was shown that the robust designs outperform the deterministic designs under real-world situations that are accompanied with uncertainties.     

Topology optimization of phononic crystals with uncertainties

Topology optimization of phononic crystals (PnCs) is generally based on deterministic models without considering effects of inherent uncertainties existed in PnCs. However, uncertainties presented in PnCs may significantly affect band gap characteristics. To address this, an interval Chebyshev surrogate model-based heuristic algorithm is proposed for topology optimization of PnCs with uncertainties. Firstly, the interval model is introduced to handle the uncertainties, and then the interval Chebyshev surrogate model (ICSM), in which the improved fast plane wave expansion method (IFPWEM) is used to calculate the integral points to construct the ICSM, is introduced for band structure analysis with uncertainties efficiently. After that, the sample data, which is randomly generated by the Monte Carlo method (MCM), is applied to the ICSM for predicting the interval bounds of the band structures. Finally, topology optimization of PnCs is conducted to generate the widest band gaps with uncertainties included by utilizing the genetic algorithm (GA) and the ICSM. Numerical results show the effectiveness and efficiency of the proposed method which has promising prospects in a range of engineering applications.     

Aeroelastic tailoring considering uncertainties in material properties

The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.     

Topology optimization of trusses considering static and dynamic constraints using the CSS

Topology optimization of structures reveals outstanding advantages when compared to sectional optimization. Many unnecessary members and nodes may exist in a structure and a topology optimization provides an opportunity to remove them. This advantage will specially become apparent when comparatively large cost of the nodes is taken into account. Fundamental frequencies of a structure are important, easily obtained characteristics which allow the designer to keep out from the dangerous resonance phenomenon. When dynamic excitations are critical, these characteristics cannot be neglected. In this paper, topology optimization of truss structures is investigated considering stress, displacement, buckling and frequency constraints. To perform such an optimization is not simple because of large, highly nonlinear and non-convex search space. Here the newly developed charged system search algorithm is used to accomplish this optimization.     

Topology optimization of continuum structures under hybrid uncertainties

The aim of this paper is to study the topology optimization for mechanical systems with hybrid material and geometric uncertainties. The random variations are modeled by a memory-less transformation of random fields which ensures their physical admissibility. The stochastic collocation method combined with the proposed material and geometry uncertainty models provides robust designs by utilizing already developed deterministic solvers. The computational cost is decreased by using of sparse grids and discretization refinement that are proposed and demonstrated as well. The method is utilized in the design of minimum compliance structure. The proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling by using adaptive sparse grids method.     

Topology optimization of couple-stress material structures

Conventional topology optimization is concerned with the structures modeled by classical theory of mechanics. Since it does not consider the effects of the microstructures of materials, the classical theory can not reveal the size effect due to material’s heterogeneity. Couple-stress theory, which takes account of the microscopic properties of the material, is capable of describing the size effect in deformations. The purpose of this paper is to investigate the formulation for topology optimization of couple-stress material structures. The artificial material density of each element is chosen as design variable. Based on the basic idea of SIMP (Solid Isotropic Material with Penalization) method, the effective material stiffness matrix of couple-stress material is related to the artificial density by power law with penalty. The structural analysis is implemented by finite element method for couple-stress materials, and a 4-noded quadrilateral couple-stress element is formulated in which C ₁ continuity requirement is relaxed. Some typical problems are solved and the optimal results based on the couple-stress theory are compared with the conventional ones. It is found that the optimal topologies of couple-stress continuum show remarkable size effect.     

Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design

Additively manufactured components often require temporary support material to prevent the component from collapsing or warping during fabrication. Whether these support materials are removed chemically as in the case of many polymer additive manufacturing processes, or mechanically as in the case of (for example) Direct Metal Laser Sintering, the use of sacrificial material increases total material usage, build time, and time required in post-fabrication treatments. The goal of this work is to embed a minimum allowable self-supporting angle within the topology optimization framework such that designed components and structures may be manufactured without the use of support material. This is achieved through a series of projection operations that combine a local projection to enforce minimum length scale requirements and a support region projection to ensure a feature is adequately supported from below. The magnitude of the self-supporting angle is process dependent and is thus an input variable provided by the manufacturing or design engineer. The algorithm is demonstrated on standard minimum compliance topology optimization problems and solutions are shown to satisfy minimum length scale, overhang angle, and volume constraints, and are shown to be dependent on the allowable magnitudes of these constraints.     

Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.     

Topology optimization of rubber isolators considering static and dynamic behaviours

A topology optimization for the design of rubber vibration isolators is proposed. Many vibration isolators are made of rubbers and they operate under small oscillatory load superimposed on large static deformation. Vibration isolators must have a certain degree of static stiffness in order to endure the static loading due to large gravitational and inertial forces. On the other hand, isolators must have a small dynamic stiffness in order to reduce the force transmission from vibrating systems to base structures. Therefore both the static and dynamic behaviours of rubber should be simultaneously considered in the design process. The static behaviours of rubber under large and slow loads are generally treated with hyperelastic constitutive models. Rubber under fast dynamic loads can be modelled as a viscoelastic material. In this paper, the steady state viscoelastic model, which is suggested by Kim and Youn and correctly predicts the influence of the pre-strain on the relaxation function, is applied for the dynamic analysis. The continuum-based design sensitivity analyses (DSA) of both the static hyperelastic model and dynamic viscoelastic model are developed. The topology optimization formulation is proposed in order to generate the system layouts considering both the static and dynamic performance. The density distribution approach and sequentially linear programming (SLP) are used as the optimization algorithms. Some design examples are presented in order to verify the proposed approach.     

Topology optimization of channel cooling structures considering thermomechanical behavior

Structural and Multidisciplinary Optimization - A topology optimization method is presented to design straight channel cooling structures for efficient heat transfer and load carrying capabilities....     

Topology optimization considering fracture mechanics behaviors at specified locations

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.     

Robust multiphase topology optimization accounting for manufacturing uncertainty via stochastic collocation

Structural and Multidisciplinary Optimization - This paper presents a computational framework for multimaterial topology optimization under uncertainty. We combine stochastic collocation with...     

Topology optimization of structures with gradient elastic material

Topology optimization of structures and mechanisms with microstructural length-scale effect is investigated based on gradient elasticity theory. To meet the higher-order continuity requirement in gradient elasticity theory, Hermite finite elements are used in the finite element implementation. As an alternative to the gradient elasticity, the staggered gradient elasticity that requires C ⁰-continuity, is also presented. The solid isotropic material with penalization (SIMP) like material interpolation schemes are adopted to connect the element density with the constitutive parameters of the gradient elastic solid. The effectiveness of the proposed formulations is demonstrated via numerical examples, where remarkable length-scale effects can be found in the optimized topologies of gradient elastic solids as compared with linear elastic solids.     

Effect of dimensional and material property uncertainties on thermal flexure microactuator response using probabilistic methods

Micromachining of microelectromechanical systems such as other fabrication processes has inherent variation that leads to uncertain dimensional and material properties. In this paper, the effect of material and feature dimension uncertainties due to fabrication process on electrothermal microactuator tip deflection is investigated. A simple and efficient uncertainty analysis method is used based on direct linearization method (DLM); uncertainty analysis is performed by creating second-order metamodel through Box-Behnken design and Monte Carlo simulation. The standard deviations of tip deflection obtained by these two probabilistic methods are very close. Simulation results have been validated by a comparison with experimental results in literature. Experimental results fall within 95% confidence boundary obtained by DLM method. Also, sensitivity analysis of microactuator has been explored; the results show that microactuator performance has been affected more by thermal expansion coefficient and microactuator gap uncertainties.     

Topology optimization using B-spline finite elements

Topology optimization algorithms using traditional elements often do not yield well-defined smooth boundaries. The computed optimal material distributions have problems such as ??checkerboard?? pattern formation unless special techniques, such as filtering, are used to suppress them. Even when the contours of a continuous density function are defined as the boundary, the solution can still have shape irregularities. The ability of B-spline elements to mitigate these problems are studied here by using these elements to both represent the density function as well as to perform structural analysis. B-spline elements can represent the density function and the displacement field as tangent and curvature continuous functions. Therefore, stresses and strains computed using these elements is continuous between elements. Furthermore, fewer quadratic and cubic B-spline elements are needed to obtain acceptable solutions. Results obtained by B-spline elements are compared with traditional elements using compliance as objective function augmented by a density smoothing scheme that eliminates mesh dependence of the solutions while promoting smoother shapes.     

Topology optimization of continuum structures with material failure constraints

This work presents an efficient strategy for dealing with topology optimization associated with the problem of mass minimization under material failure constraints. Although this problem characterizes one of the oldest mechanical requirements in structural design, only a few works dealing with this subject are found in the literature. Several reasons explain this situation, among them the numerical difficulties introduced by the usually large number of stress constraints. The original formulation of the topological problem (existence/non-existence of material) is partially relaxed by following the SIMP (Solid Isotropic Microstructure with Penalization) approach and using a continuous density field as the design variable. The finite element approximation is used to solve the equilibrium problem, as well as to control through nodal parameters. The formulation accepts any failure criterion written in terms of stress and/or strain invariants. The whole minimization problem is solved by combining an augmented Lagrangian technique for the stress constraints and a trust-region box-type algorithm for dealing with side constraints (0<_min1) . Numerical results show the efficiency of the proposed approach in terms of computational costs as well as satisfaction of material failure constraints. It is also possible to see that the final designs define quite different shapes from the ones obtained in classical compliance problems.