Robust Topology Optimization Research Articles

Topology optimization of structures with infill-supported enclosed voids for additive manufacturing

This paper proposes a novel topology optimization procedure for designing structures with infill-supported enclosed voids for additive manufacturing (AM). In such structures, the open and enclosed regions are separately treated, where the open regions are the standard voids but the enclosed ones are filled by porous materials. The applied porous infill can work as supports in the AM process, but at the same time be part of the final structures and will never be removed. In this way, the difficulty of removing the supports in enclosed voids can be naturally avoided. However, whether the enclosed voids should be removed or kept if the porous material must be filled inside becomes a challenging problem, and thus this work develops a rational procedure to automatically determine it, and optimize the layout of these kept enclosed voids with infill. In the proposed procedure, a nonlinear virtual temperature (N-VTM) filter is utilized to separate the enclosed voids and open regions. Then, by combining the N-VTM filter, PDE filter and projection, a novel interpolation for structures with infill-supported enclosed voids is proposed, which at the same time can be used to provide structures without enclosed voids. The robust topology optimization formulation is applied to enhance the performance of the proposed procedure, and the sensitivity equations are deduced based on the adjoint method. Several numerical and experimental examples in two and three dimensions illustrate the effectiveness of the procedure. The results show that the procedure is capable of automatically determining whether to directly eliminate an enclosed void or to use infill to support it.

Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

Simultaneous topology and fiber path optimization of composite structures with MAC constraints

The pursuit of stiffer lightweight structures has led to the development of new concepts such as the Variable Stiffness Composite Laminates (VSCL), which further enhance the design potential of these highly tailored materials. As the number of design variables rise, more robust topology optimization formulations are necessary to achieve optimal solutions. In this work, a design optimization framework is developed by assigning material and fiber orientation on discrete patches of the VSCL structures. A gradient-based optimization method is used in the framework by applying the Discrete Material Optimization (DMO) formulation. The main goal is to design structures to meet dynamic performance objectives, with particular emphasis on the synthesis of modes with a target design. Accordingly, the objective function depends on the eigenvalues (natural frequencies), while one set of constraints depends on the eigenvectors (mode shapes). The Modal Assurance Criterion (MAC) is simultaneously used to perform mode tracking between the different design iterations and to evaluate the correlation between the synthetized modes and the pre-defined objective shapes. In this study, the calculation of the weight derivatives is also revised for the DMO formulation to show the contribution of the missing terms in the final optimal results. The optimization framework is examined for some potential applications related to dynamic decoupling and dynamic synthesis. It is shown that the design approach tailors the dynamic response of the composite structure and it can effectively be used to design resonators based on a reference model.

Topological synthesis of fluidic pressure-actuated robust compliant mechanisms

This paper presents a robust density-based topology optimization approach for synthesizing pressure-actuated compliant mechanisms. To ensure functionality under manufacturing inaccuracies, the robust or three-field formulation is employed, involving dilated, intermediate and eroded realizations of the design. Darcy’s law in conjunction with a conceptualized drainage term is used to model the pressure load as a function of the design vector. The consistent nodal loads are evaluated from the obtained pressure field using the standard finite element method. The objective and load sensitivities are obtained using the adjoint-variable approach. A multi-criteria objective involving both the stiffness and flexibility of the mechanism is employed in the robust formulation, and min–max optimization problems are solved to obtain pressure-actuated inverter, gripper, and contractor compliant mechanisms with different minimum feature sizes. Limitations of the linear elasticity assumptions while designing mechanisms are identified with high pressure loads. Challenges involved in designing finite deformable pressure-actuated compliant mechanisms are presented.

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

An interval-oriented dynamic robust topology optimization (DRTO) approach for continuum structures based on the parametric Level-Set method (PLSM) and the equivalent static loads method (ESLM)

Robust topology optimization of biodegradable composite structures under uncertain degradation rates

Biodegradable implants have the potential to serve as next-generation temporary medical devices as they can safely dissolve in the human body upon bone regeneration. Degradation uncertainty of the biodegradable material can remarkably affect the mechanical performance of biodegradable composite implant structures. It is necessary to consider this issue when designing resorbable metallic composite structures. To this end, this study introduces a novel robust topology optimization approach for designing biodegradable composite structures considering the degradation rate uncertainty of the biomaterial. The density-based topology optimization method is used to track the evolving of biomaterial layout during the optimization process, the Expansion Optimal Linear Estimation (EOLE) method is used to model the degradation uncertainties, and the Polynomial Chaos Expansion (PCE) based uncertain propagation analysis is implemented to predict the stochastic response. Then the robust topology optimization problem is formulated, in which a weighted function that achieves a trade-off between the expected mean and standard deviation of the performance function of interest was used as the objective function. The sensitivities of the design variables were deduced by considering the material degradation over time. Several numerical examples were presented to demonstrate that the proposed method could generate meaningful optimal topologies with the desired mechanical performance.

Robust topology optimization of structures under uncertain propagation of imprecise stochastic-based uncertain field

This study introduces a novel computational framework for Robust Topology Optimization (RTO) considering imprecise random field parameters. Unlike the worst-case approach, the present method provides upper and lower bounds for the mean and standard deviation of compliance as well as the optimized topological layouts of a structure for various scenarios. In the proposed approach, the imprecise random field variables are determined utilizing parameterized p-boxes with different confidence intervals. The Karhunen–Loève (K–L) expansion is extended to provide a spectral description of the imprecise random field. The linear superposition method in conjunction with a linear combination of orthogonal functions is employed to obtain explicit mathematical expressions for the first and second order statistical moments of the structural compliance. Then, an interval sensitivity analysis is carried out, applying the Orthogonal Similarity Transformation (OST) method with the boundaries of each of the intermediate variable searched efficiently at every iteration using a Combinatorial Approach (CA). Finally, the validity, accuracy, and applicability of the work are rigorously checked by comparing the outputs of the proposed approach with those obtained using the particle swarm optimization (PSO) and Quasi-Monte-Carlo Simulation (QMCS) methods. Three different numerical examples with imprecise random field loads are presented to show the effectiveness and feasibility of the study.

A robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method

In this paper, a robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method is proposed. Firstly, a variable time step strategy for level set function evolution based on the gradient of the level set function that uses radial basis function interpolation to realize the separation of time and space is developed, which can achieve better design results during topology optimization. Beyond that, the dimension reduction method and dimension-wise method based on the polynomials are used for characterization and quantification of bounded field parameters with uncertainties. Finally, the sensitivity of the robust optimization model is derived based on the shape derivative principle, which provides the basis for the implementation of gradient based optimization algorithm. Three examples illustrate the effectiveness, necessity and influence of important parameters of the robust topology optimization method considering bounded field parameters with uncertainties based on the variable time step parametric level-set method from different aspects.

Size-controlled cross-scale robust topology optimization based on adaptive subinterval dimension-wise method considering interval uncertainties

Size-controlled cross-scale robust topology optimization based on adaptive subinterval dimension-wise method considering interval uncertainties

Phase-field based robust topology and orientation optimization under fiber orientation uncertainty

Recently, there has been an increased usage of fiber-reinforced composites in aerospace and automobile due to their celebrated mechanical properties. Thus, a massive research effort has hitherto been done to develop a method for concurrent optimization of structural topology and fiber orientation. Topology-optimized continuous fiber-reinforced structures tend to show complex morphologies with spatially varying fiber orientation. Due to this nature, it is nearly possible to manufacture the results only through additive manufacturing technology However, a gap must exist between prescribed optimized fiber orientation and fiber orientation of manufactured composites under a limited degree of precision of the manufacturing process. Robust anisotropic topology optimization under such uncertainty, therefore, needs to be developed. This presentation aims to present a novel robust topology optimization methodology for fiber-reinforced materials under fiber orientation uncertainty. Our method is built upon Phase-Field based topology optimization which is suitable for this type of optimization method. The effectiveness of our method is validated through the stiffness maximization problem and the fundamental eigenfrequency maximization problem.

Robust topology optimization of negative Poisson’s ratio metamaterials under material uncertainty

Metamaterials are synthetic materials designed to have unique properties like negative Poisson ratio (NPR). NPR metamaterials, also known as auxetics, offer significant value in applications that require high energy absorption, e.g., packing materials, medical knee pads, footwear. However, material uncertainty arising out of manufacturing tolerance, inhomogeneity of material properties, and others could lead to significant variations in the response of the metamaterials. Thus, a SIMP based robust topology optimization (RTO) design for the NPR metamaterials under material uncertainty is investigated. The weighted mean and variance of the deterministic objective function is utilized to form a robust objective function. The variation in effective Poisson’s ratio with respect to the lower bound goes from 15.40% to 105% with deterministic topology optimization. In contrast, RTO produces more stable designs and shows the variation of only 1.72% to 2.54%. Several parametric studies are used to demonstrate the feasibility of the proposed RTO methodology.

スペクトラルレベルセット法を用いた設計変数削減によるCMA-ESに基づくトポロジー最適化の高速化

Inthispaper, weattempttospeeduptopologyoptimizationbasedonCMA-ESbyusingthespectrallevelsetmethod (SLSM). Although CMA-ES provides robust topology optimization for design problems with intractable properties, it suffers from the curse of dimensionality. SLSM uses Fourier series expansion to generate level set functions to represent the boundary between structures and voids, which significantly reduces the number of design variables for topology optimization. We aim to use SLSM to speed up topology optimization with CMA-ES and evaluate it on several design problems.

Robust topology optimization for structures under thermo-mechanical loadings considering hybrid uncertainties

Robust topology optimization for structures under thermo-mechanical loadings considering hybrid uncertainties

Robust topology optimization of a flexural structure considering multi-stress performance for force sensing and structural safety

Robust topology optimization of a flexural structure considering multi-stress performance for force sensing and structural safety

Effects of design sensitivity schemes for incorporating loading uncertainty in robust topology optimization

The effects of different design sensitivity schemes for an appropriate incorporation of the loading position uncertainty into a gradient-based topology optimization procedure are explored intensively, so that the robust optimal design of a continuum structure can be efficiently performed over the prescribed uncertain region. Three compliance sensitivity schemes are tested and the distinctive strategy offering the most reliable and competitive material layout of the load-bearing components is obtained from comprehensive comparisons. Three benchmark examples are used to demonstrate the diverse configurations of the material layouts. The numerical results show that the scheme with the largest absolute design sensitivity can most adequately integrate the load uncertainties into the topology optimization algorithm to make the structural performance insensitive to, or have less degradation in the presence of perturbations in the loading points.

Concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties

AbstractA concurrent topology optimization for thermoelastic structures with random and interval hybrid uncertainties is discussed in this work. A robust topology optimization method is proposed for structures composed of periodic microstructures under thermal and mechanical coupled loads. The robust objective function is defined as a linear combination of the mean and standard variance under the worst case for the robust optimization model. An efficient hybrid orthogonal polynomial expansion (HOPE) method is developed to evaluate the robust objective function. The sensitivities for the robust topology optimization are then calculated based on the uncertainty analysis. Three numerical examples are provided to verify the effectiveness of the proposed method, and the Monte Carlo scanning test is used to validate the numerical accuracy of our proposed method. For comparison purpose, the topology optimizations under deterministic assumptions are also provided for these examples to show the importance of considering hybrid uncertainties.

Robust Topology Optimization of Graphene Platelets Reinforced Functionally Graded Materials Considering Hybrid Bounded Uncertainties

Abstract An efficient scheme for the robust topology optimization considering hybrid bounded uncertainties (RTOHBU) is proposed for the graphene platelets (GPLs) reinforced functionally graded materials (FGMs). By introducing the concept of the layer-wise FGMs, the properties of the GPLs reinforced FGMs are calculated based on the Halpin-Tsai micromechanics model. The practical boundedness of probabilistic variables is naturally ensured by utilizing a generalized Beta distribution in constructing the robust topology optimization model. To address the issue of lacking the information of critical loads in existing topology optimization approaches considering hybrid uncertainties, a gradient-attributed search is carried out at first based on the hypothesis of linear elasticity to determine the critical loads leading to the worst structural performance. Subsequently, the statistical characteristics of the objective structural performance under such critical loads are efficiently evaluated by integrating the univariate dimension reduction method and the Gauss–Laguerre quadrature, the accuracy of which is verified by the comparison analyses utilizing the results of Monte Carlo simulation as references. Furthermore, a novel realization vector set is constructed for the bounded probabilistic uncertainties to parallelize the sensitivity analysis and accelerate the optimization process. All the proposed innovations are integrated into the robust topology optimization scheme, the effectiveness and efficiency of which are verified by both numerical and realistic engineering examples.

Robust topology optimization for thermoelastic hierarchical structures with hybrid uncertainty

This paper presents a robust topology optimization (RTO) framework for thermoelastic hierarchical structures with hybrid uncertainty. Firstly, the thermoelastic concurrent optimization model is established and the uncertainties with interval random parameters are integrated into the thermoelastic hierarchical structure. Then, a reliable and cost-effective hybrid uncertainty perturbation analysis method (HUPAM) is derived for a quick estimate of the robust objective function subject to the mechanical and thermal loads. Finally, by calculating the design variables sensitivities of macroscale and microscale, the robust topological design can be generated efficiently. To obtain clear and optimal topologies for both macro- and micro- structures, the bi-directional evolutionary structural optimization (BESO) method is adopted. Some 2D and 3D numerical examples are presented to demonstrate the influences of the hybrid uncertainties on the final designs. The results also show that the proposed method can effectively improve the thermoelastic structural performance when it comes to uncertainties.

Topology optimization of structures subject to self-weight loading under stress constraints

PurposeThe purpose of this paper is to present an approach for structural weight minimization under von Mises stress constraints and self-weight loading based on the topological derivative method. Although self-weight loading topology has been the subject of intense research, mainly compliance minimization has been addressed.Design/methodology/approachThe resulting minimization problem is solved with the help of the topological derivative method, which allows the development of efficient and robust topology optimization algorithms. Then, the derived result is used together with a level-set domain representation method to devise a topology design algorithm.FindingsNumerical examples are presented, showing the effectiveness of the proposed approach in solving a structural topology optimization problem under self-weight loading and stress constraint. When the self-weight loading is dominant, the presence of the regularizing term in the formulation is crucial for the design process.Originality/valueThe novelty of this research work lies in the use of a regularized formulation to deal with the presence of the self-weight loading combined with a penalization function to treat the von Mises stress constraint.

Topology optimization of imperfect frame structures with improved manufacturability

We present a framework for robust topology optimization of frame structures under uncertainty in member geometry. Imperfect geometry is defined as a general out-of-straightness. The uncertainty in response is quantified using a second-order stochastic perturbation that allows for accurate estimation of first and second-order response statistics in the presence of geometric random imperfections. The adjoint method is employed to compute the sensitivities, allowing for the use of gradient-based optimizers. A novel approach based on the augmentation of the set of physical design variables with an “indicator set” and a SIMP-like strategy is utilized to control the complexity of the final design and arrive at optimized topologies with improved manufacturability. Numerical examples are provided to demonstrate the performance of the proposed framework where comparison is made with the deterministic designs to highlight the improvements and differences both in terms of overall topology, e.g., load path diversification, and volume re-distribution and the impact of the manufacturability considerations on the topology of the final design.