Reliability-based Topology Optimization Research Articles

Reliability-Based Topology Optimization for Optimal Layout of Active Controllers of Structures under Random Excitation

Reliability-Based Topology Optimization for Optimal Layout of Active Controllers of Structures under Random Excitation

Reliability-Based Topology Optimization with a Proportional Topology for Reliability

This research proposes an efficient technique for reliability-based topology optimization (RBTO), which deals with uncertainty and employs proportional topology optimization (PTO) to achieve the optimal reliability structure. The recent technique, called proportional topology optimization for reliability (PTOr), uses Latin hypercube sampling (LHS) for uncertainty quantification. The difficulty of the double-loop nested problem in uncertainty quantification (UQ) with LHS can be alleviated by the power of PTO, enabling RBTO to be performed easily. The rigorous advantage of PTOr is its ability to accomplish topology optimization (TO) without gradient information, making it faster than TO with evolutionary algorithms. Particularly, for reliability-based topology design, evolutionary techniques often fail to achieve satisfactory results compared to gradient-based techniques. Unlike recent PTOr advancement, which enhances the RBTO performance, this achievement was previously unattainable. Test problems, including an aircraft pylon, reveal its performances. Furthermore, the proposed efficient framework facilitates easy integration with other uncertainty quantification techniques, increasing its performance in uncertainty quantification. Lastly, this research provides computer programs for the newcomer studying cutting-edge knowledge in engineering design, including UQ, TO, and RBTO, in a simple manner.

A novel decoupled approach combining invertible cross-entropy method with Gaussian process modeling for reliability-based design and topology optimization

Design optimization considering the presence of uncertainties in parameters poses an extremely challenging problem. The source of difficulties comes with reliability-based formulations, where addressing the probabilistic problem exhausts the large computing efforts for failure estimations of the structure violating limit-state functions (LSFs). This paper proposes a novel decoupled approach for effectively solving reliability-based design optimization (RBDO) problems, namely an invertible cross-entropy (iCE) method advantageously combined with a Gaussian process regression (GPR) model, termed as iCE-GPR. The GPR model is applied to approximate the spectrum of LSFs under random parameters. Furthermore, to enhance the accurate prediction of the system failure probability, an active learning process is applied to systematically refine the GPR model by adding new learning points in the region with the largest uncertainty and high-reliability sensitivity through the maximization of an expected feasibility function (EFF). Based on the updated GPR model, the failure probability is estimated by a cost-effective cross-entropy (CE) method without any calls to the actual performance function. To perform the decoupling optimization process with the reliability analysis, the novel iCE, based on the CE method, is developed to update the most probable point (MPP) assigned for the next deterministic optimization process in determining the new optimal design. The method iteratively performs the deterministic optimization process based on the MPP underpinning LSFs sequentially updated by the active learning process. The proposed iCE-GPR method fast-converges the optimal design and significantly alleviates computational burdens associated with reliability analyses. The proposed method is also applied to solve a reliability-based topology optimization (RBTO) problem. Four numerical examples for both the RBDO and RBTO problems are provided to illustrate efficiency and robustness of the proposed iCE-GPR method.

Reliability-based thermal-fluid-structural topological optimization of three-dimensional coolant channels in proton exchange membrane fuel cells

In order to minimize the flow resistance in designing the cooling multi-channels of proton exchange membrane fuel cells (PEMFCs) under the double constraints of target thermal reliability index and target structural performance, this work proposes a three-dimensional (3D) thermal-fluid-structural reliability-based topology optimization (RBTO) method for the first time. In the RBTO for PEMFCs’ coolant channels, the position uncertainty of the multi-heat loads on the bipolar plates caused by irregular metal foam skeleton is considered, and the multi-heat load positions are treated as the interval uncertain variables within a certain range. To solve the problem of low computational efficiency caused by the coupling of topological parameter optimization and reliability constraint analysis, the RBTO is equivalently decoupled to a sequence of cycles of deterministic topology optimization (DTO) and inverse reliability assessment, and the latter is completed by performance measure approach (PMA) based on the target reliability index. Results show that RBTO introduces more flow resistance cost than DTO, and the additional flow resistance in RBTO increases with the increase of the target reliability index. The heat concentration caused by a smaller fiber diameter of foam skeletons results in more flow resistance cost of RBTO compared to DTO. When the inlet velocity of the coolant channel varies in the region of 0.05–0.3 m/s, the flow resistance of the multi-channel obtained by RBTO is 1.29–0.96 times as that of the straight multi-channel. In addition, the uneven reactant gas concentration in PEMFC gas channels leads to the uneven distribution of heat flow, and it affects the temperature distribution on bipolar plates, which in turn affects the results of RBTO. The above method and findings can be applied to design the cooling channels of PEMFCs with irregular metal foam gas channels.

Reliability-based topology optimization of large-tonnage crane boom head with stress constraint

This study proposes a reliability-based topology optimization model reconstruction method considering stress constraints to solve the problem of high stress on the crane boom head in the working process. Firstly, the empty area of the crane boom head model is filled as the design region. Next, random variables and reliability indexes are set, and the reliability-based topology optimization considering stress constraints is performed for the design region. Finally, the filled boom head model is reconstructed in professional software. The irregular uplift and entity parts are designed as ribbed plates and support structures to obtain the regular optimized boom head model. Meanwhile, finite element analysis is performed on the reconstructed model to determine its maximum equivalent stress during the working process. The results show that the optimized boom head has better mechanical properties than the original.

Plastic-limit probabilistic structural topology optimization of steel beams

This work presents a novel geometrically nonlinear analysis with imperfections reliability-based topology optimization (RBTO) approach, considering the volume fraction and geometric imperfections as random variables due to their crucial connections to the manufacturability of web openings in steel I-beams. The objective is achieved by controlling the plastic behavior through limit analysis, which imposes a limit on the plastic ultimate load multipliers. The suggested method is developed by imposing constraints related to the available material volume. The bi-directional evolutionary structural optimization (BESO) method is utilized to fulfill the objectives of this paper. The adequacy of the proposed technique is demonstrated by comparing the results of the algorithm with benchmark steel beams that have conventional web openings. Based on the numerical examples, it appears that the selected technique has the potential to increase the load capacity of steel beams while utilizing the same quantity of material within the design domain. Moreover, the improved beams also exhibit better performance in terms of stress levels. The results of this research suggest that the proposed technique enhances the load capacity of steel beams while maintaining the same quantity of material and improving their performance in terms of stress levels.

Accelerating reliability-based topology optimization via gradient online learning and prediction

In reliability-based topology optimization (RBTO), both quantification and maximization of structure reliability considering uncertainty of various quantities are essential. In this study, A novel gradient online learning and prediction (GoLap) method is proposed to accelerate the double-looped RBTO and reduce total computational time by using a one-hidden-layer neural networks (NN) and gradient and non-gradient based reliability quantification. The total computational time saving is achieved by (i) skipping the finite element analysis (FEA) runs and sensitivity analyses and (ii) updating the design and random variables using the gradients predicted by the NN model in selected inner and outer loops. 2D and 3D numerical examples, including an aircraft engine pylon, are presented to demonstrate the total computational time saving of up to 84.4%, and the prediction accuracy of the reliability of optimum structures for the GoLap implemented RBTO methods.

A level set reliability-based topology optimization (LS-RBTO) method considering sensitivity mapping and multi-source interval uncertainties

With the diversification of engineering structure performance requirements and the continuous development of structural design refinement, structural design methods are facing more and more factors to be considered. It is necessary to develop advanced design technology. In this paper, a sensitivity mapping technique is proposed to improve the effect of topology optimization based on a gradient optimization algorithm. The applicability of this technology is analyzed. In addition, considering the influence of multi-source uncertainties in the whole life cycle of the structure, a reliability-based topology optimization strategy based on the level set method is proposed. The sensitivity of displacement constraint and reliability constraint with pseudo time is derived. Numerical examples and Monte Carlo verification results fully illustrate the applicability, effectiveness, and efficiency of the proposed method. This method has been successfully applied to the topology optimization design of the rocket skid structure.

Reliability-based topology optimization of fractionally-damped structures under nonstationary random excitation

This contribution proposes an effective reliability-based topology optimization (RBTO) framework for the layout of viscoelastic dampers of energy-dissipating structures under nonstationary seismic excitation, in which the constitutive behavior of viscoelastic damper is described by the novel fractional derivative models. To formulate the RBTO problem, the installation number of fractional viscoelastic dampers is minimized under the constraints of dynamic reliability, and the design variables are chosen as the damper parameters and the existence variables of the potential fractional viscoelastic dampers. The explicit responses are first established for the fractionally-damped structure, and the explicit response sensitivities with respect to the design variables are further derived through the adjoint variable method (AVM). On this basis, an explicit time-domain method (ETDM) is developed for explicit formulation of the statistical moments of responses and the moment sensitivities of the fractionally-damped structure, which are further employed to analytically derive the first-passage dynamic reliabilities and the reliability sensitivities of the structure based on the classical level-crossing process theory. Finally, the RBTO problem for the layout of fractional viscoelastic dampers is solved with the gradient-based method of moving asymptotes (MMA), in which the existence variables of fractional viscoelastic dampers can be driven to binary solutions by the solid isotropic material with penalization (SIMP) technique. An engineering application involving a building frame structure installed with fractional viscoelastic dampers is presented to validate the feasibility of the proposed ETDM-based RBTO framework.

STO-DAMV: Sequential topology optimization and dynamical accelerated mean value for reliability-based topology optimization of continuous structures

In deterministic topology optimization (TO), due to not taking into account the uncertainties of the structural system, explicitly, resulting optimal layouts may conclude in low reliable levels or unreliable optimum design conditions. The objective of reliability-based topology optimization (RBTO) is to integrate the reliability concept into topology optimization, finding the optimized structures priori by the designer through a mathematical framework guaranteeing that the least reliability level is achieved. In the present investigation under a constant volume fraction, the bisection algorithm is implemented for approximating the optimal feature of the structure in the TO framework by applying the moving iso-surface threshold (MIST) topology optimization approach. The results from the previous step are employed to consider uncertainties by a reliability analysis algorithm. A reliability analytical method based on dynamical accelerated formulation is combined with the TO-based bisection algorithm as a novel sequential topology optimization and reliability analysis (STORA) for RBTO. The descent condition is guaranteed in the dynamical accelerated mean value method to search the most probable point which is applied in TO for finding optimum layouts of continuous structure problems. A dynamical step factor in the reliability loop of the STORA and the MIST integration are applied to find the optimum layouts in the optimization loop. Illustrative examples are presented for validation and to maintain the capability of the proposed sequential topology optimization and dynamical accelerated mean value (STO-DAMV). Based on the results, it is demonstrated that the application of reliability analysis with the integration of bisection in topology optimization can produce a robust reliability analysis with stable results in RBTO which facilitates a more reliable design than that acquired by TO. It is also concluded that, under identical accuracy, STO-DAMV for STORA is improved in view of computational robustness.

Non-probabilistic reliability-based topology optimization for two-material structure based on convex set and bounded field mixed model

Non-probabilistic reliability-based topology optimization for two-material structure based on convex set and bounded field mixed model

Efficient two-phase approach to reliability-based discrete variable topology optimization of continuum structures with multimodal distributions

In practical applications, some random variables follow multimodal distributions. However, conventional reliability analysis methods in reliability-based topology optimization (RBTO), such as the first order reliability method, often result in considerable computational errors when dealing with problems involving multimodal distributions. Consequently, the corresponding RBTO design is incredible. Moreover, the RBTO problem also faces the challenge of high computational cost. To this end, this paper proposes an efficient two-phase approach for RBTO of continuum structures with multimodal distributions combining sequential approximate integer programming with trust region (SAIP-TR) and direct probability integral method (DPIM). Firstly, DPIM is advanced to address the difficult problems of failure probability estimation and efficient sensitivity analysis under multimodal distributions. Secondly, a reliability-based discrete variable topology optimization framework based on SAIP-TR and DPIM is established, which yields clear topology configurations and facilitates engineering manufacturing. Owing to the merit of SAIP-TR, the original RBTO process is divided into two phases: the first phase performs deterministic topology optimization, and the second phase focuses on RBTO. Moreover, an adaptive selection strategy of representative points, considering structural compliance as performance function, is devised to further enhance computational efficiency. Finally, several examples illustrate high efficiency and accuracy of the proposed approach. The multimodal random variable and the random field are employed separately to describe global and local uncertainties in materials. In contrast, the optimized result considering global material uncertainty is more suitable for additive manufacturing. The proposed approach also presents potential in handling complex RBTO problems with random fields.

Novel methodology of fail-safe reliability-based topology optimization for large-scale marine structures

Novel methodology of fail-safe reliability-based topology optimization for large-scale marine structures

Non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the strength constraint parameterized level set method and interval mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized level set method (PLSM) is conducted. The P-norm is used to replace the local stress constraint with a single comprehensive stress constraint that approximates the maximum stress to optimize the structure model with minimum volume. In the interval-set model, the stress constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). In the NRBTO scheme, the unknown-but-bounded (UBB) uncertainties of the material and external load of the optimized structure are considered. By interpolating the level set function (LSF) with the multiquadric (MQ) splines, the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). In addition, the interval parameter vertex method is employed to obtain the stress constraint interval, and the shape derivative and adjoint vector methods are used to obtain the evolution velocity field of LSF. The optimality criterion (OC) method is used to solve the optimization problem. Finally, three examples are given, and the optimization model of stiffness constraint is introduced, which not only verifies the applicability of the topology optimization scheme based on non-probabilistic reliability but also compares the difference between the topology optimization with strength constraint and the topology optimization with stiffness constraint.

An Investigation of the Recent Developments in Reliability-based Structural Topology Optimization

Optimizing a structure's topology involves finding the best possible distribution of material and connections within a given design space. It has a significant effect on its performance, which is why there has been a meteoric rise in the number of articles published on the topic over the last two decades. This work offers an investigation of reliability-based topology optimization of structures, in light of the recent development of several topology optimization techniques for both linear and nonlinear systems. Therefore, the emphasis of this study is on the latest advancements, enhancements, and applications of reliability-based topology optimization. This paper's primary objective is to provide an overview of the latest advancements in the reliability-based topology optimization of structures, with a particular emphasis on the recent improvement of integrating reliability-based design into the bi-directional evolutionary structural optimization (BESO) method, which accounts for the topological optimization of geometric and material nonlinearity as well as thermoelastic problems.

Uncertainty-oriented multi-scale topology optimization of coupled thermo-mechanical continuum structures

Multi-scale and multi-physical field topology optimization (TO) design has attracted large of attention, especially in aerospace and mechanical engineering, for the complex loadings and increasing performance requirements. Moreover, the unavoidable uncertainties, such as environment change, load variation, manufacturing tolerance, and elastic modulus fluctuation, exist in both macro- and micro-scale under coupled thermo-mechanical environment, which poses a significant danger for the safety of topological design of multi-scale structure in practical engineering. Therefore, a 2D multi-scale reliability-based topology optimization (MSRBTO) model is constructed to account for the uncertainty behavior of multi-physical TO problems, in which mechanical and thermal loadings are considered simultaneously. In addition, the solid isotropic material with penalization method is adopted at both macro- and micro-scales. Sensitivities of the compliance and temperature constraints with respect to the macro design variables, micro design variables, and random variables are derived simultaneously. Then, a single-loop method is introduced to address the MSRBTO problem to guarantee computational efficiency. Several numerical examples are solved by the developed MSRBTO method and traditional mono-scale RBTO method. The results show that the optimized results of the MSRBTO method show better performance than the mono-scale RBTO method, and the validity and effectiveness of the proposed method are validated.

Robust reliability-based topology optimization for stress-constrained continuum structures using polynomial chaos expansion

Robust reliability-based topology optimization for stress-constrained continuum structures using polynomial chaos expansion

Interval reliability-based topology optimization of piezoelectric structures under single-loop sequential strategy and negative feedback control theory

Interval reliability-based topology optimization of piezoelectric structures under single-loop sequential strategy and negative feedback control theory

Reliability-based topology optimization for fundamental frequency maximization with frequency band constraints

Resonance can be avoided by preventing the structural natural frequencies from falling within the operating frequency range. However, structural natural frequencies are influenced by uncertain parameters, such as material properties and geometric sizes. Therefore, it is essential to consider these uncertainties during the conceptual design phase. To address this problem, a novel reliability-based eigenvalue topology optimization model under frequency-band constraints is established, in which the uncertainties of the material properties and structural size are considered. Owing to the high computational costs of dealing with frequency-band reliability constraints, a frequency-band constraint shifting method (FBCSM) is proposed to ease the heavy computational burden without sacrificing accuracy. In addition, the sensitivities of simple and multiple eigenfrequencies with respect to the design and random variables are derived. The “bound formulation” and “robust formulation” are used to reduce the gray elements and checkerboard phenomenon. Four examples are used to determine the effectiveness of the proposed reliability-based eigenvalue topology model and FBCSM. The results confirmed their superiority in terms of validity and efficiency.

An Efficient Decoupled Reliability-Based Topology Optimization Method Based on a Performance Shift Strategy

AbstractIn this paper, a new efficient reliability-based topology optimization (RBTO) method is proposed for structures, in which the double-loop optimization is equivalently decoupled into a sequential process based on a performance shift strategy. First, a volume minimization RBTO formulation is built for structures considering displacement or compliance reliability constraints. Second, an efficient decoupling scheme is proposed to turn the double-loop RBTO into a series of deterministic topology optimization and reliability analysis. For the reliability analysis, the reliability probability is calculated based on the probability distribution function of the performance function, and the probability distribution function can be solved by the maximum entropy method based on the raw moments calculated by the multiplicative dimension reduction method. A performance shift strategy is then applied to build an equivalence between probabilistic constraint and deterministic constraint to formulate the deterministic topology optimization. Thirdly, the adjoint variable method is applied to obtain the sensitivity information for topological design variables, and a gradient-based optimization algorithm is used to update the design variables. Finally, four typical numerical examples are used to verify the advantage of the proposed method. Compared with the traditional sequential optimization and reliability assessment (SORA) method, the proposed RBTO method results in a design with a 5.4% smaller volume value to reach the same reliability index requirement for the cantilever beam, and the function calls are 354 times less than that of the SORA method.