Linear versus nonlinear reliability-based topology optimization with application to bridge structures

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

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.

Reliability-based topology optimization using inverse optimum safety factor approaches

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.

Since deterministic topology optimization (TO) does not consider the uncertainties in the structure, including materials, loading and geometric dimensions, it may provide the optimal designs with the lowest state of reliability and safety. To solve this problem, reliability-based topology optimization (RBTO) is used, which is actually a combination of TO methods with reliability-based design methods (RBDO) based on a mathematical framework and process. In this article, by considering four TO methods including moving iso-surface threshold (MIST), SIMP, evolutionary structural optimization-extended finite element (XFEM-ESO) and level-set (LS), and considering a constraint or objective function, an optimal volume fraction (Vf) is obtained for TO by the bisection method. Then, with the aid of mean volume fraction, TO is performed and its optimized results are applied by an advanced reliability analysis method, i.e. accelerated dynamical mean value (ADMV), taking into account the uncertainties and their standard deviations to extract the most probable probability point (MPP). Having the MPP and the constraint, the bisection algorithm is used again and the optimized volume fraction for the RBTO model, and thereby the optimal layout for the RBTO solution, is achieved. Several examples are presented to validate and highlight the optimization capability of the RBTO method using a structural model and the mentioned TO methods, and the results are compared together. Based on the results, it is shown that the combination of RBDO and TO approach is able to result in powerful, stable, safe, and reliable structures completely different from the TO results.

Uncertainties are inherent and prevalent throughout the life cycle of engineering structures, necessitating their explicit incorporation in the design process. This article presents a reliability-based topology optimization (RBTO) framework using the phase-field method to minimize the structural compliance under the condition of meeting the reliability requirement considering the uncertainty of the dimensions of the structure, loads, material parameters, etc. Based on the idea of decoupling, the reliability-based topology optimization model is divided into a reliability analysis problem and a deterministic topology optimization (DTO) problem. In the reliability analysis stage, the structure's failure probability is approximated using the first-order reliability method. In the DTO stage, the phase-field function (representing the structure state) is updated by the Allen–Cahn equation, driven by sensitivity-based optimization. Several examples are presented to demonstrate the performance of the proposed method.

Electro-thermal-compliant mechanisms are mechanisms onto which electric voltage is applied as input. This mechanism is based on the thermal expansion properties of the materials. This study considered all of the convection effects according to layouts produced after each topology optimization process. A new material mixing method based on bi-directional evolutionary structural optimization (BESO) was suggested to obtain optimal topologies for compliant mechanisms composed of two materials. Reliability-based topology optimization (RBTO) based on BESO was also applied to the topology optimizations of the mechanisms. Performance measurement approach (PMA), which estimates has probabilistic constraints that are formulated in terms of the reliability index, was adopted to evaluate the probabilistic constraints. The procedure has been tested in numerical applications, and the results were compared with those obtained by the level set and solid isotropic material with penalization (SIMP) methods for validation of the proposed approaches.

In order to solve the application restrictions of deterministic-based topology optimization methods arising from the omission of uncertainty factors in practice, and to realize the calculation cost control of reliability-based topology optimization. In consideration of the current reliability-based topology optimization methods of continuum structures mainly based on performance indexes model with a power filter function. An efficient probabilistic reliability-based topology optimization model that regards mass and displacement as an objective function and constraint is established based on the first-order reliability method and a modified economic indexes model with a composite exponential filter function in this study. The topology optimization results obtained by different models are discussed in relation to optimal structure and convergence efficiency. Through numerical examples, it can be seen that the optimal layouts obtained by reliability-based models have an increased amount of material and more support structures, which reveals the necessity of considering uncertainty in lightweight design. In addition, the reliability-based modified model not only can obtain lighter optimal structures compared with traditional economic indexes models in most circumstances, but also has a significant advantage in convergence efficiency, with an average increase of 44.59% and 64.76% compared with the other two reliability-based models. Furthermore, the impact of the reliability index on the results is explored, which verifies the validity of the established model. This study provides a theoretical reference for lightweight or innovative feature-integrated design in engineering applications.

This research proposes a reliability-based topology optimization (RBTO) using the finite element method. RBTO is a topology optimization based on probabilistic (or reliability) constraints. Young's modulus, thickness, and loading are considered as the uncertain variables and RBTO is applied to static and eigenvalue problems. The RBTO problems are formulated and a sensitivity analysis is performed. In order to compute probability constraints, two methods-RIA and PMA-are used. Several examples show the effectiveness of the proposed method by comparing the classical safety factor method.

This paper presents the optimum load-speed diagram evaluation for a linear micromotor, including multitude cantilever piezoelectric bimorphs, briefly. Each microbeam in the mechanism can be actuated in both axial and flexural modes simultaneously. For this design, we consider quasi-static and linear conditions, and a relatively new numerical method called the smoothed finite element method (S-FEM) is introduced here. For this purpose, after finding an optimum volume fraction for piezoelectric layers through a standard numerical method such as quadratic finite element method, the relevant load-speed curves of the optimized micromotor are examined and compared by deterministic topology optimization (DTO) design. In this regard, to avoid the overly stiff behavior in FEM modeling, a numerical method known as the cell-based smoothed finite element method (CS-FEM, as a branch of S-FEM) is applied for our DTO problem. The topology optimization procedure to find the optimal design is implemented using a solid isotropic material with a penalization (SIMP) approximation and a method of moving asymptotes (MMA) optimizer. Because of the higher efficiency and accuracy of S-FEMs with respect to standard FEMs, the main micromotor characteristics of our final DTO design using a softer CS-FEM are substantially improved.

In geotechnical engineering, stabilization of slopes is one of the significant issues that needs to be considered especially in seismic situation. Evaluation and precise prediction of factor of safety (FOS) of slopes can be useful for designing/analyzing very important structures such as dams and highways. Hence, in the present study, an attempt has been done to evaluate/predict FOS of many homogenous slopes in different conditions using Monte Carlo (MC) simulation technique. For achieving this aim, the most important parameters on the FOS were investigated, and finally, slope height (H), slope angle (α), cohesion (C), angle of internal friction ( $$\varnothing$$ ) and peak ground acceleration (PGA) were selected as model inputs to estimate FOS values. In the first step of analysis, a multiple linear regression (MLR) equation was developed and then it was used for evaluation and prediction by MC technique. Generally, MC model simulated FOS of less than 1.18, lower and higher than measured and predicted FOS values, respectively. However, the results of MC simulation for the FOS values of more than 1.33, is higher than those measured and predicted FOS values. As a result, the mean of FOS values simulated by MC was very close to the mean of actual FOS values. Moreover, results of sensitivity analysis demonstrated that the ( $$\varnothing$$ ), among other parameters, is the most effective one on FOS. The obtained results indicated that MC is a reliable approach for evaluating and estimating FOS of slopes with high degree of performance.

In many countries, slope failure is a complex natural issue that can result in serious natural hazards, such as landslide dams. It is associated with the challenge of slope stability evaluation, which involves the classification problem of slopes and the regression problem of predicting the factor of safety (FOS) value. This study explored the implementation of machine learning to analyze slope stability using a comprehensive database of 880 homogenous slopes (266 unstable and 614 stable) based on a simulation model developed as a surrogate model. A classification model was developed to categorize slopes into three classes, including S (stable, FOS > 1.2), M (marginally stable, 1.0 ≤ FOS ≤ 1.2), and U (unstable, FOS < 1.0), and a regression model was used to predict the target FOS value. The results confirmed the efficiency of the developed classification model via testing, achieving an accuracy of 0.9222, with 96.2% accuracy for the U class, 55% for the M class, and 95.2% for the S class. When U and M are in the same class (i.e., the U + M class), the test accuracy is 0.9315, with 93.3% accuracy for the S class and 92.9% accuracy for the U + M class. The low accuracy level for class M led to minor inaccuracies, which can be attributed to a data imbalance. Additionally, the regression model was found to have a high correlation coefficient R-square value of 0.9989 and a low test mean squared error value of 5.03 × 10−4, which indicates a strong relationship between the FOS values and the selected slope parameters. The significant difference in the elapsed time between the traditional method and the developed surrogate model for slope stability analysis highlights the potential benefits of machine learning.

This research applies the reliability-based topology optimization (RBTO) method to a heat transfer problem under variation on coefficient of heat transfer. The design problem is formulated to minimize the structural volume subject to the reliability constraint concerning thermal problem under variation on coefficient of heat transfer. The topology optimization uses level set boundary expressions based on the concept of the phase field method for obtaining clear boundary shapes. The topology optimization is combined with the reliability-based optimization method that is formulated as a single loop algorithm by using Single Loop Single Variable (SLSV) method to reduce computational time. Through numerical example, difference between the optimum configurations on deterministic method and reliability-based method is demonstrated and the effectiveness of this method is discussed.

The jig production process at PT IMS focuses on strength, function and production speed. The known problem is the higher strength and cost of producing jigs because the production process is not much at the trial and error stageThis study analyzes the static strength of the jig side frame bogie of the Jabodebek LRT with the parameters of the safety factor and deformation values.Static loading simulation method testing process used is conducted by using Finite Element Method software. It is counted manually to one of conditions to obtain the voltage value compared to the simulation results. This analysis is in form of measured jig design with safety factor and deformation value. The redesigning aim is weight decreasing efficiency. The simulation obtained result in jig early design is the safety factor value 25 and deformation value 0,045 mm. After jig dimension decreasing has been conducted, the redesigning result for safety factor value is 32 and the maximum deformation is 0,114 mm. The results of both simulations are considered as safe because the safety factor values are more than 2 and the deformation value are less than 0,5 mm. Redesigning by decreasing dimension gives better safety factor yet the deformation value increases.

Seepage criteria based optimal design of water retaining structures with reliability quantification utilizing surrogate model linked simulation-optimization approach