Reliability-based Topology Optimization Research Articles

Reliability-Based Topology Optimization of Thermo-Elastic Structures with Stress Constraint

Traditional topology optimization of thermo-elastic structures is based on deterministic conditions, without considering the influence of uncertainty factors. To address the impact uncertainty on structural strength, a reliability-based topology optimization of thermo-elastic structure with stress constraint is proposed. The probabilistic uncertainty quantities are associated with the structural material property, mechanical loads and the thermal stress coefficient with the topology optimization formulation considering volume minimization and stress constraint. The relaxation stress method combined with normalized p-norm function is adopted to condense whole element stresses into the global stress measurement that approximates the maximum stress. The adjoint variable method is utilized to derive the sensitivity of the stress constraint and the optimization problem is solved by the method of moving asymptote (MMA). Finally, several numerical examples are presented to demonstrate the effectiveness and validity of the proposed approach. Compared with the deterministic design, the reliability design has distinct topological configurations and the optimized structures maintain a higher reliability level.

Convexity-oriented reliability-based topology optimization (CRBTO) in the time domain using the equivalent static loads method

Owing to the wide existence of multi-source uncertainties in practical engineering, the reliability-based design optimization for structural dynamics is particularly important and meaningful. However, compared to the size and shape optimization, it has rarely been considered in topology optimization with the lack of reasonable uncertainty cognition and the difficulty in the sensitivity solution for reliability constraints calculated from dynamic characteristics. In this research, a new design strategy, which combines the equivalent static loads (ESLs) algorithm and the set-theoretical convex method, is proposed for dynamic response topology optimization under convex reliability constraints. Time-varying boundary rules of the layout response are determined using the state-space transformation and the set perturbation theory. As the safety measurement, a new non-probabilistic reliability index (related to the function of dynamic compliance near the peaks) is defined under the hypervolume-ratio principle and its design sensitivity is further analyzed by solving central finite difference (CFD) equations. Three typical optimization cases, including a rectangular plate, an equivalent control surface and a win rib, are presented to demonstrate the validity and effectiveness of the developed methodology.

Quantile-based topology optimization under uncertainty using Kriging metamodel

This paper investigates reliability-based topology optimization (RBTO) and proposes a novel quantile-based topology optimization (QBTO) method. With this method, the traditional RBTO model is transformed into an equivalent quantile-based formulation, which can well avoid the issues existing in RBTO with Monte Carlo simulation (MCS), i.e., stagnation of optimizer due to near zero sensitivities of probabilistic constraint with regard to element densities, huge computational cost on calculating sensitivities, and the discontinuity of failure indicator function. Specially, in QBTO, the sensitivities only require to be calculated at the sample corresponding to the quantile instead of all MCS samples, which can drastically reduce the computational effort. Furthermore, a sequential update strategy of Kriging metamodel is developed to efficiently calculate the quantile by evaluating the true constraint at fewer samples, rather than all MCS samples. The high accuracy and efficiency of QBTO are validated by truss, beam and bridge problems.

Time-dependent reliability-based optimization for structural-topological configuration design under convex-bounded uncertain modeling

In this work, a novel convexity-oriented time-dependent reliability-based topology optimization (CTRBTO) framework is investigated with overall consideration of universal uncertainties and time-varying natures in configuration design. For uncertain factors, the initial static ones are quantified by the convex set model and nodal dynamic responses are then expressed by the convex process model, where both the boundary rules and time-dependency properties are revealed by the full-dimensional convex-set collocation theorem. Unlike the original deterministic constraints in topology optimization schemes, a new convex time-dependent reliability (CTR) index is defined to give a reasonable failure judgment of local dynamic stiffness and impel the overall CTRBTO strategy. In addition, the gradient-based iterative algorithm is utilized to guarantee the computational robustness and the CTR-driven design sensitivities are explicitly analyzed by the Lagrange multiplier method. Several numerical examples are used to illustrate the effectiveness of the proposed method, and numerical results reflect the significance of this study to a certain extent.

Reliability-based topology optimization of piezoelectric smart structures with voltage uncertainty

Currently, most of the piezoelectric structures are designed under deterministic conditions, where the influence of uncertain factors on the output motion accuracy is ignored. In this work, a probabilistic reliability-based topology optimization method for piezoelectric structure is proposed to deal with the working voltage uncertainty. A nested double-loop optimization algorithm of minimizing the total volume while satisfying the reliability requirement of the displacement performance is established, where the PEMAP-P (piezoelectric material with penalization and polarization) model is used for parameterization of stiffness matrix, piezoelectric coupling matrix, and polarization direction. This strategy consists of an inner loop for reliability analysis and an outer loop for topology optimization. The reliability index approach based on most probable point (MPP) is used for realizing the evaluation of reliability constraint in reliability analysis. The sensitivities of reliability constraint with respect to the random variables and design variables are detailed using the adjoint variable method. Typical examples are performed to illustrate the effectiveness of the proposed RBTO method. A comparison of the optimization results for different reliability indexes, standard deviations of the voltage, spring stiffnesses, and displacement limits are conducted, as well as the deterministic topology optimization results.

Two decades review of reliability-based topology optimization developments

This review seeks to classify the different developments of the Reliability-Based Topology Optimization (RBTO) according to their formulations and methods. According to the RBTO formulations, two (...)

A New Design for the Humerus Fixation Plate Using a Novel Reliability-Based Topology Optimization Approach to Mitigate Stress Shielding Effect

A New Design for the Humerus Fixation Plate Using a Novel Reliability-Based Topology Optimization Approach to Mitigate Stress Shielding Effect

Reliability-based topology optimization using stochastic gradients

This paper addresses the computational challenges in reliability-based topology optimization (RBTO) of structures associated with the estimation of statistics of the objective and constraints using standard sampling methods. The aim is to overcome the accuracy issues of traditional methods that rely on approximating the limit-state function. Herein, we present a stochastic gradient-based approach, where we estimate the probability of failure at every few optimization iterations using an efficient sampling strategy. To estimate the gradients of the failure probability with respect to the design parameters, we apply Bayes’ rule wherein we assume a parametric exponential model for the probability density function of the design parameters conditioned on the failure. The design parameters and the parameters of this probability density function are updated using a stochastic gradient descent approach requiring only a small, e.g., $${\mathcal {O}}(1)$$ , number of random samples per iteration, thus, leading to considerable reduction of the computational cost as compared to standard RBTO techniques. We illustrate the proposed approach with a benchmark example that has an analytical solution as well as two widely used problems in structural topology optimization. These examples illustrate the efficacy of the approach in producing reliable designs.

Elasto-Plastic limit analysis of reliability based geometrically nonlinear bi-directional evolutionary topology optimization

This paper presents elasto-plastic limit analysis of reliability-based geometrically nonlinear topology optimization. For this purpose, by reason of uncertainties the volume fraction is considered randomly during optimization. Thus reliability-based design has been considered for solving the problems. To perform reliability-based topology optimization design, the Monte-Carlo simulation method has been applied to calculate the probability of failure, thus the reliability index. Besides, bi-directional evolutionary structural optimization (BESO) method is used to consider the effect of geometrically nonlinear design for elasto-plastic analysis. Plastic behavior is controlled by applying a bound on the plastic limit load multipliers using limit analysis. The adequacy of the proposed method is exhibited by three 2D benchmark problems. 2D models of L-shape beam and U-shaped plate are considered for reliability-based design and geometrically nonlinear analysis topology optimization in case of elastic material. Additionally, 2D and 3D elasto-plastic material models have been considered to demonstrate that the proposed method can find the optimal topology of elasto-plastic models for reliability-based design and geometrically nonlinear analysis.

A novel dynamic reliability-based topology optimization (DRBTO) framework for continuum structures via interval-process collocation and the first-passage theories

In view of the general inertia and damping features as well as the inevitable uncertainty factors in engineering structures, a novel dynamic reliability-based topology optimization (DRBTO) strategy is investigated for time-variant mechanical systems with overall consideration of material dispersion and loading deviation effects. The static interval-set model is first utilized to quantify multi-source uncertainty inputs and the transient interval-process model is then established to characterize unknown-but-bounded response results, which can be readily solved through the proposed interval-process collocation approach combined with a classical Newmark difference scheme. Different from the traditional deterministic design framework, the present DRBTO scheme will directly consider new reliability constraints, for which the non-probabilistic time-variant reliability (NTR) index is mathematically deduced using the first-passage principle. In addition, the issues related to uncertainty-oriented design sensitivity and filtering method are discussed. The usage and effectiveness of DRBTO are demonstrated with three numerical examples.

Reliability-based topology optimization for heterogeneous composite structures under interval and convex mixed uncertainties

With the rapid development of intelligent manufacturing and smart design, the optimal design conception for composite laminates is arising more attentions in both academic and engineering fields. Note that most of the current researches focus on the deterministic laminate design whereas the uncertainty-oriented topological design scheme for heterogeneous composites are rarely considered. In view of this, this study will present a new layout optimization process which combines the hybrid uncertainty quantification analysis and non-probabilistic reliability-based topology optimization (NRBTO). Owing to the sample limitation of multi-source uncertainties in material properties and external loads, the non-statistical convex-theoretical set theory is first explored to determine displacement response bounds of multi-layer structures under anisotropic constitutive relationship. As the safety judgment, a novel reliability index corresponding to the local stiffness features is defined via interval and convex mixed uncertainties, and its explicit expression is further deduced by the set-interference principle. To ensure the robustness and convergence of the large-scale iteration in NRBTO framework, the adjoint equations are also analyzed for the design sensitivity updating and the gradient algorithm is involved as well. Several typical optimization cases are presented to demonstrate the validity and effectiveness of the developed methodology.

Reliability-based acoustical topology optimization of mufflers under noise frequency and temperature uncertainties

A reliability-based topology optimization method is proposed to optimally design mufflers under noise frequency and temperature uncertainties. The optimal mufflers designed with deterministic noise frequency and temperature frequently fail to reduce duct noise effectively when the noise frequency or temperature varies. To resolve this problem, a reliability-based acoustical topology optimization problem is formulated for a two-dimensional expansion chamber muffler. The partition volume inside the muffler is selected as the objective function, and its acoustical reliability and transmission loss are used as constraints so as to design a simple but highly reliable muffler. To expedite finding an optimal solution, a gradient-based optimizer is used; further, to calculate the acoustical reliability and its sensitivity, the first-order reliability method and the weighted uniform sampling method are selectively employed. The formulated design problem is solved for various design conditions, and the acoustical reliabilities of optimal mufflers designed according to the proposed method and the deterministic muffler method are compared. Finally, the noise attenuation performance of the optimally designed muffler is experimentally validated.

Reliability-based topology optimization with stochastic heterogeneous microstructure properties

With advanced manufacturing processes, many modern structural designs need to include microscale uncertainties of material properties and microstructures that can affect overall performances, such as microelectromechanical systems, micro–opto–electro–mechanical systems, and micro-optical electronics systems. Topology design optimization at this scale must consider uncertainties from material microstructures as the scale of the structure and material microstructure is comparable. Very few studies consider microstructure uncertainties in topology design due to their scales are much different in classical mechanical/civil engineering. A novel framework of reliability-based topology optimization is proposed to specifically address this gap for the design. The microscale uncertainties of heterogeneous materials are quantified by the explicit mixture random field model. Then, the material distribution is optimized by a heuristic updating scheme, and sensitivities of reliability constraints are calculated using the adjoint design-point-based importance sampling method. The proposed methodology simultaneously considers microscale hierarchical uncertainties in microstructures and material property variations for each phase. The feasibility of the framework is demonstrated by several numerical examples with different multi-phase materials. Compared with deterministic topology optimization and classical topology optimization with first-order reliability method approximations, the optimal design obtained from the proposed method can achieve accurate target reliability with minimized limit state function evaluations.

Reliability-based topology optimization for freely vibrating continuum structures with unknown-but-bounded uncertainties

This study investigates a novel reliability-based topology optimization framework to determine optimal material configurations for freely vibrating continuum structures with unknown-but-bounded uncertainties. Firstly, the optimization formulation for freely vibrating continuum structures with fundamental eigenfrequency and eigenfrequency gap constraints is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the natural eigenfrequencies under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization model and the performance measure approach is proposed to overcome the convergence difficulties. Meanwhile, the sensitivity analysis is further discussed based on the chain rule, in which the sensitivities of the target performance measure to the bounds of eigenfrequencies are calculated and the sensitivities of the eigenfrequencies with respect to the design variables are deduced. Numerical examples are eventually given to demonstrate the validity of the developed reliability-based topology optimization methodology.

An efficient approach to reliability-based topology optimization for the structural lightweight design of planar continuum structures

Abstract 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.

Reliability-based topology optimization framework of two-dimensional phononic crystal band-gap structures based on interval series expansion and mapping conversion method

• The NRBTO scheme is presented for PnCs where the interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions. • A mapping conversion method is proposed to eliminate the gray-scale elements when relatively coarse grids are used. • The maximum band-gap with a specific frequency region and a filling fraction is obtained while ensuring reliability requirements considering the uncertainty in the optimization parameters. In this study, a non-probabilistic reliability-based topological optimization (NRBTO) method is investigated for the optimized design of the band-gap structure of two-dimensional phononic crystals (PnCs), in which unknown but bounded (UBB) uncertainty issues from the material uncertainty are taken into account. A measurement metric is first defined based on the interval set-theoretical model, which may transform the band-gap setting into a non-probabilistic reliability format. The interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions between frequency measures and design variables. The method of moving asymptote (MMA) is adopted as well to iteratively solve the reliability-oriented band-gap optimization problem for PnCs. To effectively tackle the gray-scale units, a mapping conversion method is involved eventually. The usage and validity of the present methodology are verified through three examples, and numerical results show that considering the uncertainty effect of UBB throughout the topology optimization process has an important impact on the final layout scheme. Additionally, the optimization of the out-of-plane modes of PnCs (both the band-gap upper bound and the filling rate constraint are considered) indicates that the topological configuration may always change accompanied with the filling rate increases.

Nonlinearity investigation of reliability-based topology optimization strategies with application to total hip replacement

In literature, the topology optimization can be divided into two main models. The first model is called Deterministic Topology Optimization (DTO) producing a single configuration for a given design space. The second one is called Reliability-Based Topology Optimization (RBTO) generating several layouts. In our previous work, two approaches considering the concept of Inverse Optimum Safety Factors (IOSF) were elaborated and only applied to the normal distribution being linear distribution. In this work, a nonlinearity investigation is presented to compare between the linear and nonlinear distribution. The RBTO developments are applied to the total hip replacement to provide suitable hollow stems at the conceptual design stage. The nonlinearity presented here, is related to the types of the random variable distributions. The most common distributions such as normal, lognormal, uniform and Weibull are considered here to perform the investigation. The results show the nonlinearity effect on the output parameter values, but lead to almost similar layouts of the resulting hollow stems in several cases. In certain types of distributions such as Weibull, the changes on the input parameters are very variant. At certain values of the reliability index, some input parameters of material properties exceeded their limits and the algorithm stopped.

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

In general, two types of topology optimization models can be found in literature. The first type is called Deterministic Topology Optimization (DTO) leading to a single layout when considering a given design space. The second type is called Reliability-Based Topology Optimization (RBTO) leading to various solutions. In our previous work, two strategies based on the Inverse Optimum Safety Factors (IOSF) were established and applied simply to the normal (Gaussian or linear) distribution law. In this work, linear and nonlinear (normal and lognormal) distribution RBTO cases are compared. A bridge structure is considered a numerical application to perform this comparison where different layouts at the same level of reliability can be found. The numerical results show that regarding the distribution laws, certain output parameters and safety factor values are affected. This change can affect the resulting topology layouts as well as the output parameter such as the compliance. When rising the values of the reliability index, the values of the compliance become larger and the volume values become lesser for the lognormal distribution when comparing to the normal one.

Reliability-based topology optimization as effective strategy for additive manufacturing: Influence study of geometry uncertainty on resulting layouts

The effectiveness of Additive Manufacturing (AM) is to fabricate very complex configurations that can be resulting layouts from topology optimization. The topology optimization can be classified into two models: Deterministic Topology Optimization (DTO) and Reliability-Based Topology Optimization (RBTO). The DTO may lead to unrealistic designs since it provides a single solution for a given design space. However, the RBTO is considered as a topology generator which provides several layouts. This way the designer can overcome the manufacturing constraint problems such as overhang, minimum feature size, anisotropy …, that may lead to a failure during the production process. Two RBTO strategies based on the Inverse Optimum Safety Factor (IOSF) are presented in this work: Objective-Based IOSF Approach and Performance-Based IOSF Approach. These approaches can be efficiently used as a topology regularization procedure. In addition, to consider the geometry uncertainty, different layouts can be obtained when increasing the reliability index values in order to provide more choices for designers. The geometry uncertainty is applied here in a manner to not affect the boundary conditions which can lead to change the structure’s functioning. The resulting RBTO layouts in this work, are too organic which leads to manufacturability problems. To use the conventional manufacturing procedures, classic CAD primitives must be used to simplify the geometry, that may lead to lose the optimality. Because of this manufacturability difficulty, AM is then obligatory. The RBTO being a good design tool for AM, paves the way to the next developments to overcome other fabrication constraints considering the reliability index values as a topology regulator.

Topology optimization of elasto-plastic structures under reliability constraints: A first order approach

The objective of this study is to propose a relatively simple and efficient method for reliability based topology optimization for structures made of elasto-plastic material. The process of determining the optimal topology of elasto-perfectly plastic structures is associated with the removal of material from the structure. Such a process leads to weakening of structural strength and stiffness causing at the same time increase the likelihood of structural failure. An important aspect of engineering design is to track this probability during the optimization process and not allow the structure safety to exceed a certain level specified by the designer. The purpose of this work is to combine the previously developed yield-limited topology optimization method with reliability analysis using first order approach. Effectiveness of the proposed methodology is demonstrated on benchmark problems proposed by Rozvany and Maute, and the elasto-plastic topology design of L-shape structure which is frequently used in different approaches for stress constrained topology optimization.