Deterministic Topology Optimization Research Articles

A decoupled probabilistic constrained topology optimization method based on the constraint shift

AbstractTopology optimization (TO) has recently emerged as an advanced design method. To ensure practical reliability in the design process, it is imperative to incorporate considerations of uncertainty. Consequently, performing reliability analysis (RA) during the design phase becomes necessary. However, RA itself constitutes an optimization problem. Combining these two optimization problems can result in inefficiency. To address this challenge, we propose a decoupled approach that integrates deterministic topology optimization (DTO) and RA cycles. The reliability‐based stress‐constrained TO (RBSCTO) problem is considered in this paper. The DTO constraint is derived based on shifting vectors derived from the previous cycle's RA outcomes, enabling low‐reliability constraint shift towards the feasible direction. The DTO is solved based on solid‐isotropic‐material‐with‐penalization (SIMP) and augmented Lagrangian method. Meanwhile, the optimization problem in RA is addressed using finite differences and the interior point method. To reduce the errors resulting from linear approximation and optimization in RA when the target reliability is very low, an outlier handling method is employed. Meantime, we utilize a probabilistic neural network to enhance the efficiency of reliability assessment. Comparative studies against traditional methods across four RBSCTO tasks are demonstrated to validate its effectiveness. Monte Carlo simulations are used to validate the reliability of results.

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.

Uncertainty‐oriented thermoelastic topology optimization with stress constraint

AbstractThe material redistribution abilities of traditional deterministic topology optimization are effective in addressing stress‐related design issues in thermal elastic structures. However, uncertainties are inevitable in real‐world. The structural strength of a design, achieved through deterministic topology optimization, is highly susceptible to these uncertainties, which may result in failure. This article introduces a novel method for topology optimization in stress‐constrained thermoelastic structures, taking into account the uncertainties associated with heat sources and loads. We employ the Kieisselmeier–Steinhauser function to aggregate the stress constraint when constructing the performance function. In order to improve optimization efficiency, the sequential optimization and reliability assessment method is used to decouple the double‐layer loop reliability‐based topology optimization. Initially, we derive the derivative of stress‐based performance functions with respect to heat source and load uncertainty variables, thereby facilitating the use of modified chaos control for assessing structural reliability and imposing constraints. The adjoint method and chain rule are utilized to obtain the derivative information of the performance function with respect to density variables, guiding the topology updates. We present five design examples to demonstrate the effectiveness of the presented method. Monte Carlo simulations for the optimized results are performed to show that the presented method can obtain a structure that meets reliability requirements.

Robust Topology Optimization of Coated Structures with Surface Layer Thickness Uncertainty Considered

The rapid development of additive manufacturing has made coated structures an innovative configuration with high design flexibility. However, poor forming accuracy and surface roughness during manufacturing will cause uncertainty in surface layer thickness, which results in structure performance deviation and failure to achieve the expected goals. This paper proposes a robust topology optimization method for coated structures considering the surface layer thickness uncertainty to obtain high-quality designs that can resist disturbance by uncertainties. First, an erosion-based approach is used to establish the model of the coated structure surface layer. Second, modeling the surface layer thickness uncertainty applies a random field whose dimensionality of the random fields is reduced by the Expansion Optimal Linear Estimation (EOLE) method. Then, minimizing the weighted sum of the mean and standard deviation of structural compliance is taken as the optimization objective, and robust topology optimization considering uncertainty is established. Finally, estimate the stochastic response by the perturbation technique, then the sensitivity of the objective function with respect to the design variables is derived. Numerical examples show that the structural design obtained with the proposed method has a stronger resistance to uncertainty than the deterministic topology optimization method, proving the method’s effectiveness in this paper.

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.

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.

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.

Experimental characterisation and data-driven modelling of unsteady wall pressure fields induced by a supersonic jet over a tangential flat plate

This work deals with the investigation and modelling of wall pressure fluctuations induced by a supersonic jet over a tangential flat plate. The analysis is performed at several nozzle pressure ratios around the nozzle design Mach number, including slightly over-expanded and under-expanded conditions, and for different radial positions of the rigid plate. Pitot measurements and flow visualizations through the background oriented schlieren technique provided a general overview of the aerodynamic interactions between the jet flow and the plate at the different regimes and configurations. Wall pressure fluctuations were measured using a couple of piezoelectric pressure transducers flush mounted over the plate surface. The spectral analysis has been carried out to clarify the effect of the plate position on the single and multivariate wall pressure statistics, including the screech tone amplitude. The experimental dataset is used to assess and validate a surrogate model based on artificial neural networks. Sound pressure levels and coherence functions are modelled by means of a single fully connected network, built on the basis of a recently implemented fully deterministic topology optimization algorithm. The metamodel uncertainty is also quantified using the spatial correlation function. It is shown that the flow behaviour as well as the screech and broadband noise signatures are significantly influenced by the presence of the plate, and the effects on spectral quantities are correctly reproduced by the proposed data-driven model that provides predictions in agreement with the available data.

An Efficient Sequential Strategy for Non-probabilistic Reliability-based Topology Optimization (NRBTO) of Continuum Structures with Stress Constraints

In the study, the sequential strategy is proposed for non-probabilistic reliability-based topology optimization (NRBTO) of continuum structures with stress constraints. Firstly, the qp-relaxation criterion is involved to circumvent the stress singularity. Then the deterministic topology optimization formulation by Stress-Influence-Function (SIF) approach is described. Uncertainty quantification analysis is conducted to determine the feasible bounds of the elemental stresses under unknown-but-bounded uncertainties. For safety reasons, the non-probabilistic reliability concept is introduced in the optimization formulation. The sequential strategy is proposed to handle the NRBTO problem and thus the original nested process of topology optimization and reliability analysis was decoupled, in which the update of design variables and the shift of the strength feature in the stress influence function are conducted alternately in the optimization process. The Performance Measure Approach (PMA) is applied to calculate the shift value of the strength feature and a shift value function is presented to avoid the numerical oscillations caused by the large variation of strength feature. Three numerical examples are eventually given to illustrate the applicability and validity of the developed sequential strategy for NRBTO with stress constraints.

Reliability‐based topology optimization of continuum structure under buckling and compliance constraints

AbstractBuckling failure mainly occurs in slender rod structure under pressure, which incurs great concern and should be addressed in topology optimization (TO). However, due to the internal defects of material and the external environment interference of engineering system, the practical engineering structure inevitably produces many uncertain factors in the manufacturing and service stages. Therefore, a reliability‐based topology optimization (RBTO) model under buckling and compliance constraints is established to operate the structural lightweight design considering reliability and stability requirements. To reduce the number of probabilistic constraints, the Kreisselmeier–Steinhauser aggregation function is adopted for combining the multiple constraints into a single smooth and differentiable constraint. Then, a single‐loop approach combining modified chaos control strategy is applied to guarantee the robustness, effectiveness, and computational efficiency of RBTO. In addition, the sensitivities of the probabilistic constraint with respect to design and random variables are deduced, and the accuracy of sensitivities is verified by finite difference method. Finally, the validity of the RBTO method is verified by comparing with deterministic TO, in which three numerical examples are employed to demonstrate the robustness and computational accuracy.

Reliability-Based Topology Optimization: An Extension of the SESO and SERA Methods for Three-Dimensional Structures

This study takes an approach to reliability-based topology optimization (RBTO) for 3D structures by applying an expansion of smoothing evolutionary structural optimization (SESO) and sequential element rejection and admission (SERA) for three-dimensional optimization. In the search for the stable optimal solution, and a more reliable structure, we present a performance index with the ability to monitor the evolutionary optimization procedure and adopt the filtering scheme usually applied in solid isotropic material with penalization (SIMP). The limit state functions are the maximum displacement constraints imposed in the topology optimization procedure and a structure’s performance control; a comparative analysis of the deterministic topology optimization (DTO) with the RBTO models is also explored. The obtained results suggest the importance of using the RBTO concept in 3D structures as part of the design analysis process.

Reliability based geometrically nonlinear bi-directional evolutionary structural optimization of elasto-plastic material

The aim of this paper is to integrate the reliability-based analysis into topology optimization problems. Consequently, reliability-based topology optimization (RBTO) of geometrically nonlinear elasto-plastic models is presented. For purpose of performing (RBTO), the volume fraction is considered reliable since that the application of (RBTO) gives different topology in comparison to the deterministic topology optimization. The effects of changing the prescribed total structural volume constraint for deterministic designs and changing the reliability index for probabilistic designs are considered. Reliability index works as a constraint which is related to reliability condition added into the volume fraction and it is calculated using the Monte-Carlo simulation approach in the case of probabilistic design. In addition, bi-directional evolutionary structural optimization (BESO) method is utilized to study the effect of geometrically nonlinear elasto-plastic design. The plastic behavior can be controlled by defining a limit on the plastic limit load multipliers. The suggested work's efficiency is demonstrated via a 2D benchmark problem. In case of elastic material, a 2D model of U-shape plate is used for probabilistic design of linear and geometrically nonlinear topology optimizations. Furthermore, a 2D elasto-plastic model is considered for reliability-based design to demonstrate that the suggested approach can determine the best topological solution.

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.

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.

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.

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.

Reliability-Based Topology Optimization Using Two Alternative Inverse Optimum Safety Factor Approaches: Application to Bridge Structures

Introduction. The Deterministic Topology Optimization model provides a single solution for a given design space, while the Reliability-Based Topology Optimization model provides several reliability-based topology layouts with high-performance levels. The objective of this work is to develop two strategies that can provide the designer with two categories of resulting topologies. Materials and Methods. Two alternative approaches based on the Inverse Optimum Safety Factor are developed: the first one is called the Objective-Based IOSF Approach and the second one is called Performance-Based IOSF Approach. When dealing with bridge structures, the uncertainty on the input parameters (boundary conditions, material properties, geometry, etc.) and also output parameters (compliance, etc.) should not be ignored. The sensitivity analysis is the fundamental idea of both developed approaches, identifies the role of each parameter on the structural performance. In addition, the optimization domain choice is important when eliminating material that should not affect the structure functioning. Results. Two numerical examples on a 2D bridge structure are presented to demonstrate the efficiency of the developed approaches. When considering a certain reliability level, the Reliability-Based Topology Optimization leads to two different configurations relative to the Deterministic Topology Optimization one. When increasing the reliability levels, the quantity of materials decreases that leads to an increase in the number of holes in the structures. Discussion and Conclusion. In addition to their simplified implementation, the developed alternative approaches can be considered as two generative tools to produce two different categories (families) of solutions where an alternative choice between two functions (objective/performance) is presented.

Topology optimization under uncertainty using a stochastic gradient-based approach

Topology optimization under uncertainty (TOuU) often defines objectives and constraints by statistical moments of geometric and physical quantities of interest. Most traditional TOuU methods use gradient-based optimization algorithms and rely on accurate estimates of the statistical moments and their gradients, e.g., via adjoint calculations. When the number of uncertain inputs is large or the quantities of interest exhibit large variability, a large number of adjoint (and/or forward) solves may be required to ensure the accuracy of these gradients. The optimization procedure itself often requires a large number of iterations, which may render TOuU computationally expensive, if not infeasible. To tackle this difficulty, we here propose an optimization approach that generates a stochastic approximation of the objective, constraints, and their gradients via a small number of adjoint (and/or forward) solves, per optimization iteration. A statistically independent (stochastic) approximation of these quantities is generated at each optimization iteration. The total cost of this approach is only a small factor larger than that of the corresponding deterministic topology optimization problem. We incorporate the stochastic approximation of objective, constraints, and their design sensitivities into two classes of optimization algorithms. First, we investigate the stochastic gradient descent (SGD) method and a number of its variants, which have been successfully applied to large-scale optimization problems for machine learning. Second, we study the use of the proposed stochastic approximation approach within conventional nonlinear programming methods, focusing on the globally convergent method of moving asymptotes (GCMMA). The performance of these algorithms is investigated with structural design optimization problems utilizing a solid isotropic material with penalization (SIMP), as well as an explicit level set method. These investigations, conducted on both two- and three-dimensional structures, illustrate the efficacy of the proposed stochastic gradient approach for TOuU applications.