Sequential Approximate Programming Research Articles

Active set strategy-based sequential approximate programming for reliability-based design optimization

To improve the evaluation efficiency of failure probability in RBDO models with uncertainty, many RIA-based, PMA-based methods have evolved as a powerful procedure, including the modified reliability index approach (MRIA), PMA two-level, PMA with sequential approximate programming (SAP). However, MRIA may encounter inefficiency and instability when applied to complex concave performance functions, and so does PMA two-level, not for PMA with SAP. The active set strategy-based SAP (ASS-based SAP) for PMA is proposed to accelerate computational efficiency through establishing an active set strategy and a deciding factor. The active set strategy defined by using inequality is to identify the feasible most probable target point (MPTP) in the inner loop. The decision factor integrates the reliability index and the active set strategy to quickly renew the active constraints in the outer loop. The reliability assessment and outer optimization are driven simultaneously, thereby the computational efficiency is strengthened. Numerical examples are compared with other reliability methods to demonstrate the excellent performance of the proposed method in efficiency and robustness. Results also show that the proposed method has the ability to solve complex RBDO problems.

Structural optimization considering dynamic reliability constraints via probability density evolution method and change of probability measure

The present paper presents a framework for solving dynamic reliability–based design optimization (DRBDO) problems. The proposed approach is based on a sequential approximate programming technique, where the solution of the original design problem is obtained by converting it into a sequence of sub-optimization problems with a simple explicit algebraic structure. The technique is combined with the probability density evolution method (PDEM) for the purpose of estimating the reliability and/or the failure probability at the different designs in an efficient manner. The change of probability measure (COM), a strategy based on the Radon-Nikodym derivative, is employed to estimate the sensitivity information required by the optimization scheme. Based on this strategy, the sensitivity information can be obtained without any additional function evaluation. Thus, the numerical efforts associated with the reliability assessment and sensitivity analysis can be considerably reduced. The results of the numerical examples indicate that the proposed method is an effective and efficient tool for solving a class of DRBDO problems.

A second order SAP algorithm for risk and reliability based design optimization

This paper presents a decoupling approach for the efficient solution of Risk Optimization (RO) and Reliability Based Design Optimization (RBDO) problems. The proposed approach, SAP2nd, is a Sequential Approximate Programming (SAP) technique including second order terms obtained with the BFGS (Broyden-Fletcher-Goldfarb-Shanno) approximation for the Hessian. A first advantage of SAP2nd is that any reliability analysis method can be employed for the evaluation of the probabilities of failure and its sensitivities. Here, Polynomial Chaos Expansion (PCE) is employed for this purpose. Several benchmark problems are solved to study the efficiency, robustness and accuracy of SAP2nd. It is demonstrated that the inclusion of second order terms leads to: (i) a much more stable algorithm in comparison to a first order SAP algorithm, i.e. it was able to avoid convergence issues arising from cycling, and (ii) a more efficient algorithm since SAP2nd reduced the computational effort, in both RO and RBDO problems, when compared to the coupled PCE algorithm previously proposed by the authors. The use of PCE for the evaluation of the probabilities of failure and its sensitivities allowed SAP2nd to achieve much more accurate results when compared to FORM based approaches, requiring the same order of computational effort. Finally, SAP2nd using PCE for reliability and sensitivity analysis is well suited for RO and RBDO problems where the drawbacks of FORM based approaches prevail, especially cases with highly nonlinear limit state function and non Gaussian random variables.

Topology optimization via sequential integer programming and Canonical relaxation algorithm

The mathematical essence of structural topology optimization is large-scale nonlinear integer programming. To overcome its huge computational burden, a popular way is to relax the 0–1 variable constraints and transform the integer programming problem into a continuous variable programming problem which can be solved by using gradient based mathematical programming methods. To cope with the variable transformation, the well-known SIMP (Solid Isotropic Material with Penalty) method introduces interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programming is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory (CDT) by Gao and Ruan (2010) together with Sequential Approximate Programming approach under the classic structural topology optimization formulations.Following the Sequential Approximate Programming (SAP) frame from the structural optimization, the present paper firstly utilizes sensitivity information to construct the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems for the topology optimization. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Kuhn–Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients assert that the Canonical relaxation algorithm can obtain approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases.Because move limit strategies play a key role in many search algorithms of structural optimization, this paper combines one of two different move limit strategies within the new method. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem. And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way. Beyond that, the new method can achieve integer solutions when combined with the move limit strategy of controlling volume fraction parameter. It can deal with much more design variables than the general branch-and-bound method and makes no use of any sensitivity threshold or heuristic stabilization scheme during the iterative process in comparison with BESO method. Finally, the new method in this paper can be further developed as a general solver for these large-scale discrete variable structural topology optimization problems.

Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization

• The sequential approximate programming (SAP) is enhanced using SORM for RBDO. • The proposed enhanced SAP-based SORM has the same efficiency as well as the FORM-based RBDO. • ESAP is applied several nonlinear, applicable RBDO problems and reliability-based topology optimization. • Results demonstrated that the ESAP shows the superiority of performance compared to RIA, PMA, SAP, and SORA. Second-order reliability method (SORM) can provide sufficient accuracy for evaluating the probabilistic constraints in reliability-based design optimization (RBDO). However, the application of SORM in RBDO significantly increases the computational burden, as it is necessary to calculate the second-order sensitivities of the performance function. In order to achieve equal efficiency to that of the first-order reliability method-based RBDO approach, enhanced sequential approximate programming (ESAP) is proposed by implementing the SORM-based RBDO method. Based on the diagonal quadratic approximation method, the Hessian matrix is calculated without generating additional computational costs for providing the design sensitivity analysis of probabilistic constraints within the same iterations. Furthermore, ESAP is applied to the reliability-based topology optimization domain, and five numerical benchmark RBDO problems with two complex engineering examples are studied. The proposed ESAP is compared with other RBDO methods, including the reliability index approach, performance measure approach, sequential optimization and reliability assessment method, and SAP, and the results demonstrate the superiority of the proposed ESAP.

A hybrid sequential approximate programming method for second-order reliability-based design optimization approach

The second-order reliability method (SORM) is an effective tool to evaluate the reliability, but its application for reliability-based design optimization faces unbearable computational cost. In this study, a new hybrid sequential approximate programming (HSAP) method is developed to calculate the optimum efficiently by developing a distance-checking criterion and a convex approximate method. Since the distance-checking criterion identifies the feasibility of the probabilistic constraint effectively, the proposed method combines the efficiency of the sequential approximate programming method and the accuracy of SORM. The convex approximate method is also constructed using the sensitivity and function value of the probabilistic constraint. So no additional computational cost is required in the optimization process. Five illustrative examples, including two mathematical examples and thee practical engineering examples, demonstrate the efficiency and accuracy of the proposed HSAP.

Simultaneous parameter and tolerance optimization of structures via probability-interval mixed reliability model

Both structural sizes and dimensional tolerances strongly influence the manufacturing cost and the functional performance of a practical product. This paper presents an optimization method to simultaneously find the optimal combination of structural sizes and dimensional tolerances. Based on a probability-interval mixed reliability model, the imprecision of design parameters is modeled as interval uncertainties fluctuating within allowable tolerance bounds. The optimization model is defined as to minimize the total manufacturing cost under mixed reliability index constraints, which are further transformed into their equivalent formulations by using the performance measure approach. The optimization problem is then solved with the sequential approximate programming. Meanwhile, a numerically stable algorithm based on the trust region method is proposed to efficiently update the target performance points (TPPs) and the worst case points (WCPs), which shows better performance than traditional approaches for highly nonlinear problems. Numerical results reveal that reasonable dimensions and tolerances can be suggested for the minimum manufacturing cost and a desirable structural safety.

Efficient algorithm for probability-based design optimisation of complex structures and related issues

The probabilistic structural design optimisation problem can be carried out using either the conventional reliability index approach (RIA) or the performance measure approach (PMA). The direct approach for either RIA or PMA is to form a two-level optimisation by simply connecting algorithms of probabilistic calculation and design optimisation, which is computationally prohibitive for complex structures. The authors used the sequential approximate programming (SAP) strategy for RIA and extended to PMA. Examples show that SAP for both RIA and PMA reduce the total number of function evaluations notably, whereas PMA with SAP is more efficient and less dependent on probabilistic distributions and thus serves as a promising approach for complex structures. This study introduces the two SAP approaches in a unified way and shows that in these approaches, Taylor linear approximation is applied to both design variables and random variables. Two kinds of variables are treated in the same way from the mathematical viewpoint.

Reliability-based design optimization of adhesive bonded steel–concrete composite beams with probabilistic and non-probabilistic uncertainties

It is meaningful to account for various uncertainties in the optimization design of the adhesive bonded steel–concrete composite beam. Based on the definition of the mixed reliability index for structural safety evaluation with probabilistic and non-probabilistic uncertainties, the reliability-based optimization incorporating such mixed reliability constraints are mathematically formulated as a nested problem. The performance measure approach is employed to improve the convergence and the stability in solving the inner-loop. Moreover, the double-loop optimization problem is transformed into a series of approximate deterministic problems by incorporating the sequential approximate programming and the iteration scheme, which greatly reduces the burdensome computation workloads in seeking the optimal design. The validity of the proposed formulation as well as the efficiency of the presented numerical techniques is demonstrated by a mathematical example. Finally, reliability-based optimization designs of a single span adhesive bonded steel–concrete composite beam with different loading cases are achieved through integrating the present systematic method, the finite element analysis and the optimization package.

Non-probabilistic Reliability Based Topology Optimization Design of Two-material Structures

Using the convex model description and the quantified definition of the non-probabilistic reliability,topology optimization of two-material structures considering uncertainties in material properties,geometry and loads is studied.Based on the extended penalization scheme of relative density,a continuous minimax optimization problem,which satisfies the material volume constraint and the reliability requirement,is presented for finding the united distribution of two candidate materials.The minimax problem is solved in the sequential approximate programming manner by embedding a direct iterative formula and the method of moving asymptotes(MMA).The computation cost is thus greatly decreased since the original problem is transformed into a series of deterministic ones.Numerical examples show that system uncertainties may have considerable effects on the optimal united layout of two-material structures.The results prove that a powerful approach for the topology design of two-material structures is furnished by the proposed numerical technique.

Convergence control of the iterative procedure for performance-measure-based probabilistic structural design optimization

The advanced mean value (AMV) iterative scheme is commonly used to evaluate probabilistic constraints in the performance measure approach (PMA) for probabilistic structural design optimization (PSDO). However, the iterative procedure of PSDO may fail to converge. In this article, the chaotic dynamics theory is suggested to investigate and attack the non-convergence difficulties of PMA-based PSDO. Essentially, the AMV iterative formula forms a discrete dynamical system with control parameters. If the AMV iterative sequences present the numerical instabilities of periodic oscillation, bifurcation, and even chaos in some control parameter interval, then the outer optimization loop in PSDO cannot converge and acquire the correct optimal design. Furthermore, the stability transformation method (STM) of chaos feedback control is applied to perform the convergence control of AMV, in order to capture the desired fixed points in the whole control parameter interval. Meanwhile, PSDO is solved by the approaches of PMA two-level and PMA with the sequential approximate programming (SAP)—PMA with SAP. Numerical results of several examples illustrate that STM can smoothly overcome the convergence failure of PSDO resulting from the periodic oscillation, bifurcation, and chaotic solutions of AMV iterative procedure for evaluating the probabilistic constraints. Moreover, the probabilistic optimization with uniform random variables, which is widely recognized as a highly nonlinear and fairly difficult problem, can be attacked through introducing the strategy of chaos control. In addition, the approach of PMA with SAP combining with STM is quite effective and efficient.

Further study on efficiency of sequential approximate programming for probabilistic structural design optimization

Compared to the traditional deterministic optimization based on safety factors, the probabilistic structural design optimization (PSDO) is considered to be a more rational design philosophy because of reasonable account of uncertainties in material properties, loading, boundary condition and geometry, and even mathematical representation of the system model. However, it is well known that the computation for PSDO can be prohibitive when the associated function evaluation is expensive. As a result, many approximate PSDO methods have been developed in recent literatures. In previous works, we developed two sequential approximate programming (SAP) strategies for PSDO based on reliability index approach (RIA) and performance measure approach (PMA). In PMA with SAP, a sequence of approximate programming of PSDO was formulated and solved before the final optimum was located. In each subprogramming, rather than relying on direct linear Taylor expansion of the probabilistic performance measure (PPM), we developed a formulation for approximate PPM at the current design point and used its linearization instead. The approximate PPM and its sensitivity were obtained by approximating the optimality conditions in the vicinity of the minimum performance target point (MPTP). The present paper further elaborates the SAP for PMA. In addition to detailed description of the algorithm, we present error analysis and show that in the ɛ-vicinity of optimum design and corresponding MPTP, the difference between the Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order of ɛ. Four examples are optimized by six algorithms appearing in recent literatures for efficiency comparison. The effect of target reliability index and statistical distribution of random variables on the comparison is discussed. The third example shows that PMA with SAP performs well even for the problem for which reliability index calculation by first order reliability method (FORM) fails. Finally, the fourth example with 144 probabilistic constraints is shown to demonstrate the effectiveness of PMA with SAP. All example results illustrate that with the algorithm PMA with SAP, we get concurrent convergence of both design optimization and probabilistic performance measure calculation, which agrees well with the error analysis.

A sequential approximate programming strategy for performance-measure-based probabilistic structural design optimization

The evaluation of probabilistic constraints in the probabilistic structural design optimization (PSDO) problem can be carried out using either the conventional reliability index approach (RIA) or the more recently proposed performance measure approach (PMA). The latter is sometimes regarded as more efficient and stable with less dependence on probabilistic distribution types of random variables. Herein we apply PMA to evaluate probabilistic constraints and solve the PSDO using the sequential approximate programming (SAP) strategy. The sequential linear programming (SLP) approach in structural optimization achieves optimum design by solving a sequence of sub-programming problems iteratively. Each sub-programming consists of a linear objective subjected to a set of linear constraints, all based on approximation of the original objective and constraints at the current design. Implementing this approach to PSDO in a straightforward manner will require linear approximation of probabilistic performance measure and its sensitivities, thus implying large number of iterations and huge computational cost. In our new approach, rather than using the linear expansion of the probabilistic performance measure, we propose a formulation for an approximate probabilistic performance measure and its linearization. Obtained based on optimality conditions in the vicinity of the minimum performance target point (MPTP), the approximate measure and its sensitivity enables efficient sub-programming step. We update MPTP simultaneously at each step using iterative formula from the advanced mean-value (AMV) method and apply it as the initial estimate for the next step. As the sub-programming steps are no longer the linear approximation of the original problem, this is essentially a sequential approximate programming (SAP) approach. Through the application of this method to the most relevant examples frequently cited in similar studies, we compare its efficiency to other existing approaches and illustrate the concurrent convergence of both optimization and probabilistic performance measure calculation.

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.

Tolerance synthesis by a new method for system reliability-based optimization

The performance–cost ratio of a product may be significantly improved by a better tolerance assignment. Current practice guided by experience usually leads to a conservative and costly design. Because yield in mechanical engineering is analogous to system reliability in structural engineering, this article applies a system reliability-based optimization formulation to tolerance synthesis and presents a sequential approximate programming method. This method is an extension of the approach for the optimization with component reliability constraints. Two often-quoted examples are analyzed in detail to illustrate its effectiveness and a strategy for tolerance assignment is recommended.