Decomposition Approach for Reliability-Based Multidisciplinary Design Optimization

An efficient approach for reliability-based design optimization (RBDO) of series systems is presented. A modified formulation of the RBDO problem is employed in which the reliabilities of the failure modes of a system are included in the set of the design variables. This allows for an optimal apportionment of the reliability of a system among its failure modes. A sequential optimization and reliability assessment method is used to efficiently determine the optimum design. Here, the constraints on the reliabilities of the failure modes of the RBDO problem are replaced with approximate deterministic constraints. The proposed approach is demonstrated on two example problems that have been solved in previous studies without optimizing the required reliability levels of the failure modes. The first example performs RBDO to a cantilever beam with a rectangular cross section under lateral and vertical loads. The constraints are on the strength and the maximum allowable displacement. The second example performs RBDO to a cable-stayed box girder with five failure modes. Compared to the designs found by previous studies, the new approach finds designs with lower mass but without reducing the system reliability.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

The low order Taylor’s series expansion was employed in this study to estimate the reliability indices of the failure criteria for reliability-based design optimization of a linear static structure subjected to random loads and boundary conditions. By taking the advantage of the linear superposition principle, only a few analyses of the structure subjected to unit-loads are needed through the entire optimization process to produce acceptable results. Two structural examples are presented in this study to illustrate the effectiveness of the proposed approach for reliability-based design optimization: one deals with a truss structure subjected to random multiple point constraints, and the other conducts shape design optimization of a plane stress problem subjected to random point loads. Both examples were formulated and solved by the finite element method. The first example used the penalty method to reformulate the multiple point constraints as external loads, while the second example introduced an approach to propagate the uncertainty linearly from the nodal displacement vector to the nodal von Mises stress vector. The final designs obtained from the reliability-based design optimization were validated through Monte Carlo simulation. This validation process was completed with only four unit-load analyses for the first example and two for the second example.

Based on the reliability-based optimization design theory, the reliability sensitivity technique and the robust design method, the reliability-based robust design of rubbing rotor system is extensively discussed and a numerical method for reliability-based robust design is proposed. The reliability sensitivity is added to the reliability-based optimization design model and the reliability-based robust design is described as a multi-objection optimization. On the condition of known first four moments of basic random variables, the respective program can be used to obtain the reliability-based robust design information of rubbing rotor system accurately and quickly using the fourth moment technique. According to the numerical results, the approach proposed is a convenient and practical reliability-based roust design method.

An adaptive decoupling approach for reliability-based design optimization

On Probabilistic Approaches for Reliability-Based Design Optimization (RBDO)

To reduce the cost of prototype and physical test, CAE analysis has been widely used to evaluate the vehicle performance during product development process. Combining CAE analysis and optimization approach, vehicle design process can be implemented more efficiently with affordable cost. Reliability based design optimization (RBDO) formulation considers variations of input variables, such as component gauges and material properties. As a result, the design obtained by using RBDO is more reliable and robust compared to those by deterministic optimization. The RBDO process starts from running simulation at DOE sampling data points, generating surrogate models (response surface) and performing robust and reliability based design optimization on the surrogate models by using Monte Carlo simulation. This paper presents a RBDO framework in Excel enviroment. Within this framework, the engineer can perform DOE sampling, surrogate modeling, robustness assessment, Monte Carlo simulation, robust design and reliability-based optimization with any type of distribution.

A novel reliability-based design optimization (RBDO) method using simulation-based techniques for reliability assessments and efficient optimization approach is presented in this paper. In RBDO, model-based reliability analysis needs to be performed to calculate the probability of not satisfying a reliability constraint and the gradient of this probability with respect to each design variable. Among model-based methods, the most widely used in RBDO is the first-order reliability method (FORM). However, FORM could be inaccurate for nonlinear problems and is not applicable for system reliability problems. This paper develops an efficient optimization methodology to perform RBDO using simulation-based techniques. By combining analytical and simulation-based reliability methods, accurate probability of failure and sensitivity information is obtained. The use of simulation also enables both component and system-level reliabilities to be included in RBDO formulation. Instead of using a traditional RBDO formulation in which optimization and reliability computations are nested, a sequential approach is developed to greatly reduce the computational cost. The efficiency of the proposed RBDO approach is enhanced by using a multi-modal adaptive importance sampling technique for simulation-based reliability assessment; and by treating the inactive reliability constraints properly in optimization. A vehicle side impact problem is used to demonstrate the capabilities of the proposed method.

This study performed reliability-based robust design optimization (RBRDO) of stator and rotor shape of the IPM type electric motor to reduce the torque ripple considering uncertainties. A metamodel based on the design of experiments results, which was calculated by using 2-D electromagnetic finite element analysis, was used to calculate manufacturing uncertainties and material property distribution. As a result, the mean torque ripple of the optimum electric motor design was reduced by 56.4 % and the robustness was increased by 61.8 % compared with the initial design. The RBRDO design was compared with the designs of deterministic design optimization and reliability-based design optimization, and it was shown that, by using the RBRDO method, the optimum IPMSM can be designed to satisfying not only the minimum torque ripple but also the robustness and the reliability of output torque and torque ripple.

Traditional Reliability-based design optimization (RBDO) requires a double loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. It makes even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test the RBDO methods, including single and double loop methods. In addition its constraint size, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.

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.

In this research two comparative studies are conducted in application to the concurrent design optimization method of Simulta neous ANalysis and Design (SAND) In one study three different forms of equality constraints referred to as compatibility constraints in the SAND procedure are compared with respect to their impact on algorithm efficiency. Results indicate that the form of the compatibility con straints can have a significant impact on performance in the SAND based procedure. In a second study a Generalized Reduced Gradient (GRG) optimizer and a Sequential Quadratic Programming (SQP) optimizer are compared with respect to their efficacy as the internal opti mizer within the SAND based optimization procedure. Comparative studies indicate that the SQP method is preferred over the GRG method in application of SAND based design. The comparative studies were conducted in application to two Multidisciplinary Design Optimization (MDO) test problems. The method of SAND was originally developed as an optimization strategy for structural design problems. Recent studies have focussed on applying the principles of SAND to multidisciplinary design environments In this research we focus on the concurrency that can be achieved in multidisciplinary design environments using SAND. In a multidisciplinary design environment the SAND approach facilitates each of the seven principles of concurrent engineering as described in Prasad, 1996. The SAND based method provides for: parallel work groups, parallel product decomposition, concurrent resource scheduling, concurrent processing, minimal interfaces, transparent communi cation and quick processing. Application studies confirm that a significant savings in the number of analyses required for design optimiza tion are observed when using the SAND approach of concurrent design. SAND based design delivers on the promise of concurrent engi neering, namely to develop optimal designs, working concurrently, while reducing design cycle time.

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

<div class="htmlview paragraph">This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the hybrid mean value (HMV) first-order method for <i>ε</i>-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO while maintaining numerical accuracy and stability. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, several numerical examples are provided, including a large-scale multi-crash application.</div>

In this paper, an approach for reliability-based design optimization of reinforced concrete cantilever retaining wall is described. A parametric study is conducted to assess the effect of uncertainties in design parameters on the probability of failure of cantilever retaining walls. In total, ten modes of failure are considered, viz. overturning of the wall about its toe, sliding of the wall on its base, eccentricity, bearing capacity failure below the base slab, and shear and moment failure in the toe slab, heel slab, and stem. The analysis is performed by treating backfill and foundation soil properties, geometric properties of wall, and reinforcement and concrete properties as random variables. These results are used to develop a set of reliability-based design charts for different coefficients of variation of friction angle of backfill soil (5 and 10%) and targeting reliability index (βt) in the range of 3–3.2 for all failure modes. A comparative study is also presented, which shows that optimized sections have less areas of cross section compared to those obtained from specifications on dimensioning of retaining walls available in literature.