Facilitating Probabilistic Multidisciplinary Optimization Using Kriging Approximation Models

Load stiffening ability in the flight load direction and complaint nature in the opposite direction are important structural behaviors of the bendable unmanned air vehicle (UAV) wing, developed at the University of Florida. The complaint nature of the wing enables UAV storage inside smaller packing volumes. The present paper discusses deterministic and probabilistic design optimization of a bendable wing having 24 inch span and 7 inch root chord. In deterministic optimization, the wing shape definition parameters and the layup scheme used to manufacture the wing are treated as design variables. Aerodynamic and structural performances of the wing are studied to determine wing aerodynamic efficiency and a maximum flying velocity that the wing can withstand without buckling or failing under flight loads. The Tsai-Wu criterion is used to check failure of the wing due to rolling stresses to determine minimum safe storage diameter. Multidisciplinary (aerodynamics and structural) shape and layup optimization is performed using an elitist non-dominated sorting genetic algorithm: NSGA-II. Simultaneous maximization of wing aerodynamic efficiency and maximum flying velocity a wing can withstand without in-flight buckling are used as two design objectives. The design points on the deterministic Pareto optimal front achieved are compared with a baseline design to observe some designs with improved performance. A mixed approach is tried during probabilistic optimization. The design variable and random parameter uncertainties are included explicitly and a modeling uncertainty is included through a factor (similar to factor of safety). We found that when uncertainties in modeling, design variables and random parameters are taken into account, deterministic optimization designs show high probability of failure. In the probabilistic design optimization a single constraint of satisfying target probability of failure is used. To reduce computational cost, we use surrogate models for objectives and constraints. After several surrogates were evaluated for each of the objectives and the constraints, the best surrogates based on cross validation or RMSE were selected for the probabilistic optimization work. Finally, the effect of the target probability of failure on the Pareto front was studied. It was observed that as the target probability of failure becomes stringent, all of the Pareto front designs use almost the same leading edge curvature indicating the original storage diameter constraint drives the designs when higher reliability is desired. By relaxing the target probability of failure, stiffer wings can be obtained. Some designs with acceptable probability of failure are identified which have higher aerodynamic efficiency than a baseline wing having higher probability of failure.

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.

Probabilistic analysis and optimization of energy absorbing components made of nanofiber enhanced composite materials

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

This work integrates developments in simplified dynamic modeling of unanchored fluid-filled tanks, stochastic modeling of earthquake loading time histories, probabilistic structural analysis tools, cost analyses, and multi-criteria optimal design formulations for the design optimization of unanchored fluid-filled cylindrical tanks operating in an uncertain seismic environment. An appropriate simplified mechanical model is developed which includes effects from the sloshing at the liquid free surface, the soil flexibility and the separation of the base plate of the tank from its foundation, which occurs during strong ground motion. A class of stochastic processes with frequency content and time-varying intensity characteristics determined by seismological parameters is used to model the earthquake loading time history. The reliability of the cylindrical tank against failure of the shell structure is computed using latest developments in efficient Monte Carlo simulation methods. A multi-criteria design optimization methodology is then used to obtain the optimal design characteristics of the tank system that meet cost and reliability constrains. Issues related to reliability estimation and optimization are addressed. The methodology is useful for optimally designing fluid-filled structures that operate in a seismic environment to withstand earthquakes with minimal social and economical losses during the lifetime of the structure.

Model-Based Systems Engineering (MBSE) supports the system-level design of complex products effectively. Currently, system design and optimization for complex products are two distinct processes that must be executed using different software or platforms, involving intricate data conversion processes. Applying multidisciplinary optimization to validate system optimization often necessitates remodeling the optimization objects, which is time-consuming, labor-intensive, and highly error-prone. A critical activity in systems engineering is identifying the optimal design solution for the entire system. Multidisciplinary Design Optimization (MDO) and reliability analysis are essential methods for achieving this. This paper proposes a SysML-based multidisciplinary reliability design optimization modeling method. First, by analyzing the definitions and mathematical models of multidisciplinary reliability design optimization, the SysML extension mechanism is employed to represent the optimization model based on SysML. Next, model transformation techniques are used to convert the SysML optimization model generated in the first stage into an XML description model readable by optimization solvers. Finally, the proposed method’s effectiveness is validated through an engineering case study of an in-vehicle environmental control integration system. The results demonstrate that this method fully utilizes SysML to express MDO problems, enhancing the efficiency of design optimization for complex systems. Engineers and system designers working on complex, multidisciplinary projects can particularly benefit from these advancements, as they simplify the integration of design and optimization processes, facilitating more reliable and efficient product development.

In order to design a vehicle body with high strength and high stiffness, a multi-disciplinary design process should include careful consideration of multi-disciplinary design constraints to properly account for vehicle static stiffness (bending/torsional), durability, Noise/Vibration/Harshness (NVH), crash worthiness, light weight vehicle structure during the early stage of vehicle design process. With this approach, fast development of new vehicle body structures can be achieved with minimal number of iterations to match the conflicting design goals from each discipline. In the current research, a multi-disciplinary design optimization (MDO) based on a meta model is developed and refined to apply for the design of body structure. In an effort to apply the MDO for vehicle body structure, 4 phase procedures were established in the current research. In Phase I, a base model is created. In Phase II, an effect analysis is carried out. In Phase III, a meta model is created. Finally in Phase IV, using the optimization algorithm, the meta model created in Phase III is eventually refined through the process of optimization. In this research, static stiffness (bending / torsional), dynamic stiffness (1st torsion mode) were used for constrained conditions and the mass minimization was the object function for optimization.

A methodology is presented for probabilistic design optimization of aircraft structures subject to a set of multidisciplinary constraints including reliability, manufacturability, and manufacturing cost. Using the advanced firstorder second-moment method, reliability of each structural component is evaluated by considering its primary static failure mode. Metrics-based analytical models are used in manufacturability analysis, and semi-empirical models are used for manufacturing cost estimation. The described design methodology is developed into a software tool and applied to probabilistic sizing optimization of a moderately loaded built-up wing spar. Using two alternative web design concepts, one limited by buckling and the other by diagonal semitension field action, the spar model is optimized for weight considering uncertainties in the material properties, structural sizing, and loading parameters. The optimization problem is solved using the method of sequential quadratic programming for different values of target reliability, with or without any limit on manufacturing cost. The optimization results are used to highlight the interactions between reliability, weight, and cost, whereas the probabilistic sensitivities are used to examine the influence of each random variable on structural reliability.

A number of studies have been carried out in past years on optimum design under the condition of probabilistic constraints. For example, Switzky derived a probabilistic optimality criterion for the structural design with stress constraints. However, few attempts have been made to formulate the probabilistic optimality criterion for the displacement constraint. In the present study, an attempt was made first to derive the probabilistic optimality displacement criterion for the minimum weight design by using the Lagrange multiplier method. Then, a new probabilistic optimality criteria method was proposed, which unifies Switzky's probabilistic optimality stress criterion and the present probabilistic optimality displacement criterion by using the envelope method. Two design examples were presented to demonstrate the efficiency and convergency of the method developed herein.The results shows that the proposed method can be used to optimize the structure having probabilistic stress and displacement constraints only in few cycles of trials, and hence it has a good convergency in comparison with other probabilistic optimum design methods using mathematical programming.

The optimal design of multidisciplinary systems under uncertainty involves three types of iterative analyses- multidisciplinary analyses, reliability analyses, and optimization. If these processes are carried out in a nested manner in reliability based multidisciplinary design optimization (RBMDO), the computational expense can be enormous. Computationally efficient simultaneous analysis and design (SAND) or All-AtOnce methods have been developed earlier for deterministic MDO. Single loop RBDO methods have also been developed to integrate reliability analysis and optimization in a single loop. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints. The formulation allows for solution with a single design optimization loop by using the KKT optimality conditions of the first-order reliability method (FORM) as constraints in the design optimization along with disciplinary consistency constraints. Alternative solution processes which remove the FORM MPP’s from the decision space and many constraints from the problem are developed. The alternative solution methods are more efficient and more robust than using nonlinear programming to solve the single loop formulation. A numerical example is solved with the two proposed solution algorithms. I. Introduction In the design of multidisciplinary systems it is often necessary to make decisions under uncertainty. This requires that multidisciplinary systems modeling be integrated with uncertainty analysis and optimization. A common example of such an engineering design application would be the design of an aircraft wing under uncertain aerodynamic loading conditions, with multiple structural components with uncertain capacities, such that the entire structural system achieves a prescribed level of reliability. The problem has coupled physics, in that the pressure distribution is dependent upon the deflection, and the deflection is dependent on the pressure distribution. Such designs have been performed using multidisciplinary optimization (MDO). Other examples where MDO may be applied include electronics packaging, in which the disciplines of heat transfer and circuit analysis are coupled, the design of new types of aircraft and integrating them into an existing fleet of aircraft, and even the management of interconnected homeland infrastructures. While MDO has matured with the development of different types of algorithms for various types of problems and design organizations, there has been little work to incorporate reliability analysis into MDO. However, system and component reliabilities are often a major concern in the design of multidisciplinary systems. MDO can be computationally expensive due to the need to achieve consistency across multiple disciplinary analysis codes; each disciplinary analysis code may be computationally expensive to run. Further, the optimal design of multidisciplinary systems under uncertainty involves three types of iterative procedures- multidisciplinary analyses, reliability analyses, and optimization. The integration of these three procedures is referred to in this paper as reliability based multidisciplinary design optimization (RBMDO). If these processes are carried out simultaneously, the computational expense can be enormous. Computational savings can often be achieved with simultaneous analysis and design (SAND) or All-At-Once methods, which allow for the solution of the multidisciplinary analyses and the optimization in one single loop. Single loop RBDO methods allow for the integration of reliability analysis and optimization in a single loop. Applying SAND and single-loop RBDO together may allow for large computational savings when applied to problems in RBMDO. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints based on the SAND method for MDO and by use of the KKT conditions of the first order reliability method (FORM). The resulting formulation may be solved as a standard nonlinear programming problem. While the single loop formulation presented in this paper is a solution of the RBMDO problem, robustness and efficiency are improved through two improved algorithms which can remove the MPP’s from the decision space, thereby significantly reducing the optimizer’s workload.

Around the engineering background of the probabilistic design of high-pressure turbine (HPT) blade-tip radial running clearance (BTRRC) which conduces to the high-performance and high-reliability of aeroengine, a distributed collaborative extremum response surface method (DCERSM) was proposed for the dynamic probabilistic analysis of turbomachinery. On the basis of investigating extremum response surface method (ERSM), the mathematical model of DCERSM was established. The DCERSM was applied to the dynamic probabilistic analysis of BTRRC. The results show that the blade-tip radial static clearance δ = 1.82 mm is advisable synthetically considering the reliability and efficiency of gas turbine. As revealed by the comparison of three methods (DCERSM, ERSM, and Monte Carlo method), the DCERSM reshapes the possibility of the probabilistic analysis for turbomachinery and improves the computational efficiency while preserving computational accuracy. The DCERSM offers a useful insight for BTRRC dynamic probabilistic analysis and optimization. The present study enrichs mechanical reliability analysis and design theory.

The design of multidiscipli nary systems (such as aircraft, automobiles, and others) often requires an iterative cycle that includes a design initialization, a system analysis, a sensitivity analysis, and design optimization. This design cycle is standard in the field of Multidisciplinary Optimization (MDO) and has often been referred to in literature as the Multiple Discipline-Feasible (MDF) approach. The name stems from the fact that complete multidisciplinary feasibility is maintained each and every design cycle. The drawback of MDF is that it can be a timely and a costly procedure. Numerous researchers have developed alternate means for posing and subsequently solving the multidisciplinary design problem. One such solution procedure has been referred to both as Simultaneous Analysis and Design (SAND) and All-at-Once (AAO), and treats the entire multidisciplinary design cycle as one large optimization problem. Another alternate solution procedure lias been referred to as Individual-Discipline-Feasible (DDF); this procedure exhibits characteristics which lie in between the two extremes exemplified by MDF and AAO. IDF assures that each individual discipline is feasible on every design cycle, while driving the entire system (all disciplines) towards multidisciplinary feasibility. The present work will present a rigorous numerical comparison of these solution strategies over a wide variety of problem sizes and complexities. The multidisciplinary design test problems that are used for these comparisons are generated by a robust simulation tool called CASCADE. * Research Assistant, Student member, AIAA | Associate Professor, Member AIAA Background and Motivation Concurrent engineering is a systematic approach to the integrated, simultaneous design of products and their related processes. Many of the recently developed capabilities to address concurrent engineering have stemmed from the emerging area of Multidiscipli nary Optimization, or MDO. The origins of MDO can be traced back to the early 1980's, where Sobieski14 used a linear decomposition approach to subdivide the design of a large engineering system into a grouping of related and more manageable subsystems. However, such a decomposition often results in a grouping of subsystems which cannot be placed into a definitive top-down hierarchy. The resultant decomposition grouping is typically hybrid-hierarchic in nature as shown in the example decomposition of Figure 1.

A blowout preventer (BOP) is a large valve that encases an oil well on the surface. The valve may be closed while drilling if overpressure from a reservoir causes formation fluids such as oil and natural gas to back up within the wellbore and endanger the rig. A pipe ram is one of the critical components in the BOP system, which is designed to seal around a drill pipe, restricting the flow in the annulus between the outside of the drill pipe and the wellbore. The pipe ram design family, which contains hundreds of pipe ram designs, intends to cover different configurations to suit a variety of dill pipe sizes. To assure their structural integrity under service loads, these pipe rams need to be designed to meet the stress requirements per the American Petroleum Institute (API) Specification 16A and ASME (American Society of Mechanical Engineers) Boiler and Pressure Vessel Code Section VIII, Division 2. With a conventional manual design workflow, it would take a significant amount of time and effort to complete all the designs for the product family with code compliance. Therefore, a scalable solution is highly desirable for delivery of customer-needed ram blocks with much shorter lead time. MDO (Multi-Disciplinary Optimization) involves multi-code integration, CAE (Computer-Aided Engineering) workflow automation, design space exploration, and optimization. MDO-enabled, automated design optimization is becoming increasingly popular in both scientific and engineering communities. In this paper, a methodology for integration of CAD (Computer Aided Design), FEA (Finite Element Analysis), cost, and optimization packages to enable FEA automation and design optimization is presented. A BOP pipe ram is adopted as the use case. The ram block geometry was parameterized before being imported into an FEA package. The FEA workflow was automated such that once a set of geometric parameters are given, the preprocessing, solving, and postprocessing steps can be automatically completed. As part of the FEA postprocessing, stress linearization analysis per the API and ASME BPVC codes has also been automated, which had never been done in the past. A manufacturing cost analysis package can also be used to consume the parameterized geometry for automatic manufacturability assessment and cost predictions. The stress analysis and cost analysis workflows are conducted separately but also orchestrated by an MDO package. Reports that contain the analysis results are sequentially generated for various design permutations. The MDO-enabled automated design and cost analysis approach could substantially enhance efficiency and consistency in performing FEA and cost studies and producing analysis reports. It is also the backbone for automated design optimization, which could significantly improve the product performance and reliability and, meanwhile, minimize the development and product cost.

The design of space systems is a complex and multidisciplinary process with multiple conflicting objectives, large number of design variables, and constraints that limits application of the existing multidisciplinary design optimization architectures to this class of design problems. This paper presents an enhanced multidisciplinary design optimization architecture to concurrent holistic design optimization of a satellite system. The proposed multidisciplinary design optimization architecture extends concepts of multidiscipline feasible and bi-level integrated system synthesis into a unified architecture using metamodels. The proposed architecture was evaluated and compared with the existing multidisciplinary design optimization architectures that include all-at-once, bi-level integrated system synthesis, and multidisciplinary design optimization using a remote sensing small satellite in low earth orbit. The satellite design optimization problem deals with the minimization of the total mass of the satellite, involving disciplines of mission analysis, payload, structures, attitude determination and control, communication, command and data handling, power and thermal. The computational performance and accuracy of the proposed architecture were compared with multidisciplinary design optimization benchmark problems. Then the proposed architecture is successfully applied to the satellite system design problem. The results obtained show that metamodel-based bi-level integrated system synthesis-multidisciplinary design optimization architecture presented in this paper provides an effective way of solving large-scale design problems.

Microelectromechanical systems (MEMS) are a highly multidisciplinary field and this has large implications on their applications and design. Designers are often faced with the task of balancing the modelling, simulation and optimisation that each discipline brings in order to bring about a complete whole system. In order to aid designers, strategies for navigating this multidisciplinary environment are essential, particularly when it comes to automating design synthesis and optimisation. This paper outlines a new multi-objective and multidisciplinary strategy for the application of engineering design problems. It employs a population-based evolutionary approach that looks to overcome the limitations of past work by using a non-hierarchical architecture that allows for interaction across all disciplines during optimisation. Two case studies are presented, the first focusing on a common speed reducer design problem found throughout the literature used to validate the methodology and a more complex example of design optimisation, that of a MEMS bandpass filter. Results show good agreement in terms of performance with past multi-objective multidisciplinary design optimisation methods with respect to the first speed reducer case study, and improved performance for the design of the MEMS bandpass filter case study.