Aerospace Science and Technology 28 (2013) 297–304 Contents lists available at SciVerse ScienceDirect Aerospace Science and Technology www.elsevier.com/locate/aescte Latin hypercube sampling applied to reliability-based multidisciplinary design optimization of a launch vehicle Jafar Roshanian 1 , Masoud Ebrahimi ∗,2 Faculty of Aerospace Engineering, K.N. Toosi University of Technology, East Vafadar Bld., 4th Tehranpars Sq., Postal Code: 16569-83911, Tehran, Iran article info abstract Article history: Received 20 March 2012 Received in revised form 15 October 2012 Accepted 29 November 2012 Available online 12 December 2012 Keywords: Multidisciplinary design optimization Reliability-based design Latin hypercube sampling Solid propellant launch vehicle In this paper, Reliability-Based Multidisciplinary Design Optimization (RBMDO) of a two-stage solid propellant expendable launch vehicle (LV) is investigated. Propulsion, weight, aerodynamics (geometry) and trajectory (performance) disciplines are used in an appropriate combination. Throw weight minimization is chosen as objective function. Design variables for system level optimization are selected from propulsion, geometry and trajectory disciplines. Mission constraints contain the final velocity, the height above ground, and flight path angle. The constraints that appear during the flight are also considered. Assuming a normal distribution for the uncertain variables, Latin Hypercube Sampling (LHS) method selects the sample values for simulation runs which are eventually utilized for calculating probability density function of constraints and their reliability at each design point. Sequential Quadratic Programming (SQP) technique is used to achieve the optimal solution. Although the launch vehicle throw weight is increased negligibly in comparison with deterministic optimization, results show that the reliability-based method satisfied desired reliability of the constraints. © 2012 Elsevier Masson SAS. All rights reserved. 1. Introduction The design of complex systems such as aerospace launch ve- hicles (ALVs) requires making appropriate compromises to achieve the optimal design among many coupled objectives such as high performance, safety, simplicity, ease of operation and low cost. A complex interrelation exists among mission trajectory, aerody- namics, propulsion, weight and structure with conflicting goals which have to be matched by an appropriate optimization strat- egy. Multidisciplinary Design Optimization (MDO) involves the co- ordination of multidisciplinary analyses to realize more effective solutions during the design and optimization of complex systems. It will allow system engineers to systematically explore the vast trade space in an intelligent manner and consider many more sys- tem architecture during the conceptual design phase before con- verging to the final design [1,3,11,15,23]. Many successful examples of MDO applications have been found in many fields, such as aerospace (aircraft, launch vehicle, satellite, etc.), automobile, electronic, and structures [5,7,8,16–18, 20]. Traditional optimization methods which are applied in MDO problems generate optimal designs using deterministic design * Corresponding author. Tel.: +98 21 77791043; fax: +98 21 77791045. E-mail addresses: roshanian@kntu.ac.ir (J. Roshanian), ebrahimim@dena.kntu.ac.ir (M. Ebrahimi). 1 Professor. 2 Ph.D. variables and parameters. However, considering some degrees of uncertainty in characterizing any real engineering system is in- evitable. In different cases, the objective function and constraints may be highly sensitive to these uncertainties which will lead to constrain violation or performance reduction. Customary design procedures for dealing with uncertainties in structural design are based on combinations of factors of safety and knockdown factors. Unfortunately these procedures have several shortcomings (diffi- cult to apply to aerospace vehicles which have new configurations and use new material, measures of reliability and robustness are not provided in the design process) and are not suitable for com- plex problems such as aerospace design problem [10,25]. Two major classes of uncertainty-based design problems, robust design problems and reliability-based design problems have been proposed. A robust design problem is one in which a design is sought that is relatively insensitive to small changes in the uncer- tain quantities. A reliability-based design problem is one in which a design is sought that has a probability of failure that is less than some acceptable (invariably small) values [10,25]. Methods for structural reliability analysis have been developed in last decades to incorporate uncertainties associated with ge- ometrical and material properties, loading and boundary condi- tions, and operational environment into structural analysis. These methods can easily be used in system design. The design reliabil- ity is then defined as the probability of satisfying a design con- straint [10,25]. Many methods for determining the probability of failure or re- liability (estimating the areas inside and outside the constraints) 1270-9638/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ast.2012.11.010