Matrix-free algorithm for the optimization of multidisciplinary systems
- 336 Downloads
Multidisciplinary engineering systems are usually modeled by coupling software components that were developed for each discipline independently. The use of disparate solvers complicates the optimization of multidisciplinary systems and has been a long-standing motivation for optimization architectures that support modularity. The individual discipline feasible (IDF) formulation is particularly attractive in this respect. IDF achieves modularity by introducing optimization variables and constraints that effectively decouple the disciplinary solvers during each optimization iteration. Unfortunately, the number of variables and constraints can be significant, and the IDF constraint Jacobian required by most conventional optimization algorithms is prohibitively expensive to compute. Furthermore, limited-memory quasi-Newton approximations, commonly used for large-scale problems, exhibit linear convergence rates that can struggle with the large number of design variables introduced by the IDF formulation. In this work, we show that these challenges can be overcome using a reduced-space inexact-Newton-Krylov algorithm. The proposed algorithm avoids the need for the explicit constraint Jacobian and Hessian by using a Krylov iterative method to solve the Newton steps. The Krylov method requires matrix-vector products, which can be evaluated in a matrix-free manner using second-order adjoints. The Krylov method also needs to be preconditioned, and a key contribution of this work is a novel and effective preconditioner that is based on approximating a monolithic solution of the (linearized) multidisciplinary system. We demonstrate the efficacy of the algorithm by comparing it with the popular multidisciplinary feasible formulation on two test problems.
KeywordsMultidisciplinary design optimization Individual discipline feasible Inexact-Newton-Krylov Matrix-free Second-order adjoint Preconditioning
This material is based upon work supported by the National Science Foundation under Grant No. 1332819. The authors gratefully acknowledge this support.
- Conn AR, Gould NI, Toint PL (2000) Trust region methods. SIAMGoogle Scholar
- Heinkenschloss M, Ridzal D (2008) An inexact trust-region sqp method with applications to pde-constrained optimization. In: Numerical mathematics and advanced applications. Springer, pp 613–620Google Scholar
- Heroux MA, Bartlett RA, Howle VE, Hoekstra RJ, Hu JJ, Kolda TG, Lehoucq RB, Long KR, Pawlowski RP, Phipps ET, Salinger AG, Thornquist HK, Tuminaro RS, Willenbring JM, Williams A, Stanley KS (2005) An overview of the trilinos project. ACM Trans Math Softw 31(3):397–423. doi: 10.1145/1089014.1089021 CrossRefzbMATHMathSciNetGoogle Scholar
- Jameson A (1989) Aerodynamic design via control theory. In: Recent advances in computational fluid dynamics. Springer, pp 377– 401Google Scholar
- Kennedy GJ, Martins JRRA (2010) Parallel solution methods for aerostructural analysis and design optimization. In: 13th AIAA/ISSMO multidisciplinary analysis optimization conference, p 9308Google Scholar
- Kreiss HO, Scherer G (1974) Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor C (ed) Mathematical aspects of finite elements in partial differential equations. Mathematics Research Center, the University of Wisconsin. Academic Press, New YorkGoogle Scholar
- Martins JRRA, Lambe AB (2013) Multidisciplinary design optimization: a survey of architectures. AIAA J 51(9):2049–2075Google Scholar
- Özkaya E, Gauger NR (2009) Single-step one-shot aerodynamic shape optimization. In: Optimal control of coupled systems of partial differential equations. Springer, pp 191–204Google Scholar
- Ta’asan S, Kuruvila G, Salas M (1992) Aerodynamic design and optimization in one shot. In: 30th aerospace sciences meeting and exhibit, p 25Google Scholar
- Turner M (1959) The direct stiffness method of structural analysis. Boeing Airplane CompanyGoogle Scholar
- Wright S, Nocedal J (2006) Numerical optimization, 2nd edn. Springer ScienceGoogle Scholar