Publications

Multidisciplinary models are composed of numerous implicit systems that are solved in a coupled manner. A common challenge arises when an individual discipline’s Jacobian is non-invertible, which results in the coupled multidisciplinary model representing a saddle-point problem. In general, the small-scale multidisciplinary saddle-point problems are solved with the coupled Newton’s method. Nevertheless, coupled solver approaches are difficult to build and have robustness problems. In addition, they necessitate solving large coupled linear systems in large-scale applications. In this work, we use the newly developed nonlinear and linear Schur complement (SC) solvers, appropriate for multidisciplinary models based on computational fluid dynamics (CFD) to solve these saddle-point problems. The SC solvers are not susceptible to the robustness problems of conventional coupled solutions since they leverage specialized CFD solvers. Thanks to these solvers, the conventional constrained optimization, which has a saddle-point system in its Jacobian, can be transformed into unconstrained optimizations. In this work, we demonstrate several applications of Schur-complement-based optimizations of CFD-based saddle-point problems: aerodynamic shape optimization (ASO) of a wing and coupled aeropropulsive design optimization of a podded propulsor. In the ASO cases, the Schur complement solvers-based optimizations outperform the conventional constrained optimizations. Although the conventional optimization in the coupled aeropropulsive design optimization problem outperforms the SC solvers-based optimization, it requires an effective scaling of the design variables of the boundary conditions and the constraints. SC solvers-based optimizations always provide a feasible design at each design iteration, whereas the conventional approach does not. Furthermore, in large-scale nonlinear saddle-point problems, the nonlinear SC solver offers an alternative way to obtain a feasible solution for a given design instead of carrying out a modest optimization in order to get feasible solutions. In the CFD-based saddle-point applications, SC solvers will play an important role in the future because they are robust and can solve the saddle-point problem using the native solvers of the specialized CFD model.