حلال Schur نیوتن-Krylov برای تجزیه و تحلیل ائروالاستیک حالت پایدار و تجزیه و تحلیل حساسیت طراحی

This paper presents a Newton–Krylov approach applied to a Schur complement formulation for the analysis and design sensitivity analysis of systems undergoing fluid–structure interaction. This solution strategy is studied for a three-field formulation of an aeroelastic problem under steady-state conditions and applied to the design optimization of three-dimensional wing structures. A Schur–Krylov solver is introduced for computing the design sensitivities. Comparing the Schur–Newton–Krylov solver with conventional Gauss–Seidel schemes shows that the proposed approach significantly improves robustness and convergence rates, in particular for problems with strong fluid–structure coupling. In addition, the numerical efficiency of the aeroelastic sensitivity analysis can be typically improved by more than a factor of 1.5, especially if high accuracy is required.

Today, the utility of high-fidelity analysis methods for the design of engineering devices and systems is widely recognized [1] and [2]. This particularly applies to problems, which are dominated by nonlinear effects and subject to multi-physics phenomena, such as fluid–structure interaction (FSI) and electrostatic–mechanical coupling. However, high-fidelity analysis methods are most often used to only verify the final design. Large computational costs prevent these tools from being applied in the iterative design development process, requiring multiple analyses and sensitivity analyses. This paper is concerned with computational strategies that increase the computational robustness and efficiency of high-fidelity analysis and sensitivity analysis for FSI problems, and the implementation of these strategies within a formal design optimization framework. In the past decade, computational FSI methods for aeroelastic and hydro-elastic problems have significantly advanced. The predominant amount of work has focused on algorithms for transient analysis in order to simulate, for example, the behavior of realistic aircraft configurations [3]. Recently, these high-fidelity computational models have been augmented by sensitivity analysis methods, and transient solution techniques have been adopted for design optimization purposes [4], [5], [6] and [7]. For practical reasons, however, design optimization problems are typically based on quasi-static models, approximating dynamic load effects by equivalent static load conditions. So far, most analysis and sensitivity analysis algorithms for solving quasi-static FSI design optimization problems rely either on direct solvers [7] and [8], which are therefore only applicable to idealized and small-scale problems, or are based on loosely coupled and segregated solvers, which have been proven to be accurate and efficient for transient problems but may be inefficient for quasi-static problems. The latter methods follow nonlinear block Gauss–Seidel schemes typically using an approximate aeroelastic Jacobian, due to implementation and computational cost issues, and under-relaxation techniques for stability purposes. These schemes allow for solving the FSI problem by algorithms and software modules, which are tailored to the structure and fluid subproblems. Applications of such schemes in the context of aeroelastic design optimization have recently been reported in the literature [6], [9], [10] and [11]. However, this approach lacks robustness and efficiency, in particular when the FSI problem is strongly coupled and the aeroelastic system is close to static divergence, that is the aeroelastic Jacobian becomes singular. In these cases, the convergence highly depends on the under-relaxation factor, whose optimal value is in general unknown. Only few approaches have been presented for the analysis of nonlinear aeroelastic problems and applied to design sensitivity analysis that overcome the shortcomings of block Gauss–Seidel schemes. Ghattas and Li [12] present a modified Newton scheme for a two-field formulation of the quasi-static FSI problem. The linear subproblem in each Newton step and the global sensitivity equations are solved by a generalized minimal residual (GMRES) algorithm [13]. The authors propose to precondition the linearized system with an approximate Jacobian by dropping the off-diagonal terms that couple the structural displacements and the fluid variables. This approach is applied to the analysis and sensitivity analysis of a two-dimensional flow around an infinite elastic cylinder with variable flow and structural parameters. As the discretization is rather coarse, that is the total number of unknowns is less than 10,000, the applicability to realistic FSI problems cannot be assessed. Heil [14] follows a similar Newton approach that simultaneously advances the state variables of a two-field formulation of the quasi-static FSI problem. The author studies different preconditioners, based on block-triangular approximations for the aeroelastic Jacobian, for solving the linearized system by a GMRES algorithm. This approach is applied to an incompressible, low Reynolds number, two-dimensional flow inside a tube with flexible walls. Kim et al. [15] present a general framework for the analysis of coupled multi-physics problems using existing solvers. Their approach is based on a multi-level Newton scheme for simultaneously solving the set of coupled nonlinear equations. While in the upper level all variables are simultaneously advanced by a Newton correction, in the lower level only the variables of the related subproblems are solved for by Newton’s method. A Krylov method is used to solve the Jacobian system of the multi-level Newton residual. The authors apply this approach to a three-field formulation of the quasi-static FSI problem. The robustness and numerical performance are demonstrated with a laminar, low Reynolds number, two-dimensional flow around a soft structure undergoing large displacements. However, as the total number of state variables is less than 25,000, no conclusions can be drawn towards the applicability of the multi-level approach to realistic three-dimensional FSI problems. Summarizing the above work, Newton schemes appear to be a promising method for solving quasi-static FSI problems. However, most of the proposed algorithms require substantial modifications to existing software, which was typically developed for transient FSI simulations. Furthermore, the potential numerical advantage of Newton schemes versus nonlinear block Gauss–Seidel methods has been shown only for small-scale problems, but questions concerning the applicability and performance of Newton schemes for realistic three-dimensional FSI problems remain open. The purpose of this work is to develop a new method for solving quasi-static FSI problems by combining a Schur complement formulation with a Newton method and Krylov solvers. The Schur complement formulation is derived from a fixed-point strategy applied to the interface variables. The proposed method requires only local modification to existing FSI software, while featuring the robustness and efficiency of the Newton methods discussed above. This method is referred to as Schur–Newton–Krylov (SNK) method. While similar fixed point strategies in combination with quasi-Newton methods have been presented in the context of transient FSI formulation, for example by Gerbeau and Vidrascu [16], the SNK method is based on an exact Jacobian resulting in a true Newton scheme and tailored towards quasi-static FSI design optimization problems. The performance of the proposed method is studied in the context of the design optimization of realistic, three-dimensional, aeroelastic wing structures. The structure is represented by a linear finite element model, and the aerodynamic loads are predicted by the discretization of the nonlinear Euler equations. For the purpose of design optimization, an extension of the SNK method is presented for efficiently computing the design sensitivities of aeroelastic systems with respect to structure and fluid parameters. The remainder of this paper is organized as follows: In Section 2, a generic optimization framework is outlined, identifying the role of FSI analysis and coupled design sensitivity analysis. The three-field formulation of the steady-state aeroelastic problem is summarized in Section 3. The conventional block Gauss–Seidel approach and the proposed SNK method are presented in Section 4. The extension of the Schur complement formulation for design sensitivity analysis is discussed in Section 5. The robustness and numerical efficiency of the proposed approach for aeroelastic analysis and sensitivity analysis is studied by numerical examples in Section 6. The proposed algorithm and the outcomes of the numerical studies are summarized in Section 7. In Appendix A the relation of the proposed SNK algorithm to a variation of a nonlinear block Gauss–Seidel method is discussed.

A Schur–Newton–Krylov (SNK) method for aeroelastic analysis and design sensitivity analysis for steady-state conditions has been presented and studied in the context of aeroelastic design optimization problems. The proposed method is based on the condensation of the aeroelastic steady-state equations in one set of equations governing the mesh motion on the fluid–structure interface. The condensed equations can be viewed as the Schur complement of the steady-state equations. In comparison with other Newton schemes for coupled multi-physics problems, the Schur complement formulation allows for easy implementation, as only local modifications to existing software are required. Two numerical examples have been studied showing that the SNK algorithm efficiently solves aeroelastic problems with weak and strong fluid–structure coupling. Conventional staggered algorithms, even when using adaptive relaxation factor strategies, show an unsatisfactory performance in particular for strongly coupled problems. The design optimization of a rather stiff realistic aeroelastic wing, which is characterized by weak coupling, reveals that the SNK approach outperforms Gauss–Seidel schemes with respect to the total computational costs. Although the SNK solver spends in average more CPU time for computing the steady-state aeroelastic response, these costs are compensated by applying a Schur–Krylov solver to the global sensitivity equations. In conclusion, the SNK solver features an improved robustness in comparison with conventional nonlinear block Gauss–Seidel algorithms. However, for weakly coupled problems the SNK algorithm leads to larger average costs for computing the aeroelastic steady-state response when compared to staggered schemes with appropriately chosen under-relaxation factors. Applying the SNK approach to the design sensitivity analysis improves both the robustness and efficiency for both weakly and strongly coupled problems. For optimization problems requiring a large number of sensitivity evaluations this feature becomes the most promising aspect of the proposed SNK method.