As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based [Formula: see text] method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar–turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint–RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.
Pandemic with mutation and permanent immune spreading in a small-world network described is studied by a modified SIR model, with consideration of mutation-immune mechanism. First, a novel mutation-immune model is proposed to modify the classical SIR model to simulate the transmission of mutable viruses that can be permanently immunized in small-world networks. Then, the influences of the size, coordination number and disorder parameter of the small-world network on the spread of the epidemic are analyzed in detail. Finally, the influences of mutation cycle and infection rate on epidemic transmission in small-world network are investigated further. The results show that the structure of the small-world network and the virus mutation cycle have an important impact on the spread of the epidemic. For viruses that can be permanently immunized, virus mutation is equivalent to making the immune cycle of human beings from infinite to finite. The dynamical behavior of the modified SIR epidemic model changes from an irregular, low-amplitude evolution at small disorder parameter to a spontaneous state of wide amplitude oscillations at large disorder parameter. Moreover, similar transition can also be found in increasing mutation cycle parameter. The maximum valid variation mutation decreases with the increase of disorder parameter and coordination number, but increase with respect to system size. In addition above, as the infection rate increases, the fraction of the infected increases and then decreases. As the mutation cycle increases, the time-average fraction of the infected and the infection rate corresponding to the maximum time-average fraction of the infected also decrease. As one conclusion, the results could give a deep understanding Pandemic with mutation and permanent immune spreading, from viewpoint of small-world network.
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