This work presents results from the application of an aerodynamic shape optimization code, Jetstream, to a suite of benchmark cases defined by the Aerodynamic Design Optimization Discussion Group. Geometry parameterization and mesh movement are integrated by fitting the multi-block structured grids with B-spline volumes and performing mesh movement based on a linear elastic model applied to the control points. Geometry control is achieved through two different approaches. Either the B-spline surface control points are taken as the design variables for optimization, or alternatively, the surface control points are embedded within free-form deformation (FFD) B-spline volumes, and the FFD control points are taken as the design variables. Spatial discretization of the Euler or Reynolds-averaged Navier-Stokes equations is performed using summation-by-parts operators with simultaneous approximation terms at boundaries and block interfaces. The governing equations are solved iteratively using a parallel Newton-Krylov-Schur algorithm. The discrete-adjoint method is used to calculate the gradients supplied to a sequential quadratic programming optimization algorithm. The first optimization problem studied is the drag minimization of a modified NACA 0012 airfoil at zero angle of attack in inviscid, transonic flow, with a minimum thickness constraint set to the initial thickness. The shock is weakened and moved downstream, reducing drag by 91%. The second problem is the liftconstrained drag minimization of the RAE 2822 airfoil in viscous, transonic flow. The shock is eliminated and drag is reduced by 48%. Both two-dimensional cases exhibit optimization convergence difficulties due to the presence of nonunique flow solutions. The third problem is the twist optimization for minimum induced drag at fixed lift of a rectangular wing in subsonic, inviscid flow. A span efficiency factor very close to unity and a near elliptical lift distribution are achieved. The final problem includes single-point and multi-point liftconstrained drag minimizations of the Common Research Model wing in transonic, viscous flow. Significant shape changes and performance improvements are achieved in all cases. Finally, two additional optimization problems are presented that demonstrate the capabilities of Jetstream and could be suitable additions to the Discussion Group problem suite. The first is a wing-fuselage-tail optimization with a prescribed spanwise load distribution on the wing. The second is an optimization of a box-wing geometry.
Two aerodynamic shape optimization geometry control methods, B-spline surface control and free-form deformation (FFD), are applied to three optimization problems and compared on the bases of optimal shape performance and problem setup ease of use. For both methods, the geometry is parameterized using B-spline surfaces, with mesh movement accomplished using an e cient integrated technique. Gradients for the optimization algorithm are computed using the adjoint method. The first problem is a wing twist optimization under inviscid, subsonic flow, achieving an elliptical load distribution. The second is a lift-constrained drag minimization of a wing under transonic flow based on the Reynolds-averaged Navier-Stokes (RANS) equations. The third involves lift-to-drag ratio maximization, based on the RANS equations, beginning from a classically-shaped blended wing-body aircraft and converging to a lifting-fuselage configuration. B-spline surface control is often found to result in slightly better performance; however in general both methods perform equally well. FFD provides a more general approach to problem setup, decoupling geometry control from parameterization. Overall, the results suggest that B-spline surface control is better suited for simple geometries such as wings, while FFD is better suited for complex geometries such as unconventional aircraft and for implementation with multistart algorithms and adaptive geometry control approaches.
Piezoelectric energy microgenerators are devices that generate continuously electricity when they are subjected to varying mechanical strain due to e.g. ambient vibrations. This paper presents the mathematical analysis, modelling and validation of a miniaturized piezoelectric energy harvester based on ambient random vibrations. Aluminium nitride as piezoelectric material is arranged between two electrodes. The device design includes a silicon cantilever on which AlN film is deposited and which features a seismic mass at the end of the cantilever. EulerBernoulli energy approach and Hamilton's principle are applied for device modeling and analysis of the operation of the device at various acceleration values. The model shows good agreement with the experimental findings, thus giving confidence into model. Both mechanical and electrical characteristics are considered and compared with the experimental data, and good agreement is obtained. The developed analytical model can be applied for the design of piezoelectric microgenerators with enhanced performance. List of symbolsL Length of cantilever beam (m) B Width of cantilever beam (m) H Thickness of structural layer (m) T Thickness of piezoelectric layer (m) V Voltage across piezoelectric element (m) Z Coordinate parallel to beam thickness (m) X Coordinate parallel to beam length or axial coordinate (m) T k Kinetic energy (J) U Internal energy (J) W e Electrical work (J) W External work (J) q s Density of structural layer (kg m -3 ) q p Density of piezoelectric layer (kg m -3 ) U Displacement (M) S Applied strain (None) T Developed stress (Pa) EElectric field (V m -1 ) DElectric displacement (C m -2 ) dV s Differential volume of structural layer (m 3 ) dV p
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