In this edition, these methods have been treated in separate chapters, but the book remains primarily one on differential shape optimization. This book is essential reading for engineers interested in the implementation and solution of optimization problems using commercial packages or in-house solvers and graduates and researchers in applied mathematics, aerospace, or mechanical engineering, fluid dynamics, and CFD. More generally, anyone needing to understand and solve design problems or looking for new exciting areas for research and development in this area will find this book useful, especially in applying the methodology to practical problems.
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Amazon Global Store US International products have separate terms, are sold from abroad and may differ from local products, including fit, age ratings, and language of product, labeling or instructions. Manufacturer warranty may not apply Learn more about Amazon Global Store. Review 'I found it very stimulating and furthermore convincing in its basic tenet that gradient based optimization is workable and highly competetive for shape optimization. No customer reviews. We will introduce and investigate methods for efficient numerical solutions to shape optimization problems in fluid mechanics and questions in optimal active flow control.
In the tutorial classes we will apply such methods to an airfoil design problem within the open-source solver SU2. Max Sagebaum Dr. Beckett Y. Objective: We will introduce and investigate methods for efficient numerical solutions to shape optimization problems in fluid mechanics and questions in optimal active flow control.
Thus, they do not guarantee that a solution, for example, of the inverse problem for a certain target pressure distribution will necessarily be attained. Traditionally, finite-difference methods have been used to calculate sensitivities of aerodynamic cost functions. The computational cost of the finite-difference method for problems involving large numbers of design variables is both unaffordable and prone to subtractive cancellation error. In order to produce an accurate finite difference gradient, a range of step sizes must be used, and thus the ultimate cost of producing N gradient evaluations with the finite-difference method is the product mN , where m is the number of different step sizes used to obtain a converged finite difference gradient.
The used number of flow solvers is equal to mN. An estimate of the first derivative of a cost function is as:. A small step size is desired to reduce the truncation error O h , but a very small step size would also increase subtractive cancellation errors.
But, in the adjoint method, according to Eq. The advantage is that Eq. Since the cost of the adjoint approach is independent of the number of design variables, it is feasible to use the surface points as design variables, whereas the cost would be prohibitive, if the gradients were computed by the traditional finite difference method. Figure 2 is a graphical representation of the design procedure using adjoint method in present work.
The flow is subsonic with Mach number of 0. Both the initial and target airfoils are at zero degree angle of attack. Surface points, parameters of the NACA 4-digit airfoil series Airfoil camber m and its thickness t and the Hicks-Henne bump functions are used as the design variables. A x 80 cells O-Type grid is employed in this calculation. The initial value for m, t are 0. The value of the parameter t 2 which controls the width of the Hicks-Henne bump functions was set to values of 30 and 32 bump functions for all the test cases. Figure 3 compares the final airfoil shape after 85 design cycles for first method surface points , second approach parameters of the NACA 4-digit airfoil series and third approach Hicks-Henne bump functions.
The gradient in first and third approaches is smoothed and the smoothed steepest descent algorithm acts as a preconditioner effectively reducing the number design cycles required to achieve a converged solution. The figure illustrates that all of the approaches achieve the final shape airfoil with a high degree of accuracy after 85 design cycles. The point-to-point match between the target airfoil and the first approach as well as the second and third approaches is clearly visible.
Figure 5 shows convergence history of the norm of cost function gradient. The figure demonstrates that with using of third method, the convergence of the objective function drops 5 orders of magnitude to a level of 2. The negative slope of the convergence plot indicates that a further drop in the objective function is still possible. The case with meshes points first method as the design variable attained a convergence level of 1.
This figure validates the convergence of optimization program.
Applied shape optimization for fluids
Figure 6 compares the final pressure distribution for the various cases against the target pressure. However, a slight discrepancy is observed at the shock, as shown by the inset in Fig. The figure demonstrates the effectiveness of the mesh points, parameters of the NACA 4-digit airfoil series and Hicks-Henne bump functions that accurately capture the geometry of the NACA airfoil.
In this case, we have implemented the method in the constrained drag minimization. In first test case, the surface points are used as the design variables and, in the second test case, m and t are adopted as design variables. The design is started by a NACA airfoil at 3.
The flow is transonic with Mach number of 0. We performed computations on a x 80 O-grid.
It is noticeable that the optimization has been performed in an inviscid and compressible transonic flow filed. Therefore, drag force is type of wave drag and is due to the produced shock wave on the surface due to compressible effects. Table 1 represents the design results. The reduction in the drag coefficient is considerable. We obtained Figure 7 represents the convergence of the cost function. This figure shows that full convergence of aerodynamic optimization is obtained after design iterations and we reach to the optimum design. And Note that during final cycles obtained only 1.
Figure 8 shows the convergence history of the norm of gradient during the design process. According to this figure, the trend is similar to that of the cost function variations.
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Figure 9 compares the initial and optimal pressure contours. Figure 10 represents the initial and optimal pressure coefficients. The figure shows that the strong shock on the initial airfoil surface has been weakened strongly and drag coefficient has been reduced but the surface area under the curve, which is represented the value of the lift coefficient, has remained constant and consequently this coefficient is nearly the same for both the initial and optimal airfoils.
Figure 11 shows the geometry of initial and optimal airfoils. The change in the upper surface and around the trailing edge is considerable, whereas the change in the lower surface is very small. Figures 10 and 11 show that the upper surface of the optimal airfoil has approached to a flat geometry.
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The flat surface has weakened the strength of the shock wave. Furthermore, the geometry of the airfoil at its trailing edge has curved downward to compensate the reduction of the lift coefficient due to the weakening the strength of the shock. It should be noted that for purpose of fixing the lift coefficient, the angle of attack is considered as an extra design variable.
Table 2 represents the design results. Figure 12 gives the variation of the cost function with design cycle. For this problem, the design cycle has iterations. The drag coefficient reduction is This coefficient has only 1. Figure 13 shows the variation of the norm of the gradient vector with design cycle.
The trend is similar to that of the cost function. With regard to Figs. Figure 14 shows the pressure contours around the initial and optimal airfoils. Regarding this figure, the shock wave has moved toward the leading edge and its strength is considerably reduced. Figure 15 gives the pressure coefficients on upper and lower surface of initial and optimal airfoils.
It is seen that the location of the shock wave has changed and its strength is reduced. But the surface under the curve is remained constant, which reveals no change in lift coefficient.
It is clear that the drag coefficient reduction has been achieved due to reduction of thickness, whereas increase of the camber has led to increase of the lift coefficient. In fact, the reduction of the lift due to the reduction of the thickness has been compensated by increase of the camber.
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More over the variation of the angle of attack is such that the lift coefficient remains constant. If the step size was taken larger, it increased the convergence rate. Sometimes, larger step size caused oscillatory behavior of the gradients. It is known that the location and strength of a shock wave in transonic regime are main parameters in drag calculation. In fact, the drag reduction is carried out via the variation of the curvature only at region of the shock wave. And the lift coefficient is recovered via the increase in curvature only at the trailing edge region. When the thickness and camber of NACA four digits were considered as design variables, the shock wave moves toward the leading edge and the strength is reduced.
In fact, the drag reduction is carried out via reduction of the thickness on all of the surface points and recovery of the lift coefficient is achieved by increasing the camber on all of the surface points. Table 3 summarizes the required runtime and number of adjoint and flow solvers to achieve the convergence of the optimization program for the constrained drag minimization problem.
In this paper, we implemented the adjoint equations method for the inverse pressure design and the constrained drag minimization problems. In the inverse design problem, surface points, parameters of the NACA 4-digit airfoil series Airfoil camber m and its thickness t and the Hicks-Henne bump functions are used as the design variables. In this problem, values of design variables were obtained successfully. The results of the test case show that we can use the adjoint approach as an efficient tool in airfoil inverse design problem.
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