###### Abstract  
The upgraded elastic surface algorithm (UESA) is a physical inverse design method that was recently developed for a compressor cascade with double-circular-arc blades. In this method, the blade walls are modeled as elastic Timoshenko beams that smoothly deform because of the difference between the target and current pressure distributions. Nevertheless, the UESA is completely unstable for a compressor cascade with an intense normal shock, which causes a divergence due to the high pressure difference near the shock and the displacement of shock during the geometry corrections. In this study, the UESA was stabilized for the inverse design of a compressor cascade with normal shock, with no geometrical filtration. In the new version of this method, a distribution for the elastic modulus along the Timoshenko beam was chosen to increase its stiffness near the normal shock and to control the high deformations and oscillations in this region. Furthermore, to prevent surface oscillations, nodes need to be constrained to move perpendicularly to the chord line. With these modifications, the instability and oscillation were removed through the shape modification process. Two design cases were examined to evaluate the method for a transonic cascade with normal shock. The method was also capable of finding a physical pressure distribution that was nearest to the target one. inverse design; ESA; normal shock; cascade; physical pressure; target pressure; compressor cascade +  
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To obtain the performance curve of a blade cascade, inlet total pressure and outlet static pressure with periodic boundary conditions were applied to the numerical solutions. The inlet boundary condition for the inlet flow angle was also changed for each numerical solution to calculate the total pressure loss and the lift and drag coefficients at each incidence angle. Figures 29-32 compare the performance curves for the current blade cascade and each design case. A total of 13 inlet flow angles from 44 to 60 degrees were considered for calculating the performance curves. +++  
Figure 31: Variation of loss coefficient and deviation flow angle versus inlet flow angle for the current and third designed blade cascades. Figure 30: Variation of lift and drag coefficients versus inlet flow angle for the current and the second designed blade cascades. Figure 32: Variation of lift and drag coefficients versus inlet flow angle for the current and the third designed blade cascades. Figure 29 shows the variations of the deviation flow angle and loss coefficient versus the inlet flow angle. The design point was placed at 80% of the maximum deviation angle. According to this figure, the design point is equivalent to the inlet flow angle of 53 degrees. Compared to the current case, there is a loss reduction for the inlet flow angles greater than 52.5 degrees and a loss increase for the inlet flow angles less than 52.5 degrees. A 2% loss reduction at the design point is observed for the second designed blade relative to the current blade. Figure 30 compares the variations of the lift and drag coefficient for the current and second designed blade cascades. According to this figure, as the inlet flow angle increases, the lift coefficient increases before reaching the stall point, and the drag coefficient significantly increases when approaching the stall point. In the second design case, the lift coefficient increases for flow angles more than 51 degrees compared to the current blade. For inlet flow angles from 53 to 57 degrees, the drag coefficient also decreases for the second design case. A 2% load increment (relative increase in the lift coefficient) at the design point is observed for the second designed blade relative to the current blade. Figure 31 compares the variations of the loss coefficient versus the inlet flow angle for the current and third designed blade cascades. The loss coefficient for the third designed blade relative to the current blade decreases at inlet flow angles of 52-55 degrees.