The inverse problem of determining how to shot peen a plate such that it deforms into the desired target shape is a challenge in the peen forming industry. While peening thick plates uniformly on one side results in a spherical shape, with the same curvature in all directions, complex peening patterns are required to form other shapes, such as cylinders and saddles found on fuselages and wing skin panels. In this study, we present an optimization procedure to automatically compute shot peening patterns. This procedure relies on an idealized model of the peen forming process, where the e ect of the treatment is modeled by in-plane expansion of the peened areas, and on an o -the-shelf optimization algorithm. For validation purposes, we peen formed three 305 3054.9 mm and two 7627624.9 mm 2024{T3 aluminium alloy plates into cylindrical and saddle shapes using the same peening treatment. The obtained shapes qualitatively match simulations. For 305 3054.9 mm plates, the relative di erences had the same distribution and were of the same order of magnitude as initial out-of-plane deviations measured on the as-received plates. Keywords: Shot peening, Peen forming, Optimization, Eigenstrains, Finite element method, Residual stresses, 2024-T3 aluminium alloy Email address: frederick.gosselin@polymtl.ca (Fr ed erick P. Gosselin) URL: https://www.polymtl.ca/lm2 (Fr ed erick P. Gosselin) Preprint submitted to engrXiv June 23, 2021 1. Introduction During the shot peen forming process, small shot propelled at several tens of meters per second are used to plastically deform the surface layer of a thin part, causing it to deform, as illustrated in Figure 1. Miao et al. (2010) experi- mentally demonstrated that uniformly shot peening a plate of uniform thickness 5 with the same intensity and coverage deforms it into a spherical shape, as il- lustrated in Figure 1(e). However, most industrial applications demand more complex geometries such as combination of cylindrical or saddle shapes. One way to do so is to vary the intensity and/or the density of impacts over the peened surface. Another technique, called stress peen forming, consists in elas- 10 tically pre-straining the part in speci c directions. The process results in larger plastic strains, hence larger curvatures, in the direction in which the part is stretched. Hornauer and Kohler (1990) peen formed tank bulkheads of the AR- IANE 5 launcher with a tolerance of 1.5 mm from a nominal radius of 3 000 mm using the stress peen forming technique. When compared to conventional 15 peen forming at room temperature, warm peen forming at a temperature below the aging temperature of the material increases the bending capability of the treated panel to achieve further curvature. Ramati et al. (1999) combined the (a) (b) (c)(d) (e) Figure 1: Schematic illustration of the shot peen forming process. (a) A high speed shot stream bombards a metallic surface. (b) Shot impacts on the surface create a plastically deformed layer. (c) Average in-plane plastic strain distribution through the thickness of the surface layer. (d) The plastically deformed layer stretches the surface of the material. (e) Uniformly shot peening the surface induces a spherical shape from an initially at plate. 2 conventional peen forming, the stress peen forming and the warm peen form- ing techniques to form complex industrial parts such as a wing skin panel with 20 di erent curvatures, chordwise and spanwise. However, in the previously cited studies, the peen forming sequences were developed by trial and error attempts, which are time consuming and which severely restrict the number of parts that are economically viable to peen form. Most existing shot peen forming studies seek to numerically simulate the 25 de ections of a given part under a speci ed shot peening treatment, which are called the direct problem. Wang et al. (2006) developed a nite element model to simulate the de ection of unconstrained Aluminium 5251-H22 specimens uni- formly peen formed with S660 media. Three di erent equivalent thermal loads were applied to represent the equivalent plastic strain after peen forming with 30 di erent peening pressures. However, the treated specimens were limited to di- mensions of 76193 mm and a treated area of 19 19 mm. Faucheux et al. (2018) simulated the peen forming process on larger panels having dimensions of 1 0001 000 mm and thickness of 5, 10 and 15 mm. Eigenstrain pro les after shot peening were derived from experimentally measured residual stress 35 pro les and were reconstructed by two parameters: depth of idealized eigen- strain pro le heqand its amplitude  eqto simplify the nite element analysis. The simulations successfully explained the elastic instability which causes the transition from spherical to cylindrical shapes as the panel thickness decreased from 15 mm to 5 mm. 40 In practice, however, it is the inverse problem that process engineers are faced with|i.e., determining the optimal peening parameters, such as intensity, coverage, peening trajectory, and peening pattern, etc., to achieve a prescribed target shape. To solve this inverse problem, it is necessary to develop a numeri- cal optimization model that minimizes the deviation of the computed deformed 45 shape from the desired shape subject to certain shot peening treatments. Few investigations on process optimization procedures were reported in the shot peening literature and most of them were only validated on relatively small plates. VanLuchene and Cramer (1996) developed an optimization model to de- 3 termine the optimal peening intensity pattern required to produce an arbitrary 50 aerodynamic contour. Their approach was based on a nite element shell model of the part to be formed and used the amplitude of stretching and bending loads in each element as optimization variables. Nevertheless, although it was applied to form actual wing skins, neither typical patterns nor experimental re- sults were reported. Kopp and Schulz (2002) applied their previous established 55 optimization tool to establish a direct relationship between the objective shapes and the process parameters to form three-dimensional contours by peening both sides of a plate. Using the optimized peening parameters, seats with wave or saddle contours were produced as a demonstrator. However, little detail about the optimization was provided in their report. Wang et al. (2008) developed 60 an optimization method to minimize the di erence between the target shape and peen formed shape based on a simple non-linear relationship assumption between the deformed arc height and the peening time. Since the model was de ned as a one-dimensional problem, the model was validated on a cylindrical target shape on a small aluminium 5251-H22 sheet with dimensions of 76 76 65 3 mm. Siguerdidjane et al. (2020) developed a method to nd the peen form- ing patterns based on a neural network. The authors trained a neural network with thousands of nite element simulations of the direct problem. Once the costly o -line task of generating the simulation data and training was done, the neural network could predict in a fraction of a second the shot peening pattern 70 which would yield the wanted shape. However, this approach has not yet been validated and the o -line task of generating thousands of simulation is costly. Inverse methods have been developed for laser peen forming applications. Thomson and Pridham (1997) presented an application of a closed-loop con- trol system for the laser peen forming process. When compared to a predictive 75 approach that uses empirical and theoretical method to determine the pro- cess parameters, the feedback approach monitors the deformation of the plate with the attached sensors, compares the real time resulting de ections with the target shape, and nally adjusts the process parameters. Experimental tests showed that the proposed closed-loop control approach produced better results 80 4 than the predictive approach. However, the system produced some variabilities, especially when the de ections were close to the target shape. A maximum tolerance of 0.4 mm above the target de ection were presented in a series of 18 forming tests with target de ections between 1 to 6 mm. Therefore, fur- ther improvements were required to reach a greater degree of precision to meet 85 industrial applications requirements. Recently, Luo et al. (2020) developed a partial di erential equation constrained optimization method, combined with a distributed eigen-moment, to compute the laser shot peening process parameters yielding a target saddle shape. The process was validated by forming a 272  2724 mm 2024-T351 aluminium alloy plate into a saddle shape. Experimen- 90 tal results qualitatively matched the desired shape with a maximum di erence of 0.35 mm over a maximum deformation of 4.5 mm. Although the literature on optimizing peening patterns is scarce, many ro- bust optimization procedures were proposed to address closely related problems encountered in other elds of engineering. For example, Koconis et al. (1994) 95 developed an analytical model to determine the required voltages to achieve a speci ed target shape on ber-enforced composite beams, plates and shells with embedded piezoelectric actuators. Paradies and Hertwig (1999) presented an adaptive composite mirror with integrated piezoelectric actuators to compen- sate their out-of plane displacements due to their self-weight. Irschik (2002) 100 reviewed studies for the shape control of smart structures with piezoelectric control actuation. There are striking similarities between peen forming and shape control of thin structures with piezoelectric actuators: in both cases, a thin multilayered structure deforms in response to the incompatible expansion of its outer layers and the objective is to nd an optimal loading for the plate 105 to achieve the desired target shapes. Pajot et al. (2006) presented one example of such a system where a patterned gold lm was deposited onto a polysilicon substrate such that the di erence of thermal expansion between the two layers caused the structure to deform into a prescribed target shape when cooled down. The optimal patterns were computed with a procedure derived from topology 110 optimization - similar to the solid isotropic material with penalization (SIMP) 5 method described in Bendsoe and Sigmund (2013) - where a nite element model of the polysilicon-gold substrate was used to compute deformed shapes and a gradient-based optimizer was used to generate patterns. In this paper, we propose a procedure to automatically compute shot peening 115 patterns to achieve the pre-de ned shapes. Our procedure is adapted from the multilayered structure presented in Pajot et al. (2006), the SIMP method of Bendsoe and Sigmund (2013) and combined with the eigenstrain approach for modeling the e ect of shot peening by Faucheux et al. (2018). Section 2 describes a numerical procedure to automatically compute peening patterns required to 120 form at plates into desired target shapes. Section 3 presents materials and methods. Section 4 presents the simulated optimized shot peening patterns to achieve cylindrical and saddle shapes and the validations with experimental results. Finally, Sections 5 and 6 discuss key results and conclude this work. 2. Overview of the numerical optimization procedure 125 2.1. Computing deformed shapes for speci ed peening patterns: the direct prob- lem To optimize peening patterns, we rst need a model to predict how a shot peened plate deforms. Here, we used an idealized model similar to that described in Faucheux et al. (2018), where the plates are meshed with thin-shell elements 130 and composite section properties | with a passive inner layer and two active outer layers that can expand | to mimic the e ect of the peening treatment. The amount of expansion was controlled by adjusting the value of the in-plane thermal expansion coecient in each element and the whole model were loaded by applying a unit temperature increment. 135 2.1.1. Finite element model All simulations were done with an in-house nite element code using the discrete Kirchho triangle DKT18 thin-plate elements described in Batoz and Dhatt (1992). As illustrated in Figure 2(a), the plates were discretized with 6 (a) (b) (c)(d)Figure 2: Schematic illustration of the optimization procedure that identi es the peening pattern to form a thin at plate into a 3D surface. (a) The plate is meshed with rectangular patches of 4 superimposed DKT18 elements. (b) Peening-induced loads are imposed as an in- plane expansion in the top and bottom surface layers with thickness of heq, which are active layers. The middle of the plate is considered as a passive layer. (c) The peening patterns during the optimization process for the top and bottom faces. The expansion on each face of each patch is varied between 0 and an upper bound to represent unpeened, partially peened, and totally peened areas. (d) A model of an initially at plate deforms as loads representative of the peening process are applied to it. The optimization procedure attempts to minimize the z-coordinate error between the deformed shape and a prescribed target shape. 7 Plastic strains Residual stressesYield stress(a) (b) In-plane plastic strains Peening induced pro�ile Idealized pro�ile Depth0Figure 3: Surface eigen strain for the calculation of de ection of the plate. (a) Plastic strains and a balanced compressive residual stress pro le are produced in the depth hunder the surface of the plate. (b) An idealized `step' pro le with an equivalent depth heqand in-plane expansion" eqare de ned to impose the peening induced plastic strain pro le in the nite element simulation model that yields the same stretching and bending as the real plastic deformation pro le. rectangular patches and each patch was meshed with four triangular DKT18 140 elements. We model the peening pattern by controlling the expansion of the three layers. The upper and lower layers of all elements in a patch are linked to a single optimization variable , a vector of peening intensities for all patches i= 1, . . . , n, where i= 1 corresponds to a fully peened area and i= 0 corresponds to an unpeened area. The amplitude of the expansion pro le input 145 in the upper and lower layers of those elements was equal to the original input eigenstrain pro le, introduced in Section 2.1.2, multiplied by i. Only a quarter of the plate was simulated using two symmetry boundary condition. All translations and rotations of the node located at (0,0,0) were set to zero to prevent rigid body motions. 150 For nonlinear simulations, we implemented the small strain corotational for- mulation presented in Caselli and Bisegna (2013) using the same DKT18 element as a core element| similarly to what was done in Pajot et al. (2006). Corota- tional formulations are appropriate for the problem at hand because they make the computation of gradients with respect to the magnitude of follower loads 155 (which are the optimization variables used here) much easier than with total or updated Lagrangian formulations (Pajot and Maute, 2006). 8 2.1.2. Loads We reconstructed the eigenstrain pro les for the studied material and the peening treatment from the measured residual stress using the inverse procedure 160 described in Faucheux et al. (2018). However, because we are only interested in the deformed shape of the plates, it is simpler to input an idealized expansion pro le that induces the same axial load and bending moment. When thin-shell elements are used, pro les having the same in-plane and bending resulting loads yield identical deformed shapes. Therefore, it is not necessary to input the full 165 expansion pro le in the model since loads are integrated over the thickness of the element. In what follows, Faucheux et al. (2018) used idealized `step' pro les, as shown in Figure 3(b), which are fully characterized by an equivalent depth heq and an equivalent in-plane expansion " eq. The active upper and lower layers of thicknessheqcould expand, while, the central passive layer could not expand. 170 The depth heqand the amplitude " eqof the equivalent eigenstrain pro les used for simulations were obtained by requiring that they yield the same result- ing eigenstrains and the same rst eigenstrain moment as pro les computed from the experimentally measured residual stress pro le. This condition can be written as heq= 2 = ; " eq= 2=2 : (1) 2.2. Computing peening patterns for speci ed target shapes: the inverse problem The goal of the optimization procedure is to nd a combination of optimiza- tion variables that best- ts a given target shape, as shown in Figure 2(d). We introduce the vector of peening intensities , and we seek to optimize: "(h) =8 >< >:" eq ifhheq; 0 ifh>h eq;(2) The optimization problem is de ned as, minimizekz(x;y; )ztarget (x;y)k2 2 subject to 0 i1; i = 1;:::;n AT Amax;(3) 9 wherezis the vector of current nodal coordinates in the z-direction,ztarget is a vector that contains coordinates of the target shape in the z-direction evaluated at the location of the nodes, is the vector of peening intensities for all patches, introduced in Section 2.1.1, Ais a vector that contains the area of the patches 175 associated with each optimization variable, and Amaxis a user-de ned upper bound that limits the total area that can be peened. It is implicitly understood that deformed nodal coordinates are computed with the nite element model of the structure and that the computed solution satis es equilibrium, compatibil- ity, the constitutive law, and boundary conditions. The constraint AT Amax 180 was used to steer the solution towards easy-to-peen [0-1] patterns, as progres- sively decreasing the value of Amaxforces the optimization algorithm to allocate peened areas only where this has the most impact on the shape of the plate. This penalizes pattern with areas with intermediate values of or checkerboard. Matlab's (R2018a) fmincon interior point algorithm was used for the opti- 185 mization with the default termination criteria. The cost function and second constraint were scaled by max ( ztarget )=100 andAmax, respectively. In all the examples presented hereafter, simulations were systematically initialized from several initial random guesses of as multiple optimal solutions could exist. In all cases, the procedure converged towards the same solution irrespective of the 190 initialization. 3. Experimental tests and Materials 3.1. Peen forming process Initially at plates were laid horizontally and shot peened in the Canablast compressed-air cabinet shown in Figure 4. The shot peening machine is equipped 195 with a 6-axis robotic arm for accurate positioning of the nozzle (M{20 iA sup- plied by Fanuc), with a particle velocity sensor to measure the average shot velocity (Shotmeter G3 supplied by Progressive Technologies), and with a Go- Pro digital camera to record the tests. The plates were secured with double 10 (a) (c) (b)Figure 4: Shot peening setup and treated plates with masks for certain peening patterns. (a) Compressed air peening cabinet used for peen forming tests. During the peen forming tests, the plate was free to deform. (b) The bottom face of plate P1-S-Sphere was masked by tapes according to a simulated peening pattern to achieve a cylindrical shape. The four corners of the plate was xed to prevent the movement of the plate (c) The bottom face of plate P5-L-Saddle was masked by tapes according to a simulated peening pattern to achieve a saddle shape. Two sides of the plate were xed with holders to prevent its movement. sided tape on their bottom surfaces and with holders on their corners or sides 200 to prevent their movement during shot peening. The plates were peened with SCCW28 media having a hardness of 55{62 HRC and a nominal diameter of 0 :71 mm, an air pressure of 0.172 MPa, a mass ow of 0.12 kg/s, a shot speed of 41 m/s, and the nozzle was at a stand-o distance of 41 mm. An Almen intensity of 0.3 mmA (12A), as de ned in SAE 205 standard SAEJ443, and a coverage of 100%, as de ned in SAE standard SAE standard J2277, were applied during all the peen forming tests (AMS Standard 2430). According to the simulated [0{1] peening pattern, 1 represents the surface peened with 100% coverage, 0 represents the surface without peening and was masked with duct tape during the peening process, as shown in Figure 4 (c). 210 3.2. Peen forming residual stress and calculation of eigenstrain pro le A 2024{T3 aluminium alloy plate having dimensions of 1016 2544:9 mm was shot peened at an Almen intensity of 0.3 mmA (12A) and 100% coverage 11 using SCCW28 media. A coupon having dimensions of 254 254 mm was removed from the center of peen formed plate with a lubricated jigsaw and was 215 sent to Hill-Engineering, Rancho Cordova, California to measure the residual stresses with a hole drilling strain gage method as per ASTM standard E837-08. Residual stresses were measured at the center of the coupon, far from the edges, to minimize the e ect of cutting-induced plastic deformations and heating on measurement. The holes were 2 mm in diameter and were drilled in 0 :051 mm 220 increments to a nal depth of 1 :020 mm. Figure 5(a) presents the measured residual stress pro le along the plate's rolling (L) and transverse (T) directions. Peen forming was performed using SCCW28 media with an Almen intensity of 0.3 mmA (12A) and a coverage of 100 %. Figure 5(b) reports the calculated peening induced eigenstrains using the 225 method introduced in 2.1. Using the average eigenstrain pro les of the L and T directions in Figure 5(b), heq= 0:31 mm and " eq= 4:1103were computed from Equation 1. The computed eigenstrain pro les were then modelled as anelastic in-plane expansions of the surface in peened areas. 3.3. Materials samples for the validation of the optimized peening pattern 230 Three 2024{T3 aluminium alloy plates having dimensions of 305 3054:9 mm and two plates having dimensions of 762 7624:9 mm were peened according the optimized peening patterns computed to form them into the target shapes. The dimensions of the smaller plates were chosen such that the peening treat- ment induced small out-of-plane displacements compared to the thickness of 235 the plate, and that they could be predicted with a linear model. On the other hand, the dimensions of the larger plates were chosen such that out-of-planes de ections were larger than the thickness of the plate. Modeling the response of these plates therefore required a geometrically nonlinear model. The target shape of the plate having dimensions of 305 3054:9 mm (P1-S-Cylinder) was a cylinder. The two plates having dimensions of 305  3054:9 mm (P2-S-Saddle and P3-S-Saddle) and the two larger plates having dimensions of 762 7624:9 mm (P4-L-Saddle and P5-L-Saddle) were to be 12 Fit Fit(a) (b)Figure 5: Residual stress and reconstructed eigenstrain pro les in peen formed 2024{T3 alu- minium alloy specimen along rolling (L) and transverse (T) directions. Peen forming was performed using SCCW28 media with an Almen intensity of 0.3 mmA (12A) and a coverage of 100%. (a) Residual stresses measured with the hole drilling method at the center of the coupon having dimensions of 254 254 mm, which was removed from the center of a peen formed specimen having dimension of 1016 (L) 254 (T) 4:9 mm. The tting curves| shown only to guide the eye|are polynomial ts. (b) Peening-induced eigenstrains computed from the residual stress pro le using the inverse method introduced in (Faucheux et al., 2018). formed as saddles and required respectively a linear and a nonlinear code. The target cylindrical and saddle shapes were de ned as z(x;y) =1 2 xx2+yy2 ; (4) wherexandyare the curvatures of the target shape in the x- andy-directions, 240 respectively. The coordinate (0, 0, 0) was de ned at the middle of the plate. Table 1 lists the information of the ve plates considered for the optimization procedure validation. A Young's modulus of 71 :5 GPa and a Poisson ratio of 0:33, taken from the MMPDS-08 handbook, were used for all simulations. 3.4. 3D scans and curvature measurements 245 All ve peen formed plates were scanned with a coordinate measuring ma- chine (Mitutoyo, Crysta-Apex 163011) equipped with a REVO®5-axis mea- surement system. Measurements were taken every 4 mm in continuous scanning mode along several lines parallel to the xandydirections of the plate. 13 Table 1: The plates dimensions, target shapes, curvatures of the target shapes, the maxi- mum de ection of the target shapes z(x;y)tarand the code types used for the optimization procedure. Plate name Plate dimensions (mmmmmm)Target shape x (104mm1)y (104mm1)Maximum z(x;y)tar(mm)Linear/Nonlinear Code P1-S-Cylinder 305 3054:9 Cylinder -1.50 0.00 1.74 Linear P2-S-Saddle 305 3054:9 Saddle -1.08 1.08 2.51 Linear P3-S-Saddle 305 3054:9 Saddle -1.08 1.08 2.51 Linear P4-L-Saddle 762 7624:9 Saddle -1.00 1.00 14.52 Linear P5-L-Saddle 762 7624:9 Saddle -1.00 1.00 14.52 Nonlinear 3.5. Evaluation of the di erences and relative di erences between target shapes, 250 simulation and experiments Several di erences and relative di erences were de ned to compare the re- sulting shapes to the target shapes, namely: Dst=z(x;y)simz(x;y)tar; RDst=jDstj z(x;y)tar100%; (5) Dit=z(x;y)iniz(x;y)tar; RDit=jDitj z(x;y)tar100%; (6) Des=z(x;y)expz(x;y)sim; RDes=jDesj z(x;y)sim100%; (7) Deo=z(x;y)expz(x;y)tar; RDet=jDetj z(x;y)tar100%; (8) wherez(x;y)tar,z(x;y)sim,z(x;y)ini, andz(x;y)expare the target shape, the simulated optimal shape, the as-received initial shape, and the experimental shape of the plate respectively. D st, RD st, Dit, RD it, Des, RD es, and Det, RD etare the absolute di erences and relative di erences between simu- 255 lated optimal and target shapes, initial and target shapes, experimentally mea- sured and simulated optimal shapes, and experimentally measured and target shapes, respectively. 14 4. Results 4.1. Plates having dimensions of 305 3054:9mm 260 4.1.1. Comparison between the target shape and that predicted from the opti- mized peening pattern Figure 6 shows the target cylindrical (a) and saddle (e) shapes, the simulated optimal [0{1] peening patterns (b, f), the simulated optimization shapes (c, g), and the di erences between the simulated optimal shape and the target 265 shape D st(d, h) for cylindrical (P1-S-Cylinder) and saddle (P2-S-Saddle and P3-S-Saddle) target shapes, respectively. The green color area on the plate represents the peening region and the white area on the plate represents the unpeened region in the experiments. The light green area in Figure 6 (b), where = 0:27 or shot peening coverage of 27%, represents a `partially peened 270 areas' area resulting from the optimization code. To achieve the pattern with a single treatment, the area with <0:4 were masked during the experiments. In Figure 6 (d, h), the di erence between the simulated optimal shapes and the target shapes D stis smaller than 0.08 mm for both the cylindrical and the saddle shapes, for most of the plate surfaces except the edges. The maximum 275 di erences D stof0.22 mm and0.4 mm were concentrated on the four edges of the plate for both the cylindrical and the saddle shapes. 4.1.2. Comparison between experimentally peened plates and simulated optimal shapes Figure 7 presents the three-dimensional scans of the four plates peened with 280 the simulated peening patterns of Figure 6 and their di erences with respect to the simulated target shapes D es. The three-dimensional scans revealed that all the plates qualitatively exhibited the desired target shape. It should be noted that, for the purpose of peening pattern optimization, the plates were considered as perfectly at. However, Figs. 7 (b, f, j) show that the plates were 285 not perfectly at prior to peening. Plate P1-S-Cylinder had an initial de ection Diniof 0.23 mm and the maximum di erence between peen formed shape and 15 P1-S-Cylinder P2-S-Saddle, P3-S-Saddle (a) (b) (c) (d)(e) (f) (g) (h) 01 01 Ds-t Figure 6: Peening patterns to achieve a cylindrical shape for plate P1-S-Cylinder and saddle shapes for plates P2-S-Saddle and P3-S-Saddle. (a) Target cylindrical shape having a target de ection D tarof 1.74 mm for plate P1-S-Cylinder. (b) Optimized peening patterns on the top and bottom faces of the plate to achieve a cylindrical shape. The light green area in the middle of the plate, where = 0:27, represents a surface with 27% coverage. (c) Simulated optimal cylindrical shape z(x;y)sim. (d) Di erence between the simulated optimization shape and the target cylindrical shape D stwith a maximum value of 0.22 mm. (e) Target saddle shape having a target de ection D tarof 2.51 mm for plates P2-S-Saddle and P3-S-Saddle. (f) Optimized peening patterns on top and bottom faces of the plate to achieve a saddle shape. (g) Simulated optimal saddle shape z(x;y)sim. (h) Di erence between the simulated optimization shape and the target saddle shape D stwith a maximum value of 0.40 mm.16 As received After peening Difference (in mm) xy -0.22 -0.11 00.110.22 xy -1 -0.5 00.5 1 1.83 mm 1.83 mm 0.24 mm1.07 mm -0.1 -0.5-0.025 0.025 1.69 mm xy -0.24-0.1200.120.24 -0.10.10.23 mmP1-S-Cylinder P2-S-Saddle P3-S-Saddle (b) (c)(e)(a) (d)(f) (g) (h)(i) (j) (k) (l)Deformed shape (Simulation)Figure 7: Comparison between the peen formed shapes and simulated optimal shapes for plates P1-S-Cylinder, P2-S-Saddle and P3-S-Saddle. (a) Simulated cylindrical shape with a de ection D simof 1.69 mm. (b) Initial de ection D iniof 0.23 mm in the as-received plate before peening. (c) Experimental scan of peened shape z(x;y)expafter shot peening. (d) Di erence between the peened shape and the simulated cylindrical shape with a maximum Desof0:24 mm. (e) Simulated saddle shape with a de ection D simof 1.83 mm. (f) Initial de ection D iniof 0.24 mm in the as-received plate before peening. (g) Scan of the peened shape z(x;y)expafter shot peening. (h) Di erence between the peened shape and the simulated saddle shape with a maximum D esof0:22 mm. (i) Simulated saddle shape with a de ection D simof 1.83 mm. (j) Initial de ection D iniof 1.07mm in the as-received plate before peening. (k) Scan of the peened shape z(x;y)expafter shot peening. (l) Di erence between the peened shape and the simulated saddle shape with a maximum D esof1:0 mm. 17 the simulated optimal shape D eswas0.24 mm. Plate P2-S-Saddle had an initial de ection D iniof 0.24 mm and the maximum di erence D eswas 0.22 mm. Plate P3-S-Saddle had an initial de ection D iniof 1.07 mm and the 290 maximum di erence D eswas1.0 mm. These observations suggest that the initial de ections have a signi cant impact on the nal peen forming process outcome. This situation is further emphasized by the fact that plates P2-S- Saddle and P3-S-Saddle, which had di erent initial geometries, were peened with the same patterns and were supposed to achieve the same saddle shapes, 295 presented di erent di erences in their nal geometries. 4.2. Plates having dimensions of 7627624:9mm The results presented so far were limited to small de ections and linear nite element models. We now turn to experiments conducted on 762 7624:9 mm plates that exhibited suciently large deformations so that geometrically nonlin- 300 ear e ects had to be accounted for. For the plates P4-L-Saddle and P5-L-Saddle, the target de ection D tarwas 14.52 mm and was about 3 times their thickness (4.9 mm), which corresponds to the beginning of the geometrically nonlinear regime. Therefore, the optimization procedure was extended to large-de ection scenarios by replacing the linear plate model with one that includes geometric 305 nonlinear e ects, as introduced in Section 2. Figure 8 shows the computed peening patterns for plates P4-L-Saddle and P5-L-Saddle, with the same peening treatment, the same target saddle shape, but using linear and nonlinear plate models, respectively. Figure 8(b) presents the peening pattern computed from the linear model, which was the same as 310 that of Figure 6(e) to achieve the saddle shape for plates P2-S-Saddle and P3- S-Saddle. Figure 8(e) shows the peening pattern computed from the nonlinear model, where the optimal pattern features an outer ring peened on both faces surrounding an unpeened region. Note the similarity between these patterns and the experiments reported by Pezzulla et al. (2015) in the context of soft 315 matter physics, where the assembly of two non-conforming concentric polymer rings causes the system to pop out of plane and assume a saddle shape. Because 18 14.52 mm 0.55 mm -0.55 mm 1.97 mm - 1.97 mm(a) (b) (c) (d)(e) (f) (g)P4-L-Saddle -0.6 mm 2.0 mm -1.5 mm 0 1.4 mmP5-L-SaddleFigure 8: Optimal peening patterns computed with the linear and nonlinear optimization codes for plates P4-L-Saddle and P5-L-Saddle. (a) Target saddle shape with a target de ection Dtarof 14.52 mm for both plates. (b) Optimized peening patterns on the top and bottom faces of plate P5-L-Saddle to achieve saddle shape with the linear code. (c) Di erence between the simulated optimal shape and the target shape with a maximum D stof1:97 mm using the linear code. (d) Di erence between the peened shape and the target with a maximum D etof 1:4 mm. (e) Optimized peening patterns on the top and bottom faces of plate P6-L-Saddle to achieve saddle shape with the nonlinear code. (f) Di erence between the simulated shape and the target shape with a maximum D stof0:55 mm using the nonlinear code. (g) Di erence between the peened shape and the target shape with a maximum D etof 2:0 mm. 19 the linear code cannot capture this instability, it returns a markedly di erent pattern as in Figure 8(b). However, since out-of-plane de ections are moderate, the deformed shape obtained with this pattern stays close to the target shape, 320 except near the rim of the plate where it causes the edges to curl. The maximum di erences on z-displacements between the simulated optimal shape and the target shape D stwere1:97 mm for plate P4-L-Saddle with the linear pattern and were 0:55 mm for plate P5-L-Saddle with the nonlinear pattern. The maximum di erences between the peened shape and target shape 325 Detwere 1:4 mm for plate P4-L-Saddle and were 2 :0 mm for plate P5-L-Saddle. Part of this larger di erence for plate P5-L-saddle could come from its initial de ection D ini, which was not measured before the peen forming test. 5. Discussion 5.1. The di erences between the optimization process and resulting peened ge- 330 ometry Table 2 lists the maximum initial de ection D ini, the maximum relative di erence between the initial shape and the target shape RD it, the maximum simulated de ection D sim, the maximum relative di erences between the simu- lated shape and the target shape RD st, the maximum peened plate de ection 335 Dexp, the maximum relative di erence between the peened shape and the simu- lated shape RD es, and between the peened shape and the target shape RD et for the ve studied plates. For the three plates having dimensions of 305 mm 305 mm4.9 mm, smaller target de ections were expected and the linear optimization code was 340 applied and validated. The maximum relative di erences between the simulated shape and target shape RD stwas 12.6% for plate P1-S-Cylinder, and was 15.9% for plates P2-S-Saddle and P3-S-Saddle. Experimentally formed shapes, using the predicted peening patterns, presented consistent nal shapes with a relative di erence RD etof 9.2% for plate P1-S-Cylinder, 10.6% for plate P2- 345 S-Saddle, and 25.5% for plate P3-S-Saddle. However, the larger di erence in 20 Table 2: Initial de ections D ini, relative di erences between the initial shapes and the target shapes RD it, simulated de ections D sim, relative di erences between the simulated shape and the target shape RD st, peened plate de ections D exp, relative di erence between the peen shape and the simulated shape RD es, and relative di erence between the peened shape and the target shape RD etfor the ve plates considered in this study. Plate name DiniRDitDsimRDstDexp RDesRDet (mm) (%) (mm) (%) (mm) (%) (%) P1-S-Cylinder 0.23 6.6 1.69 12.6 2.06 13.8 9.2 P2-S-Saddle 0.24 4.8 1.83 15.9 1.98 8.8 10.6 P3-S-Saddle 1.07 21.3 1.83 15.9 1.23 39.8 25.5 P4-L-Saddle n/a n/a 10.88 13.6 9.98 3.1 9.6 P5-L-Saddle n/a n/a 13.65 3.8 10.62 10.5 13.8 plate P3-S-Saddle could be related to its larger initial relative de ection RD it of 21.3%. For the two larger plates with dimensions of 762 mm 762 mm4.9 mm, a nonlinear model was required and applied. Using a nonlinear code led to pat- 350 terns that signi cantly di ered from those obtained from a linear code. While de ection measurements on peened plates showed that the patterns calculated with both linear and nonlinear codes produced similar nal shapes, with max- imum relative di erences RD etof 10.5% for plate P4-L-Saddle and of 13.8% for plate P5-L-Saddle. This is mainly due to the fact that the target de ection 355 was still considered as moderate deformations for the plates. The discrepancy between the target shape resulting from the optimization procedures and that experimentally measured could be explained by two main reasons: 1. The initial deformations of the aluminium plates present before peening 360 were neglected in the optimization procedure. The plates were assumed to be initially perfectly at in the optimization procedure. Figure 7 shows similarities between the as-received initial de ection D iniand the maxi- mum di erence between the experimentally measured shape and simulated saddle shape D es. 365 2. In the models presented in this paper, it is assumed that the amplitude of the plastic deformations induced by the treatment is independent of the 21 position on the plate. In practice, the treatment is not instantaneous. The assumption that we made was to assume that the plate was held at during shot peening, and that all stresses are released at the end. The progressive 370 deformation of the plates causes the amplitude of the eigenstrains to vary in the plate. However in the whole optimization procedure, a consistent eigenstrain pro le was used in the simulation, which could bring certain di erences. 5.2. Automating the identi cation of peening patterns and process parameters 375 In the optimization process introduced in this work, the peening treatment was selected beforehand and target shapes were chosen such that they could be approached with this treatment. This was done iteratively by increasing the curvatures of the target shapes until the optimal shape could not keep pace with the target. In practice, however, it is the opposite problem that needs 380 to be solved: the target shape is a requirement and the objective is (i) to determine whether the prescribed shape can be peen formed and, when this is the case, (ii) to nd a peening pattern and process parameters that yield the desired shape within acceptable tolerances. In addition, the treatment must be compatible with production requirements. The latter usually include using 385 as few di erent shot types as possible, ensuring that the maximum allowed intensity and coverage are not exceeded, and maximizing the smoothness of the patterns|as intricate patterns with many details and sudden variations of intensity or coverage are harder to manufacture than smoother, simpler patterns. To achieve this, we propose the following four-step procedure: 390 1. Run the optimization procedure presented in the paper without any con- straint apart from the upper and lower bound on and with a very large value for the amplitude of the expansion in the active layer. For example, use thefheq;" eqgfor the most intense treatment that the available peening equipment can deliver. If, at the end of this analysis, the optimal shape 395 matches the target shape within tolerances, proceed to step 2. Otherwise, the target shape can likely not be peen formed. 22 2. Re-run the analysis by progressively decreasing the intensity of the treat- ment until the di erence between the optimal shape and the target shape becomes unacceptable|using the same metrics as those used to accept 400 or reject a part to de ne `acceptable'. This gives a lower bound on the intensity of the peening treatment. 3. Identify process parameters that (i) deliver an intensity within the upper bound of step 1 and the lower bound of step 2, and (ii) are compatible with production requirements. For example, only a nite number of shot 405 type might be available, and certain combinations of intensity and shot size might be prescribed if they result in a poor surface nish. 4. Choose one treatment, re-run the analysis by progressively adding or tight- ening constrains to steer the optimal solution towards a peening pattern that can be easily manufactured. For example, decreasing the upper limit 410 on the area that is allowed to be peened tends to force the algorithm to- wards [0{1] designs that can be manufactured by masking part of surface and indiscriminately blasting the entire surface. Similar [0{1] designs can be obtained with appropriate penalty schemes, as described in Bends e and Sigmund (2004) and Pajot et al. (2006). When automated peening 415 equipment is available that can continuously adjust process parameters, less severe constraints that allow for smooth transitions between peened and unpeened areas can be used, such as those listed in VanLuchene and Cramer (1996). Note that, although they can improve the manufactura- bility of a pattern, constraints inevitably degrade the agreement between 420 the optimal solution and the desired shape. If, at the end of this step, the optimal shape does match the target shape within tolerances, return to step 3 and select a di erent treatment. To illustrate the third step, suppose that a perfect [0{1] pattern has been computed, that is, that all components of the optimal are either zero or 425 equal to the same value, . We now wish to nd a treatment that induces the same axial loads and bending moments as a step of depth heqand of magnitude 23 " eq. Unless fast and accurate impact models are available, this must be done experimentally. To narrow down the search, a preliminary step would be to build databases offheq;" eqgversus process parameters using available experimental 430 data sets. Most companies and research teams involved with peening processes already have databases of residual stress pro les at their disposals from which it is possible to extract eigenstrain pro les using the procedures introduced in Faucheux et al. (2018). 6. Conclusions 435 In this article, we demonstrated that it is possible to procedurally generate peening patterns for given complex target shapes. In particular, we managed to form a cylindrical shape and a saddle shape from thick plates, which are usually manufactured by costly stress peen forming operation. Our strategy relied on the combination of (i) an idealization of peening induced loads as 440 anelastic expansion (Faucheux et al. (2018)) and (ii) an optimization algorithm (Pajot et al. (2006)). The formulation of applying eigenstrains into a nite element model proved to be ecient for the peening pattern optimization procedure. The simulated peening patterns [0{1] were easily performed by just masking the unpeened sur- 445 face of the plate without changing other peening parameters. The optimization procedure could predict the peening patterns to achieve cylindrical or saddle shapes with a maximum relative di erence of 15.9%. Experimental validations show that using the predicted peening patterns produced consistent target cylin- drical or saddle shapes with a maximum relative di erence of 13.8%, but with 450 an exception of a maximum relative di erence of 25.5%, in which case, the plate P3-S-Saddle had a large initial relative de ection of 21.3%. For the future improvement of the optimization model, the initial de ec- tion of the plate and the progressive deformation of the plates during the peen forming process should be considered. 455 24 Declaration of Competing Interest The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to in uence the work reported in this paper. Acknowledgment 460 The authors gratefully acknowledge nancial support from the Canada Re- search Chairs program, and from Discovery Grants program by Natural Sciences and Engineering Research Council of Canada (NSERC).The authors would also like to thank Dr. P. Faucheux for original ideas behind this work and the ex- perimental results. 465