**2. Methods and Materials**

The implemented corrective approach combines a semi-analytical model, which exhibits significant deviations in predictions outside its calibration parameter space, with a data-driven machine learning model correcting those deviations towards the solution of the high-fidelity model. The corrective model is required to be less complex, for solely representing a corrective component, compared to a purely data-driven prediction model mapping the more complex and complete relationships that are relevant. Additionally, this hybrid approach shows good generalization ability and also exhibits low prediction errors in an expanded input parameter space outside the parameter space used for training, as opposed to decreased generalization ability of a purely data-driven model, which is not physics-related. For the selected use-case of LSP, the residual stress distributions intended to be corrected are calculated via the semi-analytical model from Hu et al. [23]. An FE model was used for computing the desired reference residual stress distributions, which represent the true/desired data in this work. The correction task is developed through training, validating and testing of an ANN. Both numerical and semi-analytical models will be briefly introduced in the following two sections. For more details, the reader is referred to the original publications, as the focus of this study lies on the correction task where those models are assumed as black-box models and their detailed mechanisms are deliberately not intended to be relevant for the current study. (Note: the selected use-case LSP serves only as selected example. Generally speaking, the analytical model could be replaced by any physics-based model and the data from the FE model represents the corresponding, typically scarce, experimental data). Material parameters correspond to the aluminium alloy AA2024 in T3 heat treatment condition, frequently used in the aircraft industry for fuselage structures [24].

#### *2.1. Laser Shock Peening*

One of the main goals of the transportation industry is to reach weight, fuel and CO<sup>2</sup> savings as well as increase the sustainability of engineering components [25]. For improving the fatigue life of light-weight materials such as aluminium alloys, LSP has gained attention in scientific research and industrial application developments. LSP is known as residual stress modification technique to introduce high and deep compressive residual stresses in metallic components [26]. These compressive residual stresses can be used to enhance fatigue properties of metallic structures, which is of high interest for damage tolerant design concepts, as applied in aircraft structures. However, compressive residual stresses are always accompanied by fatigue-critical tensile residual stresses due to stress equilibrium. During LSP, short-time (nanosecond regime), high-energy (Joule regime) laser pulses are used to convert material at the surface into plasma. Plasma expansion initiates mechanical shock waves that cause local plastic strains in the material. After relaxation of the dynamic process, a characteristic residual stress field is developed, which contains both: Relatively high compressive residual stresses and balancing tensile residual stresses. Experimental process observation is very challenging and requires great effort due to the magnitudes of physical quantities, such as plasma pressure as well as temperature, and the short time scale. The knowledge of the residual stress fields is essential for efficient application of LSP, motivating the development of suitable prediction tools. Modelling of the LSP process is challenging due to the short time scale of the process, which, so far, leads to imprecise experimental determination of physical quantities occuring during shock wave propagation and plasma formation, such as material strain rates up to 10<sup>6</sup> s −1 , plasma pressure of several GPa or the high plasma temperature; therefore, the utilized material model can exhibit determination inaccuracies regarding these quantities. There are various approaches to simulate the LSP process, such as FE models [27–29] or (semi) analytical models. While FE models represent the most commonly used modelling approach, to represent the three-dimensional physics involved in the complex LSP process, the considered semi-analytical model by Hu et al. [23] is computationally very efficient but does not provide any information on tensile stresses because stress equilibrium is neglected.

Other simplifications include the assumption of an infinite instead of finite specimen thickness as well as single value calculations of stresses at distinct model locations as opposed to averaged stress calculation based on extrapolation of finite element integration points towards nodes, among others. Since the considered LSP system and FE model uses quadratic pulse spots, see Keller et al. [29], the underlying assumption of a circular spot in the semi-analytical model represents a further simplification in the current case.

Ultimately, the proposed correction approach is employed to achieve low prediction errors while simultaneously using the implied physics and maintaining the computational efficiency of the analytical model. Such a hybrid modelling approach is new in the context of LSP, where the number of publications on the application of machine learning approaches for the LSP process is scarce, overall. Frija et al. [30] optimized the LSP surface conditions by using an FE model exposed to the laser-induced pressure pulse as well as Design of Experiments (DoE) to infer related laser parameters. They extended the work by the use of an ANN to efficiently predict significant characteristics of numerical compressive residual stress profiles and approximated a simplified 1st-order linear slope of residual stresses [31]. In this study, it is aimed for efficiently predicting the original non-linear distribution of compressive and tensile residual stresses, provided by an FE model, throughout the complete depth of the specimen. Wu et al. [32] also performed predictions of LSP-induced residual stresses via an ANN based on the laser profile and laser energy purely based on experimental data; thereby, not explicitly considering relevant physical relationships. Mathew et al. [33] used an ANN for the prediction and optimization of residual stress distributions induced by LSP, where the relative importance of four process parameters on residual stresses is investigated purely based on experimental data. In this work, the proposed hybrid model generates highly accurate predictions that are physics-related via the corrective approach of a physics-based analytical model.

#### *2.2. Physical Models*

In the following, the pressure pulse input definition for both physical models as well as the semi-analytical model and high-fidelity FE model, are described.

#### 2.2.1. Pressure Pulse Definition for Physical Models

The definition of the pressure pulses over time, in Figure 1, is utilized as input for the semi-analytical model, see Figure 2a, and for the high-fidelity FE model, see Figure 3a. The pressure pulse over time is uniquely defined in this work based on three pressure pulse parameters: Maximum pressure *Pmax*, the time of maximum pressure *t<sup>I</sup>* and the pulse duration *tI I*, see Figure 1. This pressure pulse function is preferred in the utilized ABAQUS solver of the FE analysis since it is differentiable and assures efficiency and stability of the FE solver [34]. Note that the original semi-analytical model by Hu et al. [23] is slightly modified by using this pressure pulse as input, instead of laser parameters. Note: The pulse duration *tI I* is not considered in the semi-analytical model, as described in the following Section 2.2.2.

**Figure 1.** Pressure pulse over time including its uniquely defining parameters: Maximum pressure *Pmax*, time of maximum pressure *t<sup>I</sup>* and pulse duration *tI I*. As additional information, the full width at half maximum is given by *tI I I*.

#### 2.2.2. Low-Fidelity Model — Semi-Analytical Model

A semi-analytical LSP process model to predict residual stress profiles depending on the plasma pressure is developed by Hu et al. [23], which is adopted in this study. In the process model, a semi-infinite space and rotational symmetry are assumed since a circular laser focus is considered. Furthermore, single laser pulse impacts are modelled instead of a laser pulse sequence. The residual stress profile is evaluated along the symmetry axis. The LSP process of a single laser pulse impact is split into two phases: Loading and relaxation. During the loading phase, the pulse pressure from *t* = 0 to *t<sup>I</sup>* is considered as input and during the relaxation phase, the resulting residual stresses are calculated (note that the pressure pulse interval from *t<sup>I</sup>* to *tI I* is not considered in this model). Plasma induced stresses that are present during the loading phase are assumed to be superposed and fully developed stress fields that are caused by time dependent surface tractions of the plasma pressure, representing the elastic solution. The stress field caused by a single traction is described by closed-form expressions corresponding to the equation found for single forces, see Timoshenko and Goodier [35]. Plastic material deformation and resulting stresses are calculated by the McDowell Hybrid Algorithm [36]. A strain-rate dependent material model, including isotropic and kinematic hardening is employed. The strain-rate dependency of the yield stress is modelled by the Johnson–Cook model, where material parameters are listed in Table 1. After the application of the plasma pressure, the residual stress field is calculated during the relaxation phase; therefore, stresses are incrementally reduced while plastic deformation is taken into account to match stress and strain boundary conditions of an axisymmetric half space. A stress equilibrium is not calculated by this algorithm, as opposed to the FE analysis, which is explained in the following Section 2.2.3. For more details on the semi-analytical model, the interested reader is referred to the original work by Hu et al. [23]. Overall, the main involved physical phenomena are considered in the semi-analytical model but to a substantially simplified extent leading to a relatively narrow parameter space, where in combination with a subsequent correction, the desired high fidelity solution of the FE model within a much wider parameter space can be reached, nevertheless.

**Figure 2.** Illustration of the semi-analytical model by Hu et al. [23] for computing residual stresses induced by pressure pulse from Figure 1. Circular pressure pulse area (i) (in red) on the half-space model, which is simplified in (ii) as a concentrated normal load (in red) in the axisymmetric half-space model. Figures (i) and (ii) are republished with permission of the American Society of Mechanical Engineers ASME from [23].

#### 2.2.3. High-Fidelity Model — FE Model

The FE LSP-process model, set up to calculate residual stresses in AA2198 [29] and adopted to AA2024 [37] in the author's previous works, is used in this work to generate a database with the plasma pressure as input and residual stress profiles as output, see Figure 3. The LSP process model consists of a cuboid with dimensions of 60 mm × 60 mm × 4.8 mm and the depth is discretized with an element size of 0.02 mm next to the surface. Sides parallel to *x-z* and *y-z* plane are modelled with fixed boundary conditions, whereas sides parallel to *x-y* plane are considered as free surfaces. The plasma pressure caused by a single laser pulse is modelled as a time dependent surface traction that is uniformly distributed within the peened area. The temporal pressure profile is varied to set up the data set for training, validation and testing. A square of 3 × 3 laser pulses is simulated without overlap, where the square focus size is 3 mm × 3 mm. Residual stresses below the centred laser pulse are averaged layer-wise to calculate a residual-stressover-depth profile, which has shown to be valid by comparison to experiments [29,37]. The LSP process model consists of approximately 1.4 <sup>×</sup> <sup>10</sup><sup>6</sup> continuum elements with reduced integration (C3D8R). The Johnson–Cook material model [38] is utilized, where the used material parameters for AA2024 are summarized in Table 1 for convenience. Nine pressure pulses are simulated in Abaqus/Explicit. A relaxation time of 50 µs is simulated between each pulse, which ensures that the dynamic process reaches a state sufficiently close to equilibrium to prevent significant interaction between two consecutive laser pulses, modelled as pressure loadings. After the simulation of all laser pulses, a final quasi-static implicit simulation (Abaqus/Standard) is conducted to determine the residual stress equilibrium. For further details on the model, the interested reader is referred to [29].


**Table 1.** Elastic and Johnson–Cook material parameter representative for aluminium alloy AA2024 in T3 heat treatment condition with an equivalent plastic strain rate *<sup>ε</sup>*˙*P*,0 <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> s <sup>−</sup><sup>1</sup> according to [39].

**Figure 3.** Finite element process model for computing residual stresses induced by pressure pulse from Figure 1.
