## 2 Thermoelastoplastic Approximate Algorithms
### Numerical Integration Scheme
We investigated the accuracy of their predictions in thermoelastoplastic plane-strain half-space problems using appropriate FE analyses to obtain reference solutions. The following main conclusions can be drawn:

* The algorithms are rather sensitive to the hardening slope and require some minimum amount of strain hardening (how much is dependent on the intensity of the prescribed loading) to robustly produce reasonable predictions. * Due to the strong dependency on the hardening slope, a careful and exact characterization of the strain-hardening behavior is indispensable in order to avoid the possible pitfall of \"adjusting\" the algorithms to experimental residual stress results by crude constitutive assumptions. * The MM algorithm does not suit anisothermal problems as studied in this paper due to its inadequate assumption of zero total strain in the moving direction of the surface loads. Specifically, this kinematic constraint inappropriately predetermines the predicted residual stress components to be equal and furthermore introduces increased plastic flow to accommodate the thermal expansion, thus leading to poor approximations especially in the hot surface-near zone. * In accordance with these limitations of the MM method, we found no obvious incentive to prefer the hybrid algorithm (which effectively reduces to the MM approach for small hardening slopes) over the JS algorithm for the considered type of problem within our investigation. While the hybrid algorithm can be somewhat readjusted to a particular loading and hardening behavior by carefully \"tuning\" the blending function, doing so requires the knowledge of the correct residual stress distributions beforehand and thus may be impractical when considering a large parameter space as e.g., in prediction or optimization applications. * The JS method was demonstrated to provide excellent residual stress approximations in anisothermal problems with thermal softening if sufficiently strong strain hardening is present. * The general approach of uncoupling the problem w.r.t. the depth coordinate \(\eta\), thereby relinquishing kinematic compatibility, and instead relying on the prescribed stress components from the elastic solution to ensure continuity along the depth, appears to be a weak point in all algorithms. Indeed, for larger loads and for smaller hardening slopes, the transient elastoplastic stresses within the loading pass increasingly deviate from the elastic solution, leading to the breakdown of the algorithms with this underlying simplification. As indicated by the last bullet point, the load regime and the hardening characteristics limit the algorithms' applicability. Given the sensitivity of the approximation quality on these two factors, it seems imperative to verify the residual stress predictions as a matter of principle with more accurate and mechanically equivalent computational methods (e.g., FE analyses) at least on a subset of the parameter space of interest. We consider this an important additional verification beyond comparing predicted and experimentally obtained residual stresses, since it provides insight into the modeling error introduced by the approximate method itself in separation from e.g., measurement errors in experimental residual stress analyses or uncertainties in the specification of the surface loads. Beyond the parameter range where the approximate algorithms work, refined strategies and algorithms are required. An obvious choice is to properly solve the initial boundary value problem using a transient elastoplastic FE analysis as we did in our study. Indeed, this seems preferable to us in most applications in the light of current computational capabilities. However, recent interest in real-time residual stress predictions, e.g., for process control, constitute a clear target application where computational efficiency demands approximate methods as the ones we studied. Therefore, further research on improved approximate methods is still worthwhile. Conceptualization, C.H.W. and C.K. ; methodology, C.H.W. ; software, C.H.W.