In the framework of solid mechanics, the task of deriving material parameters from experimental data has recently reemerged with the progress in full-field measurement capabilities and the renewed advances of machine learning. In this context, new methods such as the virtual fields method and physics-informed neural networks have been developed as alternatives to the already established least-squares and finite element-based approaches. Moreover, model discovery problems are emerging and can be addressed in a parameter estimation framework. These developments call for a new unified perspective, which is able to cover both traditional parameter estimation methods and novel approaches in which the state variables or the model structure itself are inferred as well. Adopting concepts discussed in the inverse problems community, we distinguish between all-at-once and reduced approaches. With this general framework, we are able to structure a large portion of the literature on parameter estimation in computational mechanics—and we can identify combinations that have not yet been addressed, two of which are proposed in this paper. We also discuss statistical approaches to quantify the uncertainty related to the estimated parameters, and we propose a novel two-step procedure for identification of complex material models based on both frequentist and Bayesian principles. Finally, we illustrate and compare several of the aforementioned methods with mechanical benchmarks based on synthetic and experimental data.

During manufacturing of components using wire arc additive manufacturing, specific cooling times are required to prevent overheating of the structure and geometrical distortions. Currently, these cooling times are inserted based on experience at certain interlayer temperatures — which reduces the reproducibility, leads to unwanted component properties, and increases the process time. In this contribution, instationary thermal finite element simulations are applied to compute the temperature evolution of additively manufactured components using the inactive element method. This allows to optimize the process parameters, which are — in our considerations here — the welding velocity and the cooling time of each layer, to reduce the total process time while achieving sufficient component properties. The optimization is carried out with the gradient-free Nelder–Mead simplex algorithm, where certain constraints of the process parameters are considered via penalty functions. To obtain reasonable simulation results, the temperature-dependent heat transfer of the experimental setup is modeled and calibrated with experimental data beforehand. It becomes apparent that thermal finite element simulations combined with a gradient-free optimization procedure are a suitable numerical tool to perform the optimization of process parameters for wire arc additive manufacturing. The optimized process parameters fulfill certain requirements regarding the cooling of the manufactured component. Moreover, the optimized parameters can significantly reduce the process time compared to manually chosen parameters. In our example, this is around 48 %.

Multiscale FE${}^2$ computations enable the consideration of the micro-mechanical material structure in macroscopical simulations. However, these computations are very time-consuming because of numerous evaluations of a representative volume element, which represents the microstructure. In contrast, neural networks as machine learning methods are very fast to evaluate once they are trained. Even the DNN-FE${}^2$ approach is currently a known procedure, where deep neural networks (DNNs) are applied as a surrogate model of the representative volume element. In this contribution, however, a clear description of the algorithmic FE${}^2$ structure and the particular integration of deep neural networks are explained in detail. This comprises a suitable training strategy, where particular knowledge of the material behavior is considered to reduce the required amount of training data, a study of the amount of training data required for reliable FE${}^2$ simulations with special focus on the errors compared to conventional FE${}^2$ simulations, and the implementation aspect to gain considerable speed-up. As it is known, the Sobolev training and automatic differentiation increase data efficiency, prediction accuracy and speed-up in comparison to using two different neural networks for stress and tangent matrix prediction. To gain a significant speed-up of the FE${}^2$ computations, an efficient implementation of the trained neural network in a finite element code is provided. This is achieved by drawing on state-of-the-art high-performance computing libraries and just-in-time compilation yielding a maximum speed-up of a factor of more than 5000 compared to a reference FE${}^2$ computation. Moreover, the deep neural network surrogate model is able to overcome load-step size limitations of the RVE computations in step-size controlled computations.

The calibration of constitutive models from full-field data has recently gained increasing interest due to improvements in full-field measurement capabilities. In addition to the experimental characterization of novel materials, continuous structural health monitoring is another application that is of great interest. However, monitoring is usually associated with severe time constraints, difficult to meet with standard numerical approaches. Therefore, parametric physics-informed neural networks (PINNs) for constitutive model calibration from full-field displacement data are investigated. In an offline stage, a parametric PINN can be trained to learn a parameterized solution of the underlying partial differential equation. In the subsequent online stage, the parametric PINN then acts as a surrogate for the parameters-to-state map in calibration. We test the proposed approach for the deterministic least-squares calibration of a linear elastic as well as a hyperelastic constitutive model from noisy synthetic displacement data. We further carry out Markov chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A proper statistical evaluation of the results underlines the high accuracy of the deterministic calibration and that the estimated uncertainty is valid. Finally, we consider experimental data and show that the results are in good agreement with a finite element method-based calibration. Due to the fast evaluation of PINNs, calibration can be performed in near real-time. This advantage is particularly evident in many-query applications such as Markov chain Monte Carlo-based Bayesian inference.

Background: Thermographic images provide two-dimensional information of the surface temperatures on specific selected component regions. If these components have curved surfaces, there is the question of calculating the surface temperature assigned to the material points concerned on the one hand and determining the associated temperature gradient on the other. Apart from general objects, special problems might occur with additively manufactured components as the surfaces are often rough and rippled. Objectives: In this paper, the image information from 2D-thermography as well as 3D-digital image correlation data are combined to determine both the temperature at the material points as well as the temperature gradients concerned. Thus, on the one hand, the basic theoretical equations of the transformations are provided and, on the other hand, the required steps in the experiment are discussed. Methods: Since both discrete data sets of thermography and digital image correlation have to be interpolated, radial basis functions are drawn on. In this context, both a consistent presentation of the underlying equations as well as the error propagation of the occurring uncertainties are addressed as well. First, this is demonstrated at a pure verification example to estimate the expected accuracies. Second, the concept is investigated at real samples made of 3D-printed polymer as well as a wire-arc additively manufactured steel specimen. Results: It turns out that (a) edge effects can lead to more uncertain data at the boundaries of the evaluated region, and (b) a required oblique tripod attached to the specimen are essential uncertainty factors. However, the uncertainty of the temperature determination due to the projection scheme is in the order of general temperature dispersions. Conclusions: Thus, an additional cheap and reliable experimental device in form of a oblique tripod is required which both camera systems have to detect. Then, the evaluation tool can map the 2D-data onto the curvilinear surface. Moreover, the temperature gradient calculation is possible.

Additively manufactured components, especially those produced in deposition welding processes, have a rough curvilinear surface. Strain and surface deformation analysis of such components is increasingly performed using digital image correlation (DIC) methods, which raises questions regarding interpretability of the results. Furthermore, in triangulation or local tangential plane based DIC strain analysis, the principal strain directions are difficult to be calculated at any point, which is due to the non-continuity of the approach. Thus, both questions will be addressed in this article. Apart from classical local strain analysis based on triangulation or local linearization concepts, the application of globally formulated radial basis functions (RBF) is investigated for the first time, with the advantage that it is possible to evaluate all interesting quantities at arbitrary points. This is performed for both interpolation and regression. Both approaches are studied at three-dimensional, curvilinear verification examples and real additively manufactured cylindrical specimens. It is found out that, if real applications are investigated, the RBF-approach based on interpolation and regression has to be considered carefully due to so-called boundary effects. This can be circumvented by only considering the region that has a certain distance to the edges of the evaluation domain. Independent of the evaluation scheme, the error of the maximum principal strains increases with increasing surface roughness, which has to be kept in mind for such applications when interpreting or evaluating the results of manufactured parts. However, the entire scheme offers interesting properties for the treatment of DIC-data.