Simulation approaches

As outlined in the introduction, a number of domain-specific continuum modeling approaches exist which have been developed focusing on individual materials science and engineering aspects. For many applications, e.g., static recrystallization^[ Reference Kugler and Turk ^{26 ]} or “virtual laboratory”-based materials testing,^[ Reference Pagenkopf, Butz, Wenk and Helm ^{27 –} Reference Kraska, Doig, Tikhomirov, Raabe and Roters ^{29 ]} the underlying “single mechanism” assumption is well justified as dominating mechanisms can be clearly identified. In fact, such approaches have significantly contributed to progress in materials science: an effect termed “cold-dwell fatigue”, causing significant damage in titanium turbine blades was understood with the help of crystal plasticity modeling^[ Reference Dunne, Rugg and Walker ^{30 ,} Reference Cuddihy, Stapleton, Williams and Dunne ^{31 ]}; phase-field methods have been employed to investigate the reasons for the “topological phase inversion” in nickel-based superalloys^[ Reference Goerler, Lopez-Galilea, Mujica Roncery, Shchyglo, Theisen and Steinbach ^{32 ]}; and cellular automata modeling helped to understand the influence of nucleation on the recrystallized microstructure.^[ Reference Goetz and Seetharaman ^{33 ]}

Nevertheless, the use of task- and mechanism-specific tools prevents examination of engineering processes such as hot rolling, press hardening, and Charpy testing. Therefore, different simulation techniques have been combined to investigate such processes. A combination of several simulation approaches for investigating multi-physics phenomena can be achieved by a seemingly simple data transfer between stand-alone software tools. This has the advantage that modifications on the individual model implementations are not required. Unfortunately, such data transfer is associated with significant challenges as obvious from the following example dealing with the coupled simulation of recrystallization and plastic deformation. The example^[ Reference Raabe and Becker ^{34 ]} is concerned with static recrystallization; hence, only a one-way data transfer is required to use the dislocation density computed with the crystal plasticity model as the driving force for recrystallization in the cellular automata model. More detailed: “The coupling between both methods consists of: extracting and translating the state variables of the finite-element plasticity model (texture and accumulated shear) into state variables of the cellular automaton model (texture and dislocation density); mapping these data on the cellular automaton grid; scaling the cellular automaton mesh in terms of the derived cell size, maximum occurring driving force and grain boundary mobility; and in establishing an adequate nucleation criterion, which makes use of these data.”^[ Reference Raabe and Becker ^{34 ]} Several other scientists employed similar models to study related problems.^[ Reference Güvenc, Henke, Laschet, Böttger, Apel, Bambach and Hirt ^{35 ,} Reference Han, Tang, Kou, Cheng, Li and Feng ^{36 ]} It is obvious that the interplay between plastic deformation and grain growth occuring during dynamic recrystallalization can hardly be tackled with such an approach as it requires the repeated two-way communication between the two separated simulation tools.

Above presented example illustrates how already a loose coupling entails technical challenges associated with the interpolation of exchanged data from the grid points of the originating domain to the receiving domain due to different simulation techniques^[ Reference Lim, Abdeljawad, Owen, Hanks, Foulk and Battaile ^{37 ]}: While in micro-mechanical simulations FEM- based approaches—often operating on unstructured meshes—have been challenged only recently by spectral methods^[ Reference Moulinec and Suquet ^{38 ,} Reference Lebensohn ^{39 ]} (working on regular grids), microstructure evolution tools are built on top of various numerical techniques. In phase-field modeling, finite-difference and finite volume schemes working on a fixed square grid are common; the use of spectral methods and FEM is relatively rarely reported.^[ Reference Vidyasagar, Tan and Kochmann ^{40 ]} Cellular automata also work on regular, usually square grids even though hexagonal grids are also possible.^[ Reference Hernández Encinas, Hoya White, Martín Del Rey and Rodríguez Sánchez ^{41 ]} The required interpolation is not only a technical difficulty, but, as the example of interpolating crystallographic orientations across grain boundaries illustrates, might introduce significant numerical artifacts. A further detail causing interpolation errors is the kinematic framework: small-strain formulations are widespread for microstructure evolution models, as the displacements usually do not introduce severe shape changes. In contrast, large-strain formulations are state of the art for crystal plasticity models.^[ Reference Abrivard, Busso, Forest and Appolaire ^{43 ]}

Trend: integrated multi-physics modeling

Limited computational resources, largely separated scientific communities with their distinct approaches and terms, and the high model complexity have impeded the development of strongly coupled modeling approaches. This holds not only for the case of combining microstructure evolution modeling and micro-mechanics as presented above, but also for coupling of damage mechanics and crystal plasticity and other multi-physics applications. A current trend, fueled by matured simulation techniques, improved numerical methods, and powerful computers, is the development of fully coupled approaches that broaden the range of applicable problems and enable a joint tackling of interconnected questions. In the following, examples for such models are presented.

A fully coupled simulation approach for investigating recovery, plastic deformation, and recrystallization has been presented by Abrivard et al.^[ Reference Abrivard, Busso, Forest and Appolaire ^{43 ,} Reference Abrivard, Busso, Forest and Appolaire ^{44 ]} Limited to two-dimensional (2D) simulations, the presented model—which is formulated in a thermodynamically consistent framework—allows one to investigate with spatial resolution a large part of the process chain, i.e., from hot rolling over cold rolling to materials testing. Approaches aiming at the same phenomena but realized using a spectral method to solve for numerical equilibrium have also been presented.^[ Reference Chen, Chen, Lebensohn, Ji, Heo, Bhattacharyya, Chang, Mathaudhu, Liu and Chen ^{45 –} Reference Zhao, Song En Low, Wang and Niezgoda ^{47 ]} In the latter approaches, internal data transfer was possible in a straightforward manner as the same grid, respectively a sub-grid was used to couple phase-field and mechanical quantities. Other multi-physics couplings on the continuum scale consider phase transformation and recrystallization,^[ Reference Bos, Mecozzi and Sietsma ^{48 ]} crystal plasticity and fracture,^[ Reference Shanthraj, Sharma, Svendsen, Roters and Raabe ^{49 –} Reference Bache, Dunne and Madrigal ^{52 ]} effects of chemical composition and stress on precipitate formation via concentration-dependent elastic properties,^[ Reference Schwarze, Gupta, Hickel and Darvishi Kamachali ^{53 ]} recrystallization including the modeling of Zener pinning,^[ Reference Raabe and Hantcherli ^{54 ]} and the evolution of a γ′/γ nickel-based superalloy microstructure in dependence of plastic deformation.^[ Reference Gaubert, Le Bouar and Finel ^{55 ]}

The strength of fully coupled approaches is their ability to investigate the competition or amplification among different mechanisms. As a beneficial side-effect, manual data conversion—a time-consuming process, which is a potential source of errors due to necessary interpolations or misinterpretation—is not required if, e.g., a subsequent recrystallization step after cold rolling can be performed using the same software. Above Integrated Computational Materials Engineering (ICME) examples demonstrate how multi-physics approaches allow one to investigate for example whether microstructure evolution is dominated by recovery or recrystallization, whether the macroscopic stress–strain curve shows softening due to recrystallization or hardening due to cold working, or whether dislocation glide or fracture determines the strain partitioning. For predictive simulations, accurate model parametrization is therefore imperative as the complex and non-linear interactions might amplify inaccuracies to such an extent that trends are not even qualitatively captured. Fig. 3 shows the diverse interactions in a thermo-chemo-mechanical simulation framework, illustrating the manifold and complex interactions between partly competing and partly amplifying mechanisms in such a setup.

Figure 3.

Coupling phenomena and interactions in a thermo-chemo-mechanical simulation framework suitable for modeling the process chain in steel production. (a) Direct interactions between the thermal, chemical, and mechanical field quantities. (b) Coupling between the three fields via recrystallization and grain growth (including precipitates). (c) Coupling effects due to the austenite–ferrite phase transformation.

Model parametrization

Continuum modeling requires making strong assumptions on how the collective behavior of atoms or even larger building blocks [dislocations, twins, (internal) surfaces, etc.] can be described. The a priori knowledge of the mechanism(s) relevant to model the behavior under consideration is therefore the key ingredient when designing predictive simulation tools. Based on the nature of the model parameters, models are classified as either phenomenological or physics-based. While phenomenological models are based on the reproduction of the observed behavior, physics-based models are formulated in terms of the underlying physical effects that cause a specific behavior of interest.^[ Reference Roters, Eisenlohr, Bieler and Raabe ^{2 ]} In general, physics-based models are valid for a larger parameter range in comparison with phenomenological models which often assume a relative simple formulation. However, the example of Fourier's law—a very phenomenological model for heat transfer not taking into account any physical details such as impurity atoms or crystal defects—shows that their accuracy is in many cases sufficient to model important phenomena.

When developing physics-based models, it is required to decide on the degree of sophistication needed to answer the specific question. In crystal plasticity modeling, for example, the assumption of dislocation glide on dense packed planes and the parametrization in terms of one “dislocation density” population is often very successful despite the knowledge of complex dislocation substructures, movements of dislocations on non-dense planes and local differences in chemical composition with influences on the Peierl's potential. This success can be attributed to the fact that for typical engineering materials, non-planar glide, local composition gradients, dislocation pile-up in front of obstacles, e.g., grain boundaries do not dominate the aggregate behavior. However, in consequence, these simplified models lose their predictive capabilities as soon as at least one other mechanism contributes significantly to the investigated behavior.^[ Reference Reuber, Eisenlohr, Roters and Raabe ^{56 ]} A second example is the study of precipitate formation in phase-field modeling. Even though thermodynamic databases are available for large composition ranges, the region of interest in the phase diagram can for many cases be linearly approximated, giving a fast and reliable modeling approach suitable to describe the phase changes for a specific range of concentrations.^[ Reference Schwarze, Gupta, Hickel and Darvishi Kamachali ^{53 ]} As a last example, recrystallization modeling using cellular automata and phase-field approaches is discussed. In their simplest form, both approaches neglect the reduction of the driving force for recrystallization due to recovery—an assumption that can be justified for a wide range of chemical compositions and temperatures. Moreover, curvature-driven motion is inherent to the phase-field method but absent in standard cellular automata approaches. In all three cases, it remains in the scientist's responsibility to ensure that the considered problem is within the limits of the employed simulation approach.

The parameter values of both phenomenological and physics-based models need to be determined to obtain the behavior of a certain material. This is typically associated with conducting (and designing) experiments. The distinct advantage of physics-based models is the ability to identify their parameters with measureable quantities such as stacking fault energy, dislocation density, or stored elastic energy. However, physics-based models are often parameterized by a larger number of parameters that are often harder to obtain than the simple quantities used for phenomenological models.

Trend: multi-scale informed models

Developing and employing a physics-based continuum-scale model can be separated into two tasks: understanding the relevant physical mechanisms to cast them into model equations and determine the specific parameters of the material under investigation. A current trend in continuum modeling is the use of simulation techniques at lower length-scales for both tasks to minimize experimental efforts. Such multi-scale or scale-bridging approaches require to combine expertise from different fields, i.e., physicists doing ab initio simulations, material scientists for the discrete dislocation dynamics simulations and finally mechanical engineers and computational mechanics for implementing the continuum-scale models. The following short overview briefly recapitulates how discrete simulation techniques can be used to develop and adjust continuum models.

An example where a crystal plasticity model was developed and parameterized based on kinetic Monte Carlo simulations is given by Cereceda et al.^[ Reference Cereceda, Diehl, Roters, Raabe, Perlado and Marian ^{57 ]} Similarly, key parameters for the modeling of iron have been obtained from atomistic simulations.^[ Reference Köster, Ma and Hartmaier ^{58 ]} The parameters describing different interaction types among dislocations have been successfully determined using discrete dislocation dynamics simulations for the three most relevant crystal structures.^[ Reference Queyreau, Monnet and Devincre ^{59 –} Reference Bertin, Tomé, Beyerlein, Barnett and Capolungo ^{62 ]} Discrete dislocation dynamics simulations have also been used to identify and quantify dislocation interactions relevant for continuum modeling.^[ Reference Stricker ^{63 –} Reference Groh, Marin, Horstemeyer and Zbib ^{66 ]} Based on ab initio results, crystal plasticity models^[ Reference Wong, Madivala, Prahl, Roters and Raabe ^{67 ]} and phase-field models for phase transformation have been also developed and parametrized.^[ Reference Zhao, Shen, Li and Wang ^{68 ,} Reference Shchyglo, Hammerschmidt, Čak, Drautz and Steinbach ^{69 ]} The different small-scale simulation techniques employed to develop and parametrize continuum-scale models are graphically represented in Fig. 4.

Figure 4.

Small-scale simulations used to develop and parametrize continuum-scale models. (a) Ab initio (Density Functional Theory, DFT) simulations. Figure displays and artistic sketch. Adapted with permission from T. Hickel from Ref. Reference Hickel, Kattner and Fries70, John Wiley & Sons, 2014. (b) Atomistic modeling based on empirical potentials. Figure display defect structures in Ni₃Al according to the following color code: Red and orange, Perfect L1₂ structure; green, pseudo-twin structure; blue, anti-phase boundary; white, complex stacking faults; white, other defects. Adapted from Ref. Reference Amodeo, Begau and Bitzek71, Taylor & Francis, 2014 licensed under CC BY 3.0. (c) Discrete Dislocation Dynamics (DDD). Figure display dislocation structures in aluminum single crystals under torsion load. The sample shown in the left has cross-slip enabled, the sample shown in the right has cross-slip disabled. Color code indicates glide systems according to: Turquoise, Burgers vector parallel to the torsion axis; blue, dislocation pile-ups on parallel glide planes; red and green, dislocation pile-ups on parallel glide planes. Adapted from Ref. Reference Senger, Weygand, Kraft and Gumbsch72, IOP Publishing, 2011.

The same advantages that are valid when employing continuum-scale models hold for the complementation and replacement of small-scale experiments when developing continuum-scale models: simulations are fully observable, they are often cheaper to perform and they allow systematic investigations of the parameter range. Hence, multi-scale approaches for model development are especially suited for complex coupled models with multiple parameters.

Integration of experiments

Computational materials science has been established as an important sub-discipline of materials science and engineering with a distinct community. However, a close cooperation between computational and experimental materials science is advantageous for two main reasons: firstly, to benchmark the computational models on the experimental reality and secondly for an improved understanding based on the broader view enabled by the combination of results from experiments and simulation. Ideally, starting from matching results between experiment and simulation, additional data can be extracted from the simulation that is not accessible from within the experiment. In this context, an evident advantage of simulations over experiments is the possibility to perform systematic parameter studies, which might even reach areas in the parameter space not feasible for experimental studies.

Essentially, the comparison between experimental and numerical results can be done in two ways: either by comparing average results or by comparing field quantities. While quantitative comparisons are desirable, qualitative comparison is a viable option to identify trends and associated mechanisms. The use of average results from microstructure evolution simulations allows one to discuss the time evolution of, e.g., recrystallized volume fractions. Typical examples in micro-mechanics are the comparison of yield behavior^[ Reference Cereceda, Diehl, Roters, Raabe, Perlado and Marian ^{57 ]} and stress–strain curves.^[ Reference Wong, Madivala, Prahl, Roters and Raabe ^{67 ]} An example for a qualitative comparison of field data, namely the shape and spatial arrangement of two evolving phases is given by Goerler et al.^[ Reference Goerler, Lopez-Galilea, Mujica Roncery, Shchyglo, Theisen and Steinbach ^{32 ]} For a quantitative one-to-one comparison, microstructure models are built from the measured microstructures. This approach is especially common in micro-mechanics. An early work in which the microstructure of a dual-phase steels has been used as input to an FEM micro-mechanical simulation has been presented by Karlsson and Sundström already in 1974.^[ Reference Karlsson and Sundström ^{73 ]} A direct comparison, however, was not possible due to experimental restrictions. Today, crystal plasticity modeling based on electron backscatter diffraction (EBSD) data is well established.^[ Reference Choi, Kim, Woo, Han and Kwak ^{74 ,} Reference Wang, Barabash, Yang, Bieler, Crimp, Eisenlohr, Liu and Ice ^{75 ]} In conjunction with digital image correlation, a direct comparison between experimentally obtained and simulated strain fields is possible.^[ Reference Tasan, Diehl, Yan, Zambaldi, Shanthraj, Roters and Raabe ^{76 ,} Reference Pinna, Lan, Kiu, Efthymiadis, Lopez-Pedrosa and Farrugia ^{77 ]}

Since the acquisition of real microstructures is associated with experimental efforts, computational studies in microstructure evolution^[ Reference Zhu and Militzer ^{78 –} Reference Contieri, Zanotello and Caram ^{80 ]} and micro-mechanical^[ Reference Pagenkopf, Butz, Wenk and Helm ^{27 ,} Reference Abdolvand and Daymond ^{81 ,} Reference Wan, Cuddihy, Jiang, MacLachlan and Dunne ^{82 ]} modeling are often performed on synthetic microstructures. In many cases, microstructures are generated using Voronoi tessellation approaches, which are also called “power diagrams” in the mathematics community and are known from the Wigner-Seitz cell in materials science. Current tools for generation of synthetic microstructures can take a number of statistical quantities into account,^[ Reference Pagenkopf, Butz, Wenk and Helm ^{27 ,} Reference Groeber and Jackson ^{83 ]} making artificial microstructures statistically equivalent to real ones with respect to selected features. Another suitable way is the physics-based modeling of microstructure evolution, e.g., solidification or recrystallization, for subsequent simulations. Such approaches, especially the more complex flavors, are certainly helpful for simulation series using virtual laboratories or parameter studies. However, real microstructures exhibit a tremendous complexity. This holds especially for materials containing more than one phase, e.g., dual-phase steels and α–β titanium alloys. In cases where even subtle differences between real and synthetic microstructure are of importance, i.e., when tail and head of a distributed quantity significantly alter the behavior—consider damage modeling and its dependence on stress and strain peaks—the use of real microstructures is without alternative. In this context, it should be mentioned that not even accepted measures exist to reliably discriminate all differences between microstructures.

Trend: three-dimensional microstructure models based on experimental data

The volumes accessible to simulation studies in reasonable time frames grow with increasing computation power. At the same time, the availability of advanced characterization techniques allows one to acquire 3D microstructures with high precision. Consequently, a current trend in continuum modeling is the use of such datasets as input data. By that the limitations of 2D real microstructures, i.e., the missing subsurface microstructure^[ Reference Zeghadi, Nguyen, Forest, Gourgues and Bouaziz ^{84 –} Reference Diehl, Shanthraj, Eisenlohr and Roters ^{86 ]} as well as the shortcomings of artificial microstructures, i.e., missing in-grain-scatter, simplified grain morphology, and idealized phase distribution, can be overcome. The works presented in the following make use of recently developed characterization techniques to acquire realistic microstructure models.

Synchrotron measurements allow the non-destructive acquisition of in situ data at the cost of expensive and costly experimental procedures. Different synchrotron-based acquisition techniques^[ Reference Wang, Li, Almer, Bieler and Barabash ^{87 ]} have been used to generate microstructure models for subsequent simulations: differential-aperture x-ray microscopy (DAXM),^[ Reference Zhang, Li, Eisenlohr, Liu, Boehlert, Crimp and Bieler ^{88 ]} 3D x-ray diffraction (XRD),^[ Reference Abdolvand, Majkut, Oddershede, Wright and Daymond ^{89 ]} and high-energy diffraction microscopy (HEDM).^[ Reference Pokharel, Lind, Li, Kenesei, Lebensohn, Suter and Rollett ^{90 ,} Reference Turner, Shade, Bernier, Li, Schuren, Lind, Lienert, Kenesei, Suter, Blank and Almer ^{91 ]} On the laboratory scale, x-ray tomography^[ Reference Maire and Withers ^{92 ]} is an alternative, where however most setups are not readily suitable for orientation-aware polycrystal characterization.^[ Reference Diehl, Wicke, Shanthraj, Roters, Brueckner-Foit and Raabe ^{51 ]} However, current developments in the field of XRD contrast tomography take crystallographic orientation-aware in situ measurements in three dimensions out of the realm of large-scale synchrotron facilities.^[ Reference McDonald, Reischig, Holzner, Lauridsen, Withers, Merkle and Feser ^{93 ]} As a destructive alternative that prohibits direct comparison of e.g., strain partitioning, serial sectioning (3D) EBSD^[ Reference Zaefferer, Wright and Raabe ^{94 ]} has been established. To illustrate the increased complexity of the models, Fig. 5 displays a microstructure model of a dual-phase steel in 2D from 1974^[ Reference Karlsson and Sundström ^{73 ]} and a recent 3D model of a similar material from 2017.^[ Reference Diehl, An, Shanthraj, Zaefferer, Roters and Raabe ^{95 ]}

Figure 5.

Increasing complexity of microstructure models for micro-mechanical simulations. (a) Dual-phase steel model from 1974.^[ Reference Karlsson and Sundström ^{73 ]} The phase distribution is shown in the top image where black color indicates martensite and white color indicates ferrite. The resulting distribution of plastic strain (ε _p) under uniaxial loading in horizontal direction is shown in the bottom image. Adapted from Ref. Reference Karlsson and Sundström73, Elsevier, 1974. (b) Dual-phase steel model from 2017.^[ Reference Diehl, An, Shanthraj, Zaefferer, Roters and Raabe ^{95 ]} Crystallographic orientation in terms of an inverse pole figure coloring and phase distribution (dark areas indicate martensite) are shown in the top image. The resulting distribution of equivalent total strain (ε _vM) under uniaxial loading in the horizontal direction is shown in the bottom image.

The use of highly resolved 3D datasets as input for continuum-scale models has two distinct advantages: limitations of 2D models and artificially created microstructures are surmounted and—depending on the selected characterization method—a qualitative comparison of field data is possible. This allows one to benchmark models with unpreceded accuracy and accompany experiment by valuable data for an improved understanding. Exemplarily, the local stress distribution can be contributed from micro-mechanical simulations or the influence of selected aspects of the materials behavior can be systematically investigated.

Data processing

Evaluating the data obtained from simulations is a crucial point for extracting information and gaining knowledge. A suitable evaluation and visualization is therefore imperative when simulation techniques are used to produce more valuable information than just “beautiful pictures”. In comparison with most laboratory-scale experiments, simulations are able to generate a vast amount of data as the full state of the model can be written out at any time increment. This distinct advantage of simulations, however, requires systematic analysis to separate important from unimportant information. Systematic data processing, visualization, and analysis are therefore important steps to make use of the information obtained from continuum-scale simulations.

While employed models and preprocessing techniques (e.g., for using experimental data as discussed in Integration of experiments) are discussed extensively in a large number of publications, little attention is typically paid to the post processing of simulation results. This can be understood from the fact that the small simulation sizes and the focusing on single effects simplified evaluation of grain scale simulations in the past. Typical visualization techniques include line plots to display average quantities^[ Reference Zhang, Diehl, Roters and Raabe ^{28 ,} Reference Wong, Madivala, Prahl, Roters and Raabe ^{67 ]} and spatially resolved quantities mapped on the sample geometry to present field data^[ Reference Abrivard, Busso, Forest and Appolaire ^{44 ,} Reference Tasan, Diehl, Yan, Zambaldi, Shanthraj, Roters and Raabe ^{76 ,} Reference Wan, Cuddihy, Jiang, MacLachlan and Dunne ^{82 ]}—in the latter case the often used rainbow color bars renders efforts to interpret the data extremely hard.^[ Reference Moreland, Bebis, Boyle, Parvin, Koracin, Kuno, Wang, Pajarola, Lindstrom, Hinkenjann, Encarnação, Silva and Coming ^{96 ,} Reference Borland and Taylor ^{97 ]} It can be assumed that the types of plots shown in publications also represent the visualization types used by the researchers to analyze their data. Hence, correlations not visible in typically used line plots and field maps (including maps in orientation space) might remain undiscovered in the acquired data.

While many individual phenomena are obvious in line plots,^[ Reference Liedl, Traint and Werner ^{98 ]} cross-coupling effects that are of special importance in complex multi-physics simulation approaches might be hidden if the data are presented in such simple diagrams. The same applies for spatially resolved pictures of field data: while they are important to communicate results, the possibility of easily producing them in large quantities from simulation results should not prevent a thorough search for features in the obtained data. It remains therefore a challenge to visualize, analyse, and interprete data obtained from simulations.

Trend: advanced statistical analysis

The increasing model size due to finer discretization and larger volumes (discussed in Trend: three-dimensional microstructure models based on experimental data) as well as the employment of more complex models (as outlined in Trend: integrated multi-physics modeling) render visualization-based manual data analysis an insufficient option: hand-operated inspection of field data for identification of “interesting” regions becomes cumbersome due to the sheer size of the virtual samples, especially if an unbiased view not aligned with the direction of the initial coordinate system is taken and no iso-surface filtering according to certain ad hoc rules is employed. Similarly, high-dimensional data (e.g., shear rates on different types of slip systems, crystallographic orientations) and especially the interplay between different vector- or tensor-valued fields is impossible to grasp for the human eye. Some recent studies—not only concerned with continuum simulations—that follow the trend of using advanced data analysis in materials science are presented in the following.

Advanced analysis techniques allow one to identify complex interactions to arrive at elaborated (micro)structure–property relationships. Statistical analysis, using non-spatially ordered data is a significant advancement, which gained momentum recently.^[ Reference Diehl, An, Shanthraj, Zaefferer, Roters and Raabe ^{95 ,} Reference De Geus, Peerlings and Geers ^{99 ]} Figure 6 shows how spatially arranged field data and especially heat maps provide much more information than simple one-dimensional line plots. An even larger progress can be expected from the use of machine-learning approaches that detect patterns themselves. Data science/machine-learning approaches, currently gaining traction in materials science^[ Reference Kalidindi and De Graef ^{100 ]} in the field of microstructure recognition,^[ Reference Chowdhury, Kautz, Yener and Lewis ^{101 –} Reference Azimi, Britz, Engstler, Fritz and Mücklich ^{103 ]} and feature extraction,^[ Reference Gomberg, Medford and Kalidindi ^{104 ]} can be expected to aid evaluation of continuum-scale simulations in the future. Running such analysis requires efficient data structures capable to store a large amount of data that can be accessed in a non-consecutive way. HDF5-based^[ Reference Folk, Heber, Koziol, Pourmal and Robinson ^{105 ]} file formats seem to establish themselves as the standard in experimental^[ Reference Jackson, Simmons and De Graef ^{106 ,} Reference Jackson, Groeber, Uchic, Rowenhorst and Graef ^{107 ]} and computational^[ Reference Schmitz ^{108 ,} Reference Diehl, Eisenlohr, Zhang, Nastola, Shanthraj and Roters ^{109 ]} materials science. The current process of establishing standard data descriptors suitable for fast and random access can contribute to the efficient use of tools for automated data mining.

Figure 6.

Results from data different analysis and visualization approaches applied to a dataset obtained from a micromechanical simulation. (a) Average results in terms of a stress–strain curve. (b) Equivalent stress mapped onto the deformed configuration of the probed material. (c) Heat map showing the distribution of stress–strain relations for each point in the simulation. Strain and stress in loading direction are denoted as ε and σ, respectively and σ _vM is the equivalent stress.

Employing advanced data analysis techniques to results obtained from continuum simulations enables deeper insights into phenomena predicted by the underlying models. While non-classical visualization and plotting methods can be a significant step forward in finding and quantifying expected features, machine learning approaches are especially suited to automatically extract unpredicted couplings.

Software solutions

A huge variety of models for continuum simulations has been developed in the last 20 years. However, as the development of new models or the improvement of existing models are much more prestigious in materials science than the actual implementation, only a few implementations have found their way into routine application. In fact, many developed models are only implemented in prototype software to obtain results for presenting the model. For such project-specific in-house software there is no clear differentiation between developers and users: Members of the research group develop a model, implement it, and use it. In consequence, usability aspects are of little importance as only experts with reliable personal support are expected to use the software. Moreover, there are no software solutions with commercial support that can fulfill the various demands required for continuum simulations in materials science and engineering.

Among the rare examples for implementations that have gained significant attention in micromechanics are the spectral method solver, including material models, by Lebensohn^[ Reference Moulinec and Suquet ^{38 ]} and the crystal plasticity Abaqus UMAT developed by Huang.^[ Reference Huang ^{110 ]} The latter one stands exemplarily for the common approach of using commercial FEM solutions such as Abaqus, MSC.Marc, or LS Dyna as the basis for the model development. However, these commercial tools are used for solving the governing equations and for visualization, they contain no materials science and engineering-related features. In microstructure evolution simulation, the commercial MICRESS software has gained significant attention.

The task-specific development of single-purpose tools has significant drawbacks for developers and (potential) users. Developers, one the one hand, need to re-implement certain aspects for every new model. While libraries for general-purpose functionalities such as handling of file IO (input and output) and numerical tasks exist, in the case of multi-physics modeling a true re-implementation associated with significant efforts is required even when only a single aspect of a sub-model is of scientific interest. Users, on the other hand, are handicapped by the missing portability and usability. Especially experimentally oriented scientists with little knowledge in computer programming are hence deprived of using state of the art simulation tools to further analyzing their experimentally obtained data.

Trend: established free and open source software packages

The absence of widely used software packages for continuum-scale simulations can be explained by the high and long-lasting efforts associated with developing such tools. Developing advanced multi-physics continuum simulation tools for a large community requires not only the combined interdisciplinary knowledge from physics, materials science, and mathematics, but also the knowledge how to use elaborated tools for testing and debugging. As an important consequence, during e.g., a Ph.D. project, a state of the art framework cannot be developed. Rather, many developers with their specific expertise need to improve or enhance existing tools or frameworks in a joint manner. The general trend of collaborative software development has therefore also been followed in materials science and engineering as the following examples show.

Extendable open-ended frameworks for continuum-scale simulations, which have emerged in recent years, allow to re-use aspects like storage of data, coupling to analysis tools (see Trend: advanced statistical analysis) and to experimental input data (see Trend: three-dimensional microstructure models based on experimental data), numerical solution techniques and standard models. An overview of free and open sources tools, categorized into micro-mechanics (AMITEX_FFTP, DAMASK), phase-field (OpenPhase, FiPy), and general numerical libraries (deal.II, MOOSE framework, FEniCS project) is given in Fig. 7. While the tools presented in the first two categories allow one to readily perform simulations, the software in the “general numerical solver frameworks” category can substitute above-mentioned commercial FEM solvers—with the additional benefit of having the chance to modify the source code. How beneficial such frameworks are for the community can be illustrated by a few basic statistics: DAMASK^[ Reference Roters, Eisenlohr, Kords, Tjahjanto, Diehl, Raabe and Cazacu ^{111 ]} contains more than 50,000 lines of Fortran code in 65 files and additionally 165 files of Python code for pre and post-processing. OpenPhase contains more than 50,000 lines of C++ code in almost 100 files. In contrast, the mentioned crystal plasticity routine of Huang^[ Reference Huang ^{110 ]} consists of less than 3000 lines of code (including comments).

Figure 7.

Selection of free and open source software solutions for continuum modeling classified after their main focus. (a) Crystal plasticity tools: DAMASK^[ Reference Roters, Eisenlohr, Kords, Tjahjanto, Diehl, Raabe and Cazacu ^{111 ]} and AMITEX_FFTP. (b) Phase-field solvers: OpenPhase and FiPy.^[ Reference Guyer, Wheeler and Warren ^{112 ]} (c) General solver frameworks for scientific and engineering modeling: MOOSE framework,^[ Reference Gaston, Newman, Hansen and Lebrun-Grandié ^{113 ]} FEniCS project,^[ Reference Alnæs, Blechta, Hake, Johansson, Kehlet, Logg, Richardson, Ring, Rognes and Wells ^{114 ]} and deal.II.^[ Reference Arndt, Bangerth, Davydov, Heister, Heltai, Kronbichler, Maier, Pelteret, Turcksin and Wells ^{115 ]} The DAMASK logo by Philip Eisenlohr is licensed under CC BY-ND 4.0. The AMITEX_FFTP logo is reproduced with permission by the copyright holder (Lionel Gélébart). The OpenPhase logo is reproduced with permission by the copyright holder (ICAMS). The FiPy logo is reproduced with permission by the copyright holder (NIST). The MOOSE framework logo is reproduced with permission by the copyright holder (Derek Gaston). The FEniCS project logo by Mattias Schläger is licensed under CC BY-SA 4.0. The deal.II logo is reproduced with permission by the copyright holder (Wolfgang Bangerth).

Free and open source software packages have two main advantages: existing models can be used and extended, reducing entry barriers for employing and developing continuum models. Synergy effects during implementation especially occur for complementing fields of expertise, e.g., when a new homogenization scheme or damage model can be built on existing crystal plasticity. DAMASK, for example, was used and extended in multiple Ph.D. projects in research groups not belonging to the core development team^[ Reference Stricker ^{63 ,} Reference Meier, Schwarz and Werner ^{116 –} Reference Ebrahimi and Hochrainer ^{118 ]} and one of the models developed in a cooperation made it into the standard release distribution.^[ Reference Cereceda, Diehl, Roters, Raabe, Perlado and Marian ^{57 ]} Moreover, efforts for documentation, compilation of best-practice examples, and systematic testing can be shared among the developers to provide an added value for the materials science community—including experimentally oriented scientists not interested in model development.

Conclusions and outlook

Continuum modeling approaches, located at the upper end of time and length scales, are an important tool for computational materials science. Although these approaches require strong assumptions and simplifications, their applicability to time and length scales relevant for engineering processes renders them important tools in industry and academia. Current trends related to continuum modeling are expected to increase the benefit of such approaches even more: integrated multi-physics modeling enables one to tackle ICME-related problems that have been out of scope in the past and multi-scale-informed models accelerate development of models that accurately represent the reality. The strong coupling to highly resolved experimental 3D data tightens the connection between reality and simulation and advanced statistical analysis ensures that even unpredicted phenomena are discovered. Finally, the use of established free and open source software packages enables faster development and makes state of the art simulation tools available to the broader materials science and engineering community.