1. The short version, what to know first, and where this fits in the series

• Multiscale simulation is the idea of "linking" calculations at levels of different size: electrons, atoms, microstructure and components. Linkage comes in three kinds: linking, which hands the results below to the level above; iterative coupling, which goes back and forth until convergence; and tight coupling, which solves everything at once as a single system Sourced[Ref. 1]
• When scales are linked, a small error at a lower level can swell at a higher level. In our calculation, a 0.05 eV error in an activation barrier changes a rate by about 6.9 times at 300 K Our calculation
• A digital twin is not simply a simulation. The National Academies report says "a digital twin is more than just simulation and modeling", and names the bidirectional interaction with the physical counterpart (updating by data and decision-making) as a key element Sourced[Ref. 19]. For materials there is the added difficulty that the state (the microstructure) cannot be seen directly (commentary)

2. What "scale" means: electrons, atoms, microstructure, components

Even for the same metal part, a different size of view shows a different thing: the cloud of electrons, the arrangement of atoms, the pattern of grains (clusters of small crystals), and the shape of the whole part. At each level, the tools for calculation and the size and time that can be handled are different.

A document of the European standards body CEN (CEN Workshop Agreement 17284, April 2018) first divides materials models into four by the "entity" the model describes: electron entity, atom entity, mesoscopic entity (a set of bound atoms: a molecule, bead, grain and so on) and continuum volume entity Sourced[Ref. 1]. It also says that any material can be described by any of the entity types.

View original SVG, values and sources

Original SVG reference: Concept diagram (vector drawing). Four levels: electrons → atoms → microstructure → component. The four "entities" come from CEN CWA 17284. The scales are conceptual, not strict ranges for each method.

• Electrons: first-principles calculation (DFT) gives the energy surface and forces. What it passes to the level below is how the forces between atoms are set (the potential)
• Atoms: molecular dynamics computes diffusion coefficients, interface energies, elastic constants, thermal conductivity and so on (see our explainer on molecular dynamics)
• Microstructure (mesoscale): it handles grains and clusters of molecules. The phase-field method deals with the evolution of microstructure, and Chen (2002) wrote a review of it Sourced[Ref. 10]. There are also methods for the motion of dislocations (line defects in a crystal). Thermodynamic calculation (the CALPHAD method) is used at this level too (a history of CALPHAD is given by Spencer, 2008) [Ref. 11]
• Component: the finite-element method and similar tools compute deformation, heat and current in a part. This corresponds to the "continuum volume entity" in the CEN document

3. Why we cannot compute everything at once

You may think: "Then just compute the whole part from the atoms." Let us estimate the amount of calculation. As seen in our explainer on molecular dynamics, advancing 1 second in steps of 1 fs (10⁻¹⁵ s) takes 10¹⁵ steps, and 1 mm³ of iron holds about 8.5×10¹⁹ atoms Our calculation.

• The records: Germann and Kadau (2008) report MD of one trillion atoms, and Jia et al. (2020) report, with a machine-learning potential and a dedicated code, more than 100 million atoms advanced by more than 1 ns per day Sourced[Refs. 17, 16]. One trillion atoms of iron fill a cube only about 2.3 μm on a side Our calculation
• The estimate: for 1 mm³ and 1 second with all atoms, atoms × steps is about 8.5×10³⁴. Dividing by the throughput of Jia et al. (at least about 10¹⁴ per day) gives about 8.5×10²⁰ days, or about 2×10¹⁸ years (an upper bound) Our calculation. Even the record runs are off by a huge number of orders of magnitude

So instead of computing everything at once with atoms, we need to take out "only the necessary information" at each level and pass it up. The 2013 Nobel Prize in Chemistry went to Karplus, Levitt and Warshel "for the development of multiscale models for complex chemical systems". Their achievement is described as making Newton's classical physics work side by side with quantum physics Sourced[Ref. 2].

4. Types of linkage: pass values, go back and forth, solve together

There are three main types of linkage. The CEN document defines the terms as follows (3.14 to 3.16) Sourced[Ref. 1].

• Linking: the output of model A becomes the input of model B. One-way, with an open flow. The most common form. For example: from DFT values build a potential for molecular dynamics, and pass the diffusion coefficient from molecular dynamics to a microstructure calculation (commentary)
• Iterative coupling: A and B take each other's outputs as inputs and go back and forth until they converge. The flow is closed
• Tight coupling: the equations of A and B are collected and solved at once as a single system of equations. The result comes out as one output

The standard examples of "at once" are methods that couple in space.

• QM/MM: a method that treats only the centre where a reaction occurs with quantum mechanics and the surroundings with classical mechanics. Warshel and Levitt (1976) used it in a theoretical study of an enzyme (lysozyme) reaction [Ref. 3]. It is part of the work honoured by the 2013 Nobel Prize in Chemistry [Ref. 2] Sourced
• Coupling atoms and a continuum: Rudd and Broughton (1999) report, for MEMS (tiny mechanical parts) resonators, a method in which the central region moves millions of individual atoms, the large peripheral regions are computed with finite elements, and the two methods run concurrently and pass information back and forth. They say atoms are needed in regions of strongly anharmonic forces, of large surface-to-volume ratio, and where friction due to defects is expected Sourced[Ref. 4]. The paper by Broughton et al. (1999) deals with coupling finite elements, molecular dynamics and tight-binding electronic structure [Ref. 5]
• A case of handing values up in stages and confirming by experiment: Bulatov et al. (2006) predicted the existence of three-dislocation entanglements (multi-junctions) with dislocation dynamics and atomistic simulation, and confirmed them by transmission electron microscopy of molybdenum Sourced[Ref. 6]. Calculations at several scales were combined and tested by experiment

There is also coarse-graining, which lumps several atoms into one ball to handle larger systems. Examples are entangled polymers (Kremer and Grest, 1990) and a coarse-grained force field for biomolecules (Marrink et al., 2007, MARTINI), and there is a review of multiscale simulation in polymer science (Kremer and Müller-Plathe, 2002) Sourced[Refs. 7, 9, 8].

5. How errors below propagate above

Linking brings a new problem: errors at a lower level propagate to the level above. There are two typical ways.

• Exponential amplification: the speed of diffusion or reaction is set by exp(−E/k_B T) for a barrier E. If the barrier is off by ΔE, the rate ratio is exp(ΔE/k_B T). With ΔE = 0.05 eV, it is about 6.9 times at 300 K, about 2.6 times at 600 K and about 1.8 times at 1000 K. With ΔE = 0.1 eV, about 48 times at 300 K Our calculation. A "small" energy error at a lower level (DFT) becomes a large error at the level above (rate). It is most marked at low temperatures
• Accumulation over stages: suppose each stage is "within a factor of 2 at 95%". If the errors are independent, four stages widen to about 4 times overall (2^√4). If they all lean the same way, 16 times (2⁴) Our calculation. In a real chain errors can be correlated, for example when the same DFT bias is carried up to higher levels (commentary)

6. What is a digital twin?

Next, the digital twin. As the name "twin" says, it is the idea that the real thing (the physical counterpart) and a computational model (the virtual counterpart) are kept connected through data. We check the definitions in official documents.

• NASA (2012): Glaessgen and Stargel say that certification and sustainment of future NASA and U.S. Air Force vehicles rely largely on statistical distributions of material properties, heuristic design philosophies, physical testing and assumed similitude between testing and operating conditions, and that these will be unable to meet extreme requirements; they propose the digital twin as the shift. The digital twin integrates ultra-high-fidelity simulation with the vehicle's health-management system, maintenance history and all available historical and fleet data to mirror the life of its flying twin Sourced[Ref. 18]
• National Academies (2024): it defines a digital twin as "a set of virtual information constructs that mimics the structure, context, and behavior of a natural, engineered, or social system (or system-of-systems), is dynamically updated with data from its physical twin, has a predictive capability, and informs decisions that realize value. The bidirectional interaction between the virtual and the physical is central" (adapted from a definition by AIAA). It also states that "a digital twin is more than just simulation and modeling" (Finding 2-1) Sourced[Ref. 19]
• International standards: ISO/IEC 30173:2023 (Digital twin — Concepts and terminology) sets terms and concepts for the digital twin (system context, life cycle, types, functional view, stakeholders) and was published in November 2023. ISO 23247-1:2021 gives an overview and general principles of a digital twin framework for manufacturing (terms and requirements) and was published in October 2021 Sourced[Refs. 23, 22]. The texts are paid, and we did not check the requirements

View original SVG, values and sources

Original SVG reference: Concept diagram (vector drawing). Left: the structure of a digital twin (following the NASEM account). Right: three levels of integration (the classification of Kritzinger et al.; the account follows the summary by Barbie and Hasselbring).

Kritzinger et al. (2018) surveyed the manufacturing literature, found that there is no common understanding of the term digital twin, and distinguished the digital model, digital shadow and digital twin by the level of integration. They report that literature on the most advanced stage, the digital twin, is scarce Sourced[Ref. 20]. As summarised by Barbie and Hasselbring (2024), the three levels are: digital model: no automated data exchange; digital shadow: an automated one-way flow from physical to virtual; digital twin: automated in both directions [Ref. 21]. The same paper also reports Grieves's criticism of this classification, and the view that ISO 23247 does not mention bidirectional communication and so, on this definition, describes only a digital shadow [Ref. 21]. This article simply shows that definitions vary.

7. Digital twins of materials, and their limits

Digital twins of aircraft and plants have many quantities that sensors can measure (strain, temperature, vibration). How, then, is a digital twin of the material itself thought of?

• The idea: Kalidindi et al. (2022) regard the material itself as a highly complex multiscale physical system. Its form (details of the hierarchical structure) and function (response to external stimuli, that is, properties) are captured in a digital twin. They argue that the digital twin can represent how structure, process and performance of the material evolve over time, given both process history and in-service environment. The form is captured with n-point spatial correlations and the function with homogenization and localization surrogate models, calibrated to experimental and physics-based simulation data Sourced[Ref. 12]
• Standing: the paper sets out foundational concepts and a framework; it is not a demonstration on a real material (our reading)
• What is hard: (1) The state (microstructure and so on) often cannot be seen directly in service and is inferred from what can be measured. (2) Each part has a different processing history, so each individual needs its own updating. (3) Errors of the models linking the scales propagate to the whole (Fig. 5). These are our own organisation (commentary)

The National Academies report says the virtual side of a digital twin should be built at a fidelity and resolution that is "fit for purpose", matched to the decision to be made and the computing resources (Conclusion 3-1), and that it may mix high-fidelity, simplified and surrogate models Sourced[Ref. 19]. For materials as well, what you want to decide sets the scale and accuracy needed (commentary).

8. Checking whether a result can be trusted

For linked calculations and for digital twins alike, the work of checking whether a result can be trusted makes up half of the job. The National Academies report says that verification, validation and uncertainty quantification (VVUQ) have a role in all elements of a digital twin. On the virtual side, verification and validation build trustworthiness, and uncertainty quantification gives a measure of the quality of prediction Sourced[Ref. 19]. It also says that changes in the physical counterpart, in data collection, in the distribution of data, in the prediction or decision tasks and in the models all bear on VVUQ [Ref. 19].

• Verification: are the equations solved correctly (code errors, numerical error, time step and so on)? Checking energy conservation, in our explainer on molecular dynamics, is one example
• Validation: are these the right equations and model? Compare with measurement of the same quantity under the same conditions
• Uncertainty quantification (UQ): how much do scatter in the inputs and the model affect the prediction? Give the prediction with a range
• A standard: ASME V&V 10-2019 is a standard for verification and validation in computational solid mechanics; it was reaffirmed in 2025 (R2025) and remains in effect Sourced[Ref. 24]. It bears on checking the continuum stage of a linked chain. The text is paid, and we did not check the requirements

9. A materials engineer's view

10. Limits, what we could not confirm, and a summary

What we could not confirm

• The scales (length and time) in Fig. 2 are a conceptual organisation; we did not check the numbers in individual literature
• Warshel and Levitt (1976), Broughton et al. (1999), Allison et al. (2006), Kremer and Müller-Plathe (2002), Chen (2002) and Spencer (2008) were handled within the range of the title and the publicly available abstract or description. We did not check details of the full texts
• The explanations of the CALPHAD and phase-field methods are commentary based on general understanding; we did not check the full texts of individual literature
• We could not check the text of the classification by Kritzinger et al., and the content of the three levels follows the summary by Barbie and Hasselbring (2024)
• The calculation in Fig. 5 is an example based on assumptions (Arrhenius type, independent log-normal distributions) and does not estimate the errors of any real chain
• The texts (requirements) of ISO 23247-1, ISO/IEC 30173 and ASME V&V 10 are paid and we did not check them. Titles, status and scope were checked on the official pages
• We could not find an international standard for digital twins of materials. We also did not check the number and maturity of demonstrations of digital twins in materials

11. How to read the yardsticks and standards

The yardsticks for multiscale simulation and digital twins are, first, verification of the calculation at each stage (are the equations solved correctly), then comparison with measurement (validation), uncertainty (predictions with a range) and fitness for purpose. What the yardsticks do not measure is the behaviour of the whole linked chain under unseen conditions. Checked on 22 October 2026.

The organisation of the table is Our calculation (our arrangement). Numbers, titles and status were checked on the official pages of ISO, ASME, CEN and NASEM (checked on 22 October 2026). Because the texts of standards are paid, we limit ourselves to titles and scope and do not write requirements. We did not confirm any applicable ASTM standard within the scope of this article.

Reading notes (for materials engineers)

• Judge by content, not by name: if something calls itself a "digital twin" but the updates are one-way or manual, it is a shadow or a model. Check the effect of adoption by the frequency and direction of updates, not by the name
• Prediction versus experimental confirmation: the prediction of a linked calculation can be used for decisions only after verification of each stage and comparison of the whole with measurement (the case of Bulatov et al. is one in which a calculated prediction was confirmed by experiment) [Ref. 6]
• Are standards and guidance recommended or required? The CWA is not an official standard; ISO 30173 and 23247 are terminology and frameworks; ASME V&V 10 is a voluntary standard. All of them generally have binding force only when a contract or law cites them (commentary)
• Plan versus achievement: introduction plans for digital twins and the visions of countries and companies are targets. Check whether updating actually runs

12. Glossary

13. References

1. CEN (European Committee for Standardization) "CEN Workshop Agreement CWA 17284:2018, Materials modelling - Terminology, classification and metadata" (2018) — www.cencenelec.eu/media/CEN-CENELEC/CWAs/RI/cwa17284_2018.pdf
2. The Royal Swedish Academy of Sciences "The Nobel Prize in Chemistry 2013, press release (9 October 2013)" — www.nobelprize.org/prizes/chemistry/2013/press-release/
3. Warshel A., Levitt M. "Theoretical studies of enzymic reactions: Dielectric, electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme", J. Mol. Biol. 103, 227–249 (1976), DOI 10.1016/0022-2836(76)90311-9 — doi.org/10.1016/0022-2836(76)90311-9
4. Rudd R. E., Broughton J. Q. "Coupling of length scales and atomistic simulation of MEMS resonators", SPIE Proceedings 3680, 104 (arXiv:cond-mat/9812232) (1999), DOI 10.1117/12.341254 — doi.org/10.1117/12.341254
5. Broughton J. Q., Abraham F. F., Bernstein N., Kaxiras E. "Concurrent coupling of length scales: Methodology and application", Phys. Rev. B 60, 2391–2403 (1999), DOI 10.1103/PhysRevB.60.2391 — doi.org/10.1103/PhysRevB.60.2391
6. Bulatov V. V. et al. "Dislocation multi-junctions and strain hardening", Nature 440, 1174–1178 (2006), DOI 10.1038/nature04658 — doi.org/10.1038/nature04658
7. Kremer K., Grest G. S. "Dynamics of entangled linear polymer melts: A molecular-dynamics simulation", J. Chem. Phys. 92, 5057–5086 (1990), DOI 10.1063/1.458541 — doi.org/10.1063/1.458541
8. Kremer K., Müller-Plathe F. "Multiscale simulation in polymer science", Mol. Simul. 28, 729–750 (2002), DOI 10.1080/0892702021000002458 — doi.org/10.1080/0892702021000002458
9. Marrink S. J. et al. "The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations", J. Phys. Chem. B 111, 7812–7824 (2007), DOI 10.1021/jp071097f — doi.org/10.1021/jp071097f
10. Chen L.-Q. "Phase-Field Models for Microstructure Evolution", Annu. Rev. Mater. Res. 32, 113–140 (2002), DOI 10.1146/annurev.matsci.32.112001.132041 — doi.org/10.1146/annurev.matsci.32.112001.132041
11. Spencer P. J. "A brief history of CALPHAD", Calphad 32, 1–8 (2008), DOI 10.1016/j.calphad.2007.10.001 — doi.org/10.1016/j.calphad.2007.10.001
12. Kalidindi S. R., Buzzy M., Boyce B. L., Dingreville R. "Digital Twins for Materials", Front. Mater. 9, 818535 (2022), DOI 10.3389/fmats.2022.818535 — doi.org/10.3389/fmats.2022.818535
13. National Research Council (U.S.) "Integrated Computational Materials Engineering: A Transformational Discipline for Improved Competitiveness and National Security", The National Academies Press (2008), DOI 10.17226/12199 — doi.org/10.17226/12199
14. Allison J., Backman D., Christodoulou L. "Integrated computational materials engineering: A new paradigm for the global materials profession", JOM 58 (11), 25–27 (2006), DOI 10.1007/s11837-006-0223-5 — doi.org/10.1007/s11837-006-0223-5
15. Allison J. et al. "Virtual aluminum castings: An industrial application of ICME", JOM 58 (11), 28–35 (2006), DOI 10.1007/s11837-006-0224-4 — doi.org/10.1007/s11837-006-0224-4
16. Jia W. et al. "Pushing the limit of molecular dynamics with ab initio accuracy to 100 million atoms with machine learning", arXiv:2005.00223 (SC20) (2020) — arxiv.org/abs/2005.00223
17. Germann T. C., Kadau K. "Trillion-atom molecular dynamics becomes a reality", Int. J. Mod. Phys. C 19, 1315–1319 (2008), DOI 10.1142/S0129183108012911 — doi.org/10.1142/S0129183108012911
18. Glaessgen E., Stargel D. "The Digital Twin Paradigm for Future NASA and U.S. Air Force Vehicles", NASA Technical Reports Server 20120008178 (AIAA 2012-1818) (2012) — ntrs.nasa.gov/citations/20120008178
19. National Academies of Sciences, Engineering, and Medicine "Foundational Research Gaps and Future Directions for Digital Twins (Summary chapter)", The National Academies Press (2024) — nap.nationalacademies.org/read/26894/chapter/2
20. Kritzinger W. et al. "Digital Twin in manufacturing: A categorical literature review and classification", IFAC-PapersOnLine 51 (11), 1016–1022 (2018), DOI 10.1016/j.ifacol.2018.08.474 — doi.org/10.1016/j.ifacol.2018.08.474
21. Barbie A., Hasselbring W. "From Digital Twins to Digital Twin Prototypes: Concepts, Formalization, and Applications", IEEE Access arXiv:2401.07985 (2024), DOI 10.1109/ACCESS.2024.3406510 — doi.org/10.1109/ACCESS.2024.3406510
22. ISO ISO 23247-1:2021 Automation systems and integration — Digital twin framework for manufacturing — Part 1: Overview and general principles (page checked 22 October 2026) (2021) — www.iso.org/standard/75066.html
23. ISO/IEC ISO/IEC 30173:2023 Digital twin — Concepts and terminology (page checked 22 October 2026) (2023) — www.iso.org/standard/81442.html
24. ASME V&V 10-2019 (R2025) Standard for Verification and Validation in Computational Solid Mechanics (page checked 22 October 2026) (2019) — www.asme.org/codes-standards/find-codes-standards/standard-for-verification-and-validation-in-computational-solid-mechanics

14. Claim-to-source audit

Last updated: 22 October 2026. Sources are limited to primary material (papers, reports of NASA, the National Academies and the NRC, and official pages of CEN, ISO, ASME and the Nobel Prize). Numerical examples are our own calculations and do not show the errors of any real chain. We could not confirm the numerical scales in Fig. 2, some details of the full texts of papers, the text of the classification by Kritzinger et al., the texts (requirements) of the standards, or any international standard for digital twins of materials and the number of demonstrations, so we do not assert them. Figs. 2 to 8 are illustrative concept diagrams or drawings that include our calculations; Figs. 1 and 9 are AI-generated images (only the scene to be drawn is specified).