Pith. sign in

REVIEW 2 major objections 5 minor 73 references

A Hill–Mandel condition written with Gurtin microforces yields a well-posed RVE problem and rigorous micro–macro links for both the order parameter and its gradient, so FE² can track macro phase evolution without resolving every interface.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 03:37 UTC pith:RSDBVJMB

load-bearing objection Solid first-order phase-field homogenization via Gurtin microforces; theory is clean, DNS agreement is real but only qualitative, and RVE sizing remains the open practical issue. the 2 major comments →

arxiv 2607.09356 v1 pith:RSDBVJMB submitted 2026-07-10 physics.comp-ph

Scale-Bridging Phase-Field Modeling of Microstructure Evolution by FE² Computational Homogenization

Geralf H\"utter , Fenil Lathiya , Vincent von Oertzen , Bjoern Kiefer This is my paper
classification physics.comp-ph
keywords phase-field theorymicroforce theorycomputational homogenizationmulti-scale simulationFE2Allen-Cahnmartensitic phase transformation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Phase-field models capture microstructure evolution well, yet resolving every diffuse interface on an engineering component is almost always too expensive. This paper supplies the missing consistent homogenization: by requiring power equivalence between scales in the language of Gurtin microforces, it derives a well-posed boundary-value problem on each representative volume element and the corresponding micro–macro relations for the order parameter and its gradient. The resulting first-order theory is implemented as an FE² scheme and checked against fully resolved simulations for a minimal Allen–Cahn model and for stress-driven martensitic transformation. In both cases the macroscopic averaged fields evolve with the correct spatial pattern and temporal pace. A reader who needs microstructure-informed component predictions therefore gains a thermodynamically grounded route that keeps the fine-scale physics while discarding the need for a global fine mesh.

Core claim

Enforcing a Hill–Mandel-type condition of micro-homogeneity formulated in terms of Gurtin’s microforces produces a well-posed RVE problem whose three classical boundary conditions (uniform microtraction, linear order-parameter, periodic) all satisfy energetic equivalence; the associated micro–macro maps transfer both the order parameter and its gradient, and the resulting FE² scheme recovers the spatial and temporal evolution of the macroscopically averaged fields against direct numerical simulation with reasonable accuracy.

What carries the argument

The Hill–Mandel-type micro-homogeneity condition written with Gurtin microforces (internal power equality that identifies the macroscopic microstress and microforce). It simultaneously supplies the missing energetic link and the residual microforce that appears in the microscopic balance, thereby closing a first-order phase-field homogenization theory.

Load-bearing premise

The claim assumes that a pragmatically chosen RVE size still functions as a faithful averaging window whose internal coarsening kinetics correctly stand in for the macroscopic evolution.

What would settle it

Run the same notched-structure or plate-with-hole problem with identical initial patterns but systematically varied RVE-to-interface-thickness ratios; if the FE² macroscopic order-parameter history then diverges from the fully resolved DNS outside the authors’ chosen ratio of 0.1, the claimed predictive accuracy fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Engineering components can be simulated with phase-field physics without meshing every diffuse interface.
  • Existing mechanical FE² pipelines can be extended to order-parameter kinetics by adding the microforce balance and the three classical RVE boundary conditions.
  • Model-order reduction or machine-learning surrogates become applicable at the RVE level, opening routes to further multi-order-of-magnitude speed-ups.
  • The same first-order construction immediately suggests multi-order-parameter, higher-gradient or micromorphic enrichments once the scalar theory is in place.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because coarsening time depends strongly on the RVE-to-interface ratio, practical use on polycrystals will require grain-size-based RVE selection rules before the method can be trusted for quantitative kinetics.
  • The identical microforce Hill–Mandel construction should transfer to conserved Cahn–Hilliard dynamics and to phase-field fracture, both of which still lack fully consistent gradient-scale bridges.
  • Direct embedding of RVEs already works in plane problems via out-of-plane thickness scaling; a genuine three-dimensional embedding would remove that restriction and broaden industrial applicability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript derives a first-order homogenization framework for phase-field models by enforcing a Hill–Mandel-type micro-homogeneity condition written in Gurtin’s microforces. From the associated Lagrangian with kinematic constraints on the averaged order parameter and its gradient, it obtains a well-posed RVE problem (uniform microtraction, linear, and periodic BCs) together with consistent micro–macro maps for η̄, K̄, π̄ and ξ̄, fully coupled to mechanics. The theory is implemented via DirectFE² in Abaqus (heat-transfer analogy) and illustrated on a pure Allen–Cahn model and a stress-driven martensitic transformation model, with visual comparison to fully resolved DNS on notched and plate-with-hole geometries. The authors conclude that the scheme recovers the spatial and temporal evolution of the macroscopically averaged fields with reasonable accuracy.

Significance. A rigorous scale-bridging theory for non-conserved phase-field evolution that retains the macroscopic gradient of the order parameter has been missing; prior work either treated only mechanical coupling (Kochmann et al.) or used asymptotic/heuristic treatments limited to linear fracture PDEs. The variational construction in §2.3 is clean, recovers the expected residual microforce and the three classical BC families, and correctly distinguishes zeroth- from first-order phase-field homogenization. If the accuracy claim holds under broader conditions, the framework supplies a legitimate route to MOR/ML surrogates at the RVE level and thereby to engineering-scale phase-field simulations. The dual-example validation and the explicit acknowledgment of scale-separation limitations are strengths.

major comments (2)
  1. [§3.1.4, §3.2.3, Figs. 6–8, 12] The central empirical claim (abstract; §3–4) that FE² “reliably predict[s] … with reasonable accuracy” rests solely on visual comparison of contour plots (Figs. 6, 8, 12). No quantitative error measures (e.g., L2 or H1 norms of η̄ versus the DNS average, interface-location error, or transformation-fraction histories) are reported. Without such metrics the accuracy statement cannot be assessed, especially given the known sensitivity of coarsening kinetics to the averaging window.
  2. [§3.1.3, Fig. 5; §4] Section 3.1.3 and Fig. 5 demonstrate that the macroscopic coarsening time depends strongly on the ratio ℓ_Γ/ℓ_RVE; the authors themselves note that “broader and more systematic investigations are required” (§4). Yet the full-structure FE² examples adopt the single pragmatic value 0.1 without a sensitivity study of the predicted macro fields. Because the RVE size is an essential free parameter of the theory (it sets the macroscopic length that can be resolved), the load-bearing claim that the chosen window still represents the DNS evolution needs at least a limited parametric check on the notched or plate-with-hole problems.
minor comments (5)
  1. [§3.1.1, §3.1.4] The shared stochastic initialization between DNS and every RVE is methodologically fair for a controlled comparison, but should be stated more prominently as a deliberate validation choice rather than a general predictive setting.
  2. [§2.3, Eq. (8)] Notation for the macroscopic gradient switches between K̄ and ∇_X η̄; a single consistent symbol would improve readability.
  3. [§3.2.3] The computational-cost comparison (DNS 20 600 s vs FE² 4100 s) is useful; reporting the corresponding wall-clock ratio for the pure Allen–Cahn notched example (where scale separation is lower) would complete the picture.
  4. [Abstract, Keywords] A few typographical issues remain (e.g., “andvalidatedagainst”, missing spaces after commas in the abstract and keywords).
  5. [Fig. 7] Figure 7 caption refers to “final state t/τ = 428” but does not state the macroscopic mesh sizes used; adding element counts or h/ℓ would make the convergence claim self-contained.

Circularity Check

0 steps flagged

No circularity: independent variational derivation of micro-macro maps from Hill-Mandel microforce condition, open-loop DNS validation with literature parameters.

full rationale

The load-bearing chain in §2.3 starts from the microscopic power (3) and the kinematic averages (8), postulates the Hill-Mandel-type equality (9), introduces Lagrange multipliers for the constraints, and obtains the well-posed RVE problem (14)–(17) together with the consistent macrostress definitions (22). All three classical boundary conditions are shown to satisfy the same power equivalence by direct substitution; nothing is defined in terms of the quantity later claimed as a prediction. Material constants (double-well parameters, mobility, zirconia transformation strains and stiffnesses) are taken from the external literature (Allen-Cahn standard forms; Rajendran et al.). The FE² results are compared open-loop to fully-resolved DNS that employ identical constitutive laws and matched initializations; no macroscopic free parameters are fitted to the DNS and then re-predicted. Self-citations ([35] for the DirectFE² embedding technique, [44] for optional future morphology descriptors) supply only numerical infrastructure or outlook and do not underwrite the micro-macro relations or the accuracy claim. The acknowledged pragmatic RVE-size choice is a limitation of applicability, not a circular reduction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 2 invented entities

The load-bearing content is a variational transfer of Hill–Mandel power equivalence into Gurtin’s microforce setting, plus standard phase-field constitutive choices and a pragmatic RVE-size selection. No new physical particles or forces are postulated; the “invented” objects are the macroscopic duals and the first-order homogenization statement itself. Free parameters that affect the central accuracy claim are the RVE-to-interface size ratio and the shared random initializations used for DNS/FE² comparison.

free parameters (3)
  • ℓ_Γ / ℓ_RVE scale-separation ratio = 0.1 (adopted)
    Chosen pragmatically as 0.1 after a parametric study that shows strong dependence of coarsening time on this ratio; no uniqueness or optimality criterion is given, yet all FE² claims rest on this window size.
  • RVE mesh density h_e / ℓ_RVE = 1/20
    Fixed at 1/20 after an unreported convergence study; controls interface resolution inside every RVE and therefore the macroscopic kinetics.
  • Shared stochastic initialization pattern of η
    Identical noise field is imposed on DNS and all RVEs so that comparison is fair; different patterns (A vs B) produce qualitatively different metastable paths, so the reported agreement is conditioned on this choice.
axioms (5)
  • ad hoc to paper Hill–Mandel macrohomogeneity extends to phase-field by equating averaged microscopic Gurtin power ⟨ξ·∇η̇ − π η̇⟩ to macroscopic power ξ̄·K̄̇ − π̄ η̄̇.
    Stated as the defining postulate in §2.3 Eq. (9); standard in mechanics but not previously established for phase-field microforces.
  • domain assumption Macroscopic kinematic quantities are the volume averages η̄ = ⟨η⟩ and K̄ = ⟨∇η⟩, with K̄ = ∇_X η̄.
    Eq. (8); commuting of average and gradient is cited from classical homogenization literature.
  • domain assumption Gurtin’s microforce balance ∇·ξ + π = 0 (and its macroscopic counterpart) governs the order-parameter evolution.
    §2.1; standard continuum phase-field theory.
  • domain assumption Separation of scales holds sufficiently that an RVE attached to each macroscopic Gauss point captures the relevant fluctuations.
    Introduction and §3.1.3; the paper itself documents that kinetics remain scale-dependent.
  • ad hoc to paper DirectFE² out-of-plane thickness scaling preserves the correct virtual-work measure for plane problems without altering in-plane gradient length scales.
    §2.4, following Zhi et al. / Malik et al.; required because physical RVE size cannot be rescaled when ℓ_Γ is present.
invented entities (2)
  • First-order phase-field homogenization (macroscopic microstress ξ̄ and microforce π̄ dual to K̄ and η̄) no independent evidence
    purpose: Close the macroscopic balance ∇_X · ξ̄ + π̄ = 0 and transfer both the order parameter and its gradient across scales.
    Defined via Lagrange multipliers / boundary integrals in §2.3; no independent experimental handle outside the FE² comparisons of this paper.
  • Uniform microtraction boundary condition for the order parameter (ξ·n = ξ̄·n on ∂ΔV) no independent evidence
    purpose: Provide a Neumann-type RVE BC consistent with the Hill–Mandel condition for phase-field.
    Derived in §2.3 as the phase-field analogue of uniform traction; validated only inside the paper’s single-RVE tests.

pith-pipeline@v1.1.0-grok45 · 25053 in / 3667 out tokens · 42209 ms · 2026-07-13T03:37:57.271465+00:00 · methodology

0 comments
read the original abstract

Phase-field models have become a standard tool for simulating complex microstructure evolution in materials, but their application to engineering-scale components is often hindered by prohibitive computational costs arising from the need to resolve fine-scale features. To address this challenge, we propose a consistent homogenization framework for phase-field theory. By enforcing a Hill-Mandel-type condition of micro-homogeneity formulated in terms of Gurtin's microforces, a well-posed boundary value problem is derived for the representative volume element (RVE), establishing rigorous micro-macro relations for both the order parameter and its gradient. The theory is implemented within a computational two-scale (FE$^2$) scheme and validated against direct numerical simulations. Two distinct examples are investigated: a minimal Allen-Cahn model and a mechanically-coupled model for stress-driven martensitic phase transformation. The results demonstrate that the proposed framework can reliably predict the spatial and temporal evolution of the macroscopically averaged fields with reasonable accuracy.

Figures

Figures reproduced from arXiv: 2607.09356 by Bjoern Kiefer, Fenil Lathiya, Geralf H\"utter, Vincent von Oertzen.

Figure 11
Figure 11. Figure 11: The plate is subjected to uniaxial tension via a prescribed displacement [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

73 extracted references · 35 canonical work pages

  1. [2]

    A phase field model for isothermal crystallization of oxide melts

    J. Heulens, B. Blanpain, and N. Moelans. “A phase field model for isothermal crystallization of oxide melts”. In:Acta materialia59.5 (2011), pp. 2156–2165.doi: 10.1016/j.actamat.2010.12.016

    work page doi:10.1016/j.actamat.2010.12.016 2011

  2. [3]

    Two- scale FE–FFT- and phase-field-based computational modeling of bulk microstruc- tural evolution and macroscopic material behavior

    J. Kochmann, S. Wulfinghoff, S. Reese, J. R. Mianroodi, and B. Svendsen. “Two- scale FE–FFT- and phase-field-based computational modeling of bulk microstruc- tural evolution and macroscopic material behavior”. In:Computer Methods in Ap- plied Mechanics and Engineering305 (2016), pp. 89–110.doi:10.1016/j.cma. 2016.03.001

    doi:10.1016/j.cma 2016

  3. [4]

    Three-Dimensional Landau Theory for Multivari- ant Stress-Induced Martensitic Phase Transformations. I. Austenite↔Martensite

    V. I. Levitas and D. L. Preston. “Three-Dimensional Landau Theory for Multivari- ant Stress-Induced Martensitic Phase Transformations. I. Austenite↔Martensite”. In:Physical Review B66 (2002), p. 134206.doi:10.1103/PhysRevB.66.134206

    work page doi:10.1103/physrevb.66.134206 2002

  4. [5]

    Undercooling versus stress induced martensitic phase transformation: The case of MgO – partially stabilized zirconia

    M. K. Rajendran, M. Kuna, and M. Budnitzki. “Undercooling versus stress induced martensitic phase transformation: The case of MgO – partially stabilized zirconia”. In:Computational Materials Science174 (2020), p. 109460.doi:10 . 1016 / j . commatsci.2019.109460

    arXiv 2020

  5. [6]

    Modelling of laminated microstructures in stress- induced martensitic transformations

    S. Stupkiewicz and H. Petryk. “Modelling of laminated microstructures in stress- induced martensitic transformations”. In:Journal of the Mechanics and Physics of Solids50.11 (2002), pp. 2303–2331.doi:10.1016/s0022-5096(02)00029-7

    work page doi:10.1016/s0022-5096(02)00029-7 2002

  6. [7]

    A Phase-Field Study of Martensite Formation in Fe–Mn–Al–Ni Shape Memory Alloys as Caused by Nanoscale B2-Ordered Precipitate and Matrix Phase Interplay

    V. von Oertzen, A. Walnsch, A. Leineweber, and B. Kiefer. “A Phase-Field Study of Martensite Formation in Fe–Mn–Al–Ni Shape Memory Alloys as Caused by Nanoscale B2-Ordered Precipitate and Matrix Phase Interplay”. In:Computational Materials Science258 (2025), p. 113983.doi:10 . 1016 / j . commatsci . 2025 . 113983

    2025

  7. [9]

    Aphasefieldmodelforrate-independent crack propagation: Robust algorithmic implementation based on operator splits

    C.Miehe,M.Hofacker,andF.Welschinger.“Aphasefieldmodelforrate-independent crack propagation: Robust algorithmic implementation based on operator splits”. In:Computer Methods in Applied Mechanics and Engineering199.45—48 (2010), pp. 2765–2778.doi:10.1016/j.cma.2010.04.011

    doi:10.1016/j.cma.2010.04.011 2010

  8. [10]

    A continuum phase field model for fracture

    C. Kuhn and R. Müller. “A continuum phase field model for fracture”. In:En- gineering Fracture Mechanics77.18 (2010), pp. 3625–3634.doi:10 . 1016 / j . engfracmech.2010.08.009

    2010

  9. [11]

    A simple and unified implementation of phase field and gradient damage models

    E. Azinpour, J. P. S. Ferreira, M. P. L. Parente, and J. C. de Sa. “A simple and unified implementation of phase field and gradient damage models”. In:Advanced Modeling and Simulation in Engineering Sciences5.1 (2018), p. 15.doi:10.1186/ s40323-018-0106-7

    2018

  10. [12]

    A phase field formulation for hydrogen assisted cracking

    E. Martínez-Pañeda, A. Golahmar, and C. Niordson. “A phase field formulation for hydrogen assisted cracking”. In:Computer Methods in Applied Mechanics and Engineering342 (2018), pp. 742–761

    2018

  11. [13]

    Finite-DeformationPhase-FieldChemo- mechanics for Multiphase, Multicomponent Solids

    B.Svendsen,P.Shanthraj,andD.Raabe.“Finite-DeformationPhase-FieldChemo- mechanics for Multiphase, Multicomponent Solids”. In:Journal of the Mechanics and Physics of Solids112(2018),pp.619–636.doi:10.1016/j.jmps.2017.10.005

    work page doi:10.1016/j.jmps.2017.10.005 2018

  12. [14]

    Cahn-Hilliard Phase Field Theory Coupled to Mechanics: Fundamentals, Numerical Implementation and Application to Topol- ogy Optimization

    A. Bartels, P. Kurzeja, and J. Mosler. “Cahn-Hilliard Phase Field Theory Coupled to Mechanics: Fundamentals, Numerical Implementation and Application to Topol- ogy Optimization”. In:Computer Methods in Applied Mechanics and Engineering 383 (2021), p. 113918.doi:10.1016/j.cma.2021.113918

    work page doi:10.1016/j.cma.2021.113918 2021

  13. [15]

    Chemo- Mechanical Phase-Field Modeling of Iron Oxide Reduction with Hydrogen

    Y.Bai,J.R.Mianroodi,YMa,A.K.daSilva,B.Svendsen,andD.Raabe.“Chemo- Mechanical Phase-Field Modeling of Iron Oxide Reduction with Hydrogen”. In: Acta Materialia231 (2022), p. 117899.doi:10.1016/j.actamat.2022.117899

    work page doi:10.1016/j.actamat.2022.117899 2022

  14. [16]

    Phase-Field Modeling of Hydrogen-Promoted Fracture: Natural Incorporation of Hydrostatic Stress Dependencies via a Chemical Potential-Based Variational Formulation

    V. Diddige, S. Roth, and B. Kiefer. “Phase-Field Modeling of Hydrogen-Promoted Fracture: Natural Incorporation of Hydrostatic Stress Dependencies via a Chemical Potential-Based Variational Formulation”. In:Computer Methods in Applied Me- chanics and Engineering445 (2025), p. 118143.doi:10.1016/j.cma.2025.118143

    work page doi:10.1016/j.cma.2025.118143 2025

  15. [17]

    The Phase-Field Method in Optimal Design

    B. Bourdin and A. Chambolle. “The Phase-Field Method in Optimal Design”. In: IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Ed. by M. P. Bendsøe, N. Olhoff, and O. Sigmund. Vol. 137. Solid Mechanics and Its Applications. Dordrecht: Springer Netherlands, 2006, pp. 207– 215.doi:10.1007/1-4020-4752-5_21

    work page doi:10.1007/1-4020-4752-5_21 2006

  16. [18]

    A phase field model for stress-based evolu- tion of load-bearing structures. Stress based evolution of load-bearing structures

    I. Muench, C. Gierden, and W. Wagner. “A phase field model for stress-based evolu- tion of load-bearing structures. Stress based evolution of load-bearing structures”. In:International Journal for Numerical Methods in Engineering115.13 (2018), pp. 1580–1600.doi:10.1002/nme.5909

    work page doi:10.1002/nme.5909 2018

  17. [19]

    Phase-Field Models for Microstructure Evolution

    L.-Q. Chen. “Phase-Field Models for Microstructure Evolution”. In:Annual Review of Materials Research32.1 (2002), pp. 113–140.doi:10.1146/annurev.matsci. 32.112001.132041. 25

    work page doi:10.1146/annurev.matsci 2002

  18. [20]

    Anintroductiontophase-fieldmodeling of microstructure evolution

    N.Moelans,B.Blanpain,andP.Wollants.“Anintroductiontophase-fieldmodeling of microstructure evolution”. In:Calphad32.2 (2008), pp. 268–294.doi:10.1016/ j.calphad.2007.11.003

    2008

  19. [21]

    Phase-fieldmodelsinmaterialsscience

    I.Steinbach.“Phase-fieldmodelsinmaterialsscience”.In:Modelling and Simulation in Materials Science and Engineering17.7 (2009), p. 073001.doi:10.1088/0965- 0393/17/7/073001

    doi:10.1088/0965- 2009

  20. [23]

    Free Energy of a Nonuniform System. I. Interfacial Free Energy

    J. W. Cahn and J. E. Hilliard. “Free Energy of a Nonuniform System. I. Interfacial Free Energy”. In:The Journal of Chemical Physics28 (1958), pp. 258–267.doi: 10.1063/1.1744102

    doi:10.1063/1.1744102 1958

  21. [24]

    Homogeniza- tion methods and multiscale modeling: Nonlinear problems

    M. G. D. Geers, V. G. Kouznetsova, K. Matouš, and Y. Julien. “Homogeniza- tion methods and multiscale modeling: Nonlinear problems”. In:Encyclopedia of Computational Mechanics. Ed. by E. Stein, R. de Borst, and T. J. R. Hughes. 2nd ed. Vol. 2. Solids and Structures. John Wiley & Sons, 2017, pp. 1–34.doi: 10.1002/9781119176817.ecm2107

    work page doi:10.1002/9781119176817.ecm2107 2017

  22. [26]

    A comparison of classical phenomenological, hybrid neural network, and high performance ROM FE2 models for open-cell foams: efficiency, accuracy, and flexibility

    N. Lange, A. Malik, M. Abendroth, G. Hütter, and B. Kiefer. “A comparison of classical phenomenological, hybrid neural network, and high performance ROM FE2 models for open-cell foams: efficiency, accuracy, and flexibility”. In:GAMM- Mitteilungen48 (2025), e70004.doi:10.1002/gamm.70004

    doi:10.1002/gamm.70004 2025

  23. [27]

    FE 2 computationalhomog- enization for the thermo-mechanical analysis of heterogeneous solids

    I.Özdemir,W.A.M.Brekelmans,andM.G.D.Geers.“FE 2 computationalhomog- enization for the thermo-mechanical analysis of heterogeneous solids”. In:Computer Methods in Applied Mechanics and Engineering198.3–4 (2008), pp. 602–613.doi: 10.1016/j.cma.2008.09.008

    work page doi:10.1016/j.cma.2008.09.008 2008

  24. [28]

    Computationalhomogenisationofthermo-viscoplastic composites: Large strain formulation and weak micro-periodicity

    R.BerthelsenandA.Menzel.“Computationalhomogenisationofthermo-viscoplastic composites: Large strain formulation and weak micro-periodicity”. In:Computer Methods in Applied Mechanics and Engineering348 (2019), pp. 575–603.doi: 10.1016/j.cma.2018.12.032

    work page doi:10.1016/j.cma.2018.12.032 2019

  25. [29]

    Computational homoge- nization of piezoelectric materials using FE2 to determine configurational forces

    M. Khalaquzzaman, B.-X. Xu, S. Ricker, and R. Müller. “Computational homoge- nization of piezoelectric materials using FE2 to determine configurational forces”. In:Technische Mechanik32.1 (2012), pp. 21–37

    2012

  26. [30]

    Two-scale computational homogeniza- tionofelectro-elasticityatfinitestrains

    M.-A. Keip, P. Steinmann, and J. Schröder. “Two-scale computational homogeniza- tionofelectro-elasticityatfinitestrains”.In:Computer Methods in Applied Mechan- ics and Engineering278 (2014), pp. 62–79.doi:10.1016/j.cma.2014.04.020. 26

    work page doi:10.1016/j.cma.2014.04.020 2014

  27. [31]

    FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

    R. Zabihyan, J. Mergheim, J. P. Pelteret, B. Brands, and P. Steinmann. “FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs”. In:International Journal of Solids and Structures193–194 (2020), pp. 338–356. doi:10.1016/j.ijsolstr.2020.02.015

    work page doi:10.1016/j.ijsolstr.2020.02.015 2020

  28. [32]

    A micromorphic computational homogenization frame- work for heterogeneous materials

    R. Biswas and L. H. Poh. “A micromorphic computational homogenization frame- work for heterogeneous materials”. In:Journal of the Mechanics and Physics of Solids102 (2017), pp. 187–208.doi:10.1016/j.jmps.2017.02.012

    work page doi:10.1016/j.jmps.2017.02.012 2017

  29. [33]

    Micromor- phic Computational Homogenization for Mechanical Metamaterials with Pattern- ing Fluctuation Fields

    O. Rokoš, M. M. Ameen, R. H. J. Peerlings, and M. G. D. Geers. “Micromor- phic Computational Homogenization for Mechanical Metamaterials with Pattern- ing Fluctuation Fields”. In:Journal of the Mechanics and Physics of Solids123 (2019), pp. 119–137.doi:10.1016/j.jmps.2018.08.019

    work page doi:10.1016/j.jmps.2018.08.019 2019

  30. [34]

    Direct FE2 modeling of heteroge- neous materials with a micromorphic computational homogenization framework

    J. Zhi, L. H. Poh, T.-E. Tay, and V. B. C. Tan. “Direct FE2 modeling of heteroge- neous materials with a micromorphic computational homogenization framework”. In:Computer Methods in Applied Mechanics and Engineering393(2022),p.114837. doi:10.1016/j.cma.2022.114837

    doi:10.1016/j.cma.2022.114837 2022

  31. [35]

    Micromorphic FE2 Simulation of Plastic Deformations of Foam Structures

    A. Malik, G. Hütter, M. Abendroth, and B. Kiefer. “Micromorphic FE2 Simulation of Plastic Deformations of Foam Structures”. In:International Journal of Mechan- ical Sciences282 (2024), p. 109551.doi:10.1016/j.ijmecsci.2024.109551

    doi:10.1016/j.ijmecsci.2024.109551 2024

  32. [36]

    Computation of Non-Linear Magneto-Electric Product Properties of 0-3 Com- posites

    J. Schröder, M. Labusch, M.-A. Keip, B. Kiefer, D. Brands, and D. C. Lupascu. “Computation of Non-Linear Magneto-Electric Product Properties of 0-3 Com- posites”. In:GAMM-Mitteilungen38.1 (2015), pp. 8–24.doi:10 . 1002 / gamm . 201510002

    2015

  33. [37]

    An FE2-scheme for magneto-electro- mechanically coupled boundary value problems

    M. Labusch, J. Schröder, and M.-A. Keip. “An FE2-scheme for magneto-electro- mechanically coupled boundary value problems”. In:Ferroic Functional Materials. Ed. by J. Schröder and D. C. Lupascu. Vol. 581. CISM International Centre for Mechanical Sciences (Courses and Lectures). Cham: Springer, 2018, pp. 227–262

    2018

  34. [39]

    Phase-field elasticity model based on mechanical jump conditions

    D. Schneider, O. Tschukin, A. Choudhury, M. Selzer, T. Böhlke, and B. Nestler. “Phase-field elasticity model based on mechanical jump conditions”. In:Computa- tional Mechanics55.5 (2015), pp. 887–901.doi:10.1007/s00466-015-1141-6

    work page doi:10.1007/s00466-015-1141-6 2015

  35. [40]

    Homogenization of vis- coplastic constitutive laws within a phase field approach

    V. de Rancourt, K. Ammar, B. Appolaire, and S. Forest. “Homogenization of vis- coplastic constitutive laws within a phase field approach”. In:Journal of the Me- chanics and Physics of Solids88 (2016), pp. 291–319.doi:10.1016/j.jmps.2015. 12.026. 27

    doi:10.1016/j.jmps.2015 2016

  36. [41]

    A Numerical Convergence Study Regarding Homogenization Assumptions in Phase Field Modeling

    B. Kiefer, T. Furlan, and J. Mosler. “A Numerical Convergence Study Regarding Homogenization Assumptions in Phase Field Modeling”. In:International Journal for Numerical Methods in Engineering112.9 (2017), pp. 1097–1128.doi:10.1002/ nme.5547

    2017

  37. [42]

    Anefficientandquantitativephase-field model for elastically heterogeneous two-phase solids based on a partial rank-one homogenization scheme

    S.Chatterjee,D.Schwen,andN.Moelans.“Anefficientandquantitativephase-field model for elastically heterogeneous two-phase solids based on a partial rank-one homogenization scheme”. In:International Journal of Solids and Structures250 (2022), p. 111709.doi:10.1016/j.ijsolstr.2022.111709

    work page doi:10.1016/j.ijsolstr.2022.111709 2022

  38. [43]

    A gradient regularized model for shape memory alloys

    H. Yu and C. M. Landis. “A gradient regularized model for shape memory alloys”. In:Mechanics of Materials183 (2023), p. 104689.doi:10.1016/j.mechmat.2023. 104689

    work page doi:10.1016/j.mechmat.2023 2023

  39. [44]

    Homogenization of Phase Transforming Materials: The Concept of Phase-Morphology and Variable Scale Separations

    V. von Oertzen and B. Kiefer. “Homogenization of Phase Transforming Materials: The Concept of Phase-Morphology and Variable Scale Separations”. In:Journal of the Mechanics and Physics of Solids(2024), p. 105961.doi:10.1016/j.jmps. 2024.105961

    doi:10.1016/j.jmps 2024

  40. [45]

    A phase field approach for damage propagation in periodic microstructured materials

    F. Fantoni, A. Bacigalupo, M. Paggi, and J. Reinoso. “A phase field approach for damage propagation in periodic microstructured materials”. In:International Journal of Fracture223.1–2 (2019), pp. 53–76.doi:10.1007/s10704-019-00400- x

    work page doi:10.1007/s10704-019-00400- 2019

  41. [46]

    A numerical-homogenization based phase-field fracture modeling of linear elastic heterogeneous porous media

    B. He, L. Schuler, and P. Newell. “A numerical-homogenization based phase-field fracture modeling of linear elastic heterogeneous porous media”. In:Computational Materials Science176 (2020), p. 109519.doi:10 . 1016 / j . commatsci . 2020 . 109519

    2020

  42. [47]

    Development and Calibration of a Phe- nomenological Material Model for Steel-Fiber-Reinforced High-Performance Con- crete Based on Unit Cell Calculations

    M. Pise, D. Brands, and J. Schröder. “Development and Calibration of a Phe- nomenological Material Model for Steel-Fiber-Reinforced High-Performance Con- crete Based on Unit Cell Calculations”. In:Materials17.10 (2024), p. 2247.doi: 10.3390/ma17102247

    work page doi:10.3390/ma17102247 2024

  43. [48]

    Multi-Phase-Field Method for Dynamic Fracture in Compos- ite Materials Based on Reduced-Order-Homogenization

    N. Liu and Z. Yuan. “Multi-Phase-Field Method for Dynamic Fracture in Compos- ite Materials Based on Reduced-Order-Homogenization”. In:International Journal for Numerical Methods in Engineering126.4 (2025).doi:10.1002/nme.70012

    work page doi:10.1002/nme.70012 2025

  44. [49]

    Amultiscaleanisotropicpolymernetworkmodelcoupledwithphasefieldfracture

    P. K. Arunachala, S. Abrari Vajari, M. Neuner, J. S. Sim, R. Zhao, and C. Linder. “Amultiscaleanisotropicpolymernetworkmodelcoupledwithphasefieldfracture”. In:International Journal for Numerical Methods in Engineering125.13(2024).doi: 10.1002/nme.7488

    work page doi:10.1002/nme.7488 2024

  45. [50]

    Asymptotic homogenization of phase- field fracture model: An efficient multiscale finite element framework for anisotropic fracture

    P. Ma, X. Liu, X. Luo, S. Li, and L. Zhang. “Asymptotic homogenization of phase- field fracture model: An efficient multiscale finite element framework for anisotropic fracture”. In:International Journal for Numerical Methods in Engineering125.13 (2024).doi:10.1002/nme.7489. 28

    work page doi:10.1002/nme.7489 2024

  46. [51]

    Size Effects in Martensitic Microstruc- tures: Finite-Strain Phase Field Model Versus Sharp-Interface Approach

    K. Tůma, S. Stupkiewicz, and H. Petryk. “Size Effects in Martensitic Microstruc- tures: Finite-Strain Phase Field Model Versus Sharp-Interface Approach”. In:Jour- nal of the Mechanics and Physics of Solids95 (2016), pp. 284–307.doi:10.1016/ j.jmps.2016.04.013

    2016

  47. [52]

    GeneralizedGinzburg-LandauandCahn-Hilliardequationsbasedon a microforce balance

    M.E.Gurtin.“GeneralizedGinzburg-LandauandCahn-Hilliardequationsbasedon a microforce balance”. In:Physica D: Nonlinear Phenomena92.3–4 (1996), pp. 178– 192.doi:10.1016/0167-2789(95)00173-5

    doi:10.1016/0167-2789(95)00173-5 1996

  48. [53]

    Elastic properties of reinforced solids: Some theoretical principles

    R. Hill. “Elastic properties of reinforced solids: Some theoretical principles”. In: Journal of the Mechanics and Physics of Solids11.5 (1963), pp. 357–372.doi: 10.1016/0022-5096(63)90036-X

    doi:10.1016/0022-5096(63)90036-x 1963

  49. [54]

    Diffusion and dispersion in porous media

    S. Whitaker. “Diffusion and dispersion in porous media”. In:AIChE Journal13.3 (1967), pp. 420–427.doi:10.1002/aic.690130308

    doi:10.1002/aic.690130308 1967

  50. [55]

    Some Applications of the Homogenization Theory

    C. C. Mei, J. L. Auriault, and C. O. Ng. “Some Applications of the Homogenization Theory”. In:Advances in Applied Mechanics32 (1996), pp. 277–348.doi:10.1016/ S0065-2156(08)70078-4

    1996

  51. [56]

    Cosserat overall modeling of heterogeneous materials

    S. Forest and K. Sab. “Cosserat overall modeling of heterogeneous materials”. In: Mechanics Research Communications25.4 (1998), pp. 449–454.doi:10 . 1016 / S0093-6413(98)00059-7

    1998

  52. [57]

    Generalized continua and non-homogeneous boundary conditions in homogenisation methods

    S. Forest and D. Trinh. “Generalized continua and non-homogeneous boundary conditions in homogenisation methods”. In:Zeitschrift für Angewandte Mathematic und Mechanik91.2 (2011), pp. 90–109.doi:10.1002/zamm.201000109

    work page doi:10.1002/zamm.201000109 2011

  53. [58]

    On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers

    C. Miehe and C. G. Bayreuther. “On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers”. In:International Journal of Numerical Methods in Engineering71.10 (2007), pp. 1135–1180.doi:10.1002/ nme.1972

    2007

  54. [59]

    Minimal loading conditions for higher order numerical homogenisation schemes

    R. Jänicke and H. Steeb. “Minimal loading conditions for higher order numerical homogenisation schemes”. In:Archive of Applied Mechanics82.8 (2012), pp. 1075– 1088.doi:10.1007/s00419-012-0614-8

    work page doi:10.1007/s00419-012-0614-8 2012

  55. [60]

    Homogenization of a Cauchy continuum towards a micromorphic con- tinuum

    G. Hütter. “Homogenization of a Cauchy continuum towards a micromorphic con- tinuum”. In:Journal of the Mechanics and Physics of Solids99 (2017), pp. 394– 408.doi:10.1016/j.jmps.2016.09.010

    work page doi:10.1016/j.jmps.2016.09.010 2017

  56. [61]

    Nonlocal reaction—diffusion equations and nu- cleation

    J. Rubinstein and P. Sternberg. “Nonlocal reaction—diffusion equations and nu- cleation”. In:IMA Journal of Applied Mathematics48.3 (1992), pp. 249–264.doi: 10.1093/imamat/48.3.249

    doi:10.1093/imamat/48.3.249 1992

  57. [62]

    A numerical two-scale homogenization scheme: the FE2-method

    J. Schröder. “A numerical two-scale homogenization scheme: the FE2-method”. In: Plasticity and Beyond. Ed. by J. Schröder and K. Hackl. Springer, 2014, pp. 1–64. doi:10.1007/978-3-7091-1625-8_1. 29

    work page doi:10.1007/978-3-7091-1625-8_1 2014

  58. [64]

    A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstruc- ture Evolution and Macroscopic Material Behavior

    C. Gierden, J. Kochmann, J. Waimann, B. Svendsen, and S. Reese. “A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstruc- ture Evolution and Macroscopic Material Behavior”. In:Archives of Computational Methods in Engineering29.6 (2022), pp. 4115–4135.doi:10.1007/s11831-022- 09735-6

    work page doi:10.1007/s11831-022- 2022

  59. [65]

    Direct FE2 for concurrent multilevel mod- elling of heterogeneous structures

    V. B. C. Tan, K. Raju, and H. P. Lee. “Direct FE2 for concurrent multilevel mod- elling of heterogeneous structures”. In:Computer Methods in Applied Mechanics and Engineering360 (2020), p. 112694.doi:10.1016/j.cma.2019.112694

    work page doi:10.1016/j.cma.2019.112694 2020

  60. [66]

    Interface propagation and microstruc- ture evolution in phase field models of stress-induced martensitic phase transfor- mations

    V. I. Levitas, D.-W. Lee, and D. L. Preston. “Interface propagation and microstruc- ture evolution in phase field models of stress-induced martensitic phase transfor- mations”. In:International Journal of Plasticity26.3 (2010), pp. 395–422.doi: 10.1016/j.ijplas.2009.08.003

    work page doi:10.1016/j.ijplas.2009.08.003 2010

  61. [67]

    An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure

    A. Seupel, G. Hütter, and M. Kuna. “An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure”. In:Engineering Fracture Mechanics199 (2018), pp. 41–60

    2018

  62. [68]

    A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

    S. M. Allen and J. W. Cahn. “A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening”. In:Acta Metallurgica27.6 (1979), pp. 1085–1095.doi:10.1016/0001-6160(79)90196-2

    doi:10.1016/0001-6160(79)90196-2 1979

  63. [69]

    Theoryofphase-orderingkinetics

    A.J.Bray.“Theoryofphase-orderingkinetics”.In:Advances in Physics43.3(1994), pp. 357–459.doi:10.1080/00018739400101505

    doi:10.1080/00018739400101505 1994

  64. [70]

    Metastable patterns in solutions ofut =ϵ 2uxx −f(u)

    J. Carr and R. L. Pego. “Metastable patterns in solutions ofut =ϵ 2uxx −f(u)”. In:Communications on pure and applied mathematics42.5 (1989), pp. 523–576

    1989

  65. [71]

    H. F. Weinberger.On Metastable Patterns in Parabolic Systems. IMA Preprint. Retrieved from the University Digital Conservancy. Institute for Mathematics and Its Applications (IMA), University of Minnesota, 1985

    1985

  66. [72]

    A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements

    E. N. Karatzas and G. Rozza. “A Reduced Order Model for a Stable Embedded Boundary Parametrized Cahn–Hilliard Phase-Field System Based on Cut Finite Elements”. In:Journal of Scientific Computing89.1 (2021).doi:10.1007/s10915- 021-01623-8

    doi:10.1007/s10915- 2021

  67. [73]

    Reduced order methods for the solution of solidification Phase- Field models

    E. López-Quiroga. “Reduced order methods for the solution of solidification Phase- Field models”. In:IFAC-PapersOnLine51.2 (2018), pp. 637–642.doi:10.1016/j. ifacol.2018.03.108

    doi:10.1016/j 2018

  68. [74]

    Extraction of reduced-order process-structure linkages from phase-field simulations

    Y. C. Yabansu, P. Steinmetz, J. Hötzer, S. R. Kalidindi, and B. Nestler. “Extraction of reduced-order process-structure linkages from phase-field simulations”. In:Acta Materialia124 (2017), pp. 182–194.doi:10.1016/j.actamat.2016.10.071. 30

    work page doi:10.1016/j.actamat.2016.10.071 2017

  69. [75]

    Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks

    C. L. Wight and J. Zhao. “Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks”. In:arXiv(2020), p. 2007.04542. doi:10.48550/arXiv.2007.04542

    Pith/arXiv arXiv doi:10.48550/arxiv.2007.04542 2020

  70. [76]

    Self-adaptive physics-informed neural networks

    L. D. McClenny and U. M. Braga-Neto. “Self-adaptive physics-informed neural networks”. In:Journal of Computational Physics474 (2023), p. 111722.doi:10. 1016/j.jcp.2022.111722

    arXiv 2023

  71. [77]

    PF-PINNs: Physics-informed neural networks for solving coupled Allen-Cahn and Cahn-Hilliard phase field equa- tions

    N. Chen, S. Lucarini, R. Ma, A. Chen, and C. Cui. “PF-PINNs: Physics-informed neural networks for solving coupled Allen-Cahn and Cahn-Hilliard phase field equa- tions”. In:Journal of Computational Physics529 (2025), p. 113843.doi:10.1016/ j.jcp.2025.113843

    arXiv 2025

  72. [78]

    Deep learning for simulating the evolution of condensed matter systems at the continuum scale: methods and applications

    D. Lanzoni, F. Montalenti, and R. Bergamaschini. “Deep learning for simulating the evolution of condensed matter systems at the continuum scale: methods and applications”. In:Journal of Physics: Condensed Matter37.40 (2025), p. 403003. doi:10.1088/1361-648x/ae096d

    work page doi:10.1088/1361-648x/ae096d 2025

  73. [79]

    Generalized Continua

    S. Forest. “Generalized Continua”. In:Encyclopedia of Materials: Science and Tech- nology. Ed. by K. Buschow, R. Cahn, M. Flemings, B. Ilschner, E. Kramer, and S. Mahajan. Oxford: Elsevier, 2005, pp. 1–7.doi:10.1016/B0-08-043152-6/02091- X. 31

    work page doi:10.1016/b0-08-043152-6/02091- 2005