A phase field model with arbitrary misorientation dependence of grain boundary energy

(2) George Mason University; (3) California Institute of Technology; 4); (4) Northwestern University; (5) National Institute of Standards; James A. Warren (5) ((1) University of Michigan; Peter W. Voorhees (3; Philip Staublin (1); Technology); Yuri Mishin (2)

arxiv: 2603.14660 · v2 · submitted 2026-03-15 · ❄️ cond-mat.mtrl-sci

A phase field model with arbitrary misorientation dependence of grain boundary energy

Philip Staublin (1) , Yuri Mishin (2) , Peter W. Voorhees (3 , 4) , James A. Warren (5) ((1) University of Michigan , (2) George Mason University , (3) California Institute of Technology , (4) Northwestern University

show 2 more authors

(5) National Institute of Standards Technology)

This is my paper

Pith reviewed 2026-05-15 10:51 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords phase field modelgrain boundary energymisorientationorientation fieldgrain growthKobayashi-Warren-Carter modelpolycrystal simulation

0 comments

The pith

A modified orientation-field model now permits any desired misorientation dependence of grain boundary energy.

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

Existing orientation-field models are proven unable, in general, to produce the common physical decrease of grain boundary free energy with rising misorientation angle. The authors modify the Kobayashi-Warren-Carter phase-field model so that its free-energy coefficients become explicit functions of a non-local misorientation measure. This misorientation is evaluated by interpolating the orientation field at a fixed distance along the local grain-boundary normal. The change lets any prescribed energy-versus-misorientation function be inserted directly into the simulation. The method is demonstrated by placing a sharp cusp in the energy curve and is sketched for three-dimensional extension.

Core claim

We prove that existing orientation-field models cannot reproduce a decrease in the grain boundary free energy with increasing misorientation angle. We propose a modification to the Kobayashi-Warren-Carter model wherein the coefficients of the free-energy functional become functions of the misorientation between the grains, a non-local quantity obtained by interpolating the orientation field at a fixed distance in both directions along the local grain boundary normal vector. Due to this modification an arbitrary dependence of the grain boundary free energy on the misorientation can be embedded in the model.

What carries the argument

Non-local misorientation obtained by interpolating the orientation field at fixed distance along the grain-boundary normal, used to set the coefficients of the free-energy functional in a modified Kobayashi-Warren-Carter phase-field model.

If this is right

Any measured or theoretical grain-boundary energy versus misorientation curve can be inserted directly into the evolution equations.
Special low-energy boundaries can be represented by introducing sharp cusps in the energy function.
Grain-growth simulations will now reflect realistic misorientation dependence without the previous structural limitation.
The same non-local construction extends to three-dimensional polycrystal calculations.

Where Pith is reading between the lines

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

Polycrystal texture evolution and grain-size distributions may shift noticeably once realistic energy cusps are active.
The choice of interpolation distance becomes a tunable parameter whose optimal value can be tested against molecular-dynamics grain-boundary energies.
Similar non-local interpolation could be applied to orientation-dependent mobilities or to anisotropic interface energies.

Load-bearing premise

Interpolating the orientation field at a fixed distance along the local normal accurately captures the physical misorientation without introducing numerical artifacts or violating the variational structure.

What would settle it

Simulate a prescribed decreasing energy function versus misorientation and measure the actual grain-boundary energy extracted from the phase-field profiles; mismatch between prescribed and measured energies would show the claim does not hold.

Figures

Figures reproduced from arXiv: 2603.14660 by (2) George Mason University, (3) California Institute of Technology, 4), (4) Northwestern University, (5) National Institute of Standards, James A. Warren (5) ((1) University of Michigan, Peter W. Voorhees (3, Philip Staublin (1), Technology), Yuri Mishin (2).

**Figure 2.** Figure 2: Equilibrium profiles for varying misorientation with [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗

**Figure 3.** Figure 3: Grain boundary energy as a function of misorientation calculated from simulations [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

**Figure 4.** Figure 4: Grain boundary mobility as a function of misorientation, calculated from sim [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗

read the original abstract

Grain growth in polycrystals is often simulated using orientation-field models, which employ a field to represent the local orientation of the crystal lattice. These models can be challenging to represent a realistic misorientation dependence of grain boundary free energy. We prove that existing orientation-field models, in general, cannot reproduce a decrease in the grain boundary free energy with a increasing misorientation angle, demonstrating a significant limitation of previous models in applications to polycrystalline materials. To overcome this limitation, we propose a modification to the Kobayashi-Warren-Carter model for grain growth wherein the coefficients of the free-energy functional become functions of the misorientation between the grains, which is a non-local quantity. Due to this modification, an arbitrary dependence of the grain boundary free energy on the misorientation can be embedded in the model. We propose calculating the non-local misorientation by interpolating the orientation field at a fixed distance in both directions along the local grain boundary normal vector. The capabilities of the model are demonstrated by introduction of a sharp cusp to the misorientation dependent grain boundary free energy. Finally, we propose an extension of the model to three dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows standard orientation-field models cannot produce GB energy that decreases with misorientation and offers a non-local interpolation fix, but the variational consistency of the resulting dynamics is not obvious from the given details.

read the letter

The main point is that usual Kobayashi-Warren-Carter models cannot reproduce grain boundary energy that falls as misorientation angle rises, and this work gives a concrete way to embed any desired dependence by making the free-energy coefficients functions of a non-local misorientation. They compute that misorientation by interpolating the orientation field at a fixed distance along the local normal in both directions. This lets them insert arbitrary functions, such as a sharp cusp, directly into the model. The proof that prior models are limited in this way is laid out clearly and addresses a practical barrier in polycrystal simulations. The three-dimensional extension is sketched at the end. The construction itself appears new relative to the cited orientation-field literature. The non-local interpolation step is the part that needs the most scrutiny. Because the misorientation now depends on the field values at offset locations, the first variation of the free-energy functional will contain additional integral contributions that are absent from the standard derivation. The abstract does not show whether these terms are carried through into the evolution equations or whether they cancel. If the dynamics are written without them, the model would no longer strictly descend from the stated functional, which would undercut the thermodynamic structure. The choice of interpolation distance also functions as a free parameter whose effect on the results should be checked. This work is aimed at people who run phase-field grain-growth simulations and need realistic misorientation dependence for texture or property predictions. A reader already familiar with KWC-type models will see the limitation and the proposed workaround immediately. I would send it to peer review. The identified limitation is concrete and the fix is specific enough that referees can verify the derivations and test the numerics.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that existing orientation-field models cannot, in general, reproduce a decrease in grain boundary free energy with increasing misorientation angle. It introduces a modification to the Kobayashi-Warren-Carter phase-field model in which the coefficients of the free-energy functional depend on a non-local misorientation computed by interpolating the orientation field at a fixed distance along the grain boundary normal. This allows arbitrary misorientation dependence to be embedded, as demonstrated by a sharp cusp in the energy function, and the approach is extended to three dimensions.

Significance. If the proposed non-local modification preserves the variational structure of the model, the work would provide a significant advance in phase-field modeling of polycrystals by overcoming a fundamental limitation in representing realistic grain boundary energies, with potential impact on simulations of grain growth and related phenomena in materials science.

major comments (2)

[Model modification and evolution equations] The modification makes the free-energy coefficients functions of a non-local quantity obtained by fixed-distance interpolation of the orientation field along the local normal. This non-locality implies that the first variation of the free energy will contain additional integral terms not present in the standard local KWC derivation. The manuscript does not show whether these terms are included in the evolution equations or why they vanish, which is necessary to confirm that the dynamics still descend from the free-energy functional (see abstract description of the modification and the skeptic note on variational consistency).
[Proof of impossibility for prior models] The central claim includes a proof that existing models cannot reproduce decreasing GB energy with misorientation. While stated in the abstract, the full mathematical derivation is not visible in the provided text, making it impossible to verify the generality of the impossibility result or check for hidden assumptions about the form of the orientation field.

minor comments (2)

The interpolation distance is introduced as a free parameter whose value is chosen post-hoc; a brief sensitivity analysis or discussion of its influence on numerical artifacts would improve the presentation.
Notation for the non-local interpolation operator and the orientation field should be defined more explicitly in the main text to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and will revise the manuscript accordingly to improve its clarity and completeness.

read point-by-point responses

Referee: [Model modification and evolution equations] The modification makes the free-energy coefficients functions of a non-local quantity obtained by fixed-distance interpolation of the orientation field along the local normal. This non-locality implies that the first variation of the free energy will contain additional integral terms not present in the standard local KWC derivation. The manuscript does not show whether these terms are included in the evolution equations or why they vanish, which is necessary to confirm that the dynamics still descend from the free-energy functional (see abstract description of the modification and the skeptic note on variational consistency).

Authors: We agree with the referee that the variational consistency of the modified model requires explicit verification. The non-local dependence on the interpolated orientation does introduce additional terms in the functional derivative. In the revised manuscript, we will provide a detailed derivation of the evolution equations, showing how these integral terms are handled and confirming that the dynamics still follow from the free-energy functional. This will include the full first variation calculation. revision: yes
Referee: [Proof of impossibility for prior models] The central claim includes a proof that existing models cannot reproduce decreasing GB energy with misorientation. While stated in the abstract, the full mathematical derivation is not visible in the provided text, making it impossible to verify the generality of the impossibility result or check for hidden assumptions about the form of the orientation field.

Authors: The full proof is included in Section 2 of the manuscript. However, to make it more accessible and to address the referee's concern, we will revise the presentation to include all intermediate steps explicitly, clarify the assumptions regarding the orientation field, and ensure the derivation is self-contained without requiring reference to external material. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with no circular reductions

full rationale

Walking the claimed chain: the proof of limitation in existing models is a general mathematical argument independent of the new model. The modification defines the misorientation as a non-local interpolation, making the energy coefficients functions of this quantity to allow arbitrary dependence. This is an explicit modeling choice, not a reduction of the result to its inputs by construction. No fitted parameters are called predictions, no self-citations bear the load of uniqueness or ansatz, and no known results are renamed. The model is presented as an extension preserving the variational structure, with the non-local aspect introduced directly rather than derived circularly.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on standard phase-field variational structure plus one new non-local operator; no new particles or forces are postulated.

free parameters (1)

interpolation distance
Fixed length used to sample orientation field along the boundary normal; value must be chosen by the user.

axioms (1)

domain assumption Orientation can be represented by a continuous vector field whose gradients define grain boundaries
Standard assumption of orientation-field models referenced in the abstract.

pith-pipeline@v0.9.0 · 5555 in / 1184 out tokens · 34106 ms · 2026-05-15T10:51:06.345523+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that existing orientation-field models, in general, cannot reproduce a decrease in the grain boundary free energy with increasing misorientation angle... Due to this modification, an arbitrary dependence of the grain boundary free energy on the misorientation can be embedded in the model.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose calculating the non-local misorientation by interpolating the orientation field at a fixed distance in both directions along the local grain boundary normal vector.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Kobayashi, J.A

R. Kobayashi, J.A. Warren, and W.C. Carter. Vector-valued phase field model for crystallization and grain boundary formation.Physica D: Nonlinear Phenomena, 119(3-410 4):415–423, 1998

work page 1998
[2]

Warren, R

J.A. Warren, R. Kobayashi, and W.C. Carter. Modeling grain boundaries using a phase- field technique.Journal of Crystal Growth, page 3, 2000

work page 2000
[3]

Warren, R

J.A. Warren, R. Kobayashi, A.E. Lobkovsky, and W.C. Carter. Extending phase field models of solidification to polycrystalline materials.Acta Materialia, 51:6035–6058,415 2003

work page 2003
[4]

Henry, J

H. Henry, J. Mellenthin, and M. Plapp. Orientation-field model for polycrystalline solidification with a singular coupling between order and orientation.Physical Review B, 86:054117, 2012

work page 2012
[5]

Korbuly, M

B. Korbuly, M. Plapp, H. Henry, J.A. Warren, L. Gr´ an´ asy, and T. Pusztai. Topological420 defects in two-dimensional orientation-field models for grain growth.Physical Review E, 96:052802, 2017

work page 2017
[6]

Admal, J

N.C. Admal, J. Segurado, and J. Marian. A three-dimensional misorientation axis- and inclination-dependent Kobayashi–Warren–Carter grain boundary model.Journal of the Mechanics and Physics of Solids, 128:32–53, 2019.425

work page 2019
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Staublin, A

P. Staublin, A. Mukherjee, J.A. Warren, and P.W. Voorhees. Phase-field model for anisotropic grain growth.Acta Materialia, 237, Sept 2022. 30

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Han and A

X. Han and A. van de Walle. A nonlocal orientation field phase-field model for misorientation- and inclination- dependent grain boundaries.arXiv preprint, 2025

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Chen and Y

L.-Q. Chen and Y. Wang. The continuum field approach to modeling microstructural430 evolution.JOM, 48(12):13–18, 1996

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Fan and L.-Q

D. Fan and L.-Q. Chen. Computer simulation of grain growth using a continuum field model.Acta Materialia, 45(2):611–622, 1997

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Moelans, B

N. Moelans, B. Blanpain, and P. Wollants. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems.435 Physical Review B, 78(2):024113, 2008

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[12]

Steinbach, F

I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G.J. Schmitz, and J.L.L. Rezende. A phase field concept for multiphase systems.Physica D: Nonlinear Phenomena, 94(3):135–147, 1996

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Steinbach and F

I. Steinbach and F. Pezzolla. A generalized field method for multiphase transformations440 using interface fields.Physica D: Nonlinear Phenomena, 134(4):385–393, 1999

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[14]

Nestler and A

B. Nestler and A. Wheeler. Anisotropic multi-phase-field model: Interfaces and junc- tions.Physical Review E, 57(3):2602–2609, 1998

work page 1998
[15]

Garcke, B

H. Garcke, B. Nestler, and B. Stoth. A multiphase field concept: Numerical simula- tions of moving phase boundaries and multiple junctions.SIAM Journal on Applied445 Mathematics, 60(1):295–315, 1999. 31

work page 1999
[16]

Nestler, H

B. Nestler, H. Garcke, and B. Stinner. Multicomponent alloy solidification: Phase-field modeling and simulations.Physical Review E, 71(4):041609, 2005

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[17]

Folch and M

R. Folch and M. Plapp. Quantitative phase-field modeling of two-phase growth.Physical Review E, 72(1):011602, 2005.450

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[18]

Kim, D.I

S.G. Kim, D.I. Kim, W.T. Kim, and Y.B. Park. Computer simulations of two- dimensional and three-dimensional ideal grain growth.Physical Review E, 74(6):061605, 2006

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[19]

Steinbach

I. Steinbach. Phase-field model for microstructure evolution at the mesoscopic scale. Annual Review of Materials Research, 43:89–107, 2013.455

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[20]

Kim, S.G

H. Kim, S.G. Kim, W. Dong, I. Steinbach, and B. Lee. Phase-field modeling for 3d grain growth based on a grain boundary energy database.Modelling and Simulation in Materials Science and Engineering, 22(3):034004, 2014

work page 2014
[21]

Zhang, W

J. Zhang, W. Ludwig, Y. Zhang, H.H.B. Sørensen, D.J. Rowenhorst, A. Yamanaka, P.W. Voorhees, and H.F. Poulsen. Grain boundary mobilities in polycrystals.Acta460 Materialia, 191:211–220, 2020

work page 2020
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Gelfand and S.V

I.M. Gelfand and S.V. Fomin.Calculus of Variations. Prentice-Hall, 1963

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[23]

Williamson

J.H. Williamson. Low-storage runge-kutta schemes.Journal of Computational Physics, 35(1):48–56, 1980. 32

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Carpenter and C.A

M.H. Carpenter and C.A. Kennedy. Fourth-order 2n-storage runge-kutta schemes. Tech-465 nical Memorandum NASA/TM-109112, NASA, NASA Langley Research Center, Hamp- ton VA 23681-0001, USA, June 1994

work page 1994
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Karma, Z.T

A. Karma, Z.T. Trautt, and Y. Mishin. Relationship between equilibrium fluctuations and shear-coupled motion of grain boundaries.Physical Review Letters, 109:095501, 2012.470

work page 2012
[26]

A. Qiu, C. Qiu, I. Chesser, J. Han, D.J. Srolovitz, and E.A. Holm. Mechanisms of shape evolution and rotation during crystal growth in cylindrical grains.Acta Materialia, 305, 2026

work page 2026
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Morawiec.Orientations and Rotaitons: Computations in Crystallographic Textures

A. Morawiec.Orientations and Rotaitons: Computations in Crystallographic Textures. Springer-Verlag Berlin Heidelberg, 2004.475

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Rowenhorst, A.D

D. Rowenhorst, A.D. Rollett, G.S. Rohrer, M. Groeber, M. Jackson, P.J. Konijnenberg, and M.D. Graef. Consistent representations of and conversions between 3d rotations. Modelling and Simulation in Materials Science and Engineering, 23(8):083501, 2015. Publisher: IOP Publishing

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[29]

Staublin.Orientation Field Model for Grain Growth Accounting for Five Crystallo-480 graphic Degrees of Freedom

P. Staublin.Orientation Field Model for Grain Growth Accounting for Five Crystallo-480 graphic Degrees of Freedom. PhD thesis, Northwestern University, 2024

work page 2024
[30]

Mason and C.A

J.K. Mason and C.A. Schuh. Hyperspherical harmonics for the representation of crys- tallographic texture.Acta Materialia, 56(20):6141–6155, 2008

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[31]

Moelans, F

N. Moelans, F. Wendler, and B. Nestler. Comparative study of two phase-field models for grain growth.Computational Materials Science, 46(2):479–490, 2009.485 33

work page 2009

[1] [1]

Kobayashi, J.A

R. Kobayashi, J.A. Warren, and W.C. Carter. Vector-valued phase field model for crystallization and grain boundary formation.Physica D: Nonlinear Phenomena, 119(3-410 4):415–423, 1998

work page 1998

[2] [2]

Warren, R

J.A. Warren, R. Kobayashi, and W.C. Carter. Modeling grain boundaries using a phase- field technique.Journal of Crystal Growth, page 3, 2000

work page 2000

[3] [3]

Warren, R

J.A. Warren, R. Kobayashi, A.E. Lobkovsky, and W.C. Carter. Extending phase field models of solidification to polycrystalline materials.Acta Materialia, 51:6035–6058,415 2003

work page 2003

[4] [4]

Henry, J

H. Henry, J. Mellenthin, and M. Plapp. Orientation-field model for polycrystalline solidification with a singular coupling between order and orientation.Physical Review B, 86:054117, 2012

work page 2012

[5] [5]

Korbuly, M

B. Korbuly, M. Plapp, H. Henry, J.A. Warren, L. Gr´ an´ asy, and T. Pusztai. Topological420 defects in two-dimensional orientation-field models for grain growth.Physical Review E, 96:052802, 2017

work page 2017

[6] [6]

Admal, J

N.C. Admal, J. Segurado, and J. Marian. A three-dimensional misorientation axis- and inclination-dependent Kobayashi–Warren–Carter grain boundary model.Journal of the Mechanics and Physics of Solids, 128:32–53, 2019.425

work page 2019

[7] [7]

Staublin, A

P. Staublin, A. Mukherjee, J.A. Warren, and P.W. Voorhees. Phase-field model for anisotropic grain growth.Acta Materialia, 237, Sept 2022. 30

work page 2022

[8] [8]

Han and A

X. Han and A. van de Walle. A nonlocal orientation field phase-field model for misorientation- and inclination- dependent grain boundaries.arXiv preprint, 2025

work page 2025

[9] [9]

Chen and Y

L.-Q. Chen and Y. Wang. The continuum field approach to modeling microstructural430 evolution.JOM, 48(12):13–18, 1996

work page 1996

[10] [10]

Fan and L.-Q

D. Fan and L.-Q. Chen. Computer simulation of grain growth using a continuum field model.Acta Materialia, 45(2):611–622, 1997

work page 1997

[11] [11]

Moelans, B

N. Moelans, B. Blanpain, and P. Wollants. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems.435 Physical Review B, 78(2):024113, 2008

work page 2008

[12] [12]

Steinbach, F

I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G.J. Schmitz, and J.L.L. Rezende. A phase field concept for multiphase systems.Physica D: Nonlinear Phenomena, 94(3):135–147, 1996

work page 1996

[13] [13]

Steinbach and F

I. Steinbach and F. Pezzolla. A generalized field method for multiphase transformations440 using interface fields.Physica D: Nonlinear Phenomena, 134(4):385–393, 1999

work page 1999

[14] [14]

Nestler and A

B. Nestler and A. Wheeler. Anisotropic multi-phase-field model: Interfaces and junc- tions.Physical Review E, 57(3):2602–2609, 1998

work page 1998

[15] [15]

Garcke, B

H. Garcke, B. Nestler, and B. Stoth. A multiphase field concept: Numerical simula- tions of moving phase boundaries and multiple junctions.SIAM Journal on Applied445 Mathematics, 60(1):295–315, 1999. 31

work page 1999

[16] [16]

Nestler, H

B. Nestler, H. Garcke, and B. Stinner. Multicomponent alloy solidification: Phase-field modeling and simulations.Physical Review E, 71(4):041609, 2005

work page 2005

[17] [17]

Folch and M

R. Folch and M. Plapp. Quantitative phase-field modeling of two-phase growth.Physical Review E, 72(1):011602, 2005.450

work page 2005

[18] [18]

Kim, D.I

S.G. Kim, D.I. Kim, W.T. Kim, and Y.B. Park. Computer simulations of two- dimensional and three-dimensional ideal grain growth.Physical Review E, 74(6):061605, 2006

work page 2006

[19] [19]

Steinbach

I. Steinbach. Phase-field model for microstructure evolution at the mesoscopic scale. Annual Review of Materials Research, 43:89–107, 2013.455

work page 2013

[20] [20]

Kim, S.G

H. Kim, S.G. Kim, W. Dong, I. Steinbach, and B. Lee. Phase-field modeling for 3d grain growth based on a grain boundary energy database.Modelling and Simulation in Materials Science and Engineering, 22(3):034004, 2014

work page 2014

[21] [21]

Zhang, W

J. Zhang, W. Ludwig, Y. Zhang, H.H.B. Sørensen, D.J. Rowenhorst, A. Yamanaka, P.W. Voorhees, and H.F. Poulsen. Grain boundary mobilities in polycrystals.Acta460 Materialia, 191:211–220, 2020

work page 2020

[22] [22]

Gelfand and S.V

I.M. Gelfand and S.V. Fomin.Calculus of Variations. Prentice-Hall, 1963

work page 1963

[23] [23]

Williamson

J.H. Williamson. Low-storage runge-kutta schemes.Journal of Computational Physics, 35(1):48–56, 1980. 32

work page 1980

[24] [24]

Carpenter and C.A

M.H. Carpenter and C.A. Kennedy. Fourth-order 2n-storage runge-kutta schemes. Tech-465 nical Memorandum NASA/TM-109112, NASA, NASA Langley Research Center, Hamp- ton VA 23681-0001, USA, June 1994

work page 1994

[25] [25]

Karma, Z.T

A. Karma, Z.T. Trautt, and Y. Mishin. Relationship between equilibrium fluctuations and shear-coupled motion of grain boundaries.Physical Review Letters, 109:095501, 2012.470

work page 2012

[26] [26]

A. Qiu, C. Qiu, I. Chesser, J. Han, D.J. Srolovitz, and E.A. Holm. Mechanisms of shape evolution and rotation during crystal growth in cylindrical grains.Acta Materialia, 305, 2026

work page 2026

[27] [27]

Morawiec.Orientations and Rotaitons: Computations in Crystallographic Textures

A. Morawiec.Orientations and Rotaitons: Computations in Crystallographic Textures. Springer-Verlag Berlin Heidelberg, 2004.475

work page 2004

[28] [28]

Rowenhorst, A.D

D. Rowenhorst, A.D. Rollett, G.S. Rohrer, M. Groeber, M. Jackson, P.J. Konijnenberg, and M.D. Graef. Consistent representations of and conversions between 3d rotations. Modelling and Simulation in Materials Science and Engineering, 23(8):083501, 2015. Publisher: IOP Publishing

work page 2015

[29] [29]

Staublin.Orientation Field Model for Grain Growth Accounting for Five Crystallo-480 graphic Degrees of Freedom

P. Staublin.Orientation Field Model for Grain Growth Accounting for Five Crystallo-480 graphic Degrees of Freedom. PhD thesis, Northwestern University, 2024

work page 2024

[30] [30]

Mason and C.A

J.K. Mason and C.A. Schuh. Hyperspherical harmonics for the representation of crys- tallographic texture.Acta Materialia, 56(20):6141–6155, 2008

work page 2008

[31] [31]

Moelans, F

N. Moelans, F. Wendler, and B. Nestler. Comparative study of two phase-field models for grain growth.Computational Materials Science, 46(2):479–490, 2009.485 33

work page 2009