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FE-modeling of ideal grain growth based onpreprocessed EBSD data

 ARTICLE  in  PAMM · DECEMBER 2011

DOI: 10.1002/pamm.201110227

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Tobias Kayser

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PAMM · Proc. Appl. Math. Mech.  11, 471 – 472 (2011) /  DOI 10.1002/pamm.201110227

FE-modeling of ideal grain growth based on preprocessed EBSD data

Slawa Gladkov1,∗, Tobias Kayser2, and Bob Svendsen1

1 RWTH Aachen, Chair of Material Mechanics, 52062 Aachen, Germany2 TU Dortmund, Institute of Mechanics, 44227 Dortmund, Germany

The purpose of this work is to show the use of experimentally measured micrograph data in the context of ideal grain growth

simulation which is modeled with a help of finite element method. In this regard the micrograph data is considered as initial

condition for a grain growth simulation. General remarks on preparation (preprocessing) of the micrographs and staggered

algorithmic formulation are presented.

c 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

There is a number of models available to simulate an ideal (normal) grain growth of polycrystals – both on continuum anddiscrete side. In this work the continuum phase field model developed in [4] in conjunction with numeric finite element

method is used to perform simulation of a grain growth process. To initiate the simulation there are few options available for

the choice of initial conditions: (i) random data which in a sense simulates the initially liquid phase, (ii) artificially generated

microstructures (in 2D and 3D) and (iii) real, measured preprocessed EBSD data (2D only). In this work the last option is

chosen.

The data one gets directly from the microscope (Fig. 1, left) cannot be used in the simulation because it will lead to

tremendous computational effort in the initial simulation steps due to a fact that practically each point in the micrograph have

to be associated with a certain grain orientation. This can be avoided using preprocessing stage [1, 2] where among others the

symmetries of a crystal are taken into account and grain structure is clearly visible (Fig. 1, right). Exactly this preprocessed

data is used as initial condition for the finite element simulation explained in the next sections.

Fig. 1   Steps for polishing up the micrograph data from microscop to be ready for use in finite element simulation.

2 Grain growth model

Based on the M. Gurtin’s microforce balance [3] approach one can formulatate the most general form of evolutionary partial

differential equations for the phase fields which is consistent with the second law of thermodynamics:

 pβ=1

[B]αβ φ̇β  = ∇ ·  ∂ 

∂ ∇φαψ̂ −

  ∂ 

∂φαψ̂ +̟ α   ∀α = 1, 2,...,p   (1)

with certain initial and boundary (homogeneous Neumann in this case) conditions:

φα(x, 0) =  φ0α(x),   ∇φα(x, t) · n = 0   on ∂ Ω,   ∀α = 1, 2, ...,p.   (2)

Here {φk(x, t)} pk=1

 are p  phase fields,  ψ̂({φk} pk=1

, {∇φk} pk=1

)  is a free energy functional,  {̟ k} pk=1

 are so called external

microforces, [B]({φk} pk=1, {∇φk} pk=1

, { φ̇k} pk=1) is a symmetric positive semi-definite matrix of constitutive moduli, Ω is thedomain of problem definition with its boundary  ∂ Ω = Ω \ Ω and n is a unit outward normal to  ∂ Ω.

∗ Corresponding author: e-mail svyatoslav.gladkov(AT)rwth-aachen.de, phone +49 241 80 25009

c 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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472 Section 7: Coupled problems

The particular model can be given by the proper specification of the free energy functional and the matrix of constitutive

moduli. Firstly, lets assume that the matrix of constitutive moduli is diagonal and its elements do not depend on solution (they

are constants):

[B]αβ {φk} pk=1, {∇φk} pk=1, { φ̇k} pk=1 ≡  [B]αβ  = δ αβ1

Lα

(3)

where Lα  > 0  ∀α  = 1, 2,...,p are positive constants and δ αβ  is a Kronecker’s delta operator. Secondly, assume that there isno external microforce, i.e.

̟ α = 0   ∀α = 1, 2, ...,p.   (4)

Finally, assume the following form of the free energy:

ψ̂ ({φk} pk=1

, {∇φk} pk=1

) =

 pα=1

−C 1

2  φ2α +

 C 2

4  φ4α

+ C 3

 pα=1

 pβ=1

(1 − δ αβ)φ2αφ

2β +

 pα=1

κα

2  ∇φα · ∇φα   (5)

where C 1  >  0, C 2  >  0, C 3  >  1

2C 2  are positive constants and {κk}

 pk=1

 >  0  are positive gradient energy coefficients.

Plugging all these into the kinetic equations (1) one recovers the grain growth model from [4], i.e.  ∀α = 1, 2,...,p

φ̇α =  Lα

κα∇ · ∇φα + C 1φα − C 2φ3α − 4C 3φα

 pβ=1

(1 − δ αβ)φ2β

.   (6)

This is a system of  p  coupled quasilinear partial differential equations of second order in space and first order in time.

3 Finite element formulation

There are two possibilities to discretize the problem (6) in space: monolithic and staggered.  Monolithic approach means that

the problem is being solved in the fully coupled setting. This leads to a definition in each finite element node p  degrees of 

freedom, what in the end will lead to enormous system matrix size and make problem numerically non-treatable. On contrary

staggered  scheme means that the problem is being solved in decoupled fashion, i.e. equation by equation holding all the otherfields constant (from the previous time step or equation step). This leads to definition of only p scalar finite element fields and

sequential update of them. This scheme is employed here.

Explicit time discretization for a problem (6) leads to very small time steps. Fully implicit scheme leads to very long

computation times due to re-assembly of the system right hand side vector on each step of the Newton-Raphson procedure

and for each phase field (staggered scheme is employed). Thus semi-implicit scheme is used here. In this case only the linear

part (the "mass" matrix and discrete Laplacian which are constant for all time steps and all phase fields) is taken on the new

time step while the local nonlinear part is taken from the previous time step. This formulation allows firstly to build up system

matrix only once and secondly to avoid Newton-Raphson iterations.

Discretized in space and time the weak form of equations (6) will take a form:

1

∆t Ω φ

∗

φ

t+1

α   dx + 2 Ω ∇φ

∗

· ∇φt+1

α   dx = Ω φ

∗

φα 1

∆t  + 1 + 3φ

2

α − 4

 p

β=1

φ

2

β dx   (7)

where now φt+1α   are unknown solutions on the new time step, φα  are known from the previous time step,  ∆t is the time stepand to simplify the notation it was assumed that  C 1 =  C 2 =  C 3 = 1 and Lα = 1, κα = 2 ∀α = 1, 2,...,p.

Variational equation (7) is solved using Bubnov-Galerkin method where as a basis functions bi-linear finite elements are

taken. Numerical results obtained using the open source finite element library  deal.II  [5] will be reported in the future

works.

References

[1] F. J. Humphreys, J. Mater. Sci., 36, 3833–3854 (2001).

[2] F. J. Humphreys, J. Microsc., 213, 247–256 (2004).[3] M. E. Gurtin, Physica D, 92(3-4), 178–192 (1996).

[4] D. Fan and L.-Q. Chen, Acta. Mater. 45, 611–622 (1997).

[5] W. Bangerth, R. Hartmann and G. Kanschat, ACM Trans. Math. Soft. 33, 24/1–24/27 (2007).

c 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim   www.gamm-proceedings.com