Multi-phase-field simulations with OpenPhase

Multi-phase-field simulations
with OpenPhase
Marvin Tegeler, Johannes Görler, Alexander Monas, Oleg Shchyglo and Ingo Steinbach
Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, Bochum, Germany

2
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Physics-based microstructure
simulation framework.
• 100 man-years of research
and development since 2009
• Code is based on the multi-
phase field multi-component
PF-model
• Object oriented code written
in C++
• Modular structure allowing for
easy extension by addition of
new modules

3
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Simulation of coarsening and
ripening.
• Minimization of interfacial
energy leads to a realistic
grains size distribution:
• Examples:
• Tempering of steels.
• Coarsening of carbides.

4
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Simulation of multi-
component diffusion fluxes.
• Data from thermodynamic
and kinetic databases.
Supported interfaces to
.
• Examples:
• Segregation profiles
after solidification.
• Diffusion controlled
transformations.
g

5
OpenPhase Toolbox
g
• Simulation of deformations
up to several hundred
percents.
• Examples:
• Hot rolling of steels.
• Virtual tensile tests.
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow

6
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Simulation of plastic
deformation.
• Reduction of elastic stresses
by plastic deformation. Full
implementation of a crystal
plasticity model.
• Examples:
• Creep deformation in Ni-
base superalloys
• Martensite in metals.
g

7
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Simulation of the effect of
mesoscopic damage in the
microstructure.
• Reduction of stiffness and
yield strength depending on
plastic strain.
• Examples:
• Virtual tensile tests.
• Simulation of creep
deformation.

8
Coarsening
Diffusion
Large
Deformation
Plasticity
Damage
Flow
OpenPhase Toolbox
• Simulation of liquid flow.
• Solving Lattice-Boltzmann
method under consideration
of solid grains.
g

9
Introduction to the Multiphase Field Method
The Multiphase Field Method
 A tool for the simulation of
microstructure evolution
 A set of continuous scalar functions
 1 in bulk, 0 when phase does not exist.
 Description of regions of different
properties (crystallographic structure,
orientations, etc)
 Motion of interfaces between regions

10
Introduction to the Multiphase Field Method

 compact 27-point stencil
 N = Number of active phase fields in a
grid point
 Active phase fields or

11
Overview
• Mg-Al alloy solidification
• OpenPhase Tutorial
• Parallelization

12
Mg-Al Microstructure and Thermodynamics
 Mg-Al alloys microstructure consist
of α-phase (HCP-Mg dendrites)
surrounded by closed shell Mg17Al12
β-phase
 properties as corrosion resistance
depend on the microstructure
Goal:
 Modeling as cast microstructures
using the phase field model
200µm

13
Experimental as-cast Mg-Al Microstructure
0.3 K/s 1.0 K/s 25 K/s
1 mm
(a): furnace control (b): air cooling (c): water cooling

14
Nucleation Modeling
 Create a list of virtual particles, that satisfies a given
nucleation density
 At every time step plant nuclei by creating a phase field
with value zero at the corresponding grid point, if the
parent phase is correct
 Compute driving force , if remove
the corresponding virtual particle from the list, otherwise
remove phase field.

15
Initial temperature: 875 K
Cooling rate: 10 K/s
Initial concentration: 5 at. % Al
System size: 200x200x200 μm
Grid spacing: 1.0 um
Solidification of MgAl

16
Temperature Curves Simulation

17
Effect of cooling rate on solidification microstructure
5 K/s
15 K/s
25 K/s
200µm
Mg-5at.%Al
α-Mg
β-Mg

18
α- and β-phase nucleation during solidification
α-Mg
β-Mg
 Primary α-phase dendrites fill
most of the volume.
 Secondary β-phase fills
interdendritic region.

19
Evolution of microstructure during solidification
α-Mg
β-Mg
100µm
300³ µm = 27 mio cells
one week calculation
time on 96 cores

20
100µm
Evolution of microstructure during solidification
100µm

21
3D Secondary β-phase Nucleation and Growth Modeling
α
β
100µm

22
3D Secondary β-phase Nucleation and Growth Modeling
20µm

23
α
α
β
20µm
Rapid coverage of α-liquid-interface with β-phase
 β-phase covers the α-liquid-interface
 Further β-phase growth depletes enclosed melt
 Nucleation of tertiary α-phase is neccessary for complete
solidification
Monas A et al.; “Divorced Eutectic Solidification of Mg-Al Alloys”, JOM (2015)

24
Mg-Al β-phase formation: experiment vs simulation
100µm 5µm
SEM-image of Mg-5%Al microstructure and enlarged view on the eutectic region.
dark: α-phase, bright: β-phase
Se-Jong Kim, Chang Dong Yim, KIMS, Korea

25
Tutorial
OpenPhase can be downloaded at
http://www.openphase.de/upload/content/OpenPhase.V0.9.1.zip
The tutorial will use the SolidificationMgAl example.
In order to compile the OpenPhase library just run 'make' in the main directory.
To compile an example call ‘make’ in the corresponding directory.

26
Load Modules
int main(int argc, char *argv[])
{
Settings OPSettings;
OPSettings.Initialize();
OPSettings.ReadInput();
PhaseField Phi(OPSettings);
DoubleObstacle DO(OPSettings);
InterfaceMobility Mu(OPSttings);
InterfaceEnergy Sigma(OPSettings);
InterfaceEnergySolidLiquid IE(OPSettings);
EquilibriumPartitionDiffusion DF(OPSettings);
Composition Cx(OPSettings);
Temperature Tx(OPSettings);
DrivingForce dG(OPSettings);
BoundaryConditions BC(OPSettings);
Nucleation Nuc(OPSettings);

28
Load Modules
dG.ReadInput();
Cx.ReadInput();
DF.ReadInput();
Tx.ReadInput();
BC.ReadInput();
Nuc.ReadInput();
Mu.ReadInput();
Sigma.ReadInput();
IE.ReadInput();
int id = Initializations::Single(Phi, 0, BC, OPSettings);
Phi.FieldsStatistics[id].State= Liquid;
DF.SetInitialComposition(Phi, Cx);
Tx.SetInitial(BC, Phi, 0);

30
Work Loop
for(int tStep = OPSettings.tStart + 1; tStep <
OPSettings.nSteps + 1; tStep++)
{
Nuc.GenerateNucleationSites(Phi, Tx);
Nuc.PlantNuclei(Phi, tStep);
Sigma.CalculateHex(Phi);
Mu.CalculateHex(Phi, Tx);
DF.SetPhaseFractions(Phi);
DO.CalculatePhaseFieldIncrements(Phi, Sigma, Mu);
DF.GetDrivingForce(Phi, Cx, Tx, dG);
dG.Average(Phi, BC);
Nuc.CheckNuclei(Phi, Sigma, dG, tStep);
dG.MergePhaseFieldIncrements(Phi, Sigma, Mu);
Phi.NormalizeIncrements(BC, dt);
DF.Solve(Phi, Cx, Tx, BC, dt);
Phi.MergeIncrements(BC, dt);
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}

31
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Generates nucleation sites with a random size distribution the
first time this function is called.

32
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Plant nuclei at the generated sites.
Add storage for a phase field with value zero.

33
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Calculate interface stiffness

34
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Calculate interface mobility

35
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Calculate the sum of phase fields belonging to the same thermodynamic
phase in each grid point.

36
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Calculate the double obstacle potential for αβ-interfaces

37
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Calculate driving force

38
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
The correct phase field profile across the interface can only be obtained if the
driving force is constant across the interface and varies only along the
interface.
Therefore we calculate an average driving force , that is constant along
the normal direction of the interface.

39
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt)}
The correct phase field profile across the interface can only be obtained if the
driving force is constant across the interface and varies only along the
interface.
Therefore we calculate an average driving force , that is constant along
the normal direction of the interface.
The average is calculated in two steps
with

40
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Check driving force of nuclei. Sucessfully planted nuclei are removed from
the list of nucleation sites.

41
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Compute interface fields for the chemical driving force

42
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Limit interface fields , so that the phase fields stay within bounds [0,1].

43
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Solve diffusion equation for the phase concentrations

44
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Compute the phase field update

45
Work Loop
{
Tx.Set(BC, Phi, 6.2e8, 1.773e6, 0, dt);
}
Compute temperature update

46
OpenPhase Solutions GmbH
• A company has been founded that offers support
• Focused on industry
• GUI is in development
• Creation of ProjectInput.opi
• Creation of the .cpp file
• Sanity check on input values
• GUI will be commercial software
• OpenPhase will stay free
Johannes Görler
Matthias Stratmann

47
Parallelization
MPI-Parallelization

48
Parallelization
Thread (OpenMP)
 Simultaneous execution of
sequences of instruction
 Shared address space
Process (MPI)
 Instance of a program
 Own address space
 Message Passing
All cores should be
used efficiently

49
Parallelization
 Active phase fields or
 Dynamic storage for active phase field
 Contributions to the evolution equation only by
active phase fields
 Locally and temporally different number of active
phase fields
 Number of operations in each grid point can be
different and can change over time.

50
Parallelization
• OpenPhase uses stencil operations
• Domain decomposition to divide the work onto multiple processes

51
Parallelization
Use a domain decomposition to divide the work onto multiple processes.

52
Parallelization
The wide halo avoids communication at the expense of additional work.
This is used to limit us to one communication step per time step.

53
Load-balancing by Graph-partitioning
Over-decomposition
 Division of the domain into many more sub-domains
(blocks) than processes.
 Assignment of these sub-domains to processes
 Goal: minimize communication and load-imbalance.

54
Assignment of sub-
domains to processes.
 Simultaneous
minimization of imbalance
and communication
 Graph-partitioning
problem with vertex
weights
 Coloring of the graph with
minimal edge cut
 Constrainted sum of the
vertex weights with the
same color
Block ↔ Vertex
Communication ↔ Edge
Load ↔ Weight
Problem:
No accurate current polynomial algorithm

55
Greedy Graph-Partitioner
1. Start with color n = 1
2. Choose an uncolored vertex that
increases the edge cut the least
3. Color it with color n
4. If the sum of vertex weights with
color n is larger than set
n = n+1
5. If there are any uncolored vertices,
go to 2

56
Greedy Graph-Partitioner
1. Start with color n = 1
2. Choose an uncolored vertex that
increases the edge cut the least
3. Color it with color n
4. If the sum of vertex weights with
color n is larger than set
n = n+1
5. If there are any uncolored vertices,
go to 2

57
Graph Partitioning
 The Greedy Graph-Partitioner fulfills
 Because the last vertex added to each color fulfills
 Total work on process i.
 Average work .
 Maximum work among blocks.
Reducing limits the maximum load-imbalance.
This can be achieved by splitting blocks into smaller blocks.

58
Splitting Blocks, however, creates additional work
After splitting the most costly block the inequality becomes

59
Repeated splitting of the largest block gives
Allowing the same block to be split multiple times gives
Problem: The cost of splitting is unknown.

60
Problem: The cost of splitting is unknown.
 Self-Profiling in order to approximate the cost.
 Measure time for computations on each block.
 Fit parameters in a cost function
 that maps the number of phase fields in
each grid point to the computation time.
 This information is also used to determine the
optimal cutting plane.

61

62

63
Load-imbalance
Tolerance
Maximum Block
Maximum Process
Wall Time

64
Load-imbalance
Tolerance
Maximum Block
Maximum Process
Wall Time

65
Weak-Scaling Benchmark
Pure MPI
 96x48x48 grid
points per
core.
 2 particles per
core
 Gaussian
distribution

66
Weak-Scaling Benchmark
Hybrid
MPI+OpenMP
 96x48x48 grid
points per
core.
 2 particles per
core
 Gaussian
distribution

67
Hybrid-Parallelism reduces the
number of MPI-processes.
The number of blocks is reduced,
the blocks become larger.
Less overhead is created by the
wide halo.
A large number of threads loses
efficiency.
Only one thread per process
participates in communication.
Threads created on different sockets
can have slow memory access.
The best performance was seen with 6 or 12 threads per process.

68
Load-balancing using Phase Field
Load-balancing aims at simulatenous
reduction of idle time and communication.
Sub-domains need to
have the same
amount of work.
Surface between
sub-domains should
be minimized.
Idle time Communication
With an appropriate driving force the
multiphase field method can achieve this.

69
 Domain decompostion with one sub-domain per process .
 Each process is associated with a phase field .
 The associated sub-domain is determined by the phase field .
 Balancing of the work between adjacent domains and by a driving force
that drives the interface into the direction of the higher computational load.
 An isotropic interface energy reduces the surface area between domains.

70
Application to the Phase Field Method
 Two instances of phase field
 Simulation
 Load-balancing
 Using Wide Halo

71
 Simulation
 Load-balancing
 Using Wide Halo
 Both on the same rectangular sub-
domain
 Interior points of the sub-domains can
overlap.

72
Comparison with the previous load-
balancing scheme.

73
Convergence of load-balance
 96 MPI processes
 20% Tolerance

74
Convergence of load-balance
 96 MPI processes
 5% Tolerance

75
 Simulation
 Load-balancing
 Using Wide Halo
 Both on the same rectangular sub-
domain , in which is the majority
phase.
 Interior points of the sub-domains can
overlap.

76
Applied to Molecular Dynamics
 Cell-based molecular dynamics
 Lennard-Jones potential
 Start with a simple domain decomposition
 Colors indicate processes
 The phase fields are only updated with a probability of
With the maximum load among processes 𝜔 𝑚𝑎𝑥, the number of processes 𝑁 𝑝 and the
average time for load-balancing 𝑇𝐿𝐵.

77

78
Two Spheres Move Over Periodic Boundary

79

80
Summary
 OpenPhase is a flexible multi-phase-field framework,
that can handle an arbitrary number of phase fields
for a variety of applications
 MPI and OpenMP parallelization allow an efficient
usage of computational resources.

81
Summary
 OpenPhase is a flexible multi-phase-field framework,
that can handle an arbitrary number of phase fields
for a variety of applications
 MPI and OpenMP parallelization allow an efficient
usage of computational resources.
Thank you!

Multi-phase-field simulations with OpenPhase

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