Model¶

In this section the basic models have to be specified. I.e. each model contributes an equation or a set of equations to be solved.

Type of coupling¶

If concentration is selected, the phase-field will be coupled to concentration-fields, and solute partitioning and solute diffusion is simulated. This option is the most common for the analysis of alloys. If temperature is selected instead, a temperature field will be coupled, and local latent heat release and heat transfer are calculated.

Important: concentration coupling should not be applied for the simulation of pure metal solidification!

Example: Only phase-field model

...
# Type of coupling?
# Options:  phase  concentration [volume_change] temperature  temp_cyl_coord
#    [stress] [flow] [flow_coarse] [dislocation]
phase
...

In addition to the basic coupling options, several optional parameters can be specified: The optional parameter volume_change can only be used in combination with concentration and allows considering the volume change due to phase-transformation or temperature change. In order to obtain a homogeneous expansion of the whole simulation domain, local volume changes are compensated by internal material transport which is assumed to be sufficiently fast that the system can always relax to a stress-free state.

The comprehensive calculation of an elastic stress field can be activated by the optional keyword stress. This option requires a license for the MICRESS® elastic module.

The dislocation parameter activates an alternative recrystallization (RX) model based on a local dislocation density field. The dislocation field, which may for example stem from a Crystal Plasticity-Finite Element analysis (CP-FEM), can be read from a specific VTK file.

For phase or concentration simulations the optional flow or flow_coarse parameter can be given to enable the flow module, which allows simulation of fluid flow for phase 0. For the flow_coarse option an additional coarsening factor must be provided. An in depth description of the flow module and its usage is found under the flow solver topic.

Example: Enabling flow on a coarse grid

...
# Model
# =====
# Type of coupling?
# Options:  phase  concentration [volume_change] temperature  temp_cyl_coord
#    [stress] [flow] [flow_coarse] [dislocation]
concentration flow_coarse
# Grid coarsening factor ? (integer,even)
2
...

Thermal Conditions¶

Latent heat describes the amount of energy released or absorbed during phase transitions. The latent heat option should be applied to phase transformation processes where latent heat is dominating the temperature trend. With this option, latent heat is released averaged over the calculation domain. This is ideally the case for small samples where temperature gradients are negligible (DTA approximation). MICRESS® also allows the use of latent heat in connection with thermal gradients, but in such cases rather the use of the 1d_temp option is recommended.

In this input section, latent heat can be switched on or off by choosing one of the keywords no_lat_heat, lat_heat and no_lat_heat_dsc. Alternatively, the 1d_temp option can be selected.

Optionally, in the same line with the selected option, the keyword rotate can be used to indicate that the direction of the temperature gradients is rotated with respect to the z-axis. Otherwise, per default a T-gradient in z-direction is assumed. If rotation was activated, the user is asked to specify the location of the reference point, which constitutes the center of rotation of the T-gradient direction, as well as the direction vector of the T-gradient. The reference point is given by three coordinate numbers (x_coord, y_coord, z-coord) with respect to the grid of the microstructure domain, while the direction vector of the T-gradient is specified as a not normalized vector (e.g. 0.0 0.0 1.0 for no rotation). Please note that for both inputs always 3 coordinates must be specified in a line (even in 2D-simulations). Using a rotated temperature field has implications for the choice of Boundary Conditions as well as for the definition of diffusion segments (Diffusion).

Important: The rotation of the direction of the temperature gradient is limited to moderate angles (<~60°)

If latent heat is used, the enthalpy of each phase (including the phase 0) as well as its specific thermal capacity need to be specified when giving the phase properties (see Phases). It has further implications for the definition of the Boundary Conditions, Process Conditions, and Numerical Parameters.

If latent heat is included in the simulation, an extra output file with extension dTLat is written which contains the amounts of latent heat, enthalpy data etc.

If no_lat_heat is specified, no latent heat is taken into account. Instead, the temperature trend must be specified explicitly in section Process Conditions. no_lat_heat_dsc is identical to no_lat_heat, but triggers the output of latent heat data (enthalpies, specific heat, heat flow) in the dTLat file. This permits conduction of virtual DSC experiments displaying the heat flow for a specified temperature-time curve.

1d_temp¶

If concentration coupling is activated, an explicit 1D temperature field in z-direction can optionally be defined by the keyword 1d_temp. This field can be used as an improved thermal boundary condition for the microsimulation domain and thus replaces the normal definition of thermal boundary conditions (see section Process Conditions. With the 1D temperature field, heat flow and release of latent heat are solved explicitly. For numerical reasons, temperature in the 1D temperature field is calculated using a direct explicit solver with a default kinetic coefficient (Boettger J. Comput Phys 2009). The default value is preconfigured for typical casting processes and needs not to be changed in most cases. A different value can specified in section Numerical Parameters.

The temperature field will be evaluated in a linear (cartesian), cylindrical (cylindrical), or polar (polar) coordinate system. All of these options require further inputs for the field size (number of cells) and the field resolution (cell width in micrometer) as well as further input of thermal diffusivities (see process conditions: Thermo-physical properties for 1D temperature solver) and boundary conditions. The total size of the temperature field must be equal or larger than the height of the microstructure simulation domain (in z-direction). In case of a rotated direction of the temperature field, the projection of the 1D temperature field onto the simulation domain must cover the whole domain (depending on gradient direction and selection of reference point). As temperature is solved explicitly in the 1D temperature field, the effects of latent heat are included automatically. Next, the user has to specify the relative position of the bottom of the 1d-temperature field relative to the bottom of the microstructure domain. The 1d-temperature field must be positioned such that the 2D/3D simulation domain is located completely inside this field. Finally, in case a cylindrical (cylindrical) or polar (polar) coordinate system has been specified, the position of the center (pole point) of the coordinate system, which has to be above the 1d-temperature field, has to be specified as distance from the top of the 1d-temperature field (>0, in micrometer).

Important: the initial position of the bottom of the 1d temperature field is always negative with respect to the micro-domain.

Example: Type of coupling: concentration with 1D_temp_field

...
# Model
# =====
# Type of coupling?
# Options:  phase  concentration [volume_change] temperature  temp_cyl_coord
#    [stress] [flow] [flow_coarse] [dislocation]
concentration
# Thermal Conditions
# Options: no_lat_heat no_lat_heat_dsc lat_heat 1d_temp [rotate]
1d_temp rotate
# Reference point of simulation domain for alignment of temperature field
#   (3 integers in a line: x_coord, y_coord, z_coord)
150 1 1175
# Orientation of 1d-temperature field
#  ( direction (3 real values), default = "0.0 0.0 1.0" )
-0.839 -0.839 1.000
# Kind of coordinate system for the 1D temperature field?
# Options:  cartesian  cylindrical  polar
cartesian
# Number of cells for 1D temperature field?
500
# Cell dimension (in micrometer) for 1D temperature field ?
100.00
# Initial position (in micrometer) of botom of the 1D temperature field ?
# (distance to reference point of the simulation domain, <0!)
-500.00
...

Further settings for type of coupling stress¶

If the type of coupling stress is activated, it should be specified which type of mechanical contribution to evolution of phase- and concentration-field is involved. Option none indicates that no elastic contributions to the driving forces are calculated, and the solution of the mechanical problem is only required for the stress output. By choosing chemo_elastic, the elastic contributions are added to the phase-field driving force. By option mechano_chemical, the effects of composition on the eigenstrains are also included. Therefore, more complex interactions between stresses, diffusion and phase equilibria are analysed, which can be important in high alloyed multicomponenet systems and close to spinoidal points. For more theoretical details please refer to Böttger2020¹.

For homogenisation of stresses and strains inside the diffuse interface regions, three different models are available: Voigt-Taylor (voigt_taylor), Reuss-Sachs (reuss_sachs) and Khachaturyan (Khachaturyan). For theoretical details for these interface homgenization models please refer to Ammar2009².

If stress coupling is activated, and the dimensionality of the simulation is 2D, additional input for the 2D-abstraction of the stress/strain is required. With the plain_stress approximation, stress components perpendicular to the $xz$-plane are negligible small and assumed to be zero. This is typically the case in thin plates. However, in MICRESS® the typical 2D-assumption is that we consider a plane inside a (infinitely) thick sample. Then, the plane_strain approximation, which assumes the strains in y-direction to be constant, is more realistic.

Example: Concentration and stress coupling

...
# Model
# =====
# Type of coupling?
# Options:  phase  concentration [volume_change] temperature  temp_cyl_coord
#    [stress] [flow] [flow_coarse] [dislocation]
concentration stress
# Thermal Conditions
# Options: no_lat_heat no_lat_heat_dsc lat_heat 1d_temp [kin. Coeff.]
# kin. Coeff: Kinetics of latent heat release (default is 0.01)
no_lat_heat
# Type of mechanical contribution?
# Options: none chemo_elastic mechano_chemical
none
# Mechanical interface homgenization model?
# Options: voigt_taylor  reuss_sachs  khachaturyan
voigt_taylor
# Assumption for elastic 2D model
# Options:  plane_stress  plane_strain
plane_strain
...