SDVPN

Bases: object

Class representation of the semidiscrete variational Peierls-Nabarro dislocation model.

Extends the GammaSurface.E_gsf_surface_plot() method to plot the disregistry path on top of it.

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates. If x or disregistry are not set/given, then the disregistry path will not be added.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values. If x or disregistry are not set/given, then the disregistry path will not be added.
fmt (str, optional) – The matplotlib.pyplot.plot fmt parameter for the disregistry path line, i.e. color, marker and line style options. Default value is ‘ro-’: red with circle markers and solid line.
normalize (bool, optional) – Flag indicating if axes are Cartesian (False, default) or normalized by a1, a2 vectors (True).
smooth (bool, optional) – If True (default), then plot shows smooth interpolated values. If False, plot shows nearest raw data values.
a1vect (array-like object, optional) – Crystal vector for the a1 vector to use for plotting. Default value of None uses the saved a1vect.
a2vect (array-like object, optional) – Crystal vector for the a2 vector to use for plotting. Default value of None uses the saved a2vect.
xvect (array-like object, optional) – Crystal vector to align with the plotting x-axis for non-normalized plots. If not given, this is taken as the Cartesian of a1vect.
length_unit (str, optional) – The unit of length to display non-normalized axes values in. Default value is ‘Å’.
energyperarea_unit (str, optional) – The unit of energy per area to display the stacking fault energies in. Default value is ‘eV/Å^2’.
numx (int, optional) – The number of plotting points to use along the x-axis. Default value is 100.
numy (int, optional) – The number of plotting points to use along the y-axis. Default value is 100.
figsize (tuple or None, optional) – The figure’s x,y dimensions. If None (default), the values are scaled such that the x,y spacings are approximately equal, and the larger of the two values is set to 10.
**kwargs (dict, optional) – Additional keywords are passed into the underlying matplotlib.pyplot.pcolormesh(). This allows control of such things like the colormap (cmap).

Returns:

The generated figure allowing users to perform additional modifications.

Return type:

matplotlib.pyplot.figure

Generates a plot of the stacking fault energy, i.e. misfit energy, associated with the disregistry values for each x coordinate.

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates. If x or disregistry are not set/given, then the disregistry path will not be added.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values. If x or disregistry are not set/given, then the disregistry path will not be added.
figsize (tuple or None, optional) – The figure’s x,y dimensions. If None (default), then a figure size of (10, 6) will be generated.
length_unit (str, optional) – The unit of length to display x coordinates in. Default value is ‘Å’.
energyperarea_unit (str, optional) – The unit of energy per area to display the stacking fault energies in. Default value is ‘eV/Å^2’.

Returns:

The generated figure allowing users to perform additional modifications.

Return type:

matplotlib.pyplot.figure

property K_tensor: ndarray

Dislocation energy coefficient tensor.

Type:: numpy.ndarray

property alpha: tuple

Coefficients for nonlocal energy correction.

Type:: tuple of float

property beta: ndarray

3x3 coefficients for gradient energy correction.

Type:: numpy.ndarray

property burgers: ndarray

Burgers vector.

Type:: numpy.ndarray

property cdiffelastic: bool

Flag indicating if elastic energy used central difference for computing the dislocation density.

Type:: bool

property cdiffstress: bool

Flag indicating if stress energy used central difference for computing the dislocation density.

Type:: bool

property cdiffsurface: bool

Flag indicating if surface energy used central difference for computing the dislocation density.

Type:: bool

Prints a summary string of all computed energy components.

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.
energyperlength_unit (str, optional) – The units of energy per length to report the dislocation line energies in. Default value is ‘eV/Å’.

property cutofflongrange: float

Cutoff distance for long-range elastic energy.

Type:: float

Computes the dislocation density as the numerical derivative of disregistry with respect to x. Uses either neighboring values

ρ[i] = (δ[i] - δ[i-1]) / (x[i] - x[i-1])

or central difference

ρ[i] = (δ[i+1] - δ[i-1]) / (x[i+1] - x[i-1])

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.
cdiff (bool, optional) – Flag indicating how to compute the derivative. A value of False (default) compares nearest values while a value of True compares next-nearest (i.e., central difference).

Returns:

newx (numpy.array) – The x positions corresponding to the dislocation density values.
rho (numpy.array) – The computed dislocation density.

property disregistry: ndarray

The disregistry vector for each x coordinate.

Type:: numpy.ndarray

Creates a simple matplotlib figure showing the disregistry profiles.

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.
figsize (tuple, optional) – matplotlib figure figsize parameter. Default value is (10, 6).
length_unit (str, optional) – The unit of length to display positions and disregistry values in. Default value is ‘Å’.

Returns:

The generated figure allowing users to perform additional modifications.

Return type:

matplotlib.pyplot.figure

Computes the short-range configuration-dependent elastic energy term for the dislocation based on the dislocation density and K_tensor.

E_elastic = 1/(4π) Σ_i Σ_j χ(i,j,Δx) K_lm ρ_l[i] ρ_m[j]

χ(i,j,Δx) = (3/2) Δx² + ψ(i-1,j-1,Δx) + ψ(i,j,Δx) - ψ(i,j-1,Δx) - ψ(j,i-1,Δx)

ψ(i,j,Δx) = (1/2) (i-j)² Δx² ln(|i-j|Δx)

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The elastic energy for the dislocation.

Return type:

float

property fullstress: bool

Flag indicating which stress algorithm was used.

Type:: bool

property gamma: GammaSurface

The stacking fault map.

Type:: atomman.defect.GammaSurface

load(model: str | IOBase | DataModelDict, gamma: GammaSurface | None = None)

Load solution from a data model.

Parameters:

model (str, file-like object or DataModelDict) – The semi-discrete-Peierls-Nabarro data model to load.
gamma (atomman.defect.GammaSurface, optional) – The gamma surface to use. If not given, will check to see if the content is inside model.

Raises:

ValueError – If the gamma surface information is not given and it is not found in model.

longrange_energy() → float

Computes the long-range elastic energy term for the dislocation using the K_tensor, Burgers vector and long-range cutoff. This term is configuration-independent and thus the method takes no parameters.

E_longrange = 1/(2π) K_lm b_l b_m ln(L)

Returns:: The long-range energy for the dislocation.
Return type:: float

property min_kwargs: dict

scipy.optimize.minimize keywords used.

Type:: dict

property min_method: str

scipy.optimize.minimize method used.

Type:: str

property min_options: dict

scipy.optimize.minimize options used.

Type:: dict

Computes the misfit energy for the disregistry using the stored gamma surface

E_misfit = Σ γ(δ)Δx

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The misfit energy for the dislocation.

Return type:

float

model(length_unit: str = 'Å', energyperarea_unit: str = 'eV/Å^2', pressure_unit: str = 'GPa', include_gamma: bool = False) → DataModelDict

Generate a data model for the object.

Parameters:

length_unit (str, optional) – The unit of length to save values as. Default is ‘Å’.
energyperarea_unit (str, optional) – The unit of energy per area to save fault energy values as. Only used if the gamma surface information is included. Default value is ‘eV/Å^2’.
pressure_unit (str, optional) – The unit of pressure to save values as. Default is ‘GPa’.
include_gamma (bool, optional) – Flag indicating if the gamma surface data is to be included. Default value is False.

Returns:

The data model containing all input parameters and the current disregistry vectors.

Return type:

DataModelDict

Computes the nonlocal energy correction using alpha coefficient(s).

E_nonlocal = Σ_m α_m Σ_i δ[i] (δ[i] - (δ[i+m] + δ[i-m]) / 2) Δx

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The nonlocal energy for the dislocation.

Return type:

float

property res: OptimizeResult

scipy.optimize.minimize result.

Type:: OptimizeResult

Solves the semidiscrete variational Peierls-Nabarro dislocation disregistry through energy minimization using the set class properties. All parameters are optional keyword arguments that can be used to change any of the previous settings.

Parameters:

x (numpy.ndarray, optional) – An array of shape (N) giving the x coordinates corresponding to the disregistry solution.
disregistry (numpy.ndarray, optional) – A (N,3) array giving the initial disregistry vector guess at each x coordinate.
tau (numpy.ndarray, optional) – A (3,3) array giving the stress tensor to apply to the system using the stress energy term. Only the xy, yy, and yz components are used.
alpha (list of float, optional) – The alpha coefficient(s) used by the nonlocal energy term.
beta (numpy.ndarray, optional) – The (3,3) array of beta coefficient(s) used by the surface energy term.
cutofflongrange (float, optional) – The cutoff distance to use for computing the long-range energy.
fullstress (bool, optional) – Flag indicating which stress energy algorithm to use.
cdiffelastic (bool, optional) – Flag indicating if the dislocation density for the elastic energy component is computed with central difference (True) or simply neighboring values (False).
cdiffsurface (bool, optional) – Flag indicating if the dislocation density for the surface energy component is computed with central difference (True) or simply neighboring values (False).
cdiffstress (bool, optional) – Flag indicating if the dislocation density for the stress energy component is computed with central difference (True) or simply neighboring values (False). Only matters if fullstress is True.
min_method (str, optional) – The scipy.optimize.minimize method to use.
min_kwargs (dict, optional) – Any keyword arguments to pass on to scipy.optimize.minimize besides the coordinates, method and options.
min_options (dict, optional) – Any options to pass on to scipy.optimize.minimize.

Computes the stress energy due to the applied stress, tau. If fullstress is True, the original stress expression by Bulatov and Kaxiras will be used:

E_stress = -1/2 Σ_i (x[i]² - x[i-1]²) ρ_l τ_2l

If fullstress is False, the alternate stress expression by Shen and Cheng 10.1016/j.scriptamat.2009.04.047 will be used:

E_stress = -1/2 Σ_i τ_2l (δ_l[i] + δ_l[i+1]) Δx

Note that the Shen and Cheng expression will have a constant error associated with it giving an incorrect overall energy, but should apply a similar force on the dislocation.

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The stress energy for the dislocation.

Return type:

float

Computes the gradient surface energy correction using beta coefficients.

E_surface = Σ_j β_lj / 4 Σ_i ρ_l[i]² Δx

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The surface energy for the dislocation.

Return type:

float

property tau: ndarray

The applied 3x3 stress tensor.

Type:: numpy.ndarray

Computes the total energy for the dislocation.

E_total = E_elastic + E_misfit + E_stress + E_surface + E_nonlocal

Parameters:

x (array-like object, optional) – x-coordinates. Default value is the stored x-coordinates.
disregistry (array-like object, optional) – (N, 3) shaped array of disregistry vectors at each x-coordinate. Default value is the stored disregistry values.

Returns:

The total energy for the dislocation.

Return type:

float

property transform: ndarray

Transformation matrix from standard crystal setting to dislocation solution setting.

Type:: numpy.ndarray

property x: ndarray

The x coordinates.

Type:: numpy.ndarray

SDVPN

Table of Contents

Previous topic

Next topic