3.6. Spectral solver load definition

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Purpose

Defining the load applied to the the volume element (VE). The load file is a plain text file with default file extension *.load . A load file may contain a number of consecutive load cases; each load case corresponds to one line in the load file.

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Valid keywords

All valid keywords are given in the following table. Keywoards are case-insensitive.

Overview

Keyword	Meaning	Arguments	Comments
Fdot, dotF	deformation gradient rate ($ \dot{\bar{\tnsr F}}$)	9 real numbers or asterisks	instead of L or F; component wise exclusive with P
F	deformation gradient aim ($\bar{\tnsr F}$)	9 real numbers or asterisks	instead of L or Fdot; component wise exclusive with P
L	velocity gradient ($ \bar{\tnsr L}$)	9 real numbers or asterisks	instead of Fdot or F; component wise exclusive with P
P	Piola–Kirchhoff stress ($ \bar{\tnsr P}$)	9 real numbers or asterisks	component wise exclusive with Fdot, F, and L
t, time, delta	total time increment	1 real number
incs, N	number of increments; linear time scaling	1 integer	instead of logIncs
logIncs	number of increments; logarithmic time scaling	1 integer	instead of incs
freq, frequency, outputfreq	frequency of results output	1 integer	default value is 1, e.g. every step is written out
euler	rotation of loadcase frame by z-x-z Euler angles	keyword (optional); 3 real values	Keywords: deg, degree (default), radian; instead of rot
rot, rotation	rotation of loadcase frame by rotation matrix	9 real values	instead of euler
dropguessing, guessreset	reset guessing	None
r, restart, restartwrite	frequency of saving restart information	1 integer	default value of 0 disables writing of restart information

Deformation gradient rate (Fdot)

keyword	values
Fdot	9 real numbers or asterisks
dotF

Specifies the rate of deformation gradient evolution. See the example "Mixed Boundary Conditions" for more information about applying a deformation gradient rate in combination with stress boundary conditions.

Deformation gradient aim (F)

keyword	values
F	9 real numbers or asterisks

Specifies the deformation gradient at the end of the load case. A deformation gradient rate between initial and final deformation gradient is linearly interpolated. See the example "Mixed Boundary Conditions" for more information about applying a deformation gradient rate in combination with stress boundary conditions.

Velocity gradient (L)

keyword	values
L	9 real numbers or asterisks

Specifies the velocity gradient applying deformation to the VE. See the example "Mixed Boundary Conditions" for more information about applying a velocity gradient in combination with stress boundary conditions.

Piola–Kirchhoff stress (P)

keyword	values
P	9 real numbers or asterisks

Specifies the stress boundary conditions. See the example "Mixed Boundary Conditions" for more information about using stress boundary conditions in combination with deformation boundary conditions.

Total time increment (t)

keyword	values
t	1 real number
time

Specifies the total increment $\Delta t$ of time in seconds for the load case. Thus, the load case runs from $t_0$ to $t_0 + \Delta t$.

With linear time stepping (by keyword incs), the time at increment $n$ out of the total $N$ is given by

\[ t(n) = t_0 + \frac{n}{N} \Delta t \]

If time scaling is switched to logarithmic (by keywords logIncs) then the time of increment $n$ is given by

\[ t(n) = \begin{cases} 2^{n-N} \Delta t & \text{in the first loadcase}; \\ t_0 \left(\displaystyle\frac{t_0 + \Delta t}{t_0}\right)^{n/N} & \text{in subsequent loadcases}. \end{cases} \]

Number of increments ([log]incs)

keyword	values
[log]incs	1 integer
[log]increments
N

Specifies the number N of increments the load case is subdivided into. If prefixed by »log«, a logarithmic time scaling is used. Otherwise, linear time scaling is used. See total time increment for details on the time step calculation.

Frequency of results output (freq)

keyword	values
freq	1 integer
frequency
outputfreq

Specifies the frequency at which results are written to the output file SolverJobName.spectralOut. By default, the results of every increment are written out, e.g. freq is set to 1.

Rotate load frame (euler, rot)

The rotation of the load frame allows loading directions that are not in the direction of the periodic expansion of the VE.

z-x-z Euler angles (euler)

keyword	values
euler	optional keyword + 3 real numbers

Specifies the rotation between load frame and laboratory frame as z-x-z Euler angles. By default, or when using the keywords deg or degree, angles are given in degree; keyword radian switches to radians.

Rotation matrix (rot)

keyword	values
rot	9 real numbers
rotation

Specifies the rotation between load frame and laboratory frame as a rotation matrix $SO(3)$.

Rotation matrices requirements

A rotation matrix must be symmetric and its determinant must be 1.0 (with some allowed numerical tolerances). The following three basic rotation matrices rotate three-dimensional vectors about the $x$, $y$, and $z$ axis, respectively:

\begin{align} R_x(\theta) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin \theta & \cos \theta \\ \end{bmatrix} \\[6pt] R_y(\theta) &= \begin{bmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} \\[6pt] R_z(\theta) &= \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \end{align}

Reset guessing (dropguessing)

keyword	values
dropguessing	none
guessreset

Turns off guessing along former trajectory at start of a consecutive load case and calculation start with a homogeneous guess. By default, guessing is on for each load case except for the first one where only a homogeneous guess is possible.

Frequency of saving restart information (restart)

keyword	values
restart	1 integer
restartwrite

Specifies the frequency at which restart information is written. By default, restart information is never saved.

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Examples

Basic load case (Fdot, L, time, steps)

The most simple load case is defined by prescribing a deformation $\bar{\tnsr F}$ resulting e.g. from a constant technical deformation rate $ \dot{\bar{\tnsr F}}$, a loading time $t$ and the number of steps $n$ in which the problem should be solved.

• Fdot followed by 9 floating point values lists the deformation rate tensor components in the order 11,12,13,21,22,23,31,32,33.
• t followed by a positive floating point value specifies the total time of the load case.
• incs followed by an integer value larger than one indicates the number of steps to use.

Thus, a load case describing tension in 11 and compression in 22 direction is given by:

Fdot 1.0e-4 0.0 0.0   0.0 -1.0e-4 0.0   0.0 0.0 0.0   t 10.0   incs 10

Each increment has the duration of 1.0 second because the total deformation time of 10.0 seconds is divided into 10 increments. The resulting deformation in 11-direction is ≈ 10^-3 and in 22-direction ≈ -10^-3.

Instead of prescribing a constant technical strain rate by defining $\dot{\bar{\tnsr F}}$, it is possible to prescribe a velocity gradient $\bar{\tnsr L}$ to get a constant true strain rate. The velocity gradient is indicated by the keywords L and is also followed by 9 floating point values:

L 1.0e-4 0.0 0.0   0.0 -1.0e-4 0.0   0.0 0.0 0.0   t 10.0   incs 10

Mixed boundary conditions (P, Fdot, L)

The samples load cases above force the VE to significantly change its volume at large deformations. Except for special cases (simple shear, rotation, etc.), a load case prescribing all components of $\bar{\tnsr F}$ will lead to a non-volume preserving load. Therefore, the deformation should be undefined at at least one component of the 3x3 tensor $\bar{\tnsr F}$ and a stress must be prescribed at those components to get a unique solution.

To leave a component of the deformation undefined, use an asterisk at the corresponding position. In the following example, a deformation is described in 11 direction and the deformation in 22 direction will be adjusted to a value such that the average Piola–Kirchhoff stress (keyword P) in that direction is 0.0. All other components have 0 deformation (resulting potentially in stress).

Fdot 1.0e-4 0.0 0.0   0.0 * 0.0   0.0 0.0 0.0   P * * *   * 0.0 *   * * *   t 10.0   incs 10

Mixed boundary conditions need to fulfill the following requirements:

1. Stress and Deformation BCs are mutually exclusive.
2. The stress boundary conditions must not allow for rotation, e.g. the opposite off-diagonal elements cannot have stress components at the same time.
3. If a velocity gradient is prescribed, each row of the tensors must either contain stress or velocity gradient.

Load cases not possible due to restriction 1:

Fdot    1.0e-4 0.0 0.0   1.0 0.0 0.0   0.0 0.0 0.0   P 0.0 0.0 0.0   0.0 10.0 0.0   0.0 0.0 0.0   t 10.0    incs 40

L       1.0e-4 0.0 0.0   1.0 0.0 0.0   0.0 0.0 0.0   P 0.0 0.0 0.0   0.0 10.0 0.0   0.0 0.0 0.0   t 10.0    incs 40

Fdot     *      *   *     *   *   *     *   *   *    P  *   *   *     *    *   *     *   *   *    t 10.0    incs 40

L        *      *   *     *   *   *     *   *   *    P  *   *   *     *    *   *     *   *   *    t 10.0    incs 40

Fdot    1.0e-4  *   *     *   *   *     *   *   *    P 0.0 0.0 0.0   2.0  0.0 0.0   0.0 0.0 0.0   t 10.0    incs 40

Fdot    0.0    0.0   *   0.0 0.0 0.0   0.0 0.0 0.0   P  *   *   *    0.0   *   *     *   *   *    t 10.0    incs 40

L       0.0    0.0  0.0   *   *   *     *   *   *    P  *   *   *     *    *   *    0.0 0.0 0.0   t 10.0    incs 40

Load cases not possible due to restriction 2:

Fdot    1.0e-4  *  0.0    *  0.0 0.0   0.0 0.0 0.0   P  *  0.0  *    0.0  *   *     *   *   *    t 10.0    incs 40

L        *      *   *     *   *   *    0.0 0.0 0.0   P 0.0 0.0 0.0   0.0 0.0  *     *   *   *    t 10.0    incs 40

Fdot    1.0e-4 0.0  *    0.0 0.0 0.0    *  0.0 0.0   P  *   *  0.0    *   *   *    0.0  *   *    t 10.0    incs 40

Fdot    1.0e-4  *   *     *  0.0 0.0    *  0.0 0.0   P  *  0.0 0.0   0.0  *   *    0.0  *   *    t 10.0    incs 40

Load cases not possible due to restriction 3:

L 1.0e-4  *  0.0   0.0  *  0.0   0.0 0.0 0.0   P  *  0.0  *     *  0.0  *     *   *   *    t 10.0    incs 40

L 1.0e-4  *  0.0   0.0 0.0 0.0   0.0 0.0 0.0   P  *  0.0  *     *   *   *     *   *   *    t 10.0    incs 40

L  *     1.0  *    0.0 0.0 0.0   0.0 0.0 0.0   P 1.0  *  0.0    *   *   *     *   *   *    t 10.0    incs 40

L  *     1.0  *    0.0 0.0 0.0    *   *   *    P 1.0  *  0.0    *   *   *    0.0 0.0 0.0   t 10.0    incs 40

The following load cases do not offend any of the restrictions and are allowed:

Fdot 1.0e-4 0.0 0.0   0.0   *     0.0   0.0 0.0 0.0   P  *  * *   * 0.0 *   * * *   t 10.0    incs 40

Fdot  *     0.0 0.0   0.0 10.0e-4 0.0   0.0 0.0 0.0   P 0.0 * *   *  *  *   * * *   t 10.0    incs 40

Change of loading direction (dropguessing)

Each line in a load file specifies one load case, the load cases are subsequently applied to the VE. In the following example, uniaxial tension in 11 direction is applied at the same rate with increasing time increments.

Fdot 1.0e-4 0.0 0.0   0.0 * 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2
Fdot 1.0e-4 0.0 0.0   0.0 * 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 3.0    incs 2
Fdot 1.0e-4 0.0 0.0   0.0 * 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 5.0    incs 2

To increase the performance of the iterative scheme, the predicted deformation at the beginning of each new load case (and also during subsequent increments of the same load case) follows the rate of the last increment. However, when changing the loading direction, this strategy can lead to longer calculation times or even prevent convergence. The keyword dropguessing disables the guessing at the beginning of a new load case as shown in the following example where the deformation direction changes form the 11 to the 22 component:

Fdot 1.0e-4 0.0 0.0   0.0  *     0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2
Fdot  *     0.0 0.0   0.0 1.0e-4 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2   dropguessing

Naturally, no guessing is possible for the first increment of the first load case.

Rotation of load frame (rotation, euler)

The rotation of the load frame allows to load the VE in arbitrary directions, e.g. not following the sample x-y-z coordinate system.

Equivalent load cases (Rotation by 180°)

By rotating the VE by 180°, the sample x-y-z coordinate axes are aligned with the the lab x-y-z coordinate axes, thus the load

Fdot 1.0e-4 0.0 0.0   0.0  *     0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2

can be applied also using the following load cases (rotation of 180° around z) when using an isotropic material. For crystalline material, the orientation definition also needs to be rotated, i.e. using the keyword rotation in the texture part in material.config

Fdot  *     0.0 0.0   0.0 1.0e-4 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2   rot -1.0 0.0 0.0   0.0 -1.0 0.0   0.0 0.0 1.0

Fdot  *     0.0 0.0   0.0 1.0e-4 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2   euler 180.0 0.0 0.0

Fdot  *     0.0 0.0   0.0 1.0e-4 0.0   0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2   euler radian 3.14159265359 0.0 0.0

Rotation by 45°

The following load cases apply the same load, but for the second one load and lab frame are rotated to each other by 45°.

Fdot 1.0e-4 0.0 0.0  0.0 * 0.0  0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2
Fdot 1.0e-4 0.0 0.0  0.0 * 0.0  0.0 0.0 0.0   p * * *   * 0.0 *   * * *   t 2.0    incs 2   rot 0.70710678 -0.70710678 0.0   0.70710678 0.70710678 0.0   0.0 0.0 1.0

The shape change of the original loading direction and of the rotated one are shown in figure 1 in blue and red, respectively.