mesh_module stress_average

Purpose

Computes time averaged stresses for each mesh element and a global value for the whole mesh

Syntax

mesh_module stress_average keyword value

Keywords:

Keyword	Description
id	obligatory, user-defined name for the mesh module
exponential_averaging	value `exponential_averaging_time` time_value; value = `yes` or `no` time_value = time used for exponential averaging, units: [time] keyword `exponential_averaging_time` mandatory if value = `yes` default: `no`
calculate_mean	available options: `yes` and `no`; compute average over all time steps default: `yes`

Examples

mesh_module stress_average id avg exponential_averaging yes exponential_averaging_time 1e-2
mesh_module stress_average id avg_2 exponential_averaging yes exponential_averaging_time 1e-1 calculate_mean yes

Description

This mesh module allows to calculate time averaged normal and shear stresses for each mesh element (triangle) and the mesh as as whole. Both normal and shear time averaged stresses ( $\sigma$ ) are computed as

$\sigma_{avg}(t_n) = \left(1-w\right) \sigma_{avg}(t_{n-1}) + w \sigma(t_n),$

where $t_n$ and $t_{n-1}$ are the current and previous time levels, respectively. Furthermore, $\sigma_{avg}(t)$ and $\sigma(t)$ are the time averaged and current stresses, respectively, at time level $t$ . Finally, $w$ is the (time based) weight (a value between 0 and 1) used to balance the relative importance of the previous time averaged stress and current stress in the computation (update) of the current time averaged stress.

The exponential moving averaging, enabled by exponential_averaging yes, computes a (time) moving average (normal and shear) stresses with a time constant $\tau$ specified by the exponential_averaging_time keyword. This $\tau$ is used to set the weight to $w =\frac{\delta t}{\tau}$ , where $\delta t$ the time step. Note that $\tau \ge \delta t$ , and that smaller $\tau$ values will put more weight on the current stresses.

If calculate_mean is set to yes the time averaged (normal and shear) stresses over the whole simulation duration is computed for both normal and tangential components. In this case, the weight is set to $w =\frac{\delta t}{T}$ , where $T$ is the (steadily increasing) total simulation time. These time averaged stresses will respond to changes in the current stresses more slowly compared to the (exponential) moving average stresses.

Additional information

This mesh_module generates additional mesh properties that can be dumped, e.g. by using output_settings. The mesh properties are called moving_average_normal_stress and moving_average_shear_stress for the exponential moving averages and average_normal_stress and average_shear_stress for the mean average (total simulation time). An example output can be written as

output_settings mesh_properties {average_normal_stress, average_shear_stress, moving_average_normal_stress, moving_average_shear_stress}

It also stores a global vector for access by various output commands. This vector has four components, the first two are the global averages (over all mesh elements) of the normal and tangential stresses. The third and fourth are the global (exponential) moving averages of normal and tangential stresses, respectively. To derive these global averages, first the total normal and tangential forces are computed (by summing the normal and tangential forces of all mesh elements). Next, these total normal and tangential forces are divided by the total mesh area (sum of all mesh element areas). If calculate_mean or exponential_averaging is set to no then the respective components will return 0. The position of the data in the output depends on the additional mesh modules that are used (there are 9 components for mesh/surface/stress, the output for the average stress module comes afterwards). For easier use the property values can also be accessed via the property name. For example, for a mesh with id myMesh id_myMesh.average_normal_stress will return the (global) average normal stress. See the table below for a complete overview of the available properties and how to access them.

Mesh module property	property name (dot access)	probable array position
average normal stress	average_normal_stress	10
average shear stress	average_shear_stress	11
moving average normal stress	moving_average_normal_stress	12
moving average shear stress	moving_average_shear_stress	13