Definition
In the case of spatial domain, it calculates variance of a given variable for a given container and returns mean and variance as shown in following equations. xi is the ith element’s variable value of the corresponding container. Results will have the same type as the type of the variable specified by the user. (If it has higher dimension than a scalar, mean of each dimension will be calculated seperately resulting with a mean having same dimension as the input dimension)
$$\underline{\bar{x}} = \frac{1}{N}\sum_{i=1}^N{\underline{x}_i} $$
$$\underline{v} = \frac{1}{N}\sum_{i=1}^N{\left(\underline{x}_i - \underline{\bar{x}} \right )^2}$$
In the case of temporal domain, Variance method is the time integrated quantity’s variance for a specific variable. Mean and variance will be stored each element under user specified variables and a user specified container. xi is the ith element’s variable value of the corresponding container. Results will have the same type as the type of the variable specified by the user preserving the dimensionality as in the spatial case.
$$ \underline{\bar{x}} = \frac{1}{T_{total}}\sum_{k=1}^{P}{\underline{x}_k\Delta t_k}$$
$$Var\left(\underline{x} \right ) = \frac{1}{T_{total}}\sum_{k=1}^{P}{\left(\underline{x}_k - \underline{\bar{x}} \right )^2\Delta t_k}$$
$$ \quad where $$
$$T_{total} = T_{end} - T_{initial} \quad and \quad \Delta t_k = T_{k} - T_{k-1} \quad \forall T_k \in \left\lbrace T_{initial}, ..., T_{end} \right\rbrace$$
Examples
Spatial
Following is an example of variance calculation of non historical VELOCITY over the whole model part’s nodes
import KratosMultiphysics as Kratos
import KratosMultiphysics.StatisticsApplication as KratosStats
model = Kratos.Model()
model_part = model.CreateModelPart("test_model_part")
mean, variance = KratosStats.SpatialMethods.NonHistorical.Nodes.ValueMethods.Variance(model_part, Kratos.VELOCITY)
Temporal
Following is an example of root mean square calculation of non historical velocity. Input variable is node’s non-historical container’s VELOCITY and output variable VECTOR_3D_MEAN will store mean and VECTOR_3D_VARIANCE will store variance in same container for each node. The 0 represents echo level for this method object. Blank “” indicates that value method is used.
import KratosMultiphysics as Kratos
import KratosMultiphysics.StatisticsApplication as KratosStats
model = Kratos.Model()
model_part = model.CreateModelPart("test_model_part")
variance_method = KratosStats.TemporalMethods.NonHistorical.Nodes.ValueMethods.Variance.Array(model_part, "", Kratos.VELOCITY, 0, KratosStats.VECTOR_3D_MEAN, KratosStats.VECTOR_3D_VARIANCE)
integration_starting_time = 2.0
variance_method.InitializeStatisticsMethod(integration_starting_time)
for t in range(3, 6):
variance_method.CalculateStatistics()