About Mesh Motion Equation Controls

The mesh motion equation controls enable you to fine-tune the movement of the mesh.

This page discusses:

Stiffness Type
Stiffness Scale
Stiffness Power
Stiffness Variation Type
Solve from Undeformed Coordinates

Stiffness Type

The app calculates the deformation of the mesh by solving a linear elasticity problem. The quality of the deformed mesh depends largely on the relative stiffness distribution of the mesh elements. In general, assigning a higher stiffness to the elements located near the moving parts provides optimal results; the stiffer elements tend to maintain their original shape, while the less stiff elements absorb most of the deformation.

There are three methods for calculating the stiffness distribution: inverse distance, inverse volume, and uniform.

Inverse Distance

The inverse distance approach automatically assigns a higher stiffness to the elements close to the moving boundaries; that is, the boundaries where a moving wall condition or a fluid-structure interaction interface has been applied. If neither of these conditions have been applied, the app uses uniform stiffness. In the inverse distance approach, the stiffness, $S$ , of a cell is computed as follows:

S = {\begin{matrix} s_{d} - (s_{d} - 1) {(\frac{d_{i} - d_{\min}}{d_{l} - d_{\min}})}^{p} & , d_{i} < d_{l} \\ 1 & {, d}_{i} \geq d_{l} \end{matrix},

where

$s_{d}$: is the stiffness scale,
$d_{i}$: is the distance from the cell center of the cell to the closest point of the moving boundaries,
$d_{\min}$: is the distance to the moving boundaries of the closest cell,
$d_{l}$: is the stiffness variation distance, and
$p$: is a power coefficient.

Inverse Volume

The inverse volume approach assigns higher relative stiffness to the smallest elements in the mesh. In general, meshes tend to have smaller elements close to walls to resolve the boundary layers. Therefore, the inverse volume approach improves the quality of the deformed mesh in those areas. In the inverse volume approach, the stiffness, $S$ , of a cell is computed as:

S = {\begin{matrix} s_{v} - (s_{v} - 1) {(\frac{V_{i} - V_{\min}}{V_{l} - V_{\min}})}^{p} & , V_{i} < V_{l} \\ 1 & {, V}_{i} \geq V_{l} \end{matrix},

where

$s_{v}$: is the stiffness scale,

$V_{i}$: is the volume of the cell,

$V_{\min}$: is the volume of the smallest cell,

$V_{l}$: is the stiffness variation volume, and

$p$

is a power coefficient.

The stiffness variation volume enables you to control the range of variation of the stiffness. It is calculated based on the ratio, $r$ , using the following expression:

V_{l} = (V_{\max} - V_{\min}) \cdot r + V_{\min},

where $V_{\max}$ is the volume of the largest cell.

Uniform

The uniform distribution approach may be useful for applications such as pistons, where the whole computational domain expands and contracts in a uniform manner.

Stiffness Scale

The stiffness scale is the ratio between the stiffness of the most rigid cell and the stiffness of the most flexible cell. You can specify the stiffness scale when using either the inverse distance or inverse volume stiffness calculation methods.

Stiffness Power

The stiffness power is a parameter that determines the order of variation of the stiffness using either the inverse distance or inverse volume stiffness calculation methods.

Stiffness Variation Type

The stiffness variation type allows you to choose how to define the stiffness variation distance when using the inverse distance stiffness calculation method. You can choose between specifying the actual stiffness variation distance, $d_{l}$ , or a stiffness variation ratio, $r$ .

When using the ratio, the stiffness variation distance is calculated as follows:

d_{l} = (d_{\max} - d_{\min}) \cdot r + d_{\min},

where $d_{\max}$ is the distance to the moving boundary of the cell located at the farthest distance.

When using the value approach, if you define a distance of 0, the app automatically uses the ratio approach with a value of 1.

Solve from Undeformed Coordinates

The mesh motion solver uses the original undeformed coordinates of the model to compute the new mesh. This option is intended to be used for periodic problems in which the mesh undergoes several deformation cycles. It recovers the original mesh when the boundary returns to its original position.