## Inflation of a Passive Prolate Spheroid

Contents

### Description

- This example guides you through setting up and running a physical and material nonlinear problem of the passive (non-contracting) inflation of a truncated ellipsoid using pressure (natural) boundary conditions. This mesh is a rough approximation of a left ventricle.
- The mesh consists of three elements that are wrapped onto themselves in a prolate spheroidal coordinate system. The basis functions are linear in the circumferential direction and cubic in the longitudinal and radial directions (linear-cubic-cubic). The material chosen is a transversely isotropic exponential strain energy function. The material is stiffer in the fiber direction than perpendicular to them. The fibers whose orientation is defined by fiber angles in the node form with respect to the circumferential direction vary linearly in radial direction across the wall from -37 degrees on the epicardium to +84 degrees on the endocardium. The boundary conditions assure that all rigid body motions are suppressed. Pressure is prescribed on the endocardium to (partly) simulate the passive filling of a ventricle (diastole).
- The final (refined) model nodes and elements are defined according to the figure below:
- Stress and strain solution tables for various pressure loads and complete model files are provided at the end of this tutorial.

### Building and solving the model

#### Start Continuity

- Launch the Continuity Client
On the About Continuity startup screen

Leave the

**mesh**checkbox checked under`Use Modules:`In addition, check the

**biomechanics**checkbox

Click

**OK**to bring up the main window

#### Create mesh

Select the cont6 file for this tutorial ( prolatemesh.cont6 )

- To make sure you selected the right cont6 file, verify the following:
- Check that the list contains only:
- Linear-Linear-Linear Lagrange 3*3*3
- Linear-Linear Lagrange 3*3
- Linear-Cubic-Cubic Hermite 3*3*3
- Linear-Cubic Hermite 3*3
- Cubic-Cubic Hermite 3*3
- Cubic-Linear Hermite 3*3

- Check that the list contains only:
- Check that that there are only 8 nodes defined
Make sure that in the

**Field Vector 1**tab, under**Field Variable 1**the option**Linear-Linear-Linear Langrange 3*3*3**is selected

- Check that there are only 3 elements in the list

If the Dimensions Form appears, simply click

**Apply Marked Recommendations**and then**OK**

Click the

**lines**radio buttonClick

**Render**to display mesh lines

- The mesh should now look similar to the first screenshot

#### Refine the Mesh

To get a sufficiently converged result using linear elements, it is necessary to use multiple elements. Therefore, we will refine our single element into many elements.

Decrease the

`New Element per old element in``xi1`to 1. Increase the`New Element per old element in``xi2`and`xi3`to 3Click

**OK**to refine the mesh

Click the

**lines**radio buttonClick

**Render**to display the mesh as a wireframe

To delete or hide previously rendered objects View→Show→OpenMesh...

- Note that the element labels are preserved

#### Add biomechanics data

- Load the required biomechanics model from the database
- File→Library→Search...
In the window near the top, enter 'lagrangian' and hit

**return**.From the listed models select

**BM_TI_of_Lagrangian_strains_comp_sympy**by right-clicking on it and selecting 'Load'- When the warning window display, select the third choice: 'Retain current problem but overwrite the following objects: [dims, renderer, matEquations]'

Biomechanics→Update→Initial Conditions with undeformed nodes

- This command updates the biomechanics Boundary Conditions form with the values already inputted in the mesh nodes form.
After having opened the two forms, ensure that the

`Boundary Conditions`forms has the same basis functions as the`Nodes`forms

Biomechanics→Edit→Boundary Conditions...

Click on the

`Deformed Coordinates 1`tabClick the

`Insert Nodes`button**three**times- Edit the `Node(s)' and 'Derivative' parameters of the three nodes based on the table below
Node in

`Nodes List``Node(s)``Derivative``Value`1

APEX

wrt s(2)

0

2

APEX

wrt s(2)s(3)

0

3

APEX

Value

0

Click on the

`Deformed Coordinates 2`tabClick the

`Insert Nodes`button**six**times- Edit the `Node(s)' and 'Derivative' parameters of the three nodes based on the table below
Node in

`Nodes List``Node(s)``Derivative``Value`1

APEX

Value

0

2

APEX

wrt s(3)

0

3

APEX

wrt s(2)s(3)

0

4

BASE

Value

0

5

BASE

wrt s(3)

0

6

BASE

wrt s(2)s(3)

0

Click on the

`Deformed Coordinates 3`tabClick the

`Insert Nodes`button onceEnter

**BASE_EPI$**in the `Node(s)' text fieldFor

`Derivative`, choose**Value**. Under`Value`enter**0**.

Click on the

`External Pressure`tabFor

`Element Number`, enter**ENDO**, and hit the Enter key on your keyboardFor the

`Select Pressure Type`drop-down list, choose**Incremental**In the

`Inner surface`box on the bottom, enter**1.0**for the`Specify Pressure Increment`field

- It is a good idea to now go back through the Boundary Conditions Form to double check this parameters you just set up
Click the

**OK**button

#### Solve the biomechanics

If the Dimensions Form appears, simply click

**Apply Marked Recommendations**and then**OK**

Biomechanics→Solve Nonlinear...

For

`Time Step`, enter**0.05**Set

`Number of steps`to**40**Click the

**Solve**button, and wait for the solver to finish its job

#### Calculate and render stress and strain

Click the

**lines**radio buttonThis time select the

**deformed**radio buttonClick

**Render**to display deformed mesh lines

Click on

**3. element lines3**in the list on the left, and enter**1,0,0**in the`R,G,B`entry field to change the mesh lines from blue to red.- Press [return] and close the window
- The mesh should look like the screenshot below

Biomechanics→Render→Surface...

Change the pop-up menu choice after

`At Xi`to**1**and change`Location`to**0.5**Select the

**deformed**radio buttonunder

`Variables`select**T Cauchy Stress Tensor**Click

**OK**to render the fiber (circumferential) stress, OR select`[2,2]`to render the radial stress.

### Solution files for various pressure loads

- The variables pertinent to this problem are listed below:
- The model makes use of the following constitutive equations:
- The stress and strain solutions for various endocardial pressure loads are provided in the table:
**Pressure Load****Solution File**1.0 kPa

1.4 kPa

1.6 kPa

2.0 kPa

### Pre-built model

This cont6 file contains all data and parameters for this problem: bm2.cont6 (original), bm2_refined.cont6 (refined)