Contents
Description
 This example creates an undeformed twoelement 3D mesh, then deforms it and calculates stress and strain distributions.
 The mesh is a simple block with trilinear Lagrange elements. The deformed nodal coordinates are completely specified so no boundary value problem is solved.

The Output Variables folder in the Constitutive Model Editor contains the equations for the kinematics and stresses that are rendered and listed. An example of the rendered strains is shown below:
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 rectangular Cartesian in the Global Coordinates: popup menu

Click OK to submit Coordinate Form


Mesh→Edit→Material Coordinates…

select the variable named dY_dMatl

In the field under Equation:, type dY_dMatl = eye(3)
 This defines the transformation from global Cartesian coordinates to local material coordinates to simply be the identity matrix

The SymPy tutorial demonstrates how to define a matrix in SymPy

Select Compile… and click Compile Code as: C Double Precision

After the notice for a successful compilation, click the OK button to submit the material coordinates model



Choose Lagrange Basis Function→3D→LinearLinearLinear with 3 integration/collocation points for Xi 1, Xi 2, and Xi 3

Click Add
 Verify that the list of basis functions now contains:
 LinearLinearLinear Lagrange 3*3*3
 LinearLinear Lagrange 3*3

Click OK to submit Basis Form



Confirm that LinearLinearLinear Lagrange 3*3*3 is already selected for you under Coordinate 1, Coordinate 2, and Coordinate 3

Use the Insert Node button in the left panel to create 11 additional nodes for a total of 12

In the Value fields next to Coordinate 1, Coordinate 2, and Coordinate 3 enter the following (X,Y,Z) nodal coordinates:

Node 1:
(0., 0., 0.)
Node 2:
(1., 0., 0.)
Node 3:
(2., 0., 0.)
Node 4:
(0., 1., 0.)
Node 5:
(1., 1., 0.)
Node 6:
(2., 1., 0.)
Node 7:
(0., 0., 0.5)
Node 8:
(1., 0., 0.5)
Node 9:
(2., 0., 0.5)
Node 10:
(0., 1., 0.5)
Node 11:
(1., 1., 0.5)
Node 12:
(2., 1., 0.5)


Enable Field variable 1 by selecting the Field Vector 1 tab and choosing LinearLinearLinear Lagrange 3*3*3 from its Select Basis Number menu.

Click OK to submit Node Form


 Element 1 consists of global nodes
1 
2 
4 
5 
7 
8 
10 
11 

Enter these numbers in the Global Node Numbers boxes. Use the tab key to change the input focus to the next box. Note that the order that global node numbers are entered determines the local Xi coordinate directions in the element, as illustrated by the graphic in the input form.

Use the Insert Element button in the left panel to create another element
 The global nodes of element 2 are
2 
3 
5 
6 
8 
9 
11 
12 

enter them into the Global Node Numbers box

Click OK to submit Element Form

File→Send or click on

Mesh→Calculate Mesh… or click on

Click the OK button

Render the Mesh

Mesh→Render→Elements… or click on

Click the lines radio button

Click Render to display mesh
 The mesh should look like the screenshot below. (Use the mouse to rotate, pan and zoom the view)


File→Save→Model or click on

Specify a filename to save the model in. The file will be saved as filename.cont6

Create a Deformed 3D Bilinear Mesh

If the Biomechanics menu is not loaded, select File→Load Continuity Modules… or click on

Select biomechanics and click OK
 The menu bar should now show the Biomechanics command


Biomechanics→Update→Initial conditions with undeformed nodes

Biomechanics→Edit→Boundary Conditions…

On the Initial Conditions tab, edit Deformed Coordinate 2 of nodes 2 and 8 to change their values to 0.5

On the Initial Conditions tab, edit Deformed Coordinate 2 of nodes 5 and 11 to change their values to 1.5

Click the OK button


Biomechanics→Edit→Constitutive Model…

Rightclick on the variable dy_dx in the left panel of the Edit Equations tab

Select Insert variable here…
 Select the newly created variable and on the right panel

change its name: to F and press Enter

change its Description to something like Deformation Gradient Tensor wrt Material Coordinates

change its Type: to symbolic variable in the popup menu

in the Equation: field enter: F = dY_dMatl.T*dy_dx*dx_dMatl
 This is a matrix multiplication. .T refers to transpose.

in the left side panel, drag the new variable F to be after dy_dx and before stress



Similarly create another symbolic variable after F named C with description Right Cauchy Green Deformation Tensor

For the Equation: enter C = F.T*F


Similarly create another symbolic variable after C named E with description Lagrangian Green’s Strains wrt Material Coordinates

For the Equation: enter E = 0.5*(Ceye(3))


For the equation for the stress variable we will use as a placeholder for now: stress = eye(3)

Finally create an evaluated variable after stress named Eout with description Lagrangian Green’s Strains wrt Material Coordinates

For the Equation: enter Eout = E


Select Compile… and click Compile Code as: C Double Precision

After the notice for a successful compilation, select the OK button to submit the constitutive model


File→Send or click on

If the Dimensions Form appears, simply click Apply Marked Recommendations and then OK


Mesh→Calculate Mesh… or click on

Click the OK button


File→Save→Model or click on
 If your constitutive model compiled successfully, this is a good time to save your model again

Mesh→Render→Elements… or click on

Click the lines radio button

This time select the deformed radio button is selected

Click Render to display mesh lines


View→Show→OpenMesh… or click on

Click on 2. element lines2 in the list on the left, and enter 1,0,0 in the R,G,B entry field. Press [return] and close the window
 You should see the undeformed mesh in blue and the deformed mesh in red as shown below.

Calculate Strains

Biomechanics→List Stress and Strain…

Note that the Output Variables are calculated using the equations entered in the Output Variable folder of the submitted Constitutive model

Click OK to display a listing of the selected Output Variables in the Table Manager



Next to At Xi 3 Location enter 0.5 (this chooses the midplane of the elements in the Xi3 (here Z) direction where results will be rendered)

Check the ‘deformed radio button to render the solution deformed geometry

Select Eout – Lagrangian Green’s Strain from the Variables menu. Since strain is a tensor variable, a choice of components will be presented. Select [1,1]. Output will be determined by equations in the Output Variables folder of the Constitutive Model Editor

Click OK to create a colorcoded surface rendering of the E_yy_ component of the Lagrangian Strain
 The result should look like the screenshot below


View→Show→OpenMesh… or click on

Click on the last Textured Field entry in the list

Click on the Colors tab

Note the range of the strains corresponding to the minimum (blue) and the maximum (red). You can change these to round numbers like 0.0 and 1.3 respectively.

Press the return key and close the Open Mesh Controls window

Prebuilt model
 A prebuilt model for this tutorial is available on Continuity’s public database.

Title: bm_3D_block_strain

ID: 1385

 Note that the material coordinate model and constitutive model may still need to be compiled.