## BENG 112A Homework Assignment 5: Strain Analysis of an Artery

**Due: Tuesday February 21 ^{st}** at the beginning of class

The goal of this assignment is to perform a strain analysis on an artery and gain insights into:

- The distributions of strain in the vessel wall
- The utility of computational modeling in soft tissue biomechanics

This assignment requires a good understanding of the StrainIn3DBlock tutorial from class. If you have not worked through this independently, please do so prior to starting the homework. The InflateTube tutorial may also provide useful practice and background information.

### Create an Undeformed Mesh

Our model of a blood vessel will consist of a single 3D element in **cylindrical polar** coordinates.

Select

**cylindrical polar**as the`Global Coordinates`

Mesh→Edit→Material Coordinates...

- The image below describes the transformation from cyclindrical polar coordinates to rectangular cartesian coordinates.
**dY_dMatl**defines the coordinate system by describing the transformation from polar coordinates to rectangular cartesian coordinates for Continuity. To be more precise, dY_dMatl is the local material coordinate to global rectangular cartesian coordinate orthogonal transformation matrix. This image can help you understand the different coordinate systems in Continuity.The equation below describes the form of

**dY_dMatl**. Use this to derive the proper transformation matrix. See the SymPy tutorial on how to create a matrix in a model editor.The cylindrical polar coordinates are

**X**in Continuity (vector of undeformed world coordinates). Since Python indexes from zero (similar to C programming but dissimilar to Matlab),**X[1]**corresponds to the theta variable.Only one matrix is required to define

**dY_dMatl**. No new variables need to be defined. Use the existing Continuity coordinates (X for undeformed world coordinates and Y for undeformed cartesian coordinates).Remember to compile your material coordinate model. If your are having trouble compiling, see the Compiling Models help page.

- The image below describes the transformation from cyclindrical polar coordinates to rectangular cartesian coordinates.
Choose

**Lagrange Basis Function→3D→Linear-Linear-Linear**with**3**`integration/collocation points`for Xi 1, Xi 2, and Xi 3

Confirm that

**Linear-Linear-Linear Lagrange 3*3*3**is already selected for you under`Coordinate 1`,`Coordinate 2`, and`Coordinate 3`In the

`Value`fields next to`Coordinate 1`,`Coordinate 2`, and`Coordinate 3`enter the following (R,Theta,Z) nodal coordinates:Node 1:

(

**2., 0., 0.**)Node 2:

(

**2., 0., 6.**)Node 3:

(

**3., 0., 0.**)Node 4:

(

**3., 0., 6.**)

Select

**Radians**as the Coordinate OptionRemember to enable

`Field variable 1`by selecting Linear-Linear-Linear Lagrange 3*3*3 from its**Select Basis Number**menu.

- Element 1 consists of global nodes
**1****1****2****2****3****3****4****4**

- Element 1 consists of global nodes
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. In cylindrical polar coordinates, the`Xi1`direction is the positive theta direction.- To trace out the 3D artery model, each node wraps around to itself.
Remember to

**Send**and**Calculate Mesh**whenever you make changes to your model.

### Create a Deformed Mesh from Prescribed Displacements

- Given the below transformation of coordinates from (R,Theta,Z) to (r,theta,z) calculate the locations of the deformed nodes.
If the

`Biomechanics`menu is not loaded, select File→Load Continuity Modules... to load it.Biomechanics→Update→Initial conditions with undeformed nodes

Biomechanics→Edit→Boundary Conditions...

Enter the locations of your deformed nodes into the

`Deformed Coordinates`.Be sure to select the

**Radians**option. Depending on your screen size, this may require you to enlarge the Boundary Conditions form.

Biomechanics→Edit→Constitutive Model...

Generate the same simple constitutive model as in the StrainIn3DBlock tutorial

- If you want to calculate additional variables for saving or rendering, add them below the stress variable.
- Remember to compile your constitutive model

Mesh→Render→Elements... or click on

Render the

**undeformed**and**deformed**lines to view the mesh.

- Describe the deformation of the artery.

### Calculate Strain Computationally Using Continuity

Biomechanics→List Stress and Strain...

- Save the Lagrangian Green's Strain tensor and any other output variables you find useful.
Note that the

`Locations`allows you to select where in the mesh the results are saved at.

Biomechanics→Render Surface...

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 geometrySelect

**E-out - Lagrangian Green's Strain**from the`Variables`menu. Since strain is a tensor variable, a choice of components will be presented. Select**yy**. Output will be determined by equations in the`Output Variables`folder of the`Constitutive Model Editor`Click

**OK**to create a color-coded surface rendering of the E_yy_ component of the Lagrangian Strain- Save a picture of your Continuity screen by clicking the camera button in the bottom left corner. Be sure to enter a valid image file extension when saving (.jpeg, .png, ect.)

View→Show→OpenMesh... or click on

Click on the last

**Textured Field**entry in the listClick on the

**Colors**tabNote 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.- Be sure to define your color scale for figures used in your report.

### Calculate Strain Analytically Using SymPy

Use SymPy in the Continuity Python Shell to analytically solve for the Lagrangian strain tensor. See the SymPy tutorial for syntax help. Be careful with formulating the deformation gradient tensor in polar coordinates. The units must be consistent.

### Homework 5 Requirements

#### Model Description

- Include a matrix describing your material coordinate model. A properly formatted matrix is preferred to the raw code.
- Include a table with your undeformed and deformed nodes.
- Include a rendering (picture) showing your undeformed and deformed meshes.
- Describe the deformation that the artery undergoes.

#### Strain Analysis

- Save renderings of all non-zero components of the Lagrangian Green’s strain tensor in each axis. Be sure to include a colorbar and identify what is being depicted in each graph and where.
- Provide all 9 values of the strain tensor for at least one location in the mesh.
- Provide the result of your analytical strain calculation.
- Compare the strain results from Continuity with those calculated analytically. Describe whether these solutions agree exactly, approximately, or not at all.
- Describe the distribution of strain for each direction of the vessel. For example, how does the radial strain change with r (through the wall), theta (around the vessel), and z (along the length of the vessel)? Is this what you would expect? Is it consistent with the analytical results? Support your claims with data from Continuity. This could be the renderings of the strain field or graphs of strain data saved at multiple locations in the vessel.
- Does the prescribed deformation conserve the volume of the vessel walls? Explain your reasoning.

#### Additional Investigation

Use the transformation of coordinates from (R,Theta,Z) to (r,theta,z) below to answer the following questions:

- Include a table with your undeformed and deformed nodes.
- Include a rendering showing your undeformed and deformed meshes.
- Describe the deformation that the artery undergoes.
- Calculate the strains analytically and computationally using Continuity. Compare the strain results from Continuity with those calculated analytically. Describe whether these solutions agree exactly, approximately, or not at all.
- Describe the distribution of strain in the vessel. How does it compare to what you expect? What do your results suggest about the utility of computational modeling for understanding complex biomechanics problems?
- Does the prescribed deformation conserve the volume of the vessel walls? Explain your reasoning.