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

Due: Tuesday February 21st 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.

  • Mesh→Edit→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.

  • Mesh→Edit→Basis...

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

  • Mesh→Edit→Nodes...

    • 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 Option

    • Remember to enable Field variable 1 by selecting Linear-Linear-Linear Lagrange 3*3*3 from its Select Basis Number menu.

  • Mesh→Edit→Elements...

    • Element 1 consists of global nodes
      • 1

        1

        2

        2

        3

        3

        4

        4

  • 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

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 geometry

    • Select 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 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.

    • 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.