Contents
Description
 This example begins with a simple 2D, oneelement mesh with sharp corners. The element corners are made smooth and continuous by defining higherorder Cubic Hermite basis functions and adjusting their derivatives appropriately. The purpose of this exercise is to further explore the meaning of the derivatives that function as nodal parameters for the higherorder basis functions within Continuity.

The files needed for this exercise can be found in the Continuity Tutorials folder (if you have downloaded it) under Mesh > exercise3. The cont6 file containing the entire problem setup can also be downloaded by clicking here.
Basic Concepts
 After completing the Mesh module tutorials 1 and 2, you should understand that:
 Each node has several parameters associated with it.
 The number of associated parameters depends on the type of basis function(s) in use
 Elements are defined by specifying connections between nodes
 Nodal parameters determine the shape of an element, including the location of its corners and the shape of its edges, surfaces, and volume

After completing this tutorial, you should understand:
 The practical meaning of firstorder nodal derivatives
 How to perform simple manipulations of nodal derivatives, such as:
 Using Continuity to calculate some derivatives automatically
 Relating nodal derivatives to each other to achieve smooth, continuous mesh boundaries
 Useful points to consider:
 While analytically determining the correct values for nodal derivatives can be daunting, the good news is that with care Continuity can be made to do most of the work for you.
 When it comes to creating detailed, anatomically realistic meshes, manually adjusting derivative values to fit a data set would be enormously difficult, and in practice is never done. Instead, after defining some simple constraints on nodal parameters (the basics of which you learn in this tutorial), Continuity’s Fitting module does the job of adjusting mesh parameters in an automated way.
Stepbystep Instructions
 The following instructions will guide you in creating the mesh pictured below and then lead you through altering it to make a smooth, continuous shape.
Start Continuity
 Launch the Continuity 6.3 Client

On the About Continuity 6.3 startup screen

check the mesh box under Use Modules:

Create Mesh


Select rectangular cartesian in the Global Coordinates: popup menu

Click OK to submit Coordinate Form



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

Click Add

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

Click Add

Click OK to submit Basis Form



Click Import/Export/Graph button to open Continuity Table Manager

Continuity Table Manager→File→Open…

Select tabdelimited nodes file nodes.xls



Note that LinearLinear Lagrange 3*3 is already selected for you under Coordinate 1, Coordinate 2, and Coordinate 3

Note also that all derivative values are currently set to zero. We need to obtain correct derivative values before being able to use the higherorder bicubic Hermite basis functions.

Click OK to submit Node Form



In this case, the element definition is simply 1, 2, 3, 4, so enter this in the Global Node Numbers box

Click OK to submit Element Form



Click the lines radio button

Click Render to display mesh

Convert from LinearLinear to CubicCubic
 Switching to cubic Hermite basis functions requires the determination of 3 new parameters per coordinate direction per node. In this case, that’s a total of 24 parameters. Rather than trying to manually determine the values of these parameters, we can use Continuity’s Refine feature. Because we have defined a cubiccubic Hermite basis function, upon doing a 1 by 1 refine the necessary derivative values will be calculated and placed in the nodes form.


Enter 1 for the xi1, xi2, and xi3 fields under New Element per old element in

Click OK to submit


 Note that derivative values are no longer zero

Select CubicCubic Hermite 3*3 under Coordinate 1, Coordinate 2, and Coordinate 3

Click OK to submit Node Form
 At this point we need to resend, calculate, and render the mesh:


Click the lines radio button

Click Render to display mesh


Note that the mesh looks exactly the same. Once again, this is because during the refine 1 by 1 by 1 step, Continuity has calculated the derivative values such that the lines connecting nodes are straight.
Alter Derivatives to Give Smooth Edges
 Suppose we want this element to have smooth, continuous edges (in the global sense). Suppose in particular that we’d like:

The element edges at nodes 1 and 4 to be tangent to the x_{2} axis

The element edges at nodes 2 and 3 to be tangent to the x_{1} axis

 To understand how to alter the derivatives to achieve these conditions, consider the significance of the nodes form boxes in mathematical notation:
 Consider node 1.

In order to make the element edges tangent to the x_{2} (vertical) axis, moving from node 1 in either the ξ_{1} or ξ_{2} direction should produce no change in the x_{1} direction. In other words,

Open the nodes form (Mesh→Edit→Nodes…)

Click on node 1, and change the appropriate derivative values to zero as described above. Leave all other derivative values as they are for now.
 Now consider each of the other three nodes in turn. Carefully select which pair of derivatives should be set to zero for each node in order to create our desired tangent relationships.

Click OK when finished to submit the nodes form.


Increase the number of divisions from 10 to 25 (View→Set Divisions…)


Click the lines radio button

Click Render to display mesh


If you have made the changes correctly, the new and old meshes are now superimposed and should look like the picture below. A cont6 file containing the correct derivative values can be found here.