1. Parameterizing a Geometry using the COMSOL Moving Mesh Feature

2. This example shows how to parameterize a CAD model by deforming the corresponding finite element mesh.
The method enables geometry parameterization of non-parameterized CAD models such as:
models created by the built-in COMSOL CAD tools
imported neutral CAD files such as .igs or .step
imported .x_t, .sat, or other vendor-specific formats
imported finite element mesh files

3. The geometry is an I-beam with 7 slotted holes.

4. We will see how to simultaneously change the dimensions of the slots and …

5. … the width of the I-beam by using a single geometry parameter, here called D1 (multiple parameters may also be used).

6. This is the beam at one extreme of the geometry parameter: D1=0 mm.

7. This is the beam at the other extreme of the geometry parameter: D1=5 mm.

8. These are 6 snapshots of the von-Mises effective stress together with the finite element mesh.
The color scale is normalized with respect to the stress level of the maximum stress of all parameter values (which is at D1 = 5 mm).

9. The effective stress at D1=0 mm

10. The effective stress at D1=1 mm

11. The effective stress at D1=2 mm

12. The effective stress at D1=3 mm

13. The effective stress at D1=4 mm

14. The effective stress at D1=5 mm

15. The material is the default steel of the COMSOL Multiphysics Structural Mechanics Module.

16. The highlighted (red) boundaries are unloaded and unconstrained (free).

17. This boundary is clamped.

18. A roller boundary condition is used close to the clamped end and at the free end of the beam (highlighted surfaces). The roller boundary condition is of a flexible type – a stiff-spring-like force is acting on the beam in the positive y-direction.
This approximates the behavior of a flexible roller-type support.

19. The highlighted surface has a non-uniform distributed load in the y-direction given by: -p0*(a-z)^2, where p0 and a are constants and z is the coordinate in the direction along the length of the beam. This corresponds to a load with quadratic growth starting from zero (at the left of the highlighted surface).

20. The parameterization is handled by a so called Moving Mesh application mode which is used to create a deformable coordinate system.
This is how the geometry is being parameterized – by defining how this coordinate system should deform. The deformable coordinate system is called a Frame.

21. The Moving Mesh application mode creates a smooth deformation of the mesh on the boundaries and throughout the volume of the solid. The smoothing technique used is called Arbitrary Lagrangian-Eulerian or ALE. The Moving Mesh coordinate frame is also called an ALE-frame.

22. The Model Navigator shows that the coordinate system of the Solid, Stress-Strain application mode is defined by the ALE-frame.
The Moving Mesh (ALE) application mode handles the propagation of the mesh deformation from the boundaries and throughout the computational volume.

23. This example shows three of the most basic types of boundary conditions that you can use for parameterizing a geometry:
Fixed displacement
Parameterized normal (perpendicular) displacement
Free tangential displacement

24. These are the parameterized boundaries
These are the parameterized boundaries. They are displaced a distance given by the parameter D1 in the normal (perpendicular) direction to the surfaces. To allow for this – the mesh elements of these boundaries are allowed to stretch freely in the direction tangential to the surface.

25. These boundaries are allowed to deform freely in the tangential direction. At the same time, they are constrained in the direction normal to the surfaces.
This type of boundary condition is used at surfaces that are adjacent to the parameterized boundaries so they can “inherit” the geometry change by allowing for free deformations in the tangential direction.
Any dimensions perpendicular to these surfaces cannot change.

26. These boundaries are fixed
These boundaries are fixed. They cannot be deformed in any direction, so any dimension defined by these surfaces cannot change.
This type of boundary conditions are used furthest away from the parameterized boundaries.

27. The Moving Mesh (ALE) application mode used to create the smooth deformation defines its own equation system (the solution of which gives the smooth deformation) that needs to be solved side-by-side of the stress-strain equations. In order to minimize memory requirement, the parametric segregated solver is used. The D1 parameters solved for range from 0 to 5 mm ( m in SI units) in steps of 0.2 mm.

28. This visualization shows the von Mises effective stress at D1 = 0
This visualization shows the von Mises effective stress at D1 = m (5 mm)

29. This visualization shows an exaggerated deformation plot for the case D1 = 5 mm.

30. The next slide is an animation of the parametric sweep
The next slide is an animation of the parametric sweep. Set the presentation in Full Slide-Show Mode to watch it.

32. Further Model Implementation Details The next few slides are useful if you wish to reproduce this example.

33. This particular example is created by first making a stress-strain analysis model.
The geometry was created using COMSOL’s built-in CAD tools with two work planes. The work planes are available as Geom2 and Geom3.

34. When the initial stress-strain analysis has been made, a Moving Mesh application mode is added. This places the Stress-Strain and the Moving Mesh application modes in different frames. For the geometry parameterization we wish to have them both in the same frame.

35. To change this, select the Solid, Stress-Strain application mode and change the Frame to Frame (ale).

36. Now, the Solid, Stress-Strain application mode is defined by the Moving Mesh (ALE) frame.

37. The mesh of this example is created with the advancing front method on the boundaries. (This is not the default.)

38. These boundaries has a target Maximum element size of 0.004.

39. These boundaries has a target Maximum element size of 0.01.

40. The Global Predefined mesh size is set to Normal.

41. To get a demonstration model that is quick to run and with low memory requirements, Linear Elements are used for both application modes. Note: for reliable stress values, quadratic elements need to be used.

42. To minimize computations, the so called Weak constraints are turned off.
These are important for numerical accuracy when the ALE frame is physics driven and defined by the deformations given by a Solid, Stress-Strain application mode – such as for Fluid-Structure Interaction (FSI). However, here we are rather imposing a known deformation and Weak constraints are not needed.

43. The Weak constraints are on by default and they define additional variables solved for called lagrange multipliers: lm4, lm5, lm6 (or similar). Remove these from the ALE Segregated group so that the variables x y z are the only ones solved for.

44. For the two segregated groups, the PARDISO solver is used.

45. Depending on in what order you defined the problem, you may need to first select Update Model before solving. The reason is to get well-defined initial values for the ALE frame.

46. The Parameter values used for the visualization are controlled from the Plot Parameters dialog box, the General page.

47. To visualize the parameterized geometry solution, you need to select the Frame (ale) as your visualization frame. (Otherwise you will see the stress values mapped back into the CAD model with the original dimensions - here D1=0.)