2. Kinematic Coupling with B2000++

2.1. I-Profile Beam (Solid Model)

This report demonstrates the capabilities of B2000++ for dealing with non-matching finite element grids composed of solid elements. Coupling of non-matching grids is also referred to as 'tying'. While the grid nodes of two non-matching grids do not coincide, the surface on which the grid nodes are placed must more or less coincide. B2000++ pro couples non-matching grids with a kinematic coupling method referred to as the common mesh refinement method[32]. All solutions obtained with non-matching grids are compared to reference solutions.

The I-profile (see Figure 83) beam structure of length L is meshed with solid elements patches, an upper and a lower flange patch and a web patch. Due to symmetry in loading and boundary conditions, one half of the beam is modelled. The following 4 load cases are studied:

  • Case 1 (mechanical loads): The beam upper flange is loaded with a pressure load and is clamped at x=0 and X=L.

  • Case 2 (Thermal loads): The beam is subjected to differential thermal loading, setting the lower flange to -50° C and the upper flange to +50° C . The coupled static heat and deformation analysis is performed in 2 steps, a heat analysis followed by a deformation analysis retrieving the temperature distribution of the previous heat analysis. The beam is clamped at x=0 and X=L, thus generating stresses, but no deformations.

  • Case 3 (mechanical loads): The beam upper flange is loaded with a pressure load and is clamped at x=0 and free at x=L.

  • Case 4 (Thermal loads): The beam is subjected to differential thermal loading, setting the lower flange to -50° C and the upper flange to +50° C . The coupled static heat and deformation analysis is performed in 2 steps, a heat analysis followed by a deformation analysis retrieving the temperature distribution of the previous heat analysis. The beam is clamped at x=0 and free at x=L, creating bending with the give temperature distribution. It is interesting to note that (1) the deformations for this case, at least with beam theory, are independent of the cross section shape, the only intervening parameter being the height and (2) the beam is stress-free, since it can deform freely. This case has a simple beam theory solution for the beam tip displacement.

Beam section profile

Figure 83. Beam section profile


Two models with different mesh types are tested:

  • Mesh A: Continuous mesh model, where the element of the 3 patches match and where the patches are connected with the MDL join command. This model is the reference model. Model A is also tested with the non-matching method, replacing the coupling command join by field_transfer.

  • Mesh B: Discontinuous (non-matching) mesh model, where the 3 patches are 'tied' together with the common mesh refinement kinematic coupling method through the MDL field_transfer command.

Both meshes have a similar mesh density. The meshes A and B are obtained from the same B2000++ epatch surface description with 3 patches:

Mesh A: B2000 epatch views of lower flange (yellow), web (green), and upper flange(red). Tie interface points shown in black.

Figure 84. Mesh A: B2000 epatch views of lower flange (yellow), web (green), and upper flange(red). Tie interface points shown in black.


Mesh A is selected such that the mesh nodes at the interface are matching, thus allowing for direct comparison of the tie and join models. Figure 85 displays the continuous mesh A of the structure.

Mesh A. Continuous solid mesh.

Figure 85. Mesh A. Continuous solid mesh.


2.1.1. Mesh B

Mesh B is selected such that the mesh nodes at the interfaces are not matching and thus can be used with kinematic coupling only. The quality of the solution is then compared to mesh A. Figure 86 displays the continuous mesh A of the structure, the mesh density of the web in x-direction is 66% of the one of mesh A.

Mesh B: Non-matching solid mesh.

Figure 86. Mesh B: Non-matching solid mesh.


Results for condition C1 are summarized in the table below, demonstrating the efficiency of the kinematic coupling method, the tip displacements obtained with the 3 methods being practically identical.

Table 14. Deformation solutions for loading condition C1: Tip displacement in z direction, at y=0.

Method z-displacement
Mesh A: Join -0.00522
Mesh A: Tie (kinematic coupling, matching meshes) -0.00522
Mesh B: Tie (kinematic coupling, non matching meshes) -0.00427
Beam theory (Euler) -0.00445


The temperature distribution for mesh B and loading condition C2 resulting from the heat analysis is displayed in the figure below. Deformations in the global z-direction for the whole model are 0, as expected.

Mesh B, loading condition C2: Heat analysis solution (temperatures).

Figure 87. Mesh B, loading condition C2: Heat analysis solution (temperatures).


Mesh B, loading condition C1: Stresses Sxx (in x-direction).

Figure 88. Mesh B, loading condition C1: Stresses Sxx (in x-direction).




[32] X. Jiao, M. T. Heath, Common-refinement-based data transfer between non-matching meshes in multiphysics simulations, Int. J. Numer. Meth. Engng. 2004; 61