Method of force integration at nodes

The forces are computed by local integration of the magnetic pressures (dFmagS or dFLapV). According to the methods, these magnetic pressures are either directly integrated at nodes, or computed at the Gauss points in order to obtain forces integrated by elements which will be distributed to the nodes. Whatever the used method, the final forces are always located at the nodes in order to be able to be transferred to OptiStruct or other solvers of mechanics. In advanced mode and depending on the applications, several approaches are available:

  • Direct integration at nodes: In this method, the magnetic pressure, dFmagS or dFlapV, is computed on the nodes of the mesh and is integrated on an equivalent surface viewed by the nodes in order to obtain a force. This surface is defined by the elements surrounding this node.

    Figure 1. Simple uniform and regular first-order mesh with four elements representing three types of nodes; A: a node on a corner of the domain, B: a node in the center of the domain and C: a node on the edge of the domain.

    In figure 1, the blue frame represents the area of the ​​node A, while the green frame represents the area of the​ ​node B. As the magnetic pressure is computed on each node, the force at the node A located in a corner of the finite element can be written as follows:

    F a = P a   S 4

    The coefficient 1/4 represents the ratio between the surface viewed by the node A and the surface of the mesh elements. Noting Pi the pressure on i-th node and S the surface of the element, the forces computed on the nodes B and C are the next ones:

    F b = P b   S             ;             F c = P c   S 2

    This approach is only available for 2D applications, by default this method is set in standard mode.

  • Integration in elements and equi-distribution at nodes: For this method, the forces are initially computed in each element by integration of the magnetic pressure at the Gauss points. Then, these forces are equally distributed between all of the nodes of the elements. In the figure 2, a mesh is represented with four different forces in four elements:

    Figure 2. A mesh with four elements and one force per element.

    In the figure 2, the contribution of forces at nodes A, B and C can be written as follows:

    F a = F 1 4                                           F b = F 1 4 + F 2 4 + F 3 4 + F 4 4                                       F c = F 2 4 + F 4 4

    The node A receives the contribution of one element, the node B receives the contribution of four elements, and the node C receives the contribution of two elements, all equally distributed. This transformation keeps the fact that the sum of the forces at the elements is equal to the sum of the forces at the nodes.

    This approach can be used for all applications. This method is used by default in standard mode for Flux 3D and Flux Skew applications.

  • Integration in elements and extrapolation at nodes: In this approach, the force is also obtained by integration of the magnetic pressure in the elements however, this force is extrapolated at nodes with a statistical average established on the neighboring elements of the node. A coefficient α, equal for all the nodes, is introduced afterwards in order to guarantee that the sum of the forces at the nodes is equal to the sum of the forces at the elements.

    In the case of the figure 2, the force on the node B can be expressed as follows:

    F b = α   F 1 4 + F 2 4 + F 3 4 + F 4 4  

    Then at nodes A and B we have:

    F a = α   F 1                                                         F c =   α   F 2 2 + F 4 2