In a 3D geometry, three selected control points are given by their radius-vectors rj. Using these points, a local coordinate system is defined with the origin at point 1 and base vectors given by

The displacement vectors in the local system at point (j) are computed as

where uj are displacement vector DOF at the corresponding point.

Here, a is the translational acceleration vector and o is the angular acceleration vector.

During the training study step, unit linear or angular accelerations are applied as

In the seventh and last load case, both a and o are set to zero. Thus, the external load is present in all load cases.

at point 1, which constrains translations in directions e1, e2, and e3. The corresponding three scalar constraints are numbered as 1,2,3. At point 2,

which constrains rotations around axes e3 and e2. The corresponding two scalar constraints are numbered as 4,5. Finally,

at point 3, which constrains rotation around axis e1. This scalar constraint is numbered as 6.

where the first index (m) is the number of the constraint, and the second index (n) is the number of the load case. For load case 7, the constraint force components are represented by a 6-component vector

with the values of f and F precomputed during the training study. Applying the corresponding frame acceleration in the global system will result into zero constraint forces for all six constraints.

In the case of 2D geometry, only the constraints 1,2 and 4 are needed. The corresponding load cases are 1,2 and 6. Thus, f can be reduced to a 3x3 matrix, and F will be a 3-component vector.