Conventional methods for determining the electromagnetic force acting on magnetic materials include the equivalent current method, the equivalent magnetization method, and Maxwell's stress method. Although these methods are effective in determining the total electromagnetic force or torque acting on a magnetic material, they are physically meaningless because the spatial distribution of electromagnetic force is different for each method. Maxwell's stress method has been widely used, but it requires the definition of an integration plane, and its accuracy varies depending on where the plane is set, making it a cumbersome problem.

The energy method is one way to determine the local electromagnetic force.^[30] This method involves the variation of Energy or Co-energy with respect to the virtual displacement on an element-by-element basis to obtain a local force distribution. In this variation, the A-φ method must be performed with the magnetic flux preserved, i.e., the line-integral value of A is preserved. For edge elements, this value is assigned to the edge, so it is possible to perform the variation while preserving the magnetic flux. Conversely, there are theoretical difficulties in using the energy method with the nodal method.

Analysis of the energy method reveals that its form is the same as the Maxwell stress tensor multiplied by the nodal shape function for virtual displacements and integrated. It has the same form as the nodal force used in structural analysis (the distributed force is converted to a force on the nodes as a weighting condition for the finite element method). In this form, if the Maxwell's stress tensor is known, the local force can be derived as the force acting at the nodal point simply by integrating it. We call this the nodal force method, and it can be proved to be equivalent to the energy method in the case of the edge element method^[15]. The advantage of the nodal force method is that it is much easier to derive than the energy method and be applicable to nodal elements as well as the edge element method, but it is basically the same method.

It is important to note that in the nodal force method, the Maxwell stress tensor is determined by the magnetic field ($B$,$H$). When the magnetic material is linear, there is little objection, but when it is nonlinear, and especially when there is hysteresis, we have not obtained a clear theoretical formulation. In the future, if the distribution of electromagnetic forces in a magnetic body has an effect on deformation and other problems, this will become an issue.

EMSolution uses this nodal force method to determine the electromagnetic force of magnetic materials, but there is a problem. The magnetic field in a magnetic material is often concentrated at a corner point. In this case, the nodal force there is much larger than at other locations, and the accuracy of the electromagnetic force is dominated by the nodal force there. However, the accuracy of analysis of magnetic fields around corner points is generally not so good. In particular, when magnetic materials are pulled from both sides and almost balanced, the total electromagnetic force may be smaller than the electromagnetic force acting on a single node, and in such cases the cancellation error becomes large, making it difficult to accurately determine the total electromagnetic force. One solution to this problem is to extend the range of integration of the nodal forces to the air region around the magnetic material until the magnetic field is smooth and the calculation is accurate. This is the same as the Maxwell stress method, where the integration plane is set away from the magnetic material. However, how far to extend the integration must be determined empirically.

In the nodal force method, the force is output as a nodal quantity and is used to pass electromagnetic forces to stress and vibration analysis, as well as electromagnetic forces in plungers and actuators and torque in motors. In stress and vibration analysis, as mentioned earlier, forces are treated as nodal forces, so the nodal force method seems to be rather convenient.