General approach to prove the existence and uniqueness of the solution in vector potential formulations of 3-D eddy current problems

摘要

Gauging of vector potential formulations of 3-D eddy current problems is considered from a new viewpoint. Assuming that the formulation in terms of E and B of the eddy current problem under consideration has a unique solution, gauging can be regarded as setting conditions on potentials, say A, V and /spl psi/, that ensure a one-to-one correspondence between pairs (E,B) and triples (A,V,/spl psi/). This approach decouples two different issues which are usually mixed: gauging properly said and well-posedness of the field problem. Once the above one-to-one correspondence is proved, the existence and uniqueness of (A,V,/spl psi/) will then automatically follow from that of (E,B) whenever the field problem is well-posed. Proving that the correspondence between (E,B) and (A,V,/spl psi/) is one-to-one is a definitely easier task than proving the existence and uniqueness of (A,V,/spl psi/) directly. The approach presented applies to both A,V, and T,/spl psi/ formulations and to problems ranging from magnetostatics to high frequency electromagnetics; no additional complexity comes out when dealing with nonlinear materials. Practical impact of the proposed method is discussed.