RIA: Algebro-Topological Methods for the Formulation and Numerical Solution of Three Dimensional Magnetoquasistatic Problems

微分形式和代数拓扑的使用在理论物理中已成为不可缺少的。将这种数学形式应用于麦克斯韦方程组的边值问题，可以清晰地表达工程电磁学中的拓扑问题，并给出向量场与集总参数之间的一般关系。在最大的计算机上计算三维电磁场是一个具有挑战性的研究领域，它也受益于这种数学形式。本研究将通过研究在多重连接的无电流区域中自动寻找磁标量势“切割”的算法的实现，扩展该领域的计算工程状态；o变分原理的实际应用，产生一阶欧拉-拉格朗日磁场方程，从而避免使用矢量势；o“三维同调球的Heegard分裂”的代数拓扑，以便开发处理三维涡流问题拓扑方面的算法，其中磁标量势在电流自由区域使用。

The use of differential forms and algebraic topology have become indispensible in theoretical physics. Application of this mathematical formalism to boundary value problems of Maxwell's equations results in a clear articulation of topolocial problems in engineering electronmagnetics and gives general relationships between vector fields and lumped parameters. The calculation of three dimensional electromagnetic fields on the largest of computers is a challenging research area which also benefits from this mathematical formalism. This research will extend the state of computational engineering in this area by investigating o the implementation of algorithms which automatically find "cuts" for magnetic scalar potentials in multiply connected current free regions; o the practical use of variational principles which yield first order Euler- Lagrange equations of the magnetic field, thereby avoiding the use of a vector potential; o the algebraic topology of "Heegard Splitting of three dimensional homology spheres" in order to develop algorithms for handling topological aspects of three dimensional eddy current problems where a magnetic scalar potential is used in current free regions.