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Current Continuity Enforcement in First Order Locally Corrected Nystrom Method via RWG Moment Method
| Content Provider | Semantic Scholar |
|---|---|
| Author | Shafieipour, Mohammad |
| Copyright Year | 2012 |
| Abstract | Higher-order (HO) methods for solution have advantage over their low-order (LO) counterparts in their ability to effectively control the accuracy of solution beyond one or two digits. Such methods require maintaining the desired accuracy throughout all the stages of numerical solution such as mesh based representation of geometry, choice of basis functions approximating the unknown field, rules for integration of reaction integrals, evaluation of matrix-vector products, and possibly others. For the HO solution of the integral equations various element-based (Method of Moments (MoM) like) [Grag97] and point-based [Wand98] schemes have been proposed. As the point-based methods such as Locally Corrected Nystrom (LCN) method [Gedn03] are particularly suitable for efficient acceleration with Fast Multipole Method (FMM) they are method of choice when HO solution of large scale problems is required. The important issue in construction of an HO discretization scheme is the enforcement of continuity in the approximated field at the boundaries between the elements of the mesh. It has been reported that while the enforcement of continuity at the boundaries between elements makes only minor impact on the solution accuracy at the HOs (orders 3 and higher), incorporation of field continuity at LOs (e.g. order 1 and 2) effects the accuracy profoundly. In the class of element-based HO schemes this issue was addressed via construction of hierarchical basis function spaces which maintain field continuity at all orders [Jorg04]. The enforcement of continuity at LOs in the point-based schemes such as LCN, however, has not been addressed. As a result, a 1st order LCN scheme, for example, despite utilizing four times as many degrees of freedom per element compared to its 1st order RWG MoM counterpart, yields substantially lower solution accuracy. Such poor efficiency of LCN at LOs compared to RWG MoM hardly justifies its usage unless the orders are high. In this work we show that the loss of accuracy problem in 1st order LCN can be remedied if it is made equivalent to RWG MoM. To establish the equivalence, we first, choose barycentric coordinates for definition of position-vector and associated covariant basis vectors on each triangle in LCN. We then choose as the equivalent basis functions in LCN a complete to order one set of monomials over barycentric coordinates for expansion of the surface current over the two co-variant components of the vector basis. This method yields two benefits. First, since it is entirely based on the matrices computed in the 1st order LCN, it allows for 4x reduction in degrees of freedom and elimination of poor accuracy in 1st order LCN codes via simple pre- and post-multiplication of LCN generated matrices with sparse basis transformation matrices. Second, the method can be viewed as a method for point-based discretization of the RWG MoM offering improved efficiency in its acceleration with FMM. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://umanitoba.ca/faculties/engineering/departments/ece/pdf/2012_Mohammad_Shafieipour.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
