A Geometrical Comparison between Cell Method and Finite Element Method in Electrostatics

Cell Method, a Finite Formulation technique, is compared in detail with the Finite Element Method (FEM), a differential-based numerical technique. In the finite formulation technique, Poisson’s equation is described starting from a topological foundation. The final set of algebraic equations resulting from the two approaches are compared in matrix form. The equivalence of the coefficient matrices is proven for a Voronoi dual mesh and linear shape functions in the FEM. The difference between the source (charge) vectors in the two approaches is described. It is shown that the use of linear shape functions in the FEM is equivalent to the use of a barycentric dual mesh for charge vectors. Also, it is shown that the coefficient matrix derived from a variational technique, FEM, can be interpreted using the simple geometrical concept of a parallel plate capacitor. As an example, a Schottky barrier diode with a non-uniform doping profile is considered. The results demonstrate the differences between the two approaches in the case of a nonzero electric charge density. Keyword: Cell Method, Electrostatics, Finite Element Method, Numerical techniques, Poisson’s Equation.

[1]  L. Kettunen,et al.  Geometric Interpretation of Discrete Approaches to Solving Magnetostatics , 2003 .

[2]  Raj Mittra,et al.  A theoretical study of the stability criteria for hybridized FDTD algorithms for multiscale analysis , 2004, IEEE Transactions on Antennas and Propagation.

[3]  C. Christopoulos,et al.  The Transmission-line Modeling Method: TLM , 1995, IEEE Antennas and Propagation Magazine.

[4]  Ruben Specogna,et al.  Discrete Constitutive Equations in - Geometric Eddy-Current Formulation , 2005 .

[5]  M. Marrone,et al.  COMPUTATIONAL ASPECTS OF THE CELL METHOD IN ELECTRODYNAMICS , 2001 .

[6]  C. Mattiussi.  An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology , 1997 .

[8]  Matthew N. O. Sadiku,et al.  A further introduction to finite element analysis of electromagnetic problems , 1989 .

[9]  E. K. Miller,et al.  A selective survey of computational electromagnetics , 1988 .

[10]  Alain Bossavit,et al.  Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches , 2000 .

[11]  M. Marrone,et al.  A novel numerical approach for electromagnetic scattering: the cell method , 2002, IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No.02CH37313).

[12]  Rolf Schuhmann,et al.  解説 Discrete Electromagnetism by the Finite Integration Technique , 2002 .

[13]  P. Bettini,et al.  Electrostatic analysis for plane problems with finite formulation , 2003 .

[14]  R. Mittra,et al.  A new technique based on the cell method for calculating the propagation constant of inhomogeneous filled waveguides , 2004, IEEE Antennas and Propagation Society Symposium, 2004..

[15]  Numerical solution of transient electromagnetic scattering problems using the novel time-domain cell method , 2002, 2002 IEEE International Symposium on Electromagnetic Compatibility.

[16]  Enzo Tonti,et al.  A Direct Discrete Formulation of Field Laws: The Cell Method , 2001 .

[17]  A. Bossavit.  How weak is the "weak solution" in finite element methods? , 1998 .

[18]  Francesca Cosmi.  Elastodynamics with the Cell Method , 2005 .

[19]  Application of the Cell Method to the Simulation of Unsaturated Flow , 2006 .

[20]  H. Hernández-Figueroa,et al.  Novel numerical method for the analysis of 2D photonic crystals: the cell method. , 2002, Optics express.