FEM-based Solver for RF Structures

The finite element method (FEM) solver in RF PCB Toolbox™ computes electric and magnetic fields across RF structures. Use this solver to model and analyze PCB components with different shapes and geometries and to find a numerical solution to complex mathematical formulations of these structures. This topic explains the mathematical formulation of the FEM to electromagnetically solve RF structures in RF PCB Toolbox.

To solve time-harmonic Maxwell's equations in the computational domain  $$\Omega$$, this implementation of the FEM uses hierarchical curl-conforming finite elements (FEs). Basis functions used in these FEs belong to the $$H(curl;\Omega )$$ space. In general, curl-conforming basis function ensure tangential continuity of the field across interfaces between adjacent elements [2, 3]. Maxwell's curl equations dictate this type of field continuity. The curl-conforming functional space $$H(curl;\Omega )$$, is mathematically described as

$$H(curl;\Omega )\triangleq \{f\text{|}f\in L^2(\Omega ),curl(f)\in L^2(\Omega )\}$$

, where $$L^2(\Omega )$$ is the space of square-integrable functions defined over $$\Omega$$. For more information on functional spaces and their application in FEM, please refer to [1].

Weak Formulation

Using the $$e^{j\omega t}$$ anastaz, Maxwell's-curl equations are written as

$$\{\begin{array}{c}curl(E)=-M-j\omega \overline{\overline{\mu }}H\\ curl(H)=+J+j\omega \overline{\overline{\epsilon }}E\end{array}$$

.

Assume a total field formulation that does not separate incident and scattered field contributions in $$E$$ or $$H$$. Also, assume that an absorbing boundary condition of the form, $$\mathfrak{B}(E-E^{inc})=0$$ is imposed on the boundary of $$\Omega$$. This absorbing boundary condition truncates the computational domain while minimizing any artificial reflections from the truncating boundary. In the figure, observe that $${\gamma }^T(E)=0$$ enforces the required field condition on any of the perfect electric conductor (PEC) and $${\gamma }^T(E)=E^{src}$$ enforces any Dirichlet-type boundary-source conditions. A Dirichlet-type boundary-source condition defines a source as a known tangential $$E$$-field. In electrodynamics, when $$\omega \ne 0$$, the two curl equations completely describe the underlying physics without the need for independent enforcement of the divergence equations. To form the proper weak form corresponding to the curl equations, take the curl of the first equation and substitute the resulting $$curl\text{(}H)$$ using the second curl equation.

$${\overline{\overline{\mu }}}_r^{-1}curl(E)=-{\overline{\overline{\mu }}}_r^{-1}M-j\omega {\mu }_{0}H,$$

$$curl({\overline{\overline{\mu }}}_r^{-1}curl(E))=-curl({\overline{\overline{\mu }}}_r^{-1}M)-j\omega {\mu }_{0}curl(H),$$

$$curl({\overline{\overline{\mu }}}_r^{-1}curl(E))=-curl({\overline{\overline{\mu }}}_r^{-1}M)-j\omega {\mu }_{0}J+{\omega }^2{\mu }_0{\epsilon }_0{\overline{\overline{\epsilon }}}_{r}E.$$

Similarly, for the $$H$$-field formulation, write

$$curl({\overline{\overline{\epsilon }}}_r^{-1}curl(H))=curl({\overline{\overline{\epsilon }}}_r^{-1}J)-j\omega {\epsilon }_{0}M+{\omega }^2{\epsilon }_0{\mu }_0{\overline{\overline{\mu }}}_{r}H.$$

For an $$E$$-field formulation, test the resulting equation with an appropriate testing function $$T$$. To test this equation, multiply the equation with $$T^{*}$$ (complex conjugate of $$T$$) and take the integral over the entire problem domain, $$\Omega$$.

$$T^{*}\cdot curl({\overline{\overline{\mu }}}_r^{-1}curl(E))=-T^{*}\cdot curl({\overline{\overline{\mu }}}_r^{-1}M)-j\omega {\mu }_0{T}^{*}\cdot J+{\omega }^2{\mu }_0{\epsilon }_0{T}^{*}\cdot {\overline{\overline{\epsilon }}}_{r}E$$

$$T^{*}\cdot curl({\overline{\overline{\mu }}}_r^{-1}curl(E))-k_0^2{\langle T,{\overline{\overline{\epsilon }}}_{r}E\rangle }_{\Omega }=-{\langle T,curl({\overline{\overline{\mu }}}_r^{-1}M)\rangle }_{\Omega }-j\omega {\mu }_0{\langle T,J\rangle }_{\Omega }$$

Using the vector identity

$$\text{div(}A\times B\text{)=}B\cdot curl(A)-A\cdot curl(B)$$

and a bra-ket notation

$${\langle a,b\rangle }_{\Omega }\triangleq {\displaystyle {\int }_{{}_{\Omega }}{a}^{*}\cdot b}d\Omega$$

you arrive at

$${\langle curl(T),{\overline{\overline{\mu }}}_r^{-1}curl(E)\rangle }_{\Omega }-k_0^2{\langle T,{\overline{\overline{\epsilon }}}_{r}E\rangle }_{\Omega }+{\langle T,\widehat{n}\times {\overline{\overline{\mu }}}_r^{-1}curl(E)\rangle }_{\partial \Omega \backslash {\Gamma }_D}=-{\langle T,curl({\overline{\overline{\mu }}}_r^{-1}M)\rangle }_{\Omega }-j\omega {\mu }_0{\langle T,J\rangle }_{\Omega }$$

.

Note that the bra-ket is conjugate-linear in the first variable and linear in the second variable. The equivalent $$H$$-field formulation is

$${\langle curl(T),{\overline{\overline{\epsilon }}}_r^{-1}curl(H)\rangle }_{\Omega }-k_0^2{\langle T,{\overline{\overline{\mu }}}_{r}H\rangle }_{\Omega }+{\langle T,\widehat{n}\times {\overline{\overline{\epsilon }}}_r^{-1}curl(H)\rangle }_{\partial \Omega \backslash {\Gamma }_D}=+{\langle T,curl({\overline{\overline{\epsilon }}}_r^{-1}J)\rangle }_{\Omega }-j\omega {\epsilon }_0{\langle T,M\rangle }_{\Omega }$$

.

The test functions $$T$$ and the trial functions $$E$$ belong to the space of curl-confirming functions defined over $$\Omega$$ that is $$H(curl;\Omega )$$. Solving the weak form equations is equivalent to solving the original curl equations if and only if the weak form equations are satisfied for

$$\forall E\in H(curl;\Omega ),\forall T\in H(curl;\Omega ).$$

When forming matrix equations corresponding to the weak form, remember that the testing functions $$T$$ are chosen to have zero trace on any Dirichlet boundary where there is explicit knowledge of the $$E$$ field in the form

$$E=E_{\text{known}},{\Gamma }_D.$$

or the $$H$$ field in the form of

$$H=H_{\text{known}},{\Gamma }_D.$$

This means that the $$\langle T,\widehat{n}\times {\overline{\overline{\mu }}}_r^{-1}curl(E)\rangle$$ term in $$E$$-field formulation and the $$\langle T,\widehat{n}\times {\overline{\overline{\epsilon }}}_r^{-1}curl(H)\rangle$$ term in $$H$$-field formulation vanish by the choice of testing functions. On other types of boundaries, these terms are manipulated according to the requirements. For example, in the $$E$$-field formulation and in the presence of a Neumann boundary condition where

$$H=H_{\text{known}},{\Gamma }_N$$

,

the $$\langle T,\widehat{n}\times {\overline{\overline{\mu }}}_r^{-1}curl(E)\rangle$$ term is replaced by

$$\langle T,-j\omega {\mu }_0\widehat{n}\times H_{\text{known}}\rangle$$

and moved to the RHS of the equation.

References

[1] Leszek Demkowicz J. Tinsley Oden, Applied Functional Analysis, CRC Press, 2023, ISBN: 0-471-43818-9.

[2] Nedelec, “Mixed Finite Elements in R3”, in Numerische Mathematik, Vol. 35, 1980, pp. 315–341.

[3] Ivo Dolezel Pavel Solin Karel Segeth, Higher-Order Finite Element Methods, Chapman & Hall, CRC, 2003, ISBN: 9781584884385.