Getting Started With Finite Element Analysis In Electromagnetics
I picked up Jin Jian Ming's textbook when I was building simulation tools for microwave components. The third edition from 2014 is still useful for understanding the math behind what most people treat as black-box software. But the book has quirks that catch beginners off guard. This book covers vector finite elements for Maxwell's equations, edge elements, and boundary conditions in electromagnetics. It walks through the derivation of matrix equations from differential forms, then shows implementations for waveguides, antennas, and scattering problems. The math is solid, but you have to work through it yourself. I've seen people treat the worked examples as gospel without checking the derivations. One thing the book doesn't emphasize enough: the difference between nodal and edge elements is critical for electromagnetic problems. Nodal elements fail on problems with singularities or discontinuous fields. Edge elements fix this by placing degrees of freedom on edges rather than nodes. If you're simulating structures with sharp corners or material interfaces, you need edge elements or your solution will blow up. I learned this the hard way trying to model a microstrip discontinuity.
The edge element formulation uses basis functions defined along edges that enforce continuity of tangential field components across element boundaries. This is non-negotiable for accurate electromagnetic modeling. The book covers this in Chapter 5, but the physical intuition behind why edge elements work better isn't obvious from the math alone.
Setting Up Your First Simulation
Start with a simple waveguide problem before tackling complex geometries. Build a rectangular waveguide in TE10 mode, verify the cutoff frequency against the analytical solution, then add complexity. If you skip this step, debugging becomes a nightmare. Here's what most people miss: mesh density matters more than you think. A coarse mesh can give reasonable results for simple problems but will fail spectacularly for resonant structures. I typically use at least six elements per wavelength in regions where fields vary rapidly. Near boundaries and material interfaces, you need even finer meshing. The book mentions mesh refinement but doesn't stress how aggressive it needs to be for convergence. Another practical issue: boundary conditions. Perfect electric conductor (PEC) boundaries are straightforward. Perfect magnetic conductor (PMC) boundaries are rare but sometimes necessary for symmetry. Radiation boundaries require absorbing boundary conditions (ABC) or perfectly matched layers (PML). The book covers ABC in Chapter 8, but PML is often better for high-frequency problems. I switched to PML for most of my simulations and saw faster convergence with fewer reflections.
Get the Full Details
If you're implementing this yourself rather than using commercial software, pay attention to the assembly process. Element matrices must be assembled into the global system correctly. A common mistake: forgetting to transform local element matrices to global coordinates. The book has the transformation formulas, but it's easy to mess up the Jacobian calculation for curved elements.
Debugging Common Problems
Singularities are the biggest headache in electromagnetic FEM. Fields go to infinity at sharp corners, recessed corners, and material junctions. Standard FEM can't handle these. You either need to use specialized basis functions or accept that your solution won't converge near the singularity. I spent weeks trying to get a re-entrant corner to converge before realizing I needed to move the observation point away from the corner. Spurious modes are another issue. These are non-physical solutions that appear in edge element formulations when the mesh is too coarse or the boundary conditions are wrong. The book discusses them briefly, but the practical fix isn't always obvious. I found that refining the mesh near boundaries and using higher-order elements eliminated most spurious modes. Sometimes you need to add a small regularization term to stabilize the system matrix. Convergence testing is essential but often skipped. Run your simulation with progressively finer meshes and check that results stabilize. If they don't, something is wrong with your formulation or boundary conditions. I typically run at least three mesh densities and look for less than one percent change in the quantity of interest between the last two runs. Anything more suggests underlying issues.
Practical Tips From Experience
Use the book's worked examples as a baseline. I solved all of them before attempting original problems. This builds intuition for what correct solutions look like and helps you catch implementation errors early. The examples cover waveguides, resonant cavities, and scattering problems. Work through each one carefully before moving on. When setting up the system matrix, check that it's symmetric if your problem should produce a symmetric matrix. Asymmetric matrices often indicate boundary condition errors or incorrect element connectivity. I've spent hours debugging asymmetric systems only to find a reversed element orientation. The book mentions matrix properties but doesn't stress how much they reveal about implementation errors. For time-domain simulations, the book covers the Galerkin method in frequency domain. Time-domain FEM requires additional considerations: numerical dispersion, stability criteria, and time integration schemes. If you need time-domain results, consider supplementing this book with a text focused on transient FEM. The 2014 edition doesn't cover time-domain methods extensively.
Commercial software like COMSOL, ANSYS HFSS, and CST Studio use similar formulations under the hood. Understanding the math from this book helps you interpret results and identify when software is giving you garbage. I've seen engineers trust simulation results without questioning whether the mesh was adequate or the boundary conditions were correct. The book teaches you to question everything. One practical workaround I developed: when modeling open region problems, use a truncation boundary far enough from your structure that reflected waves don't interact with your region of interest. I typically place boundaries at least half a wavelength away and use PML to absorb residual reflections. This approach works for most antenna and scattering problems but may need adjustment for highly resonant structures. If you're working with anisotropic materials or metamaterials, the book's coverage is limited. You'll need to extend the formulation yourself or find supplementary references. I found papers by Jin himself on anisotropic FEM that built on this book's foundation. Those papers filled gaps the textbook didn't address.
The appendix with computer programs is useful but dated. I translated the Fortran code to Python for my own work. This helped me understand the implementation details and adapt the code for modern processors. If you're serious about using this book, plan to spend time rewriting or extending the examples rather than just reading them.
When This Book Falls Short
Don't expect this book to cover every electromagnetic problem. Complex geometries, nonlinear materials, and multiphysics coupling require additional techniques beyond what's presented here. I've encountered situations where I needed to combine FEM with method of moments or finite-difference time-domain methods. The book doesn't address hybrid approaches. High-frequency problems above ten gigahertz can be challenging. Mesh requirements become extreme, and computational costs skyrocket. For these regimes, you might need asymptotic methods or specialized FEM variants. I've had better luck with finite integration technique (FIT) for very high frequencies where FEM becomes impractical. The book assumes familiarity with vector calculus and partial differential equations. If you're weak in these areas, you'll struggle with the derivations. I recommend reviewing Green's theorem, Stokes' theorem, and the Helmholtz decomposition before diving in. The math moves quickly without much hand-holding.
Modern research has advanced beyond what's covered in this 2014 edition. Isogeometric analysis, adaptive mesh refinement, and GPU acceleration are active research areas. If you need cutting-edge techniques, supplement this book with recent journal articles. I subscribe to IEEE Transactions on Microwave Theory and Techniques for updates on FEM developments.
