Electromagnetic Theory and Simulation Computational Electromagnetics Informational

What is the convergence criterion for a finite element electromagnetic simulation?

Q: How many adaptive passes should I run?

Set the maximum number of adaptive passes to 10-15 and let the convergence criterion (delta-S) determine when to stop. Most well-defined problems converge in 4-8 passes. If convergence is not achieved after 10 passes: check for geometry issues, increase the convergence target (e.g., from 0.01 to 0.02), or increase the initial mesh density. For large structures: set a memory limit to prevent the simulation from exceeding available RAM. A 64 GB workstation can handle approximately 5-10 million tetrahedra.

Q: Should I verify convergence manually?

Yes, always perform at least one manual verification: (1) Check a known result: if your model includes a 50-ohm transmission line, verify that the simulated characteristic impedance is 50 ± 2 ohms. If it includes a known resonator, verify the resonant frequency matches the analytical prediction. (2) Run a convergence study: manually increase the mesh density (minimum element size, maximum element size) and re-solve. S-parameters should not change by more than 0.5% between the converged solution and the manually refined solution. (3) Compare with measurement: for an existing design, overlay simulated and measured S-parameters. Agreement within ±0.5 dB / ±5° through the operating band confirms the simulation setup is valid.

Q: What happens if I use a too-loose convergence criterion?

With delta-S = 0.1 (10%): the simulation runs faster (fewer passes, coarser mesh), but results may have 5-10% error in S-parameter magnitudes. For a filter with -20 dB return loss: the error could be ±2 dB (indistinguishable from a -18 dB or -22 dB result). This level of uncertainty makes optimization meaningless. With delta-S = 0.02 (2%): results are accurate to approximately ±0.2-0.5 dB, sufficient for most design work. With delta-S = 0.005 (0.5%): results accurate to ±0.1 dB, needed for precision calibration standards and high-performance filter design. The tighter the convergence, the more mesh elements and longer solve time: delta-S = 0.005 may require 2-4× more elements than delta-S = 0.02.

The convergence criterion for a finite element method (FEM) electromagnetic simulation determines when the solution is sufficiently accurate and further mesh refinement will not significantly change the results. In Ansys HFSS (the most widely used FEM EM solver), the primary convergence criterion is the maximum delta-S: the maximum change in any S-parameter magnitude between successive adaptive mesh refinement passes. Default: delta-S < 0.02 (2% change). For high-accuracy work: delta-S < 0.01 or delta-S < 0.005. The adaptive mesh process: (1) HFSS generates an initial mesh based on the geometry (surface and volume meshes using tetrahedral elements, seeded at approximately lambda/5 element size). (2) Solves Maxwell equations on this mesh at the solution frequency. (3) Calculates the electric field error estimator for each element. (4) Refines the mesh by subdividing elements with the highest error. (5) Re-solves on the refined mesh. (6) Compares S-parameters between the current and previous solution. If max(|delta_S_ij|) < convergence target (e.g., 0.02): the solution has converged. If not: repeat steps 3-6. Typically, 3-8 adaptive passes achieve convergence for well-defined problems. Additional convergence checks: (1) Energy-based convergence: the stored electromagnetic energy should change by <1% between passes. (2) Field convergence: maximum field magnitude change <5% in critical regions. (3) Manual verification: compare results with known analytical solutions (e.g., a rectangular waveguide mode should match theoretical cutoff frequency within 0.1%).

Category: Electromagnetic Theory and Simulation

Updated: April 2026

Product Tie-In: Simulation Software, PCB Materials

FEM Electromagnetic Convergence

Ensuring mesh convergence is the most critical step in obtaining reliable electromagnetic simulation results. An unconverged solution can produce results that are not just inaccurate, but misleading, with errors of several dB in insertion loss or return loss that lead to incorrect design decisions.

Mesh Quality and Initial Seeding

The initial mesh determines the starting point for adaptive refinement: (1) Element size: initial elements should be no larger than lambda/5 at the solution frequency (lambda in the dielectric medium). For a 10 GHz simulation on Rogers 4003C (epsilon_r = 3.55): lambda = c/(f×sqrt(epsilon_r)) = 15.9 mm. Maximum initial element size: 3.2 mm. (2) Feature resolution: the mesh must resolve all geometric features, even those much smaller than a wavelength. Thin dielectric layers (100 μm substrate), narrow traces (0.5 mm), and small gaps (0.1 mm) each need at least 2-3 elements across their smallest dimension. (3) Curved surfaces: curved boundaries (cylindrical vias, circular patches) need sufficient angular resolution. A via with 0.3 mm diameter should have at least 8-12 elements around its circumference. (4) Material boundaries: the mesh must conform to all material interfaces (dielectric-metal, dielectric-dielectric). Mixed-material elements produce large errors. HFSS handles this automatically for imported geometries, but user-defined mesh operations may be needed for complex structures.

Delta-S Convergence Behavior

Typical convergence behavior: Pass 1: coarse mesh (1000-10,000 elements for a small structure). S-parameters may have 20-50% error. Pass 2: refined mesh (2-3× more elements). delta-S = 0.1-0.3 (10-30% change). Pass 3-4: further refinement. delta-S = 0.02-0.05. Pass 5-6: fine refinement. delta-S < 0.01 (converged). Total element count at convergence: 10,000-500,000 for a passive component (connector, filter, transition). 500,000-5,000,000 for a complex module (package, antenna array). Warning signs of poor convergence: (1) delta-S oscillates instead of monotonically decreasing: may indicate insufficient initial mesh, geometry errors (overlapping objects, orphan edges), or numerical issues (port mode mismatch). (2) Mesh grows without convergence: the adaptive refinement keeps adding elements but delta-S remains high. May indicate a resonance at the solution frequency, or a structure that is too electrically large for the available RAM. (3) Negative S-parameter values or S-parameters > 0 dB: indicates a non-physical mesh (violated energy conservation). Re-check boundary conditions and port definitions.

Multi-Frequency Convergence

HFSS adaptive refinement occurs at a single frequency (the solution frequency). For broadband results: (1) Choose the solution frequency at or near the highest frequency of interest (the mesh adequate for the highest frequency automatically satisfies lower frequencies). (2) For narrowband resonant structures (filters, resonators): set the solution frequency at the resonance and verify convergence at band edges separately. (3) Use the "broadband adaptive" option (HFSS 2021+) which refines the mesh at multiple frequencies simultaneously. This is slower but ensures accuracy across the entire band. (4) HFSS interpolating frequency sweep: solves at a few discrete frequencies (adaptively chosen) and interpolates between them. The default maximum delta-S for the interpolation is 0.02. For passive structures with smooth frequency response: interpolation is accurate. For high-Q resonators: direct sweep (solve at every frequency point) is more reliable.

FEM Convergence Equations

Delta-S: max|S_ij(pass n) - S_ij(pass n-1)| < 0.02
Element Size: h < λ/(5√εᵣ)
Mesh Growth: N_elements ∝ (f/f_ref)³
Memory: RAM ≈ 1 GB per 100K tetrahedra
Solve Time: t ∝ N^1.5 to N^2

Common Questions

Frequently Asked Questions

How many adaptive passes should I run?

Set the maximum number of adaptive passes to 10-15 and let the convergence criterion (delta-S) determine when to stop. Most well-defined problems converge in 4-8 passes. If convergence is not achieved after 10 passes: check for geometry issues, increase the convergence target (e.g., from 0.01 to 0.02), or increase the initial mesh density. For large structures: set a memory limit to prevent the simulation from exceeding available RAM. A 64 GB workstation can handle approximately 5-10 million tetrahedra.

Should I verify convergence manually?

Yes, always perform at least one manual verification: (1) Check a known result: if your model includes a 50-ohm transmission line, verify that the simulated characteristic impedance is 50 ± 2 ohms. If it includes a known resonator, verify the resonant frequency matches the analytical prediction. (2) Run a convergence study: manually increase the mesh density (minimum element size, maximum element size) and re-solve. S-parameters should not change by more than 0.5% between the converged solution and the manually refined solution. (3) Compare with measurement: for an existing design, overlay simulated and measured S-parameters. Agreement within ±0.5 dB / ±5° through the operating band confirms the simulation setup is valid.

What happens if I use a too-loose convergence criterion?

With delta-S = 0.1 (10%): the simulation runs faster (fewer passes, coarser mesh), but results may have 5-10% error in S-parameter magnitudes. For a filter with -20 dB return loss: the error could be ±2 dB (indistinguishable from a -18 dB or -22 dB result). This level of uncertainty makes optimization meaningless. With delta-S = 0.02 (2%): results are accurate to approximately ±0.2-0.5 dB, sufficient for most design work. With delta-S = 0.005 (0.5%): results accurate to ±0.1 dB, needed for precision calibration standards and high-performance filter design. The tighter the convergence, the more mesh elements and longer solve time: delta-S = 0.005 may require 2-4× more elements than delta-S = 0.02.

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