Electromagnetic Theory and Simulation Computational Electromagnetics Informational

What is the proper mesh density for an accurate electromagnetic simulation at millimeter wave frequencies?

Q: How do I know if my mesh is fine enough?

The convergence test: (1) Run the simulation with the default (adaptive) mesh. Record the S-parameters. (2) Request 2-3 additional mesh refinement passes. Record the S-parameters after each pass. (3) If the S-parameters change by 0.02: continue refining. An alternative: mesh sensitivity study. Manually set the maximum mesh size to lambda/10, solve. Reduce to lambda/15, solve. Reduce to lambda/20, solve. Plot S-parameters vs mesh density. The solution is converged when further refinement produces < 0.01 change in S11 and S21.

Q: Can I reduce the simulation size?

Techniques for reducing computational cost: (1) Symmetry: if the structure has a plane of symmetry: use a symmetry boundary condition (electric wall or magnetic wall). This halves the simulation volume (and approximately halves the mesh count and computation time). (2) Port truncation: only simulate the region of interest (a few wavelengths around the DUT). Terminate the ports with proper boundary conditions. (3) 2.5D instead of 3D: for planar structures (microstrip, stripline, coupled lines): use a 2.5D solver (ADS Momentum, Sonnet). The 2.5D solver meshes only the conductor surfaces (not the volume), dramatically reducing the mesh count. Accuracy: 2.5D is very accurate for planar structures but cannot handle 3D features (vertical walls, waveguide bends, wire bonds). (4) Sub-structure analysis: simulate complex structures in sections (e.g., simulate the via transition separately from the matching network). Import the S-parameters of each section into the circuit simulator and cascade them.

Q: What about de-embedding in EM simulation?

EM simulators include access structures (port transitions, feed lines) that are not part of the DUT. De-embedding removes the effect of these access structures. Methods: (1) Port calibration (built into most simulators): the simulator automatically accounts for the port transition. (2) TRL de-embedding: simulate separate Thru, Reflect, and Line structures. Use the TRL algorithm to de-embed the port effects. (3) Port extension: specify the length of the feed line, and the simulator mathematically removes its effect (assuming a perfect transmission line). De-embedding is essential for: accurate extraction of S-parameters for circuit simulation, comparing simulated results with measured data (the measurement also requires de-embedding).

The mesh density in an electromagnetic simulation determines the accuracy and computational cost of the simulation. At millimeter wave frequencies (30-300 GHz), the short wavelengths require extremely fine meshes: (1) Fundamental rule: the maximum mesh element size should be less than lambda/10 (one-tenth of the wavelength at the highest frequency of interest). At 77 GHz (lambda = 3.9 mm in free space): mesh size < 0.39 mm. At 77 GHz on a substrate (epsilon_r = 3.55, lambda_eff = 2.07 mm): mesh size < 0.207 mm. This is the starting point, but many structures require even finer meshes. (2) Adaptive meshing vs fixed mesh: most modern EM simulators (HFSS, CST, ADS Momentum) use adaptive mesh refinement. The simulator starts with a coarse mesh, solves, refines the mesh where the fields change rapidly (near conductors, edges, vias, gaps), and re-solves until the solution converges. Convergence criterion: the S-parameter change between successive mesh refinements should be < 0.02 (or 0.01 for critical designs). The adaptive approach automatically concentrates mesh elements where they are needed (edges, vias, narrow gaps) without wasting elements in smooth-field regions. (3) Critical areas requiring fine mesh: conductor edges (the field is singular at sharp edges; the mesh must be very fine to capture this). Via barrels and anti-pads (the field varies rapidly across the via cross-section). Narrow gaps (gaps < 0.1 mm between conductors have strong capacitive fields). Thin substrates (a substrate thinner than lambda/20 requires multiple mesh elements through the thickness). Skin depth (at 77 GHz: skin depth in copper = 0.24 um; the mesh must resolve this thin layer for accurate loss calculation; however, impedance boundary conditions are typically used instead of volumetric meshing of the conductor). (4) FDTD-specific mesh density: FDTD uses a uniform Cartesian grid. The cell size is determined by the smallest feature and the highest frequency. At 77 GHz with 0.1 mm traces: cell size should be approximately 0.02-0.05 mm (the trace width divided by 5-10). For a 10 × 10 × 2 mm simulation volume: number of cells = (10/0.03)³ ≈ 37 million cells. This requires significant memory (> 10 GB) and computation time (hours).

Category: Electromagnetic Theory and Simulation

Updated: April 2026

Product Tie-In: Simulation Software, PCB Materials

EM Simulation Mesh at mmWave

Mesh quality is the most important factor in electromagnetic simulation accuracy. An insufficiently fine mesh produces inaccurate results regardless of the solver quality.

Mesh Guidelines by Structure

(1) Microstrip line: mesh along the trace width: minimum 5-10 elements across the width. Mesh through the substrate: minimum 3-5 elements through the thickness (for accurate impedance calculation). Mesh along the trace length: lambda_eff/10 element size. (2) Via transition: mesh around the via barrel: minimum 8-12 elements around the circumference. Mesh through the anti-pad: minimum 3 elements from the via to the anti-pad edge. Mesh in the z-direction: minimum 2-3 elements per dielectric layer that the via passes through. (3) Coupled lines: mesh the gap between coupled lines with minimum 3-5 elements. For a 0.1 mm gap at 77 GHz: elements < 0.02-0.03 mm in the gap. This captures the coupling capacitance accurately. (4) Patch antenna: mesh the patch area with minimum 20 × 20 elements across the patch surface. Mesh the substrate under the patch with 5+ elements through the thickness. The fringing fields at the patch edges require fine edge meshing (< 0.01 × lambda).

Computational Resources

(1) Memory: FEM (HFSS): memory scales as N^1.3 (N = number of mesh elements). For 1 million elements: approximately 8-16 GB RAM. FDTD (CST): memory scales as N (linear). For 10 million cells: approximately 3-5 GB RAM. MoM (Momentum, Sonnet): memory scales as N² (N = number of surface elements). For 50,000 surface elements: approximately 20 GB RAM. MoM is the most memory-intensive for large structures. (2) Computation time: FEM: solving a system of 1 million elements typically takes 10-30 minutes on a modern workstation. Adaptive meshing with 5 passes: 1-3 hours total. FDTD: 10 million cells with 10,000 time steps: 1-4 hours. Time-domain: includes all frequencies in one run (wideband analysis is efficient). MoM: 50,000 elements: 30-60 minutes per frequency point. 20 frequency points: 10-20 hours. (3) GPU acceleration: CST FDTD and HFSS FEM support GPU acceleration (NVIDIA CUDA). GPU can speed up simulation by 5-20× compared to CPU only. This is critical for large mmWave models.

Mesh Density Guidelines

Max mesh size: < λ_eff/10
77 GHz on substrate: < 0.2 mm
Via mesh: 8-12 elements around barrel
Gap mesh: 3-5 elements in coupling gap
Convergence: ΔS < 0.02 between passes

Common Questions

Frequently Asked Questions

How do I know if my mesh is fine enough?

The convergence test: (1) Run the simulation with the default (adaptive) mesh. Record the S-parameters. (2) Request 2-3 additional mesh refinement passes. Record the S-parameters after each pass. (3) If the S-parameters change by < 0.02 (in linear magnitude) between successive passes: the mesh is converged. If > 0.02: continue refining. An alternative: mesh sensitivity study. Manually set the maximum mesh size to lambda/10, solve. Reduce to lambda/15, solve. Reduce to lambda/20, solve. Plot S-parameters vs mesh density. The solution is converged when further refinement produces < 0.01 change in S11 and S21.

Can I reduce the simulation size?

Techniques for reducing computational cost: (1) Symmetry: if the structure has a plane of symmetry: use a symmetry boundary condition (electric wall or magnetic wall). This halves the simulation volume (and approximately halves the mesh count and computation time). (2) Port truncation: only simulate the region of interest (a few wavelengths around the DUT). Terminate the ports with proper boundary conditions. (3) 2.5D instead of 3D: for planar structures (microstrip, stripline, coupled lines): use a 2.5D solver (ADS Momentum, Sonnet). The 2.5D solver meshes only the conductor surfaces (not the volume), dramatically reducing the mesh count. Accuracy: 2.5D is very accurate for planar structures but cannot handle 3D features (vertical walls, waveguide bends, wire bonds). (4) Sub-structure analysis: simulate complex structures in sections (e.g., simulate the via transition separately from the matching network). Import the S-parameters of each section into the circuit simulator and cascade them.

What about de-embedding in EM simulation?

EM simulators include access structures (port transitions, feed lines) that are not part of the DUT. De-embedding removes the effect of these access structures. Methods: (1) Port calibration (built into most simulators): the simulator automatically accounts for the port transition. (2) TRL de-embedding: simulate separate Thru, Reflect, and Line structures. Use the TRL algorithm to de-embed the port effects. (3) Port extension: specify the length of the feed line, and the simulator mathematically removes its effect (assuming a perfect transmission line). De-embedding is essential for: accurate extraction of S-parameters for circuit simulation, comparing simulated results with measured data (the measurement also requires de-embedding).

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