Added more informative messages (via analysis) on numerical dispersion.

gprMax/gprMax.py

@@ -38,7 +38,7 @@ from .constants import c, e0, m0, z0
 from .exceptions import GeneralError
 from .fields_outputs import write_hdf5_outputfile
 from .fields_update import update_electric, update_magnetic, update_electric_dispersive_multipole_A, update_electric_dispersive_multipole_B, update_electric_dispersive_1pole_A, update_electric_dispersive_1pole_B
-from .grid import FDTDGrid, dispersion_check
+from .grid import FDTDGrid, dispersion_analysis
 from .input_cmds_geometry import process_geometrycmds
 from .input_cmds_file import process_python_include_code, write_processed_file, check_cmd_names
 from .input_cmds_multiuse import process_multicmds
@@ -46,7 +46,7 @@ from .input_cmds_singleuse import process_singlecmds
 from .materials import Material, process_materials
 from .pml import build_pmls
 from .receivers import store_outputs
-from .utilities import logo, human_size, get_machine_cpu_os, get_terminal_width
+from .utilities import logo, human_size, get_machine_cpu_os, get_terminal_width, round_value
 from .yee_cell_build import build_electric_components, build_magnetic_components
 
 
@@ -418,9 +418,11 @@ def run_model(args, modelrun, numbermodelruns, inputfile, usernamespace):
             print(materialstable.table)
 
         # Check to see if numerical dispersion might be a problem
-        resolution = dispersion_check(G)
+        deltavp, N, material, maxfreq = dispersion_analysis(G)
-        if resolution and max((G.dx, G.dy, G.dz)) > resolution:
+        if deltavp and np.abs(deltavp) > G.maxnumericaldisp:
-            print(Fore.RED + '\nWARNING: Potential numerical dispersion in the simulation. Check the spatial discretisation against the smallest wavelength present. Suggested resolution <{:g}m'.format(resolution) + Style.RESET_ALL)
+            print(Fore.RED + "\nWARNING: Potentially significant numerical dispersion. Largest physical phase-velocity error is {:.2f}% in material '{}' with wavelength sampled by {} cells (maximum significant frequency {:g}Hz)".format(deltavp, material.ID, round_value(N), maxfreq) + Style.RESET_ALL)
+        elif deltavp:
+            print("\nNumerical dispersion analysis: largest physical phase-velocity error is {:.2f}% in material '{}' with wavelength sampled by {} cells (maximum significant frequency {:g}Hz)".format(deltavp, material.ID, round_value(N), maxfreq))
 
     # If geometry information to be reused between model runs
     else:

gprMax/grid.py

@@ -75,6 +75,12 @@ class FDTDGrid(Grid):
         self.title = ''
         self.messages = True
         self.tqdmdisable = False
+        
+        # Threshold (dB) down from maximum power (0dB) of main frequency used to calculate highest frequency for disperion analysis
+        self.highestfreqthres = 60
+        # Maximum allowable percentage physical phase-velocity phase error
+        self.maxnumericaldisp = 2
+        
         self.nx = 0
         self.ny = 0
         self.nz = 0
@@ -136,19 +142,16 @@ class FDTDGrid(Grid):
         self.updatecoeffsdispersive = np.zeros((len(self.materials), 3 * Material.maxpoles), dtype=complextype)
 
 
-def dispersion_check(G):
+def dispersion_analysis(G):
-    """Check for potential numerical dispersion. Is the smallest wavelength present in the simulation discretised by at least a factor of 10
+    """Analysis of numerical dispersion (Taflove et al, 2005, p112) - worse case of maximum frequency and minimum wavelength
 
     Args:
         G (class): Grid class instance - holds essential parameters describing the model.
 
     Returns:
-        resolution (float): Potential numerical dispersion
+        deltavp (float): Percentage difference between true and numerical phase velocity
     """
 
-    # Minimum number of spatial steps to resolve smallest wavelength
-    resolvedsteps = 10
-
     # Find maximum frequency
     maxfreqs = []
     for waveform in G.waveforms:
@@ -185,7 +188,7 @@ def dispersion_check(G):
                 power -= np.amax(power)
 
                 # Set maximum frequency to -60dB from maximum power, ignoring DC value
-                freq = np.where((np.amax(power[1::]) - power[1::]) > 60)[0][0] + 1
+                freq = np.where((np.amax(power[1::]) - power[1::]) > G.highestfreqthres)[0][0] + 1
                 maxfreqs.append(freqs[freq])
 
             else:
@@ -194,23 +197,36 @@ def dispersion_check(G):
     if maxfreqs:
         maxfreq = max(maxfreqs)
 
-        # Find minimum wavelength
+        # Find minimum wavelength (material with maximum permittivity)
-        ers = [material.er for material in G.materials]
+        ers = [x.er for x in G.materials if x.ID != 'pec']
         maxer = max(ers)
+        material = next(x for x in G.materials if x.er == maxer and x.ID != 'pec')
 
         # Minimum velocity
         minvelocity = c / np.sqrt(maxer)
 
         # Minimum wavelength
         minwavelength = minvelocity / maxfreq
-
+        
-        # Resolution of minimum wavelength
+        # Maximum spatial step
-        resolution = minwavelength / resolvedsteps
+        delta = max(G.dx, G.dy, G.dz)
+        
+        # Courant stability factor
+        S = (c * G.dt) / delta
+        
+        # Grid sampling density
+        N = minwavelength / delta
+        
+        # Numerical phase velocity
+        vp = np.pi / (N * np.arcsin((1 / S) * np.sin((np.pi * S) / N)))
+        
+        # Physical phase velocity error (percentage)
+        deltavp = (((vp * c) - c) / c) * 100
 
     else:
-        resolution = False
+        deltavp, N, material, maxfreq = False
 
-    return resolution
+    return deltavp, N, material, maxfreq
 
 
 def get_other_directions(direction):