Explicit dynamics: reflection of P and S waves from a Lysmer-Kuhlemeyer absorbing boundary

Explicit dynamics (waves in a continuum)PASS · 0.39%

A Ricker pulse travels along a soil column to an absorbing boundary (viscous dashpots rho·c_p, rho·c_s of the adjacent element on every boundary face, treated implicitly in the time step). A geophone 450 m before the boundary records the incident and the reflected pulse; the reflection coefficient is the ratio of their peak velocities. An ideal absorber gives 0; the discrete one is second-order small in the mesh step (0.4 % at 23-27 cells per wavelength), the target is at most 5 %. Oblique incidence is worse (45°: about 9 %, 60°: about 24 % in the C++ test), this card is the normal-incidence case. The script shows the P run; the S run is executed by the runner.

Tolerance:: 5%

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Problem statement

Geometry: Column 1500 × 5 × 5 m (P: 300 HEX8, h = 5 m; S: 1200 HEX8, h = 1.25 m); confined laterally.
Material: Soil: E = 5 GPa, nu = 0.25, rho = 2000 kg/m³.
Boundary conditions: Face x = 0: Ricker 15 Hz pressure (P) or shear traction (S) 1 MPa on a free end; face x = 1500 m: absorbing boundary.
Measured quantity: Geophone at x = 1050 m; peak |v| in the incident window and in the window after the reflected arrival (800 m later); R = reflected / incident.

WebCAE Terminal

Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:

units mm
material E=5000000000 nu=0.25 density=2000
box dx=1500000 dy=5000 dz=5000
mesh hex nx=300 ny=1 nz=1
explicit endTime=1.425833 dtScale=0.8 saveStride=1000000 bulkQ1=0 bulkQ2=0
bc explicit_pressure surface number 1 value=1e6 ricker=15
bc explicit_absorbing surface number 2
bc explicit_symmetry surface number 3
bc explicit_symmetry surface number 4
bc explicit_symmetry surface number 5
bc explicit_symmetry surface number 6
explicit geophone at=1050000,0,0 name=g
solve explicit
probe explicit_geo_peak geophone=g component=vx from=0.506218 to=0.906218 id="inc"
probe explicit_geo_peak geophone=g component=vx from=1.02583 to=100 id="ref"

Results comparison

Reference value	R_P, R_S ≤ 5 % (ideal Lysmer–Kuhlemeyer: 0)
Our result	R_P = 0.388 %, R_S = 0.111 %
Deviation	0.39%
Tolerance	5%
Status	PASS · 0.39%
Element family	Solid element
Mesh details	P: 300 HEX8 (h = 5 m, 23 cells per wavelength); S: 1200 HEX8 (h = 1.25 m, 53 cells);: webcae_solve (native)

Updated: 2026-10-22 · version not recorded