Cut-boundary submodeling (P1 global -> P2 TET10 local) on Kirsch plate with central hole (Kt = 3.08)
Meshing / submodeling & adaptive h-refinementPASS · 3.57%Verifies the automated cut-boundary displacement submodeling (`submodel`) and Zienkiewicz-Zhu adaptive h-refinement (`refine adaptive`) workflow on a tensile plate with a central circular hole (Kirsch problem, finite width 2R/W = 0.10, analytical Kt = 3.08). A coarse global P1 tetrahedral mesh (h = 6 mm, 3112 elements) severely underpredicts the peak stress at the hole rim (1.808 MPa vs 3.080 MPa, -41.3%). Extracting a local 32 x 32 x 4 mm submodel box around the hole, remeshing locally at h = 1.2 mm with quadratic P2 (TET10) elements (10085 elements), and projecting the global displacement field onto the 1920 cut-boundary nodes via barycentric tet interpolation with nearest-element projection fallback recovers the true physical concentration at the free hole rim: sigma_vM,max = 2.970 MPa (-3.57% deviation from closed-form 3.080 MPa).Verifies the automated cut-boundary displacement submodeling (`submodel`) and Zienkiewicz-Zhu adaptive h-refinement (`refine adaptive`) workflow on a tensile plate with a central circular hole (Kirsch problem, finite width 2R/W = 0.10, analytical Kt = 3.08). A… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Global domain: rectangular plate 160 x 100 x 4 mm with a central through-hole of radius R = 5 mm at (80, 50). Local submodel domain: 32 x 32 x 4 mm box centered at (80, 50, 2) intersected with the perforated plate via OCCT boolean common.
- Material
- Isotropic linear elastic steel: E = 200 GPa, nu = 0.30.
- Boundary conditions
Global model: surface number 1 fully fixed (u = 0), uniform tensile traction sigma_0 = 1.0 MPa (pressure = -1e6 Pa) on surface number 2. Local submodel: prescribed 3-DOF displacements (ux, uy, uz) interpolated from the global solution on all nodes lying on the 4 cut planes (x = 64, x = 96, y = 34, y = 66 mm); top/botto… show full text
Global model: surface number 1 fully fixed (u = 0), uniform tensile traction sigma_0 = 1.0 MPa (pressure = -1e6 Pa) on surface number 2. Local submodel: prescribed 3-DOF displacements (ux, uy, uz) interpolated from the global solution on all nodes lying on the 4 cut planes (x = 64, x = 96, y = 34, y = 66 mm); top/bottom plate faces (z = 0, z = 4 mm) and the inner cylindrical hole wall (r = 5 mm) remain traction-free natural boundaries.
- Measured quantity
- Peak von Mises stress recovered at quadratic TET10 nodes on the free interior/hole boundary of the local submodel (`submodel_peak_vm`), compared against the analytical finite-width Kirsch peak sigma_max = 3.080 MPa.
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=200e9 nu=0.3
primitive box dx=160 dy=100 dz=4
primitive cylinder r=5 h=20 ox=80 oy=50 oz=-5
boolean cut target=body1 tool=body2
mesh tet size=6
fix surface number 1
load surface number 2 pressure=-1e6
solve
minmax vonMises
submodel at=80,50,2 box=32,32,4 size=1.2 order=2
probe submodel_peak_vm id=sub_peak target=3.08e6 tolerance=5
probe submodel_cut_node_count id=cut_nodes
probe submodel_element_count id=sub_elemsResults comparison
| Reference value | σ_max = 3.080 MPa (Kt = 3.08 при 2R/W = 0.10) |
| Our result | σ_sub = 2.970 MPa (грубая глобальная P1 сетка: 1.808 MPa) |
| Deviation | 3.57% |
| Tolerance | 5% |
| Status | PASS · 3.57% |
| Element family | Solid element |
| Mesh details | Глобальная сетка P1 TET4 (h = 6 mm, 3112 elements) → локальная субмодель P2 TET10 (бокс 32×32×4 mm, h = 1.2 mm, 10085 elements, 1920 граничных nodes разреза) |
Updated: 2026-10-27 · version not recorded