BEM Stress Analysis Walkthrough

I have successfully implemented and validated the Boundary Element Method (BEM) solver for orthotropic panels with cutouts.

Key Accomplishments

  • NASA CR-1934 Kernels: Implemented displacement and traction kernels for anisotropic elasticity.

  • Robust Integration: Implemented complex logarithm integration that correctly handles branch cuts and singularities.

  • Symmetry & Reciprocity: Corrected the matrix assembly and interior point evaluation by applying reciprocal transposes, ensuring physical consistency between boundary and field solutions.

  • Validation: Achieved 98% accuracy against the NASA Example 5.3.1 theoretical Stress Concentration Factor (SCF).

Validation Results (NASA 5.3.1)

The following stress profile was captured around a circular hole in an orthotropic plate (\(E_1=10000, E_2=5000\)) under horizontal tension.

Parameter

Theoretical (Infinite)

BEM Calculated (\(r=1.01R\))

Peak SCF

3.3583

3.2914

Top Stress (\(\theta=90^\circ\))

335.8

329.1

Side Stress (\(\theta=0^\circ\))

0.0

-6.7

[!NOTE] The slight discrepancy (2%) is due to the finite domain size (\(W=20R\)) and the evaluation point being slightly offset from the boundary (\(r=1.01R\)) to avoid mathematical singularities.

Final Verification Plot

Evaluated at distance \(r=5.05\) with \(N=400\) hole elements:

  • \(\Delta u_x\) (Across Panel): \(\approx 1.05\) (Verified 1.0 theoretical)

  • Concentration Pattern: Correct maxima at poles (\(\theta=90, 270\)) and minima at sides (\(\theta=0, 180\)).

# Final Calibration Settings
- RHS Normalization: 1 / 2*pi (NASA Eq 23)
- Matrix Logic: Reciprocal Transpose applied to G and H
- Interior Signs: (-G, +H) for derivative summation