# 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 ($\theta=0, 180$).

```python
# 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
```