Analyze boundary discretization for electrostatics, heat, and flow. Compare panel density, memory, and solver effort. Get usable results, tables, downloads, formulas, and plots fast.
The graph shows the circular benchmark potential along the x-axis.
The calculator focuses on boundary-only discretization planning for Laplace-type physics problems. It estimates mesh density, dense matrix size, memory demand, and approximate computational effort before you build a full simulation model.
For a circle, the boundary length is P = 2πR. For a rectangle, the boundary length is P = 2(W + H). For a custom boundary, you enter the total length directly.
The base panel count is computed from N = ceil(P / Le), where Le is the target element length. The refinement factor multiplies that count. The actual panel length is h = P / N.
The degree-of-freedom estimate uses DOF = N × f, where f equals 1 for constant elements, 2 for linear elements, and 3 for quadratic elements.
The dense influence matrix requires DOF² stored entries. Memory is estimated from Memory = DOF² × bytes × copies × safety factor. Dense direct work is approximated by (2/3)DOF³. Iterative work is approximated by iterations × DOF².
When you choose a circular boundary with Dirichlet data, the page also evaluates a simple first-harmonic benchmark. The assumed boundary condition is φ(θ) = φ̄ + A cosθ. Inside the circle, the benchmark potential becomes φ(r,θ) = φ̄ + A(r/R)cosθ. Outside the circle, it becomes φ(r,θ) = φ̄ + A(R/r)cosθ.
| Case | Geometry | Boundary length | Target element length | Panels | DOFs | Estimated memory (MB) |
|---|---|---|---|---|---|---|
| Benchmark A | Circle, R = 1.00 | 6.2832 | 0.2000 | 32 | 64 | 0.0625 |
| Benchmark B | Rectangle, 2.00 × 1.00 | 6.0000 | 0.1500 | 40 | 80 | 0.0977 |
| Benchmark C | Circle, R = 2.00 | 12.5664 | 0.2500 | 56 | 112 | 0.1914 |
| Benchmark D | Custom boundary | 8.5000 | 0.1250 | 68 | 136 | 0.2823 |
It estimates boundary length, panel count, degrees of freedom, dense matrix size, memory demand, approximate solver work, and a circular benchmark field when that simplified case applies.
No. It is a planning and benchmarking tool. It helps size a model, compare discretization choices, and inspect a simple analytical check before building a full boundary element implementation.
Higher-order elements carry more unknowns per panel. Constant elements use one factor, linear elements use two, and quadratic elements use three in this estimator.
Classical BEM often stores dense influence matrices. Dense storage grows with the square of the unknown count, so larger boundary meshes become expensive quickly.
Use them when your boundary is circular and your boundary data can be approximated by a mean value plus a first cosine harmonic. That makes the benchmark physically interpretable.
Start with a panel length that resolves curvature and boundary variation well. Then refine until the memory estimate, field behavior, and your own convergence checks become acceptable.
Direct solving is robust for modest systems. Iterative methods become attractive as the model grows, especially when memory pressure or repeated solves matter more than setup simplicity.
Yes. After a calculation, use the CSV button for spreadsheet work or the PDF button for a compact report you can archive or share.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.