About surfpy

surfpy is a Python package for computing surface integrals over smooth embedded manifolds.

Surface Approximation through Polynomial Interpolation

Let's consider an element $T_i$ within a reference surface $S_h$. This involves employing both an affine transformation and the closest point projection:

• ${\tau }_i:{\mathrm{\Delta }}_2\to T_i$
• ${\pi }_i:T_i\to V_i$

In this context, we define the coordinate mapping ${\phi }_i:{◻}_2\to V_i$, given by ${\phi }_i={\pi }_i\circ {\tau }_i\circ \sigma$, where $\sigma$ maps the reference square ${◻}_2$ to the reference triangle ${\mathrm{\Delta }}_2$.

We say that the mesh is order $k$ if each element has been provided as a set of nodes ${\phi }_i(p_{\alpha }),\alpha \in A_{2,k}$ sampled at $(p_{\alpha }),\alpha \in A_{2,k}$ on $S$. Consequently, we can numerically approximate the coordinate mapping on each element through interpolation using the nodes ${\phi }_i(p_{\alpha }),\alpha \in A_{2,k}$. In other words, our goal is to compute a $k^{\text{th}}$-order polynomial approximation:

$Q_{G_{2,k}}{\phi }_i\left(\mathrm{x}\right)=\sum _{\alpha \in A_{2,k}}{\phi }_i(p_{\alpha })L_{\alpha },i=1,\dots ,K$

with $G_{2,k}$ being the tensorial Chebyshev–Lobatto grid and $A_{2,k}$ a multi-index set.

To obtain partial derivatives, forming the Jacobian matrix $D{Q}_{G_{2,k}}{\phi }_i$, we utilize numerical spectral differentiation.

Surface Integral Approximation

The surface integral:

${\int }_{S}fdS\approx \sum _{i=1,...,K}{\int }_{{◻}_2}f(\phi (\mathrm{x}))\sqrt{det((D{Q}_{d,k}{\phi }_i(\mathrm{x}){)}^{T}D{Q}_{d,k}{\phi }_i(\mathrm{x}))}d\mathrm{x}$ $\approx \sum _{i=1,...,K}\sum _{\mathrm{p}\in P}{\omega }_{\mathrm{p}}f({\phi }_i(\mathrm{p}))\sqrt{det((D{Q}_{d,k}{\phi }_i(p){)}^{T}D{Q}_{d,k}{\phi }_i(p))}$

This approach provides a robust method for accurate surface integral computations in the context of surfpy.

Installation

To install, you can either download a .zip file or clone the directory with Git.

Option 1: Download a .zip file

Download a .zip of surfpy from:

https://github.com/casus/surfpy/archive/refs/heads/main.zip

Option 2: Clone with Git

To clone the surfpy repository, first navigate in a terminal to where you want the repository cloned, then type:

https://github.com/casus/surfpy

Getting started

Check out the examples folder — happy computing!

Any comment or question, send an email to: g.zavalani@hzdr.de

License

MIT