Exterior derivative operator in dxtr¶

Abstract: The exterior_derivative() operator is the corner stone of any DEC implementation. In this notebook we assess the precision of its implementation in the Dxtr library.

Note: You can download this notebook and play with it directly on your local machine.

Presentation¶

Given a k-form ($\omega\in\Lambda^k(\mathcal{M})$) defined on a n-dimentional manifold ($\mathcal{M}$), the external derivative is defined as: $$ \omega = \omega_{i_1\dots i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k} \quad \Rightarrow \quad d(\omega) = \frac{\partial\omega_{1\dots k}}{\partial x^j}dx^j\wedge dx^{i_1}\wedge\dots\wedge dx^{i_k}, $$ where the Einstein summation convention is used.

In the specific cases of 0-forms defined on a 2D domain we have: $$ \omega^{(0)} = \omega(\xi^0, \xi^1) \Rightarrow d(\omega^{(0)}) = \frac{\partial \omega}{\partial\xi^0}d\xi^0 + \frac{\partial \omega}{\partial\xi^1}d\xi^1 $$

To assess the precision of our implementation of the exterior derivative, we will consider simple differential forms defined on the 2D sphere of unit radius.

Dependencies¶

Domains definition¶

We will consider two domains of interest:

• A disk of unit radius.
• A sphere of unit radius.

While the disk is flat and feature boundaries, the sphere is curved and closed. Investigating the exterior derivative on both complexes will help us understand the influence of both boundaries and curvature on its precision.

Sphere¶

We will need to measure the position of simplices with respect to the north pole of this sphere. To that end, we define a function to compute this curvilinear abscissae.

We now compute the curvilinear positions for the 0-simplices and the 1-simplices of the primal complex as well as for the 0-cells and the 1-cells of the dual one:

Remark: Note that the precision of the circumcenters of the simplices of degrees >0 is significantly worst that the precision on the vertices. This might have an impact of the precision of our implementation.

Disk¶

Let's get the radial positions of 0-simplices and 0-cells:

Note: 1-simplices and 1-cells share the same circumcenters on the disk. That is why we do not distinguish 'primal' and 'dual' cases in the computation above.

We also define a function to compute the radial direction on every simplex circumcenter:

Finally, we define a boolean flag to mark the border of the disk on the 1-simplices.

Defining the test function and the expected results¶

To test our implementation of the exterior derivative, we propose to compute the exterior derivative of a simple 0-form: $$f(s) = \cos(\omega s), \tag{1}$$ where $s$ depicts the curvilinear abscissae with respect to a reference position ($\boldsymbol{x}_0$) defined on the considered domain, $\Omega$.

• In the case of the sphere, we choose as a reference position the "north pole" ($\boldsymbol{x}_0 = [0, 0, 1]$) and consequently, the curvilinear abscissae has the usual expression: $s = R\theta$, where $R=1$ in our case and $\theta\in [0, \pi[$ stands for the colatitude angle.
• In the case of the disk, the reference position is its center ($\boldsymbol{x}_0 = [0, 0, 0]$) and the curvilinear abscissae simply corresponds to the radial distance $s = r\in [0,1]$.

We define the test function to use, given by $\text{eq.}(1)$:

As the considered 0-form $f$ only depends on the curvilinear abscissae $s$, its exterior derivative will verify: $$ df = \frac{\partial f}{\partial s} \boldsymbol{ds}, $$ with $\boldsymbol{ds}= \boldsymbol{e}_s^\flat$ the 1-form dual of the vector field $\boldsymbol{e}_s$, tangent to the considered manifold, along the isolines $s=cste$.

Deriving $\text{eq.}(1)$ yields: $$ \frac{\partial f}{\partial s} = -\omega\sin(\omega s), \tag{2} $$ we define the corresponding partial derivative:

As mentioned previously in the documentation, in DEC, k-forms are approximated as k-cochains, whose coefficients correspond to the integration of the of the considered k-form on k-simplices (k-cells in the dual case).

In the present case, it means that the coefficients of the exterior derivative we seek verify: $$ c_i = \int_{\sigma^{(1)}_i}df, $$ where the $\sigma^{(1)}_i$ correspond to the 1-simplices (resp. 1-cells) of the considered complex.

Assuming the partial derivative, $\text{eq.}(2)$, to be constant on each 1-simplex, the sought coefficients can be approximated as: $$ c_i \approx \left.\frac{\partial f}{\partial s}\right\vert_{\boldsymbol{x}_i} w_i, \quad\text{with:}\quad w_i = \int_{\sigma^{(1)}_i}\boldsymbol{e}_s^\flat. $$ This is the formulae to estimate at every 1-simplex (1-cell) at to compare to the computed values. It is composed of two elements:

• The derivative of the considered function $f$ with respect to curvilinear abscissae, estimated at the center of the 1-simplices.
• A "weight" coefficient ($w_i$), to compute for each 1-simplex, corresponding to the circulation of the base vector $\boldsymbol{e}_s$ along the 1-simplex.

Computing these "weights" in the k=1 case, yeilds: $$ w_i = \int_{|\boldsymbol{e}_i|}ds\:\hat{\boldsymbol{e}}_s\cdot\hat{\boldsymbol{e}}_i \approx l_i\cos(\theta_i), \tag{3} $$

Note: In the dual case, the integration is performed on the 1-cells dual of the 1-simplices. The lengths ($l_i$) and angles ($\theta_i$) must be updated acordingly.

Computing the expectation, in the sphere case:

Expectation, in the disk case:

Computation of the exterior derivatives¶

On the sphere¶

We instanciate the primal and dual 0-cochain to test:

We derive these cochains using the d() function to call the exterior_derivative() function:

Then, we define an error() function to estimate the discrepency between the computed & expected values.