Glossary of Common Terms

Boundary condition

In partial differential equations, boundary conditions (BC) are constraints of the solution function \(u\) for a given domain \(D\). Thus, the values of the function are specified on the boundary (in the topological sense) of the considered domain \(D\). Neumann and Dirichlet boundary conditions are frequently used. The Python implementation of SPHARA uses the Neumann boundary condition in the solution of the Laplacian eigenvalue problem.

EEG

EEG is an electrophysiological method for measuring the electrical activity of the brain by recording potentials on the surface of the head.

Finite Element Method

The Finite Element Method (FEM) is a approach to solve (partial differential) equations, where continuous values are approximated as a set of values at discrete points. For the approximation nodal basis functions are used.

Laplace-Beltrami operator

The generalized Laplace operator, that can applied on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. For triangulated manifolds, there are several methods to discretize the Laplace-Beltrami operator.

Triangular mesh

A triangular mesh is a piecewise planar approximation of a smooth surface in \(\mathbb{R}^3\) using triangles. The triangles of the mesh are connected by their common edges or corners. The sample points used for the approximation are the verices \(\vec{c} \in V\) with \(\vec{v}_i \in \mathbb{R}^3\). A triangle \(t\) is defined by three indices to the list of vertices. Thus, a triangular grid is represented by a list of vertices and a list of triangles.