# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 3 Description of the GeoHyTE Platform
### Theoretical Fundamentals
#### 3.1.2 Hybrid-Trefftz Finite Elements  
This overview of the hybrid-Trefftz finite elements is anchored in their comparison with the conforming displacement finite elements which are widespread in commercial software and therefore coined here as 'conventional'. The comparison between conventional and hybrid-Trefftz finite elements focuses on the techniques used to enforce the domain and boundary equations; the features of the approximation bases; and the handling of the mesh and basis refinements.  
Enforcement of the Governing Equations  
The equations governing the response of a poroelastic medium to dynamic excitation are expressed in both time and space variables. Regardless of the type of finite element that is used, their solution involves the approximation of the time variation of the unknown (displacement, fluid seepage, total stress, and pore pressure) fields. In GeoHyTE, the timeapproximation is handled using the weighted residual algorithm described, for instance, in reference [17]. All time-dependent quantities are independently approximated in time as  
\[\kappa(x,y,t)=\sum_{n=1}^{N}W_{n}(t)\kappa_{n}(x,y) \tag{3}\]  
where \(\kappa\) is the generic notation of any of the unknown fields and time basis \(W_{n}\) collects Daubechies wavelets of family \(N_{f}\) and order \(O\), and \(N=2^{O}\) is the dimension of the time basis. This time discretisation process reduces the original problem in time and space to a series of \(N\) problems defined only in space. Due to favourable properties of the wavelet basis, however, it is possible to solve only half of these problems.  
As exact solutions for the time-discretised problems cannot be found, in general, some of the governing equations are enforced in an approximate (or weak) form, and some in an exact (strong) form. Most conventional finite elements are strictly compatible. They satisfy exactly the compatibility equations in the domain of the element and the Dirichlet (or kinematic) boundary conditions on the exterior boundaries where displacements (and/or seepage) are enforced. On the interior boundaries, the displacement and seepage compatibility is also exactly satisfied. The domain equilibrium equation is enforced weakly and the Neumann (or static) boundary conditions are enforced at the nodes of the mesh. As a consequence, the displacement and fluid seepage fields are typically approximated with considerably superior precision compared to the total stress and pore pressure fields.  
Hybrid-Trefftz displacement-seepage elements satisfy strongly all domain equations. All boundary conditions are enforced weakly on the exterior boundaries. Displacement and seepage compatibility is also enforced weakly on the interior boundaries of the mesh. However, the traction and pore pressure balance is not enforced at all on the interior boundaries. Since kinematic and static equations are enforced in the same way (except for the interior boundaries), the predictions of the respective fields are more balanced in terms of quality than in the case of conventional elements, although a slight bias towards the displacement-seepage field is still present.