Biomedical Engineering Reference
In-Depth Information
reproducing kernel approximation, the hp-cloud approximation function, the
polynomial interpolation, the parametric interpolation, the radial interpolation and
the Sibson interpolation. The approximation, or interpolation, function requires a
domain of applicability, outside this domain the function assumes zero values. In
the FEM this domain is the 'element' and in the meshless methods this domain is
called 'influence-domain'. In meshless methods it is necessary to determine the
influence-domain for each node within the nodal distribution discretizing the
problem domain, as a consequence the shape and the size of the influence-domain
vary with the considered node. The technique used to defined the influence-
domains varies with the used meshless method.
1.1.2 Formulation
Additionally, meshless methods can be classified in two categories, a first category
that pursues the strong form solution and another that seeks the weak form solu-
tion. The strong formulation uses directly the partial differential equations gov-
erning the studied physical phenomenon in order to obtain the solution. The weak
formulation uses a variational principle to minimize the residual weight of the
differential equations ruling the phenomenon. The residual is obtained by substi-
tuting the exact solution by an approximated function affected by a test function.
The existent distinct weak form solution methods are dependent on the used test
function. Surprisingly a differential equation may have solutions which are not
exactly differentiable and the weak formulation allows to find such solutions.
Weak form solutions are very important since many differential equations gov-
erning the real world phenomena do not admit sufficiently smooth solutions, then
the only way of solving such equations is using the weak formulation.
1.1.3 Integration
In order to obtain the integral of the residual weight of the differential equations it is
necessary to select an integration scheme. The integration can be made using a
background mesh, covering the entire problem domain, composed by integration
points. These integration points must have a defined influence area and weight
(which corresponds to the theoretical infinitesimal mass portion defined in the
integral expression) and should not overlap each other. This background mesh for
integration proposes generally is nodal independent, which jeopardises the
'meshless' denomination of such numerical methods. There are other integration
schemes often used, the point collocation and the nodal integration. In these inte-
gration schemes the node represents the integration point, the influence area is the
node influence-domain and the integration weight is the node influence volume. In
this case the integration mesh is the nodal distribution itself. Some authors have the
