Biomedical Engineering Reference
In-Depth Information
Chapter 4
Shape Functions
Abstract This chapter explicitly shows how to construct shape functions for
meshless methods. The chapter starts with the introduction of the ''support-
domain'' concept which permits to identify the field nodes contributing to the
construction of the shape function. Afterwards, the most popular approximation
function is presented: the moving least square (MLS) approximation function. The
construction procedure is presented in detail as well as the most important
numerical properties of the MLS approximation function. Additionally the
importance of the weight function used in the construction of the MLS shape
function is shown. Then, the radial point interpolation (RPI) functions are pre-
sented. Again, an exhaustive description of the RPI shape function construction is
presented supported by examples and explicative algorithms. The most important
numerical properties of the RPI shape function are demonstrated. In addition, it is
shown the relevance of the radial basis function (RBF) used to construct the RPI
shape function, as well as the influence of the RBF shape parameters on the final
solution.
4.1 Introduction
In order to obtain a numerical solution of a physical phenomenon ruled by partial
differential equations, first it is require to approximate the unknown field functions
using trial functions.
In the Finite Element Method (FEM) shape functions are obtained using the
stationary element based interpolation. Fixed sets of nodes, respecting the same
quantity and relative nodal spatial configuration, are combined to form elements.
Then, the FEM shape functions are created using interpolation techniques based on
polynomial series [
1
,
2
].
In meshless methods the problem domain is not discretized in elements as in the
FEM. Generally, the problem domain within meshless methods is discretized in a
