Biomedical Engineering Reference
In-Depth Information
Chapter 5
Solid Mechanics Problems
Abstract In this chapter basic solid mechanics benchmark examples are pre-
sented. Firstly, in order to determine the optimal shape parameters of the multi-
quadrics radial basis function (MQ-RBF), it is present an optimization test using a
standard linear patch test. Additionally, a simple study regarding the numerical
integration is performed, allowing to determine the most efficient integration
scheme for the nodal based integration. These optimization tests are extended to
the two-dimensional and three-dimensional analysis. Then, elastostatic and elas-
todynamic benchmark examples as presented. The obtained results permit to
confirm the efficiency and accuracy of the natural neighbour radial point inter-
polation method (NNRPIM).
5.1 Solid Mechanics NNRPIM Flow Chart
In Table
5.1
it is presented a flowchart to analyse a solid mechanics problem using
a meshless method, such as the element free Galerkin method (EFGM) or the
radial point interpolation method (RPIM).
The NNRPIM numerical implementation for a generic elastostatic solid
mechanics problem can be summarized with the steps suggested in Table
5.2
.
The elastodynamic analysis can be numerically implemented using the same
steps presented in Tables
5.1
and
5.2
. The numerical dynamic analysis can be
performed by substituting steps 8 and 9 by an eigenproblem solver, such as the
Jacobi solver. In addition to the free vibration elastodynamic analysis, it is possible
to modify Tables
5.1
and
5.2
to solve a transient elastodynamic analysis using the
same numerical tools used by the finite element method (FEM) [
1
-
5
].
Any type of nonlinear analysis can be implemented by assuming a recurring
process based in the basic flow charts presented in Tables
5.1
and
5.2
[
6
,
7
].
