Page 182 - Finite Element Modeling and Simulations with ANSYS Workbench

Modeling and Solution Techniques 167

(Displacement)

Exact solution

FEA solutions

No. of DOFs

FIGURE 5.5
Convergence of FEM solutions with exact solution.

That is, FEA displacement solutions approach the exact solution from below (Figure 5.5),
which can be used to monitor the FEA solutions. However, this is true only for the dis-
placement-based FEA (which is the FEA formulation discussed in this book).

5.6 Convergence of FEA Solutions

As the mesh in an FEA model is “refined” (with smaller and smaller elements) repeatedly,
the FEA solution will converge to the exact solution of the mathematical model of the prob-
lem (the model based on beam, plane stress/strain, plate, shell, or 3-D elasticity theories or
assumptions). Several types of refinements have been devised in the FEA, which include:

h-refinement: Reduce the size of the element (“h” refers to the typical size of the
elements).
p-refinement: Increase the order of the polynomials on an element (linear to qua-
dratic, and so on; “p” refers to the highest order in a polynomial).
r-refinement: Rearrange the nodes in the mesh.
hp-refinement: Combination of the h- and p-refinements (to achieve better results).

With any of the above type of refinements, the FEA solutions will converge to the ana-
lytical solutions of the mathematical models. Some FEA software can automate the process
of refinements in the FEA solutions to achieve the so-called adaptive solutions.

5.7 Adaptivity (h-, p-, and hp-Methods)
Adaptive FEA represents the future of the FEA applications. With proper error control,
automatic refinements of an FEA mesh can be generated by the program until the con-
verged FEA solutions are obtained. With the adaptive FEA capability, users’ interactions