Page 183 - Finite Element Modeling and Simulations with ANSYS Workbench

168 Finite Element Modeling and Simulation with ANSYS Workbench

are reduced, in the sense that a user only needs to provide a good initial mesh for the
model (even this step can be done by the software automatically).
Error estimates are crucial in the adaptive FEA. Interested readers can refer to Reference
[5] for more details. In the following, we introduce one type of the error estimates.
We first define two stress fields:

σ—element by element stress field (discontinuous across elements)
σ —averaged or smoothed stress field (continuous across elements)
*
Then, the error stress field can be defined as

σ = σ − σ * (5.11)
E
Compute strain energies,

M
T
U = ∑ U i , U = ∫ 1 σ E −1 σdV (5.12)
i
2
= i 1 V i
M
*
*T
*
*
*
U = ∑ U , U = ∫ 1 σ E −1 σ dV (5.13)
i
i
2
= i 1 V i
M
U E = ∑ U Ei , U Ei = ∫ 1 σ T E E −1 σ E dV (5.14)
2
i=1 V i
where M is the total number of elements and V is the volume of the element i.
i
One error indicator—the relative energy error is defined as
/
  12
η =  U E  .( 0 ≤η ≤ 1) (5.15)
 U + U E 
The indicator η is computed after each FEA solution. Refinement of the FEA model con-
tinues until, say

η ≤ 0.05

When this condition is satisfied, we conclude that the converged FE solution is obtained.
Some examples of using different error estimates in the FEA solutions can be found in
Reference [5].

5.8 Case Study with ANSYS Workbench
Problem Description: Garden fountains are popular amenities that are often found at
theme parks and hotels. As a fountain structure is usually an axisymmetric geometry with