Comparison of Explicit Finite Difference Model and Galerkin Finite Element Mo...
KrystianPaczkowski_iFEM
1. An Inverse Finite Element Strategy
to Recover Full-field, Large Displacements from
Strain Measurements
Krystian Paczkowski & H. Ronald Riggs
Department of Civil and Environmental Engineering
University of Hawai’i at Manoa
1/24
5. 5/24
Linear iFEM overview
Tessler and Spangler developed a least-squares functional:
• The first term corresponds to the membrane deformation,
whereas second and third pertain to the bending response
• As a result of functional minimization the structural
displacements can be obtained
7. 7/24
Nonlinear formulation
A similar strategy was used to develop a methodology that can
handle large displacements.
Although the ultimate focus is
on plate and shell structures,
the development only
considers the 2-D case.
8. 8/24
To minimize the functional Φ the finite element method will be
used:
and
Substitution into Green-Lagrange definitions and expansion
results in:
etc.
Nonlinear formulation
9. 9/24
The displacement field is obtained by finding the minimum of Φ,
and
These equations define pseudo-force vectors.
The second derivatives are needed, i.e. the pseudo-stiffness matrix
Nonlinear formulation
11. 11/24
The coordinates in displaced
configuration are:
where parametric mapping
φ = X1/R have been used.
Displacements are then
Analytical formulation
12. 12/24
To solve the nonlinear equations
a standard Newton-Raphson procedure was used.
The exact Green-Lagrange strains will be used for the
‘experimental’ strains.
Numerical formulation
14. 14/24
Two-node element
• The procedure was tested for h=1/40, 1/20, 1/2 and range of R
from 10 to 1.
• The tests have shown the two-node beam element works well.
For h=1/2 and any of the chosen R value, the result was always
the exact solution du=1 and dw=1.
15. 15/24
• The same holds true for the other two values of h and larger R values.
• However, as the deformation increases (R decreases), the
functional develops local minima and local maxima between (0,0)
and global minimum (1,1)
Φ functional, 3D view; R=7, h=1/40.
17. 17/24
• The local minima and maxima occur when Rh/L
2
≈1/5
or less.
• It could be necessary to use an incremental-
iterative solution procedure.
• All considered R values were beyond yield point with
maximum fiber strains between 0.025~0.625.
Two-node element summary:
Two-node element
20. 20/24
Maximum end slope rotation versus maximum aspect ratio.
The aspect ratio (AR=L/2h) impacts the performance significantly
Nodal displacements should be
predicted within 5% error.
21. 21/24
8 elements 16 elements
element aspect ratio 1 element aspect ratio 0.5
352˚ rotation 352˚ rotation
Length = 8, depth = 1.0
- exact displacements - numerical displacements
22. 22/24
Length = 8
8 elements 8 elements
element aspect ratio 10 element aspect ratio 20
49˚ rotation 24˚ rotation
- exact displacements - numerical displacements
depth = 0.1 depth = 0.05
23. 23/24
• These results demonstrate that the fundamental procedure is
valid.
• The 2-D element provides a more realistic situation, as it
could potentially be expanded to 3-D, i.e., plate and shell
structures.
• The 2-D element can also be used for beam components,
and could have practical applications in its own right.
• Issues that must be addressed in future work include:
• extend and test the method with discrete experimental
strains
• develop a methodology for plates and shells.
Conclusions
24. 24/24
THANK YOU
The authors gratefully acknowledge the financial support provided
for this work under NASA grant NNL05AA13G.
Krystian Paczkowski & H. Ronald Riggs
Department of Civil and Environmental Engineering
University of Hawai’i at Manoa
