1. By Heather Huenison and Allan Dolovich University of Saskatchewan
A Damped Least Squares Method for Finite Element Analysis of Incompressible Materials By
Heather Huenison and Allan Dolovich
University of Saskatchewan

2. Motivation Mechanical modelling of soft tissue
Chiropractic manipulation of the neck
Effects on the vertebral artery due to:
variations in technique
condition of the arteries
repeated procedures

3. Vertebral Arteries
applied
forces
vertebra
artery

4. Modelling of Deformation
Estimates from expert practitioners
Displacement data from high resolution images
synchrotron imaging
Finite element models with displacement BCs

5. Challenges Viscoelastic behaviour Dynamic Effects
Solid-fluid interaction
Nonlinearities
Incompressibility
displacement BCs

6. Incompressible Materials - Formulation
Minimise
P = U + W
+ p ∫ (J-1) dV
Potential
Energy
Strain
energy
Work
Potential
Volumetric
Strain
Lagrange Multiplier or Penalty Parameter approaches
Non-locking stable elements

7. FEM Techniques - Incompressibility
Selective Integration
Hu-Washizu variational forms
Enhanced Strain
Pressure Smoothing

8. Issues for Present Work
Indeterminate hydrostatic stress components p
No force BCs
Infinite number of solutions
Errors in displacement BCs
Volume constraint violated
No consistent solution

9. Proposed Approach Address linearized systems K BT B u p = b1 b2 oru p = b1 b2 or
A d0 = b

10. Least Squares for Consistency
Minimise Ad0 – b || 2
i.e., solve
ATA d0 = AT b or
C0 d0 = g0

11. Minimum Norm for Uniqueness
With Singular Value Decomposition (SVD)
C d = g
rank of C = number of rows
(CCT)-1 exists
Minimum norm solution:
dmn = CT (CCT)-1 g

12. Test Case – Uniform Plane Strain
s0 = 100 MPa
100 mm x 200 mm
Neo-Hookean Material
E = 70 GPa
Displacement BCs
Modified Newton-Raphson Code 9 10 11 12 s0 5 6 7 8 1 2 3 4

13. Results for Random Data Error (5% max)

14. Results for Random Data Error (5% max)

15. Results for Random Data Error (5% max)

16. Results for Random Data Error (5% max)

17. Conclusions Damped Least Squares shows promise for addressing
data noise
numerical instability
Method will be tested with other approaches, such as
selective integration
enhanced strain

18. QUESTIONS?