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

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

Vertebral Arteries applied forces vertebra artery

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

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

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

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

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

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

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

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

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

Results for Random Data Error (5% max)

Results for Random Data Error (5% max)

Results for Random Data Error (5% max)

Results for Random Data Error (5% max)

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

QUESTIONS?