Abstract

The pressure projection method [1] has been used successfully in meshfree mixed formulations to model large deformations of hyperelastic materials because it stabilises pressure oscillations and overcomes volumetric locking near the incompressibility limit. However, the current pressure projection method is limited to the solutions of nearly-incompressible problems, due to the use of a finite bulk modulus as a penalty factor to enforce incompressibility [1].

In order to solve ideally incompressible hyperelasticity, we propose the use of a pressure projection scheme based on a similar approach used in fluid mechanics called the polynomial pressure projection (PPP) [2]. A stabilisation term, similar to the PPP, is appended to the Galerkin weak form to penalise the error between the pressure field and the projected pressure field. In this case, the projected pressure field cannot be eliminated through projection onto the displacement field, because the stabilisation term vanishes for pressures within the penalisation operation [2].

We have developed a total Lagrangian meshfree framework to solve the modified Galerkin weak form with numerical differencing. This has not been possible until now because numerical differencing perturbs the fictitious field approximated from the moving least squares (MLS) shape functions. To overcome this, we performed a full transformation [3] between nodal and fictitious values, enabling the perturbation of material points for numerical differencing. Using our method, essential boundary conditions can be imposed easily.

We validated the accuracy of our framework against an analytic solution for uniaxial beam extension, and a finite element solution for a 2D annulus under inflation and radial torsion. The predicted pressure fields agreed well with the benchmark solutions. Our goal is to apply this framework to simulate large hyperelastic deformations of soft tissues. In particular, we hope to simulate breast tissue reorientation under different gravity loading conditions, as well as during mammographic compression.

In conclusion, we have developed and validated a new meshfree framework that reliably and robustly predicts large deformations of ideally incompressible soft tissues.

Keywords: Meshfree methods, Hyperelasticity, Incompressibility, Stabilisation method

References:

1. Chen, J-S., Pan C. (1998). A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory. Journal of Applied Mechanics. 63(4), 862-868.
2. Dohrmann, C.R. and Bochev, P.B. (2004). A stabilized finite element method for stokes problem based on polynomial pressure projections. International Journal of Numerical Methods in Fluids, 46: 183–201.
3. Chen, J-S., Pan C., Wu, C-T., Liu W.K. (1996). Reproducing Kernel Particle Methods for large deformations analysis of non-linear structures. Computational Methods in Applied Mechanical Engineering. 139: 195-227.