# Physiological Gels Research Group

## Computational Methods

Solving the equations that give a realistic description of the physics and chemistry underlying the behavior of physiological gels poses substantial computational challenges, and a major thrust of our research is to develop efficient and robust numerical methods to meet these challenges.

Our physiological gel models typically begin with equations for a mixture of two materials (solvent and network) and add equations for other aspects of the phenomenon of interest, such as chemistry and chemical transport, or the evolution of elastic stress. The basic equations for the solvent/network system itself describe conservation of mass and conservation of momentum of each material. They form a set of coupled, nonlinear, partial differential equations whose solution in two or three dimensions poses substantial challenges. Additional challenges arise because, for example, the material properties of the mixture may change dramatically in space and/or time; the region in which the equations are to be solved may have irregular and, perhaps, time-dependent geometry; there may be regions in which a mixture of network and solvent abuts a region of pure solvent; and the motion of the solvent/network mixture may be coupled with a containing structure (e.g., a cellular membrane) and the motion of the mixture and this structure must be determined simultaneously.

## Publications

We began our development of numerical methods for gel problems with the viscous-dominated case of the momentum equations. Some papers describing this work are:

G.B. Wright, R.D. Guy, A.L. Fogelson, An efficient and robust method for simulating two-phase gel dynamics, *SIAM J. Sci. Comput.*, **30** (2008), 2535—65.

J. Du, A.L. Fogelson, G.B. Wright, A parallel computational method for simulating two-phase gel dynamics on a staggered grid, *Int. J. Numer. Meth. Fluids*, **60** (2009) 633—649.

J. Du, A.L. Fogelson, A Cartesian grid method for two-phase gel dynamics on an irregular domain, *Int. J. Numer. Meth. Fluids*, **67** (2010) 1799—1817.

Ongoing work includes:

- Extending these methods to handle inertial terms in the momentum equations as well as viscoelastic stresses within the network phase
- Developing "interface capturing" methods for problems in which the region containing network is a subset of the computational domain
- Developing immersed-boundary type methods for capturing interactions between a gel and flexible membranes
- Developing methods for chemical and mass advective-diffusive transport in regions defined by moving surfaces