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Solving incompressible finite elasticity without tears

Solving incompressible elasticity has been quite challenging numerically. The conventional approach for handling incompressibility is the so-called penalty method. A volumetric energy term enters into the strain energy and penalizes the volumetric deformation. One straightforward issue is that the penalty parameter goes directly into the tangent matrix. The bigger the penalty parameter, the worse the condition number of the matrix. This is really a manifestation of the ill-posedness of theories based on the Helmholtz free energy, in my opinion [3]. When the problem size is not too big, direct solvers can be used, and we probably do not realize the ill-conditioning. This is not the case when the problem size gets bigger and gets coupled in the multiphysics setting (e.g. fluid-structure interaction). This drives us to think of things in a different way. Mathematically, the penalty formulation can be transformed into a saddle-point problem, just like what has been done in fluid mechanics. In thermodynamics, this is achieved through the Legendre transformation between potentials [3]. For a saddle-point problem, there are many ways of solving the equations [1]. The trick is to devise a good approximation for the Schur complement. In a recent JCP paper [2], we developed an effective way of solving the finite-strain elasticity, using an iterative algorithm for sparse matrices. The algorithm can drive the residual to machine tolerance within a few iterations; it works well for compressible and incompressible, soft and hard, isotropic and anisotropic materials; it is scalable on parallel machines. Additionally, for the thermomechanical foundation of this framework, readers are also referred to [3].

[1] Michele Benzi, Gene H. Golub, and Jörg Liesen. "Numerical solution of saddle point problems." Acta Numerica 14 (2005): 1-137.

[2] Ju Liu and Alison L. Marsden. "A robust and efficient iterative method for hyper-elastodynamics with nested blockpreconditioning." Journal of Computational Physics 383 (2019): 72-93.

[3] Ju Liu and Alison L. Marsden. "A unified continuum and variational multiscale formulation for fluids, solids, and fluid-structure interaction." Computer Methods in Applied Mechanics and Engineering 337 (2018): 549-597.

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