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# A Geometric Structure-Preserving Discretization Scheme for Incompressible Linearized Elasticity

In this paper, we present a geometric discretization scheme for incompressible linearized elasticity. We use ideas from discrete exterior calculus (DEC) to write the action for a discretized elastic body modeled by a simplicial complex. After characterizing the configuration manifold of volume-preserving discrete deformations, we use Hamilton's principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We explicitly derive the governing equations for the two-dimensional case, where the discrete spaces for the displacement field are constructed by P1 polynomials over primal meshes for incompressibility constraint, P0 polynomials over dual meshes for the kinetic energy, and P1 polynomials over support volumes for the elastic energy, and the discrete space of the pressure field is constructed by P0 polynomials over primal meshes. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible.

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## Comments

## Hi

The paper is so hard to follow. Can we say it is a pure math paper?

## without Lagrange multipler

Hi Professor Yarari,

Thanks for posting this interesting paper. In this paper, are you using finite dimensional spaces that admiss only isochoric motion and as a result no Lagrange multiplier is needed to impose the constraint (i.e., insteading of filter out volumetric deformation through the use of LM on a conventional space that admiss non-isochoric motion)? Am I understand it correctly?

Many thanks,

WaiChing Sun

## Re: without Lagrange multipler

Dear WaiChing:

Using Lagrange multipliers is just one way of imposing a constraint. Here, instead of introducing a Lagrange multiplier and working with an auxiliary action, we extremize the action (with no Lagrange multipliers) on the space of volume-preserving motions. As you can see we do it in both continuous and discrete cases. In the continuous case the manifold of isochoric motions is still infinite dimensional, while in the discrete case everything is finite dimensional. To summarize, you look for your solution in the space of volume-preserving motions directly.

Regards,

Arash