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How long should the hyperelastic equations be solved before updating the mesh? To be specific, I'm interested in the hyperelastic model with a neo-Hookean solid:

$$ \nabla\cdot\sigma + f = \rho\ddot{{u}}\\ \sigma = \mu F + (\lambda\log\det F - \mu)F^{-T}\\ F = I + \nabla u $$

where

  • $\sigma$ - Stress tensor
  • $F$ - Deformation gradient
  • $u$ - Displacement
  • $\lambda, \mu$ - Lamé parameters
  • $f$ - External force

There's another question asking about what algorithms can be used to solve this problem, but it sounded like a standard Galerkin finite element method would work. Meaning, solve the above equations for the displacements and then use the displacements to update the location of the nodes in the mesh. Certainly, there's a lot of nuance to this, but what I would like to understand better is what considerations should be taken into how long the above equations can or should be solved for the displacements before stopping the simulation and updating the mesh. Alternatively, is there a name for this kind of algorithm to make it easier to search for additional resources?

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Since there's some confusion, I don't have a definite algorithm for this problem, which is partly why I'm asking for clarification and assistance.

I'm proposing solving the hyperelastic equations using a Galerkin finite element method with piecewise linear elements to discretize the spatial components and a Runge-Kutta method to discretize and solve an initial value problem over time. Assuming that this works, we generate the displacement, $u$, which can be evaluated at any point in the solved time. Of course, having a displacement means that we can move and deform the object itself, which means move and deform the mesh. I propose moving the nodal points of the mesh based on the displacement $u$. Once we deform the mesh, we could continue to solve the hyperelastic equations with the new domain. Again, assuming this works, what's not clear to me is how long to solve the original PDE before deforming the mesh. I don't know how to generate an error measure to check, whether such algorithms actually converge, and what they are potentially called.

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    $\begingroup$ You haven't explained clearly what kind of algorithm you plan to use, so it seems impossible to answer your question. But it sounds like you are think of a sort of arbitrary Lagrangian-Eulerian (ALE) approach. Maybe that is the name you're looking for. $\endgroup$ – David Ketcheson Mar 30 at 8:12
  • $\begingroup$ In regard of the mesh updating, I haven't done such simulations but I would expect as in all these types of simulations that you have to use an error indicator to monitor convergence. If convergence starts to fail you likely need a new mesh. $\endgroup$ – Bort Mar 30 at 9:27
  • $\begingroup$ @DavidKetcheson Sorry for the confusion. I've added some clarification to the question as to a proposed algorithm. $\endgroup$ – wyer33 Mar 30 at 15:59
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    $\begingroup$ Methods in which every mesh point moves exactly according to the displacement are called Lagrangian methods. Methods with a fixed mesh are called Eulerian methods. Methods in which the mesh deforms partially depending on the deformation are (usually) called arbitrary lagrangian Eulerian methods. $\endgroup$ – David Ketcheson Mar 30 at 16:05
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Whether any mesh updating (i.e. re-meshing) is required as part of the solution of the nonlinear equations depends very much on the specific problem you are trying to solve (geometry, loading, material properties). Many problems of this type can be solved with no re-meshing.

For example, if you take a block of rubber, create a uniform mesh of rectangular elements, and apply a pressure load to the top while constraining the vertical motion of the bottom, you can compress that block to a small percentage of its original height without changing the mesh. In this case, the aspect ratio of the elements is changing significantly but they remain more or less rectangular during the loading.

One commonly-used test to determine if re-meshing is required is to check the determinant of the Jacobian during the numerical integration over the element volume. If the determinant becomes negative at any integration point, the element is so distorted that the isoparametric mapping is no longer valid and a new mesh is needed.

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