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I read a master thesis on a topic I'm interested too. This work concern the solution of the displacement equation of motion for a homogeneous, elastic, isotropic material:

$$\rho \ddot{\mathbf{u}} - \mathbf{F} = \mu \nabla^2 \mathbf{u} + (\lambda + \mu) \mathrm{grad} \; \mathrm{div} \; \mathbf{u}$$

where $\mathrm{u}$ is the displacement vector, $\lambda$ and $\mu$ are the Lamé constants (knowns with some uncertainty), and $\mathbf{F}$ an external force per unit volume (known exactly), and particularly it's steady state variant, the so called Navier-Cauchy equation.

To be more precise, the domain is not infinite and homogeneus: it's a disk with few different regions and the Lamé constants are separately constants over each sub-region; over one of these sub-region $\mu$ become very very small.

The author of the thesis uses a finite difference spatial discretization with polar coordinates and compute the coefficient matrix and the data vector of the linear system to be solved.

The author notice the system is vary badly conditioned and, analyzing the singular value decomposition of this matrix, by some features of the singular values spectrum, apparently conclude it's (or behave as) a discrete ill-posed problem, coming from the discretization of a continuous inverse ill-posed problem. So the author apply a Tickhonov regularization to get an apparently meaningful solution.

Reading that Navier-Cauchy equation can be ill-posed or that is an inverse problem surprised me a bit.

What I know about the well posedness of elasticity problem:

  • under the hypothesis of "small displacements", and if the elastic tensor is such that the deformation energy is positive-definite, the solution of the elastic problem, if exists, is unique (Kirchhoff theorem);
  • the existence of a solution "is more complex to prove but has been proved in several scenarios useful for applications".

I don't know nothing about the third requirement for well-posedness in the Hadamard sense, i.e. if the solution is continuously dependent on data.

Neglecting now this thesis, at this point I think I need to know something more about the well posedness of the linear eleasticity problem and the Navier-Cauchy equation.

  • What is a good, authoritative, reference about the well posedness of elasticity problem (not too advanced please)?

  • Are the Navier-Cauchy equation well-posed? Are there cases where the problem can become ill-posed? How the well posedness is affected by the fact that Lamé constants are non-uniform, possibly near to zero, and/or not continuous?

  • In the case the problem prove to be well-posed, but the linear still system ill-conditioned, what is a good way to handle this specific bad conditioning? A preconditioner for example? What a good one?

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    $\begingroup$ The discretized problem can sometimes be ill-conditioned even when the original is well-behaved, e.g., $u''=f$ has second-order regular grid discretization $Au=f$ with a condition number of $O(h^{-2})$. I don't think having an ill-conditioned discretization is enough to conclude the original problem is ill-posed. Especially since if you drop the $\lambda+\mu$ term you just have a regular wave equation. $\endgroup$ – Kirill Feb 14 '15 at 0:37
  • $\begingroup$ @Kirill Thanks, I know and I agree. To be honest the author analyze the SVD spectrum of $A$ and from this apparently conclude the original problem is not well posed. $\endgroup$ – unlikely Feb 14 '15 at 8:25
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    $\begingroup$ That sounds like a completely bogus inference. There are so many ways to mess up a discretization to make it ill-conditioned that have nothing to do with the well-posedness of the original problem. $\endgroup$ – Wolfgang Bangerth Feb 14 '15 at 13:46
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    $\begingroup$ Can you cite the article in question? $\endgroup$ – Bill Barth Feb 14 '15 at 21:07
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    $\begingroup$ Well-posedness of a PDE depends on domain and boundary and initial conditions. If you have a particularly simple domain, you might find ordinary Fourier transform analysis sufficient to prove well-posedness, it's found in all introductory PDE textbooks. $\endgroup$ – Kirill Feb 14 '15 at 23:58
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The constant $\mu$ is the resistance of the material to shear stresses. If this constant is zero in a region and you are considering the steady-state case, the system of PDE are singular there; i.e. they are no longer elliptic.

So it is not surprising that if you discretize the PDE using finite differences or finite elements, and define $\mu$ to be very small, the matrix equations are ill-conditioned.

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  • $\begingroup$ Thanks but not sure what this means... Lax-Milgram theorem's hypotesis of V-coercivity? Not sure I'm ready to to handle this problem with weak formulation, better a classical approach if possible. Obviously if $\mu \to 0$ the material is like a fluid an intuitively the equation of motion commonly used for a linear, isotropic, elastic solid probably cannot be used. But, which of the conditions of Hadamard well posedness fails? Existence, uniqueness, or continously dependecy from data? Why (proof/reference)? $\endgroup$ – unlikely Feb 17 '15 at 12:51
  • $\begingroup$ Yes,you are correct that $\mu=0$ implies a compressible, inviscid fluid and therefore a different constitutive relation for that region. For what its worth, I don't believe there is a non-time-dependent (i.e. velocity equal zero) solution to this problem. I don't recall seeing this problem addressed but you might take a look a good reference on continuum mechanics, e.g Malvern (amazon.com/Introduction-Mechanics-Continuous-Lawrence-Malvern/…) $\endgroup$ – Bill Greene Feb 17 '15 at 21:40
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I think I found.

If $\mu = 0$ the reverse elastic bond

$$\mathbf{E} = \frac{1}{2\mu} \mathbf{S} - \frac{\lambda}{2\mu} \frac{\mathrm{tr }\mathbf{S}}{3\lambda + 2\mu} \mathbf{I}$$

is not defined.

Moreover the Kirchoff theorem about the uniqueness of the elastic response rely on the fact the strain energy density is positve-definite. For this to be true it is required

$$\mu > 0 \quad \text{and} \quad 3\lambda + 2\mu > 0$$

So if $\mu = 0$ we cannot prove the uniqueness of the solution.

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