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We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between them? Is it that there is no difference, or one is numerically better than the other?

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When you study a deformable continuum, the current configuration $x$ means very little, unless you also know what the original configuration $X$ was. Interesting things only happen when the displacement $x-X$ is different from point to point of the continuum. More precisely, when the displacement is not the sum of a rotation and a traslation, and thus the continuum is not simply undergoing rigid body motion. So in the end, you are going to be interested not in the displacement, but in the gradient of it.

There is a more relevant question when dealing with these type of problems, and is whether you are taking a Lagrangian or Eulerian view.

In the Lagrangian description of deformation, you define the current configuration as a function of the original configuration, $x = \phi(X)$, and you track what happens to 'particles' in the original configuration as the body deforms. The displacement under such description takes the form $U(X) = x-X = \phi(X)-X$, You could compute the gradient of the displacement, which is closely related to deformation, at the point that was originally at position $X$ as $\nabla U(X) = \nabla \phi(X)-I(x)$.

In the Eulerian description, you look at a fixed point in space, and consider what is going on to the particle that now is at that position. In this description, displacement is $u(x) = x-X=x - \phi^{-1}(x)$, and the gradient of the displacement for the point currently at position $x$ is $\nabla u(x) =I - \nabla \phi^{-1}(x)$.

You may already glean that the math is harder when going Eulerian, as you are having to deal with the gradient of an inverse transformation. It is nevertheless much easier from an observational perspective to handle what is going on at a certain point in space, regardless of where that chunk of matter was in a previous time.

In most elasticity studies, only infinitesimal displacements are considered, so that material points can be considered not to move, rendering the above distinction more of a formal thing. In fluid mechanics this is of paramount importance, and generally the Eulerian view prevails, even though it adds all those 'convective' terms to the equations, to account for the fact that the point under observation is moving. Link to the relevant wikipedia page.

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