# How to implement the interface extension of fluid "displacement" in ALE?

In ALE, we first set a referenced space for fluid, then we extend the boundary fluid displacement to the whole fluid region, take harmonic extension as an example, we need $$\Delta \left ( \hat{u} \right )=0,\hat{u_{f}}=\hat{u_{s}} \ on\ \hat{I}\, and \ \hat{n}\cdot \hat{u{_f}}=0 \ on\ \partial \hat{F}-\hat{I}$$, so how to enforce the dirichlet condition of the fluid "displacement" on interface?in the variational form, we can glue test functions together to enforce that the velocity of fluid and solid are the same on the interface.what should we do here for fluid displacement?

As you correctly said, in ALE-FSI you use one shared function space for the velocity v and for a point $$x$$ in the fluid domain, we have $$v(x) = v_f(x)$$, and for a point $$x$$ in the solid domain, we have $$v(x) = v_s(x)$$. We do the same thing for the displacement in the implementation such that $$u(x) = u_f(x)$$ in the fluid domain and $$u(x) = u_s(x)$$ in the solid domain. This way the kinematic coupling condition $$v_f = v_s$$ on the interface $$\Gamma$$ and the geometric coupling condition $$u_f = u_s$$ are automatically satisfied.