Diffeomorphic approximation of piecewise affine homeomorphisms

Given any $f$ a locally finitely piecewise affine homeomorphism of $\mathrm{\Omega }\subset {\mathbb{R}}^d$ onto $\mathrm{\Delta }\subset {\mathbb{R}}^d$ (for $d=3,4$) such that $f\in W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^d)$ and $f^{-1}\in W^{1,q}(\mathrm{\Delta },{\mathbb{R}}^d)$, $1\le p,q<\mathrm{\infty }$ and any $\epsilon >0$ we construct a diffeomorphism $\tilde{f}$ such that $$\mathrm{\parallel }f-\tilde{f}{\mathrm{\parallel }}_{W^{1,p}(\mathrm{\Omega },{\mathbb{R}}^d)}+\mathrm{\parallel }{f}^{-1}-{\tilde{f}}^{-1}{\mathrm{\parallel }}_{W^{1,q}(\mathrm{\Delta },{\mathbb{R}}^d)}<\epsilon \mathrm{.}$$