In this paper we present a novel approach to efficiently simulate the deformation of highly detailed meshes using higher order finite elements (FE). An efficient algorithm based on non-linear optimization is proposed in order to find the closest point in the curved computational FE mesh for each surface vertex. In order to extrapolate deformations to surface points outside the FE mesh, we introduce a mapping scheme that generates smooth surface deformations and preserves local shape even for low-resolution computational meshes. The mapping is constructed by representing each surface vertex in terms of points on the computational mesh and its distance to the FE mesh in normal direction. A numerical analysis shows that the mapping can be robustly constructed using the proposed non-linear optimization technique. Furthermore it is demonstrated that the numerical complexity of the mapping scheme is linear in the number of surface nodes and independent of the size of the coarse computational mesh.