The adaptation of an existing volumetric simulation mesh to updated parameters of the underlying CAD geometry is a crucial component within automatic design optimization. By avoiding costly automatic or even manual (re-)meshing it enables the automatic generation and evaluation of new design variations, e.g., through FEM or CFD simulations. This is particularly important for stochastic global optimization techniques—such as evolutionary algorithms—which typically require a large number of design variations to be created and evaluated. In this paper we present a simple yet versatile method for high quality mesh morphing. Building upon triharmonic radial basis functions, our shape deformations minimize distortion and thereby implicitly preserve shape quality. Moreover, the same unified code can be used to morph tetrahedral, hexahedral, or arbitrary polyhedral volume meshes. We compare our method to several other recently proposed techniques and show that ours yields superior results in most cases.