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The burden of creating meshes increases the cost of Computational Fluid Dynamics (CFD) and slows the rate at which new engine geometries can be investigated. Internal Combustion Engines (ICEs) with moving valves and piston present a special challenge, often requiring numerous different target meshes or case-specific codes for adapting the mesh. The goal of the present paper is to facilitate remeshing by calculating vertex motion, in parallel, for hybrid tetrahedral and hexahedral meshes. The calculated vertex motion is intended to maintain good mesh quality and reduce the need for interpolation to a new mesh. The demonstrated approach uses Laplacian-based smoothing for hexahedral cells and optimization-based smoothing for tetrahedral cells. Further, planar and cylindrical surfaces in the engine geometry are automatically recognized. As the engine volume changes shape, vertices may slide along the planar and cylindrical surfaces.