Treatment of contact between a sheet and tools is one of the most difficult problems to deal with in finite-element simulations of the sheet metal forming processes. In order to obtain more accurate tool models without increasing the number of elements, this paper describes new formulations for contact problems using local interpolation proposed by Nagata for tool surfaces. A contact search algorithm between the sheet nodes and the interpolated tool surfaces and the consistent tangent stiffness matrix for the sliding nodes were developed. The developed algorithms were introduced into the static-explicit elastoplastic finite-element method code STAMP3D. Simulations of a square cup drawing process with a very coarsely discretized punch model were carried out. The simulated results showed that the proposed algorithms gave the proper deformation process, thus demonstrating the validity of the proposed formulations.