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This work presents a three-dimensional (3D) surface-to-surface contact algorithm for SPH method and its application to the simulation of material failure during high velocity impact of a projectile on a ceramic plate. In the present contact algorithm, surfaces of 3D objects represented by SPH particles are reconstructed. Based on the reconstructed surfaces, a box test followed by a crossover test is employed to detect contacts between different objects, and a penalty method is employed to enforce contact conditions between contact pairs identified by contact detection. Validation examples are presented to demonstrate the accuracy and capability of the algorithm. The calculated results show that the present contact algorithm can achieve high accuracy and it is capable of handling various contact problems, including frictionless and frictional problems as well as small and significant deformation problems. The algorithm is then used to simulate the high velocity impact of a projectile on a ceramic plate. The material failure of the projectile and the ceramic plate is reproduced, and the residual velocity of the projectile is well predicted.