This paper describes a new mesh-free algorithm for computational fluid dynamics. The algorithm is based on the CIVA method, which is a CIP-type cubie interpolation technique formulated in volume / area coordinates with tetrahedron / triangle units and can be applied to mesh-free methods such as the gridless and particle methods. The CIVA-based mesh-free method will make it possible to simulate many types of problems involving complex geometries and complex phenomena with high accuracy, because the methods are based on a simple tetrahedron / triangle unit and can treat both resting and moving particles (computing points). To achieve high accuracy with the methods, however, it is important to evaluate particle movement correctly. Therefore an improved evaluation algorithm is also described in this paper.