A global methodology for simulating multiphase flows and heat transfers interacting with complex objects or interfaces is presented. Elliptic equations or Navier-Stokes equations are resolved on a fixed structured curvilinear grid and the solid objects are initially represented by triangular surface elements. Several difficulties arise as soon as these two non-conforming grids have to interact in the same physical problem, and accurate methods are presented for each issues: Lagrangian/Eulerian grid projection, immersed boundary of interface problems, and finally visualization. Hence, a new fast point-in-solid method for curvilinear grid is presented to project Lagrangian shapes on Eulerian grid. A new immersed boundary and interface method is presented, the Algebraic Immersed Interface method. Several validation and application problems are presented to demonstrate the interest and accuracy of the method.