We show that a fluid-structure coupling method based on a fixed Eulerian mesh using the level set function is applicable to fluid-structure interaction (FSI) problems involving incompressible viscous fluids and thin elastic structures. The coupling method was originally proposed for large-deformation FSI analyses of high-speed compressible inviscid flows and thin structures such as airbags. We introduce a novel interface-treatment technique that uses virtual particles with level sets and structural normal velocities to enforce the kinematical condition at the fluid-structure interface on a fluid fixed mesh. The virtual particles also have structural tangent velocities so as to impose no-slip conditions at the interface. Application of the method to finite-deformation FSI problems, and comparison of the results with those obtained by the conventional moving ALE mesh-based scheme show the adequacy of the method. It is confirmed that the appearance of the flow and geometry of the interface are similar to those for the ALE scheme.