The problem of automatically generating unstructured tetrahedral meshes having a user specified number of elements in the thickness direction without over-refinement in the other directions is addressed in this paper. The procedures generate meshes with multiple elements through the thickness by anisotropically enriching an initial mesh, using local mesh modification procedures, wherever the mesh does not have a suffient number of elements through the thickness. A method for detecting portions of the mesh with insufficient elements through the thickness is also given. The procedure to generate multiple elements through the thickness is completely automatic with the only use input being the minimum number of elements required through the thickness and the geometric model. The initial mesh is automatically generated by Finite Octree and multiple elements are created through the thickness starting from this mesh.