One of the types of hexahedral grids that is the most commonly used for the subsurface is 3D structured grids. It is computational efficient, honoring the geometry of the faults and horizons of the geological model and the standard format of most of the current standard flow simulators. To build the grid, it requires that the faults to be distinguishable in two principal directions U and V. When it is not the case, the current technologies require that the fault network be either simplified structure-wise or approximated geometry-wise by a stair-step representation, both at the expense of honoring the complexity and precision of the input geological model. We propose a new method of building structured hexahedral grids without compromising the complexity of the structural model. Unlike current methods that generate the gridding based on a 3D parameterization UVW in which the U and V coordinates are constrained by separate families of surfaces respectively, our method does not have the limitation of classifying faults in U and V families. Our method uses a pair of parameterizations for the UV coordinates that can have local frame transformations (rotation+translation). Thus offering additional degrees of freedom during the gridding process to honor the geometry of faults on a larger family of complex geological models. These transformations are defined by transition functions that "glue" the local parameterizations back into a global one in a non-trivial way, called Global Parameterization. A hexahedral grid built this way might have singular vertices, which have valence not equal to eight. It can be used directly in next-generation unstructured flow simulators. We show through a simple treatment how to convert efficiently such a grid into a structured representation that is compatible with standard flow simulators. We call such grid Singular Structured Hexahedral (SSHex) Grid. We present the results of simulations on SSHex grids using standard flow and geomechanical simulation software. Then, we discuss the interesting potential of this new grid in the development of coupled flow-geomechanical simulation techniques.