  In isogeometric analysis, it is frequently required to handle the geometric
models enclosed by four-sided or non-four-sided boundary patches, such as
trimmed surfaces. In this paper, we develop a Gregory solid based method to
parameterize those models. First, we extend the Gregory patch representation to
the trivariate Gregory solid representation. Second, the trivariate Gregory
solid representation is employed to interpolate the boundary patches of a
geometric model, thus generating the polyhedral volume parametrization. To
improve the regularity of the polyhedral volume parametrization, we formulate
the construction of the trivariate Gregory solid as a sparse optimization
problem, where the optimization objective function is a linear combination of
some terms, including a sparse term aiming to reduce the negative Jacobian area
of the Gregory solid. Then, the alternating direction method of multipliers
(ADMM) is used to solve the sparse optimization problem. Lots of experimental
examples illustrated in this paper demonstrate the effectiveness and efficiency
of the developed method.
