Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.