A FINITE ELEMENT MIXTURE MODEL FOR HIERARCHICAL POROUS MEDIA

A nite element description of uid ow through a deforming porous solid, with a hierarchical structure of pores, has been developed and implemented in the nite element software package DIANA.1 Several standard element types can be used for 2-D, axisymmetric and 3-D nite deformation analysis. The hierarchy is dealt with as an extra dimension, quanti ed by a parameter x0. Both spatial and hierarchical uid ow is described by a Darcy equation. Fluid pressure and hydrostatic solid pressure are related via an elastic uid{solid interface. The state of the uid, the Darcy permeability tensor and the elastic interface depend on both spatial position and hierarchical level. Discretization and integration of uid related quantities are split into a spatial and a hierarchical part. The degrees of freedom of the nite element model are the displacements of the solid, the hydrostatic pressure and a number of uid pressures on di erent hierarchical levels. Blood-perfused biological tissue can be regarded as a hierarchical porous solid, where the uid represents the blood and the hierarchy corresponds to the tree-like vascular structure. As an example, a simulation of a contracting, blood-perfused skeletal muscle is presented.