Periodic finite element mesh

We introduce new ab-initio real-space method based on (1) density functional theory, (2) finite element method, and (3) environment-reflecting pseudopotentials. It opens various ways for further development and applications restricted periodic boundary conditions in a desired sub-region or in a requisite direction (e.g. for nonperiodic objects with bonds to periodic surroundings), adaptive finite-element mesh and basis playing the role of variational parameters (hp-adaptivity) and various approaches to Hellman-Feynman forces and sensitivity analysis for structural optimizations and molecular dynamics. 

The basic concept to connect both scales of simulation is illustrated in Fig, 7, The model system is a periodic box described by a continuum, tessellated to obtain finite elements, and containing an atomistic inclusion, Any overall strain of the atomistic box is accompanied by an identical strain at the boundary of the inclusion. In this way, the atomistic box does not need to be inserted in the continuum, or in any way connected (e,g, with the nodal points describing the mesh at the boundary of the inclusion). This coupling via the strain is the sole mechanism to transmit tension between the continuum and the atomistic system. The shape of the periodic cells is described by a triplet of continuation (column) vectors for each phase (see also [21]), A, B, and C for the continuous body, with associated scaling matrix H = [ABC], and analogously a, b, and c for the atomistic inclusion, with the scaling matrix h = [abc] (see Fig, 8),... 

In this example, one periodic element (a cross-over) of the laboratory scale version of Katapak -S was selected for the detailed CFD simulation with CFX-5. This solver uses the finite volume discretization method in combination with hybrid unstructured grids. Around 1,100 spherical particles of 1 mm diameter were included in the computational domain. As the liquid flows through the catalyst-filled channels at operating conditions below the load point (cf. Moritz and Hasse, 1999), permeability of the channel walls made of the wire mesh is not taken into account by this particular model. The catalyst-filled channels are considered fully wetted by the liquid creeping down, whereas the empty channels are completely occupied by the counter-current gas. It means that the bypass flow... 

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