Tham khảo tài liệu 'applied computational fluid dynamics techniques - wiley episode 1 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 190 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES a unkno rhspo ipoil ipoi2 b geoed redge ipoil ipoi2 c ipoil ipoi2 Figure . Edge-based Laplacian . First derivatives first form We now proceed to first derivatives. Typical examples are the Euler fluxes the advective terms for pollution transport simulations or the Maxwell equations that govern electromagnetic wave propagation. The RHS is given by an expression of the form ri - NiNjkdQ Fk where Fk denotes the flux in the kth dimension at node j. This integral is again separated into shape functions that are not equal to Ni and those that are equal r - E E N N kdti Fk Ị NiNikdQ Fk. As before we use the conservation property equation and get r -Ẹ EJ f NiNjkdQ Fk 1 M Ni Njk do. j i el J el j i F k. This may be restated as _ jJit k _ uk jij _ A Ji Tj JO 2 tin i r dk Fi - Fp dk NN kd j i. One may observe that - for a change in indices ij versus ji we obtain this is expected due to the unsymmetric operator 4 -dkj NjNinkdr EDGE-BASED COMPRESSIBLE FLOW SOLVERS 191 - an extra boundary integral leads to a separate loop over boundary edges adding unsymmetrically only to node j. The flow of information for this first form of the first derivatives is shown in Figure . Figure . First derivatives first form Observe that we take a difference on the edge level and then add contributions to both endpoints. This implies that the conservation law given for the first derivatives is not reflected at the edge level although it is still maintained at the point level. This leads us to a second form which reflects the conservation property on the edge level. . First derivatives second form While is valid for any finite element shape function a more desirable form of the RHS for the first-order fluxes is r ej Fj Fj e -j. In what follows we will derive such an approximation for linear elements. As before we start by separating the Galerkin integral into shape .
