Computational Physics

13.7 Concurrent coupling of length scales: FEM and MD 443 Obviously, we could take periodic boundary conditions in the transvers ...

444 The finite element method for partial differential equations Obviously, we should investigate which elasticity matrix should ...

Exercises 445 For a pair potential, this can be worked out further to yield C ̃αβ γ δ=^1 2  ∑ i=j { 1 rij^2 [ V′′(rij)− 1 rij ...

446 The finite element method for partial differential equations y a x c a' c' b' b 1 1 Figure 13.9. Archetypical triangle with ...

References 447 [13] M.-C. Rivara, ‘Design and data structure of fully adaptive, multigrid, finite element software,’ ACM Trans. ...

14 The lattice Boltzmann method for fluid dynamics 14.1 Introduction Flow problems are widely studied in engineering because of ...

14.2 Derivation of the Navier–Stokes equations 449 pioneered by LeVeque and Li[1]. Another method is to replace the macroscopic, ...

450 The lattice Boltzmann method for fluid dynamics in cases where the substance is not so dilute, such as a fluid. The Boltzman ...

14.2 Derivation of the Navier–Stokes equations 451 by multiplyingEq. (14.1)byvand integrating over the velocity. Using the indic ...

452 The lattice Boltzmann method for fluid dynamics As mentioned above, the additional currents which arise on top of the equili ...

14.2 Derivation of the Navier–Stokes equations 453 will now be calculated using(14.15): noneqαβ = ∫ mvαvβnnoneqd^3 v =−τ [∫ mvα ...

454 The lattice Boltzmann method for fluid dynamics The second term in the square brackets of(14.16)can be written, using the qu ...

14.3 The lattice Boltzmann model 455 The mass conservation equation and the momentum conservation equation together are insuffic ...

456 The lattice Boltzmann method for fluid dynamics 3 6 1 2 4 5 1 4 3 2 5 6 7 8 0 0 d2q6 d2q9 Figure 14.1. The two grids used in ...

14.3 The lattice Boltzmann model 457 whereαandβare Cartesian coordinatesxandy; the dependence on the positions is left out for c ...

458 The lattice Boltzmann method for fluid dynamics to having zero velocity in between the two particles. More accurate implemen ...

14.4 Additional remarks 459 to include the next nearest neighbours into the possible moves on the square lattice. In three dimen ...

460 The lattice Boltzmann method for fluid dynamics The extra force is included in the lattice Boltzmann equation which now read ...

14.5 Deriving Navier–Stokes equation from lattice Boltzmann model 461 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.05 0.1 0. ...

462 The lattice Boltzmann method for fluid dynamics Now we consider the moments of these equations. The zeroth moment of the non ...