| Author | Message | ||
     Wilson Chen (emptyempty) New member Username: emptyempty Post Number: 1 Registered: 09-2003 |
Hello Mr. Nelson, I have been trying to solve a heat equation, {div(D*grad(T))-a*T=0}, in a two-layer cylinder. An interesting case is when there is a heat flux jump at the interface.{ dr(T)1-dr(T)2=0.01, where dr(T)1 is the heat flux at the left hand side of interface, dr(T)2 is the heat flux at the right hand side of interface }. In other words, there are two values for NATURAL(T) at the interface, and both of them are unknown. It seems nobody did this before, and I am wondering if flexPDE can deal with this case. Thank you in advance! Sincerely, Wilson {############# Script #########################} TITLE ' NO way to create the boundary condition: {dr(T)|r=r1-} - {dr(T)|r=r1+}=Q in FlexPDE ??? ' SELECT ERRLIM=1E-3 COORDINATES XCYLINDER VARIABLES T DEFINITIONS { parameter definitions } a a1=5 a2=300 D=2800 Q=0.01 !---{dr(T)|r=interface-} - {dr(T)|r=interface+}=Q ! FLUX1 r1=25 r2=125 length=400 EQUATIONS { PDE's, one for each variable } div(D*grad(T))-a*T=0 BOUNDARIES REGION 1 a=a1 START(0,0) NATURAL(T)=0 LINE TO (LENGTH,0) LINE TO (LENGTH,R1) !!?********How to set up Natural condition**?!! !**********FLUX1=normal(-D*grad(T)) LINE TO (0,R1) LINE TO FINISH REGION 2 a=a2 START(0,R1) !!?****How to set up natural condition***?!! !******NATURAL(T)=Q-FLUX1 !******FLUX2=normal(-D*grad(T)) LINE TO (LENGTH,R1) LINE TO (LENGTH,R2) NATURAL(T)=0 LINE TO (0,R2) LINE TO FINISH PLOTS CONTOUR(T) ELEVATION(T) FROM (LENGTH/2,0) TO (LENGTH/2,R2) !REPORT(FLUX1,FLUX2) Ends END | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 31 Registered: 06-2003 |
With a divergence operator, the Natural boundary condition takes its meaning from the divergence theorem, vol_integral(Div(Vector))=surf_integral(normal<dot>Vector). The Natural BC specifies the value of (normal<dot>Vector) on the surface. Applied to two adjacent mesh cells, the total volume integral is the sum of the surface integrals, or the difference of the surface flux on the shared boundary as seen by the two cells (since the normals are oppositely directed). On an internal boundary, the Natural boundary condition specifies this difference in fluxes. If energy is conserved, the difference is zero, so the default condition at all internal surfaces is Natural()=0. You can specify a nonzero Natural BC at internal interfaces if you want, and as you see from the above arguments, its meaning is the same as Div(Vector). In other words an internal Natural BC is a surface source or sink, depending on the meaning of Vector. You do not, however, have control of the partitioning of the flux between the two sides, since this must be controlled by the PDE and the various conditions on the two sides. | ||
| Wilson Chen (emptyempty) New member Username: emptyempty Post Number: 2 Registered: 09-2003 |
Hi, Mr. Nelson, I think I got what you said. Thanks for your excellent comment. Best regards, Wilson |