Some Examples of the Poisson Equation – Ñ. (kÑu) = f
[Taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition]
Field of Application |
Primary Variable u |
Material Constant k |
Source Variable f |
Secondary Variables d, du/dx, du/dy |
Heat Transfer |
Temperature T |
Conductivity k |
Heat Source Q |
Heat Flow q [comes from conduction k ¶T/¶n and convection h(T-Tµ) |
Irrotational Flow of an Ideal Fluid |
Stream Function y
Velocity Potential f |
Density r
Density r |
Mass Production s (normally zero) Mass Production s (normally zero) |
Velocities: ¶ y /¶x = -v; ¶ y /¶y = u ¶ f /¶x = -v; ¶ f /¶y = u |
Groundwater Flow |
Piezometric Head f |
Permeability K |
Recharge Q |
Seepage: q = k¶ f/dn Velocities: u = -k¶ f/dx , v = -k¶ f/dy |
Torsion of Members with Constant Cross-Section |
Stress Function Y |
k = 1 G = Shear Modulus |
f = 2 q = angle of twist per unit length |
Gqdf/dx = -syz Gqdf/dx = -sxz |
Electrostatics |
Scalar Potential f |
Dielectric Constant e |
Charge Density r |
Displacement Flux density Dn |
Magnetostatics |
Magnetic Potential f |
Permeability m |
Charge density r |
Magnetic Flux density Bn |
Transverse Deflection of Elastic Membranes |
Transverse deflection u |
Tension T in membrane |
Transversely distributed Load |
Normal force q |