problems T(r)is given on the surface, defined by T, and therefore both problems are Dirichlet problems. For the"temperature in a tank"problem, the problem domain, Q2 is the interior of the cube, and for the "ice cube in a bath"problem, the problem domain is the infinitely extending region exterior to the cube. For such an exterior problem, one needs an additional boundary condition to specify what happens sufficiently far away from the cube. Typically, it is assumed there are no heat sources exterior to the cube and therefore limT(x)→0. For the cube problem, we might only be interested in the net heat flow from he surface. That How is given by an integral over the cube surface of the normal derivative of temperature, scaled by a thermal conductivity. It might eem inefficient to use the finite-element or finite-difference methods discussed in previous sections to solve this problem, as such methods will need to compute the temperature everywhere in Q. Indeed, it is possible to write an integral quation which relates the temperature on the surface directly to its surface normal, as we shall see shortly In the four examples below, we try to demonstrate that it is quite common n applications to have exterior problems where the known quantities and the quantities of interest are all on the surface 4 Examples 4.1 Computation of Capacitance SLIDE 4 potential given on s What is the capac Capacitance= Dielectric Permittivity J Example 1: Capacitance problem In the example in the slide, the yellow plates form a parallel-plate capacitor with an applied voltage V. In this 3-D electrostatics problem, the electrostatic potential y satisfies V-y(r)=0 in the region exterior to the plates, and the otential is known on the surface of the plates. In addition, far from the plates. 4-0. What is of interest is the capacitance, C, which satisfiesproblems T (x) is given on the surface, defined by Γ, and therefore both problems are Dirichlet problems. For the “temperature in a tank” problem, the problem domain, Ω is the interior of the cube, and for the “ice cube in a bath” problem, the problem domain is the infinitely extending region exterior to the cube. For such an exterior problem, one needs an additional boundary condition to specify what happens sufficiently far away from the cube. Typically, it is assumed there are no heat sources exterior to the cube and therefore lim
x
→∞ T (x) → 0. For the cube problem, we might only be interested in the net heat flow from the surface. That flow is given by an integral over the cube surface of the normal derivative of temperature, scaled by a thermal conductivity. It might seem inefficient to use the finite-element or finite-difference methods discussed in previous sections to solve this problem, as such methods will need to compute the temperature everywhere in Ω. Indeed, it is possible to write an integral equation which relates the temperature on the surface directly to its surface normal, as we shall see shortly. In the four examples below, we try to demonstrate that it is quite common in applications to have exterior problems where the known quantities and the quantities of interest are all on the surface. 4 Examples 4.1 Computation of Capacitance Slide 4 v + - 2 ∇Ψ= 0 Outsi Ψ is given on S potential What is the capacitance? Capacitance = Dielectric Permittivity
∂Ψ ∂n Note 2 Example 1: Capacitance problem In the example in the slide, the yellow plates form a parallel-plate capacitor with an applied voltage V . In this 3-D electrostatics problem, the electrostatic potential Ψ satisfies ∇2Ψ(x)=0 in the region exterior to the plates, and the potential is known on the surface of the plates. In addition, far from the plates, Ψ → 0. What is of interest is the capacitance, C, which satisfies q = CV 2