is determining how long signals take to get from the output of a logical gate to the input of the next gate. To compute that delay, one must determine the capacitance on each of the wires given in the slide picture. To do so requires computing charges given electrostatic potentials as noted above 5 What is common about these problems slide 8 Exterior Problems IEMS device- fluid(air)creates drag Package - Exterior fields create coupling Signal Line- Exterior fields. Quantities of interest are on surface meMS device-Just want surface traction force Signal Line-Just want surface charge Exterior problem is linear and space-invariant mEMS device- Exterior Stoke's flow equation(linear) Package -Maxwell's equations in free space(linear) Signal line- Laplace's equation in free spce(linear) But problems are geometrically very complex 6 Exterior Problems 6.1 Why not use FDM/ FEM? SLidE 9 2-D Heat Flow Example Only need a on the surface, but T is computed everywhere Must truncate the mesh, = T(oo)=0 becomes T(R)=0 Note 6 Heat conduction in 2D In this slide above. we consider a two dimensional exterior heat conduction problem in which the temperature is known on the edges, or surface, of a square Here, the quantity of interest might be the total heat flow out of the square The temperature T satisfies V2T(x)=0x∈9is determining how long signals take to get from the output of a logical gate to the input of the next gate. To compute that delay, one must determine the capacitance on each of the wires given in the slide picture. To do so requires computing charges given electrostatic potentials as noted above. 5 What is common about these problems? Slide 8 Exterior Problems MEMS device - fluid (air) creates drag Package - Exterior fields create coupling Signal Line - Exterior fields. Quantities of interest are on surface MEMS device - Just want surface traction force Package - Just want coupling between conductors Signal Line - Just want surface charge. Exterior problem is linear and space-invariant MEMS device - Exterior Stoke’s flow equation (linear) Package - Maxwell’s equations in free space (linear) Signal line - Laplace’s equation in free spce (linear) But problems are geometrically very complex 6 Exterior Problems 6.1 Why not use FDM / FEM? Slide 9 2-D Heat Flow Example T = ∞ 0 at But, must truncate t mesh Surface Only need ∂T ∂n on the surface, but T is computed everywhere. Must truncate the mesh, ⇒ T(∞)=0 becomes T(R)=0. Note 6 Heat conduction in 2D In this slide above, we consider a two dimensional exterior heat conduction problem in which the temperature is known on the edges, or surface, of a square. Here, the quantity of interest might be the total heat flow out of the square. The temperature T satisfies ∇2T (x)=0 x ∈ Ω (3) 5