Shape Function Explorer

Your tasks

1. Drag P onto node 3. Read N₁ … N₄.
   Check answer

   N = (0, 0, 1, 0). Each function is 1 at its own node and 0 at the others (Kronecker delta), so T at node 3 is exactly T̂₃: the unknowns really are nodal temperatures.

2. Press Activity 4 point: P = (1.5, 0.3). Check your hand answers from Activity 4.
   Check answer

   N = 0.175, 0.525, 0.225, 0.075; the sum is 1 (partition of unity); T = 136.0 K. Node 2 carries half the weight because P is nearest to it. Activity 5: ∂T/∂x = ∂T/∂y = 20 K/m, so q_x = q_y = −40 W/m² with k = 2.

3. Find three points where N₁ = N₃. What line joins them?
   Check answer

   The diagonal from node 2 to node 4, x/a + y/b = 1. On it (a − x)(b − y) = xy, so the two functions are equal. Try it with a different a and b: it is still that diagonal.

4. Switch to the 8-node element and show N₁. Where is it negative, and how can the weights still sum to 1?
   Check answer

   N₁ is negative over most of the element away from corner 1 (−0.25 at the centre). A corner function of the 8-node element must vanish at the two mid-side nodes beside it, so it dips below zero between them. The mid-side functions (up to 1.0) make up the difference: Σ N_i = 1 everywhere.