A four-node bilinear quadrilateral has parent coordinates (ξ,η)∈[-1,1]^2 and local…

A four-node bilinear quadrilateral has parent coordinates \((\xi,\eta)\in[-1,1]^2\) and local nodes 1 through 4 at \((-1,-1)\), \((1,-1)\), \((1,1)\), and \((-1,1)\), respectively. Its physical nodal coordinates are \((0,0)\), \((6,0)\), \((6,2)\), and \((0,2)\) m, and its nodal temperatures are \(10\), \(22\), \(30\), and \(14\ ^\circ\mathrm{C}\). Construct the four bilinear shape functions, verify their nodal interpolation and partition-of-unity properties, and at \((\xi,\eta)=(-1/2,1/3)\) determine the physical coordinates, the interpolated temperature, and the physical temperature gradient \((\partial T/\partial x,\partial T/\partial y)\).

Form each shape function by multiplying the two linear factors that equal one at its node.