Closed-Form Isoparametric Shape Functions of Four-Node Convex Finite Elements

On arbitrary plane quadrilaterals, difficulties in integrating energy densities prevented analysts from directly using shape function expressions in terms of the physical coordinate variables ( $x$ and $y$ ). With the availability of an exact integration procedure, shape functions are sought here as explicit expressions (in $x$ and $y$ ). Conventional isoparametric (indirect) representation, via canonical coordinate variables ( $\eta$ and $\xi$ ) on a unit square in the computational domain, do not reveal the presence of irrational algebraic expressions that were first elaborated on by Wachspress. Computer algebra systems demonstrate that the isoparametric shape functions are: for a general quadrilateral—linear and a square root of a quadratic (in $x$ and $y$ ); for a trapezoid—identical to the Wachspress rational polynomials; for a parallelogram—bilinear functions; and, moreover, even valid for a triangle with a side node. The failure of the isoparametric formulation for concave domains is traced to the negative argument of irrational parts. Shape functions in the physical domain ( $x$ and $y$ ) facilitate contour plotting of responses (e.g., temperature distributions) within quadrilateral elements. A subsequent paper details exact calculation of stiffness matrices where the presented shape functions (in $x$ and $y$ ) are indispensable.