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4-Node Planar Isoparametric Element — Mapping Visualization

CE/ME 532 · Section 3.5 · Drag the green probe or the red corner nodes to watch the isoparametric mapping (ξ,η) → (x,y) and every matrix in the derivation update live.

Parent element   (ξ,η)  space

→

Actual element   (x,y)  space

ξ = η = x = 0.000 y = 0.000 |J| = 0.000

The blue mesh on the right is the image of the ξ=const and η=const grid lines from the parent square. The two magenta arrows at the probe are the Jacobian columns ∂r/∂ξ and ∂r/∂η — the signed area of the parallelogram they span is |J|.
Red shading marks the region of the parent square where |J| < 0 (and its image on the actual element). Load the Near-degenerate preset and drag the green probe into the red zone — the matrix J determinant flips sign and the step-4 warning turns red.

1. Bilinear Lagrange shape functions

Four corner nodes ⇒ four shape functions, one per node. Each Ni(ξ,η) equals 1 at its own node and 0 at all three others, and together the set is bilinear (complete up to the ξη term):

N1 = ¼(1−ξ)(1−η),   N2 = ¼(1+ξ)(1−η),
N3 = ¼(1+ξ)(1+η),   N4 = ¼(1−ξ)(1+η).

At the current probe point:

∑Ni = 1.000  (partition-of-unity check).

2. Geometry mapping (“isoparametric”)

The same shape functions interpolate both geometry and displacement. The geometry map is

x(ξ,η) = ∑i=14 Ni(ξ,η) xi,    y(ξ,η) = ∑i=14 Ni(ξ,η) yi.

Matrix form:

x
y
 = ×

Numerical result at current probe:

3. Shape-function derivatives in (ξ,η)

To build strains we need derivatives of Ni with respect to the physical coords (x,y), but the Ni are written in (ξ,η). Start by differentiating in the parent space:

Ni,ξ,  Ni,η  are easy to compute from the bilinear forms.

Evaluated at the current probe:

i Ni,ξ Ni,η

Collected into the 2×4 matrix D:

N1,ξN2,ξ N3,ξN4,ξ
N1,ηN2,η N3,ηN4,η
 =

4. Jacobian matrix   J = D · X

The Jacobian of the map (ξ,η) → (x,y) is

x,ξy,ξ
x,ηy,η
 =
N1,ξN2,ξ N3,ξN4,ξ
N1,ηN2,η N3,ηN4,η
 ·
x1y1
x2y2
x3y3
x4y4

Numerical, at the current probe and element:

J = ,  |J| = 0.0000

5. Inverse Jacobian   Γ = J−1

Inverting J closes the chain rule and lets us pull derivatives into physical coords:

u,x
u,y
 = Γ
u,ξ
u,η
 ,  Γ = (1/|J|)
 J22−J12
−J21 J11

Numerical:

Γ =

6. Shape-function derivatives in (x,y)

Apply Γ to each column of D:

Ni,x
Ni,y
 = Γ
Ni,ξ
Ni,η

At the current probe:

i Ni,x Ni,y

7. Strain–displacement matrix   B = F · G · H

Block structure.

The four natural derivatives of (u,v) come from H acting on the 8 nodal DOFs:

u,ξ
u,η
v,ξ
v,η
 = H
u1
v1
u2
v2
u3
v3
u4
v4

Next G = diag(Γ,Γ) converts them to Cartesian derivatives, and F extracts the three engineering strain components:

F =
1000
0001
0110
   ⇒   εx=u,x, εy=v,y, γxy=u,y+v,x.

Live 3×8 result (at the probe):

Element stiffness integral.

The area element transforms as dx dy = |J| dξ dη, so the 8×8 element stiffness is

k = ∫−11 ∫−11 BT E B  t  |J|  dξ dη,

usually evaluated by 2×2 Gauss quadrature (four sample points at ξ,η = ±1/√3). Because B and |J| both depend on (ξ,η) for any non-rectangular element, analytic integration is not practical.

Physical meaning of |J|. A small patch dξ dη in the parent square maps to a patch of area |J| dξ dη in the actual element. When |J| > 0 the mapping is orientation-preserving and one-to-one; when |J| ≤ 0 somewhere inside the parent square the element is degenerate and its inverse Jacobian blows up — try the Near-degenerate preset and drag the probe around to see |J| flip sign.