Question

Calculate the Jacobian matrix and its determinant for the following quadrilateral element:

Answer 1

The Jacobian matrix for the given plane quadrilateral element is:

`\[\textbf{J} = \begin{bmatrix} 3 & 0 \\ 2 & 1 \end{bmatrix}\]`

The determinant of the Jacobian matrix is 3.

Step-by-step explanation:

Given coordinates:

(P₁ (5, 5) corresponds to (ξ₁,η₁) = (1, 2),

(P₂ (3, 4) corresponds to (ξ₂,η₂) = (2, 3),

(P₃ (3, 2) corresponds to (ξ₃,η₃) = (2, 1),

(P₄ (5, 2) corresponds to (ξ₄,η₄) = (1, 1).

The transformation equations from physical to natural coordinates are:

x = a₁ + a₂ ξ+ a₃ η + a₄ ξ η

y = b₁ + b₂ ξ + b₃ η + b₄ ξ η

In the given problem, the transformations between physical and natural coordinates can be expressed as:

x = c₁ + c₂ ξ + c₃ η

y = d₁ + d₂ ξ + d₃ η

The partial derivatives for x and y are calculated as:

`\[(\partial x)/(\partial \xi) = c_2, (\partial x)/(\partial \eta) = c_3\]`

`\[(\partial y)/(\partial \xi) = d_2, (\partial y)/(\partial \eta) = d_3\]`

From the given coordinates, the relations between physical and natural coordinates can be represented in matrix form:

`\[\begin{bmatrix} 5 \\ 3 \\ 3 \\ 5 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 1 \\ 2 & 3 & 1 & 1 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ d_1 \\ d_2 \\ d_3 \end{bmatrix}\]`

Solving this matrix equation yields the coefficients (c₁ = 2), (c₂ = 3), (c₃ = 0), (d₁ = 5), (d₂ = 2), (d₃ = 1).

Therefore, the Jacobian matrix representing the transformation between physical and natural coordinates is:

`\[\textbf{J} = \begin{bmatrix} 3 & 0 \\ 2 & 1 \end{bmatrix}\]`

The determinant of this matrix is calculated as follows:

det (J) = 3 * 1 - 0 * 2 = 3

Complete Question

Calculate the Jacobian matrix and its determinant for the following quadrilateral element: