# Finite element of 2D disc (circular plate)

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ahmad eldeeb on 10 Apr 2023
Answered: Rijuta on 17 May 2023
I am using FEM to solve for stresses and strains of a disc (2D circular plate) in the r-theta plane. Many references are availabel to do that for recangulr plate. However, I have done my best and there seems to be a problem with the jacobian matrix in polar coordinate system. If anyone could guide me for that. How to calculate such matrix?
* I am using Q8 element.

Rijuta on 17 May 2023
I understand that you are trying to calculate the Jacobian matrix for a 2D circular plate in the polar coordinate system. In the polar coordinate system, the Jacobian matrix is computed based on the transformation from Cartesian to polar coordinates. Here's a step-by-step guide to calculating the Jacobian matrix for a Q8 element in the polar coordinate system.
• For the Q8 element, you'll have eight shape functions corresponding to the eight nodes of the element. Define each node using polar coordinates (r, theta). You can store these in individual arrays for both r and theta.
• Calculate the derivatives of the shape functions with respect to the local coordinates using the 'diff' function. These derivatives represent the gradients of the shape functions and are used to determine the mapping between the local and global coordinates.
• The Jacobian matrix is computed by evaluating the derivatives of the shape functions with respect to the local coordinates and multiplying them by the nodal coordinates.
• For example:
J = [dN_dr * r, dN_dtheta * cos(theta); dN_dr * sin(theta), dN_dtheta * r];
• Next, the determinant of the Jacobian matrix is calculated to ensure that the element is not degenerate. A degenerate element can lead to inaccurate results or convergence issues in the finite element analysis. You can use the 'det' function to calculate the determinant of your Matrix.

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