I am a bit confused regarding how mapping from parent element to isoparametric element is done.
I am attaching a jpg file with two mapping regimes.
With reference to the photo, which one is correct? Is there any specific rule by which mapping is done? do the most exterior corners in parent element always take up the corners of isoparametric rectangle?
You should also check the validity of iso-parametric mapping for the elements. For this purpose, you can take the determinant of the jacobian matrix and check if it is zero anywhere over the element. If so, then the isoparametric mapping is invalid. The determinant of the jacobian matrix has to be either positive, or negative (only in special cases).
Regards,
George
Hi,
both are correct. The mapping between the parent element and its configuration in space is one-to-one and onto, meaning that the local ordering of nodes along the perimeter is retained (i.e. must be the same). The only difference in the figures is that in the first row node 5 is located on the perimeter between 1 and 2, whereas it is between 2 and 3 in the second row.
Btw: "iso"-parametric means that the same interpolation functions are used for both the geometry (interpolation of nodal coordinates) and the physical variables (e.g. displacement). Moreover, global node numbers are assigned to parent element (local) node numbers in the mapped element.
Regards
You should also check the validity of iso-parametric mapping for the elements. For this purpose, you can take the determinant of the jacobian matrix and check if it is zero anywhere over the element. If so, then the isoparametric mapping is invalid. The determinant of the jacobian matrix has to be either positive, or negative (only in special cases).
Regards,
George
As precised by George, you must check the uniqueness of the mapping that is, at each node of the parent element must correspond one and only one node of the real element and vice-versa. Therefore, the jacobian of the mapping must not change of sign inside the element, in other terms cannot be zero. To check that, you must define the interpolation functions used for the mapping, then calculate the jacobian matrix and at last the jacobian (determinant of the jacobian matrix).
For example, in the case of a four node quadrilateral element, the iso-parametric mapping uses bi-linear interpolation functions. In this case, it could easily be shown that to ensure the uniqueness of mapping, all the internal angles of the real element must be less than 180 degrees.
Considering now your case, I don't really know what interpolation functions you use and in particular for node five, but, considering the previous remark, I expect that mapping 1 is not valid.
Regards,
Jean-Michel
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