In 1D cases Hermite shape functions can be easily implemented. However, in 2D cases, if we want to use cubic Hermite triangle element (10 DOFs), then it is pointed out that the transformation between the physical triangle and the reference triangle is not affine-equivalent (or it is nonconforming). In this case, if calculating the gradient matrix directly then it will lead to wrong results.
The nonconforming nature of cubic Hermite triangle element is mentioned in Reddy's "An introduction to nonlinear finite element analysis" (see the attached figure), however, further discussion and examples of applying cubic Hermite triangle element are not presented in this book.
I am wondering if there are any available books/references that cover the details of the information related to this question.
Wei Ding, Yes, in finite element analysis, we often use shape functions to interpolate values within elements. In 1D, these shape functions are typically based on linear or cubic Hermite interpolation, and they work well because the mapping from the reference interval [-1, 1] to the physical element is simple and affine. This means that the relationship between the coordinates in the reference element and the physical element is a linear transformation.
However, in 2D when dealing with triangular elements, the mapping from the reference triangle to the physical triangle is not necessarily affine-equivalent. This non-affine mapping complicates the calculation of shape function gradients. In the case of cubic Hermite triangles, you have more degrees of freedom (10 DOFs), and this non-affine mapping can lead to incorrect results if you calculate the gradient matrix directly using the standard approach for affine transformations.
Thank you Engr. Tufail , yes I encountered exactly the same problem when I was trying to calculate the gradient matrix, so I am wondering what techniques should be applied in order to address this problem? Is there a standard approach to deal with situations like this and even for more complicated element types?
Wei Ding, When dealing with non-affine transformations in finite element analysis, especially for complex element types like cubic Hermite triangles, there are several techniques and approaches that can be applied to address the issue and calculate the gradient matrix correctly. Here are some common methods:
- One approach is to use an isoparametric formulation, which involves defining shape functions in terms of the natural coordinates of the reference element. By doing this, you can incorporate the non-affine mapping directly into your shape functions, ensuring that the mapping is correctly represented. This approach is commonly used for various types of finite elements, including triangular and quadrilateral elements.
- In some cases, especially for complex geometries, you may resort to subparametric mapping. This involves subdividing the physical element into smaller sub-elements, each of which can be mapped more closely to an affine shape. You then calculate the gradient matrix separately for each sub-element and interpolate the results to obtain the gradient matrix for the entire element.
- For cubic Hermite triangles, you may need to use higher-order interpolation functions that better capture the non-affine mapping. These functions can be more complex than linear or cubic Hermite interpolation but can provide more accurate results.
- When dealing with non-affine elements, you may need to use numerical integration techniques, such as Gaussian quadrature, to calculate integrals over the element accurately. This involves breaking down the element into smaller regions and approximating the integral within each region using an appropriate quadrature rule.
- It's crucial to compute and incorporate the Jacobian matrix of the mapping between the reference and physical elements. The Jacobian matrix describes how the local coordinates in the reference element map to the physical coordinates in the actual element. You can use this matrix to transform gradients correctly.
- Some finite element analysis software packages have built-in capabilities to handle non-affine elements and perform accurate gradient calculations automatically. Using such software can simplify the implementation of complex element types.
- For specialized element types and non-affine mappings, consulting relevant research papers, textbooks, or experts in the field can be invaluable. There may be specific techniques or formulations tailored to your problem.
I hope this comprehensive answer address the solution !
Engr. Tufail
Dear Professor Mabood,
Many thanks for your preceding clarification.
Can the Q8 element be used in the modeling of Kirchhoff plate?
I am trying to obtain the shape functions by assuming the displacement polynomial has the first 24 elements of the Pascal triangle. However, I got an infinity solution.
Ahmed M. Eldeeb, well, theoretically the Q8 element can be used in the modeling of Kirchhoff plates, obtaining shape functions using the first 24 elements of the Pascal triangle is not good approach and can lead to infinity solutions because of the complexity involve in it.
Engr. Tufail
I have tried but, I do not know the problem.
So, it cannot be used as I got from your answer, or there are some steps I have to follow.
Ahmed M. Eldeeb, currently I have no access to MATLAB however I recommend you not to use it, for the aforementioned objectives.
Engr. Tufail
You mean that the problem is in the software of solving?
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