A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications
Abstract
:1. Introduction
2. Materials and Methods
2.1. p-Refinement Procedure
- 1.
- Creating a set of following Equation (1), and sorting from lowest to the highest;
- 2.
- For each element associated with the sorted , :
- determining all the adjacent elements for ;
- For each adjacent element, obtaining the interelement boundary order;
- If the order of the element boundary > the order of :
- Renaming the node number of to match the interelement boundary;
- Refining the element following the protocol.
2.2. Formulation of Transition Elements
3. Results
3.1. Transition Elements
- Nodal coordinates;
- Shape functions;
- Integration quadrature.
3.2. Verification
3.3. Implementation of 3D FE Meshes
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Node No. | x | y | z |
---|---|---|---|
1 | −1 | −1 | −1 |
2 | 1 | −1 | −1 |
3 | 1 | 1 | −1 |
4 | 1 | −1 | −1 |
5 | −1 | −1 | 1 |
6 | 1 | −1 | 1 |
7 | 1 | 1 | 1 |
8 | 1 | −1 | 1 |
9 | 0 | −1 | 1 |
10 | 1 | 0 | 1 |
11 | 0 | 1 | 1 |
12 | −1 | 0 | 1 |
13 | 0 | 0 | 1 |
Integration Point No. | Weight | |||
---|---|---|---|---|
1 | −1 | −1 | −1 | |
2 | 1 | −1 | −1 | |
3 | 1 | 1 | −1 | |
4 | −1 | 1 | −1 | |
5 | −1 | −1 | 1 | |
6 | 1 | −1 | 1 | |
7 | 1 | 1 | 1 | |
8 | −1 | 1 | 1 | |
9 | 0 | −1 | −1 | |
10 | 1 | 0 | −1 | |
11 | 0 | 1 | −1 | |
12 | −1 | 0 | −1 | |
13 | 0 | −1 | 1 | |
14 | 1 | 0 | 1 | |
15 | 0 | 1 | 1 | |
16 | −1 | 0 | 1 | |
17 | 0 | 0 | 1 | |
18 | 0 | 0 | −1 |
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Element Description | Figure Number | Number of Nodes | Source File Name |
---|---|---|---|
Fourth-order element | Figure 7B | 125 | Fourth_order.txt |
Fourth-to-third-order transition element for case 1 | Figure 10A | 65 | Transition_4to3_Case1.txt |
Fourth-to-third-order transition element for case 2 | Figure 10B | 73 | Transition_4to3_Case2.txt |
Third-order element | Figure 7A | 64 | Third_order.txt |
Third-to-second-order transition element for case 1 | Figure 9A | 28 | Transition_3to2_Case1.txt |
Third-to-second-order transition element for case 2 | Figure 9B | 34 | Transition_3to2_Case2.txt |
Second-order element | Figure 6A | 27 | Second_order.txt |
Second-to-first-order transition element for case 1 | Figure 8A | 9 | Transition_2to1_Case1.txt |
Second-to-first-order transition element for case 2 | Figure 8B | 13 | Transition_2to1_Case2.txt |
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Shahriar, A.; Mostafa, A.J. A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications. Mathematics 2023, 11, 4954. https://doi.org/10.3390/math11244954
Shahriar A, Mostafa AJ. A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications. Mathematics. 2023; 11(24):4954. https://doi.org/10.3390/math11244954
Chicago/Turabian StyleShahriar, Adnan, and Ahmed Jenan Mostafa. 2023. "A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications" Mathematics 11, no. 24: 4954. https://doi.org/10.3390/math11244954
APA StyleShahriar, A., & Mostafa, A. J. (2023). A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications. Mathematics, 11(24), 4954. https://doi.org/10.3390/math11244954