Wavelet Element Modelling for Inviscid Fluid–Solid Coupling Problem based on Partitioned Approach
Abstract
:1. Introduction
2. Formulation of the Inspected Problem
2.1. The Strong Form (Physical Models)
2.2. The Weak Form (Numerical Models)
2.3. Wavelet Element Spatial Discretization
2.4. Temporal Discretization and Partitioned Approach
3. Numerical Examples
- (1)
- Cases A–C were designed for different regions (i.e., from rectangular to solid circle) and different interfaces (i.e., from straight to curved).
- (2)
- Two sub-cases were set in each case (e.g., Case A-1 and Case A-2 for Case A). These two sub-cases were used to simulate wave propagation from solid to fluid and fluid to solid, respectively.
- (3)
- Cases A–C only present some qualitative comparisons with the theoretical wavefront in snapshots. Quantitative analyses are given at the end of this section, in terms of convergence analysis.
- (4)
- Considering the similarity of Cases A–C, convergence analysis was only conducted for Case C, the most complex case among them.
3.1. Case A: Rectangular with a Straight Interface
3.1.1. Case A-1: Wave Travels from Solid to Fluid
3.1.2. Case A-2: Wave Travels from Fluid to Solid
3.2. Case B: Rectangular with a Curved Interface
3.2.1. Case B-1: Wave Travels from Solid to Fluid
3.2.2. Case B-2: Wave Travels from Fluid to Solid
3.3. Case C: Solid Circle with a Curved Interface
3.3.1. Case C-1: Wave Travels from Solid to Fluid
3.3.2. Case C-2: Wave Travels from Fluid to Solid
3.4. Convergence Analysis in Space
- (1)
- Grid-1, the densest mesh composed of 1920 elements (321,600 dofs) for the fluid part and 800 solid elements (161,604 dofs) for the solid part.
- (2)
- Grid-2, the grid used in Section 3.3, which was composed of 960 elements (160,800 dofs) for the fluid part and 400 solid elements (80,802 dofs) for the solid part.
- (3)
- Grid-3, a denser mesh composed of 648 elements (66,744 dofs) for the fluid part and 324 solid elements (66,746 dofs) for the solid part.
- (4)
- Grid-4, a sparse mesh composed of 392 elements (40,448 dofs) for the fluid part and 196 solid elements (40,490 dofs) for the solid part.
3.5. Convergence Analysis in Time Domain
4. Discussions about Relaxation
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Main Algorithm |
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#1 Loop over elements e Calculate the elemental matrices Kse, Mse, Cse and Kfe, Mfe, Cfe Assemble mass matrices Ms, Mf, interface matrices Cs, Cf, and vector ft Store all elemental stiffness matrices Kse and Kfe #1 End of the loop over element e Calculate the auxiliary vectors , , , and , , , Apply the initial condition Initialization, let , #2 Loop over t While or Invoke “The partitioned approach based on Jacobi iteration” End #2 End of the loop over instant t Output , |
The Partitioned Approach Based on Jacobi Iteration |
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Calculate and Assemble vectors and Calculate force vectors and Predict the vectors and Calculate force vectors and Update the vectors and Relaxation (alternative): |
Medium | Modulus/GPa | Density/kg·m−3 | Poisson’s Ratio | cp/m·s−1 | cs/m·s−1 |
---|---|---|---|---|---|
Solid | 70.0 | 2700 | 0.33 | 6197.82 | 3121.75 |
Fluid | - | 1020 | - | 1468.63 | - |
Medium | Modulus/GPa | Density/kg·m−3 | Poisson’s Ratio | cp/m·s−1 | cs/m·s−1 |
---|---|---|---|---|---|
Solid | 25.6 | 2500 | 0.21 | 3395.10 | 2057.00 |
Fluid | - | 1020 | - | 1468.63 | - |
Convergence Criterion | Average Number of Iterations | |||||
---|---|---|---|---|---|---|
es | ef | ω = 1 | ω = 0.98 | ω = 0.90 | ω = 0.60 | Aitken |
1 × 10−10 | 1 × 10−10 | 1 | 1 | 1 | 1 | 1 |
1 × 10−13 | 1 × 10−13 | 4 | 12 | 19 | 44 | 2.328 |
1 × 10−14 | 1 × 10−14 | 9 | 15 | 20 | 47 | 7.360 |
1 × 10−15 | 1 × 10−15 | 16 | 18 | 21 | 49 | 16.892 |
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Yang, Z.-B.; Li, H.-Q.; Qiao, B.-J.; Chen, X.-F. Wavelet Element Modelling for Inviscid Fluid–Solid Coupling Problem based on Partitioned Approach. Materials 2020, 13, 3699. https://doi.org/10.3390/ma13173699
Yang Z-B, Li H-Q, Qiao B-J, Chen X-F. Wavelet Element Modelling for Inviscid Fluid–Solid Coupling Problem based on Partitioned Approach. Materials. 2020; 13(17):3699. https://doi.org/10.3390/ma13173699
Chicago/Turabian StyleYang, Zhi-Bo, Hao-Qi Li, Bai-Jie Qiao, and Xue-Feng Chen. 2020. "Wavelet Element Modelling for Inviscid Fluid–Solid Coupling Problem based on Partitioned Approach" Materials 13, no. 17: 3699. https://doi.org/10.3390/ma13173699