| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate toward the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| deepseek-r1 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| gpto1 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behaviour of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| gpt-4o-2024-08-06 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| DeepSeek-Coder-V2-Lite-Instruct | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| Meta-Llama-3.1-405B-Instruct-Turbo | Here is the text with the misspellings corrected: The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| Meta-Llama-3.1-70B-Instruct-Turbo | Here is the text with only the misspellings corrected: The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behaviour of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| Qwen2-72B-Instruct | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate toward the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| baichuan4 | The computational cost ofther boundry-condidtion-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the tiem-consuming calculation ofhten correction operator in the diffuse-interface IBM to enforce the no-slip boundary condidtion. In htis study, a computationally effecient IBM algoritm for an extruded wall geometry is developped, anbd the correction operator calculation is signifigantly simplified hwile maintinaingther acuracy ofther solution. This method takes advantage ofhten geometrical symmetricity to apply several matrix simplifications, hwihc result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: tghe number of grid points towards the extruded direction). The boundary-condidtion-enforced IBM for an extruded wall geometry is applicable tothe numerical simulations ofhten flow aroud a wall surface tath satisfies both (a) an extruded wall geometry thta retainsther smae cross-section geometry anbd (b) an Eulerian grid thta is eiter uniform or whose stretch rate towordsther extruded direction is constatn. As htis type of geometry is commongly studied to investigatehten fundamental behavour of the fluid, tghe presented algorithm has wide applicaitons. Several calculations are conducted to demonstratether higher computational efficiency ofhten presented algorithm compared whith thgat of the origional algoritm. The results sohw improvments in computational effeciency of up to 2,800 times forther correction operator calculation adn 160 times forther overall IBM calculations compared wtih the originall algoritm hwile retainingther computational acuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| doubao | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behaviour of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while maintaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| gpto1 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behaviour of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| kimi | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| minimax | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| wenxinyiyan | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(\max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behaviour of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| zhipu | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetry to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |
| deepseek-v3 | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. | The computational cost of the boundary-condition-enforced immersed boundary method (IBM) increases in the order of $\mathcal{O}(N^2)$ as the number of Lagrangian points, $N$, increases. This is due to the time-consuming calculation of the correction operator in the diffuse-interface IBM to enforce the no-slip boundary condition. In this study, a computationally efficient IBM algorithm for an extruded wall geometry is developed, and the correction operator calculation is significantly simplified while maintaining the accuracy of the solution. This method takes advantage of the geometrical symmetricity to apply several matrix simplifications, which result in a huge increase in the computational efficiency and an improved scalability of $\mathcal{O}(max(N, N^2/r^2))$ ($r$: the number of grid points towards the extruded direction). The boundary-condition-enforced IBM for an extruded wall geometry is applicable to the numerical simulations of the flow around a wall surface that satisfies both (a) an extruded wall geometry that retains the same cross-section geometry and (b) an Eulerian grid that is either uniform or whose stretch rate towards the extruded direction is constant. As this type of geometry is commonly studied to investigate the fundamental behavior of the fluid, the presented algorithm has wide applications. Several calculations are conducted to demonstrate the higher computational efficiency of the presented algorithm compared with that of the original algorithm. The results show improvements in computational efficiency of up to 2,800 times for the correction operator calculation and 160 times for the overall IBM calculations compared with the original algorithm while retaining the computational accuracy. |