Title Page
ABSTRACT
국문 초록
Contents
CHAPTER 1. INTRODUCTION 13
CHAPTER 2. Review of the MFS and direct construction of inverse mass matrix 17
2.1. The method of finite spheres 17
2.2. Direct construction of inverse mass matrix 19
CHAPTER 3. Mass scaling for direct inverse mass matrix in the MFS 23
3.1. Element wise mass scaling 23
3.1.1. One-dimensional mass scaling 24
3.1.2. Two-dimensional mass scaling 24
3.2. Optimization of parameter matrices using meta-heuristics algorithm 25
3.2.1. 1D optimization with benchmark problem 26
3.2.2. 2D optimization with benchmark problem 33
3.3. Test for the conservation of kinetic energy 39
CHAPTER 4. Numerical examples 43
4.1. 1D impact for a elastic bar 44
4.2. 2D Ricker wavelet 48
4.3. 2D Step+Sine load 52
CHAPTER 5. Conclusion 60
REFERENCES 62
Table 1. The relative velocity error and max frequency in the one-dimensional benchmark problem when using conventional inverse method. 28
Table 2. The maximum CFL number for the explicit time integrations. 32
Table 3. The relative displacement error and max frequency in the two-dimensional benchmark problem when using conventional inverse method. 34
Table 4. The maximum CFL number for the explicit time integrations. 39
Table 5. The mass scaling parameter sets for each enrichment and time integration in one-dimensional and two-dimensional. 43
Table 6. The CFL number for the 2D Ricker wavelet problem. 49
Table 7. The CFL number for the 2D Step+Sine load problem. 49
Table 8. The computational time for linear enrichment in explicit time integration methods with the direct inverse mass matrix and conventional inverse mass matrix. 58
Table 9. The computational time for trigonometric enrichment in explicit time integration methods with the direct inverse mass matrix and conventional inverse mass matrix. 58
Fig. 1. Nodes uniformly distributed in the MFS: (a) one-dimensional (b) two-dimensional case 18
Fig. 2. Description and spatial discretization of a one-dimensional wave propagation problem with a concentrated force using finite spheres. 27
Fig. 3. Convergence curves in one-dimensional for each case, two enrichments and two different time integration methods. 29
Fig. 4. The velocity plot obtained using the mass scaling parameters obtained at iterations 1, 10, 100, and 1000. (Top left) Linear, Noh-Bathe (Top right) Linear, CDM (Bottom left)... 30
Fig. 5. Eigenfrequencies obtained through DI with optimized mass scaling parameters for both time integration methods: (a) Linear enrichment and (b) Trigonometric enrichment. 31
Fig. 6. Solution plots for the benchmark problem: (Left)using the conventional inverse mass matrix without applying DI and (Right) using the direct inverse mass matrix with the... 31
Fig. 7. Description and spatial discretization of a two-dimensional wave propagation problem with a concentrated force using finite spheres. 33
Fig. 8. Convergence curves in two-dimensional for each case, two enrichments and two different time integration methods. 34
Fig. 9. Displacement and velocity results for the scalar wave propagation problem using explicit time integration methods with maximum CFL numbers are presented in the Linear... 35
Fig. 10. Displacement and velocity results for the scalar wave propagation problem using explicit time integration methods with maximum CFL numbers are presented in the... 36
Fig. 11. Displacements and velocities along x-axis for benchmark problem are presented using linear enrichment at t=9.25s : the number of spheres (a) 51 by 51, (b) 101 by 101,... 37
Fig. 12. Displacements and velocities along x-axis for benchmark problem are presented using trigonometric enrichment at t=9.25s : the number of spheres (a) 51 by 51, (b) 101... 38
Fig. 13. Kinetic energy comparison with the exact kinetic energy for all cases (a) in one- dimensional and (b) two-dimensional. 41
Fig. 14. 1D impact of an elastic bar problem. 44
Fig. 15. Velocities of the bar at t=0.5s 46
Fig. 16. Acceleration of the bar at t=0.5s 47
Fig. 17. Displacements and velocities for 2D scalar ricker-wavelet problem at t=0.95s 50
Fig. 18. Displacements and velocities along x-axis for 2D scalar ricker-wavelet problem at t=0.95s 51
Fig. 19. Displacements for 2D Step Sine problem at t=4.36s 53
Fig. 20. Velocities for 2D Step Sine problem at t=4.36s 54
Fig. 21. Displacements along x-axis for 2D Step+Sine Load at t=4.36s: the number of spheres (a) 51 by 51, (b) 101 by 101, (c) 201 by 201 55
Fig. 22. Velocities along x-axis for 2D Step+Sine Load at t=4.36s: the number of spheres (a) 51 by 51, (b) 101 by 101, (c) 201 by 201 56
Fig. 23. Total CPU time and relative displacement error measured for each combination of time integration methods and enrichments. (a) Linear enrichment (b) Trigonometric enrichment 57