4-1 Summary
In the chapter three, we deal with the two cases, one is two free identical particles, and other is two identical particles in a cylindrical pore. In order to make sure code validity, we use the same parameter in previously literature. All of the previously works are assumes the numerical model is symmetry and reduce to two-dimensional problem. But the calculation of the interaction between two spheres closed to a planar wall [4] requires a full three-dimensional problem. Assume the problem to two-dimensional may have some uncertain. We develop this three-dimensional Poisson-Boltzmann equation just can to deal with this troublesome problem.
In the propose study, we can obtain the parallel Poisson-Boltzmann equation coupled with PAMR is very useful and convenience. It can be more closed to real situation at colloidal systems.
4-2 Recommendations for Future Work
In this propose study, we meet two difficulties, one is the convergence rate too slow, the other is refinement level can not over than five. We can use fine initial guess to overcome the convergence rate problem, but the cost of the refinement computation, we can not overcome directly. But there still have some way to overcome this situation, we can use the high order mesh and coupled different mesh such as hexahedral cell and pyramid. If we can use high order mesh to capture the EDL potential distribution, we can reduce the inaccuracy without large computationally cost.
References
[1] D.J. Shaw, Electrophoresis, Academic Press, New York, (1969)
[2] W.R. Bowen, A.O. Sharif, “Adaptive Finite-Element Solution of Nonlinear Poisson-Boltzmann Equation: A Charged Apherical Particle at Various Distances from a Charged Cylindrical Pore in a Charged Planar Surface” J. Colloid Interface Sci. 187(1997)
[3] W.R. Bowen, A.O. Sharif, “ Long-range electrostatic attraction between like-charge spheres in a charged pore” Nature 385 (1998) 663
[4] A.E. Larsen, D.G. Grier, “Like-charge attractions in metastable colloidal crystallites” Nature 385 (1997) 230
[5] P.E. Dyshlovenko, “Adaptive Mesh Enrichment for the Poisson-Boltzmann Equation” J. Comp. Phs 172 (2001) 198
[6] P.E. Dyshlovenko, “Adaptive numerical method for Poisson-Boltzmann equation and its application” CPC. 147 (2002) 335-338
[7] R. Tuinier, “ Approximate solutions to Poisson-Boltzmann equation in spherical and cylindrical geometry” J. Colloid Interface Sci 258 (2003) 45-49
[8] Prodip K. Das, Subir Bhattacharjee, “Finite element estimation of electrostatic double layer interaction between colloidal particles inside a rough cylindrical capillary: effect of charging behavior” Colloids and Surfaces A 256 (2005) 91-103 [9] R. –C. Chen, Jinn-Liang Liu. “Monotone iterative methods for the adaptive finite
solution of semiconductor equation” J. of Computational and Applied Mathematics 159 (2003) 341-364
[10] Almasi, G. and A., G., “Highly Parallel Computing, Benjamin/Cummings, Red-Wood city, CA,1989
[11] Shivaraju B. Gowda “A Comparison of Sparse & Element-by-Element Storage
Schemes on The Efficiency of Parallel Conjugate Gradient Iterative Methods for Finite Element Analysis” (2002) Graduate School of Clemeson University
[12] Yuan-Wen Jeng , NCTU MUST, ‘’Parallel Implementation of Three-dimensional Unstructured Mesh Refinement in the Direct Simulation Monte Carlo Method’
[13] Kuo, Chia-Hao, "The Direct Simulation Monte Carlo Method Using
Unstructured Adaptive Mesh and Its Applications,", MS Thesis, NCTU, Hsinchu, Taiwan 29, Jun., 2000.
[14] Wu, Fu-Yuan, "The Three-Dimensional Direct Simulation Monte Carlo Method Using Unstructured Adaptive Mesh and It Applications", MS Thesis, NCTU, Hsinchu, Taiwan , July., 2002.
[15] S. Levine, J.R. Marriot, G. Neale, N. Epstein. J. Colloid Interface Sci. 52 (1) (1975) 136-149
[16] S. Levine, G.M. Bell, Discuss. Faraday Soc. 42(1966) 69 [17] E. Bresler. J. Colloid Interface Sci. 118 (2) (1987) 326 [18] P.L. Levine, J. Colloid Interface Sci. 51(1975) 72
[19] T.A.R. Strauss, H.K. Bowen, J. Colloid Interface Sci. 118(2) (1987) 326 [20] H.K. Bowen, J. Colloid Interface Sci. 173 (1995) 388-395
[21] G.M. Mala, D. Li, C. Yang, ‘’Electrical double layer potential distribution in a rectangular micro channel,’’ Int. J. Heat Fluid Flow, (1997).
[22] Patankar, N. A., and Hu, H. H., ‘’Numerical Simulation of Electroosmotic Flow,’’ Anal. Chem. 70, 1870 (1998)
[23] Arulanandam, S., and Li, D., ‘’Liquid transport in rectangular microchannels by electroosmotic pumping,’’ Colloids and Sufaces A: Physicochemical and Engineering Aspects, 161, 89-102 (2000).
[24] J. Yang, L. M. Fu, and C. C. Hwang, “Electroosmotic Entry Flow in a
Microchannel,”J. Colloid Interface Sci., 244, 173-179, (2001).
[25] J. Yang, L. M. Fu, and Y. -C. Lin ‘’Electroosmotic Flow in Microchannels,’’ J.
Colloid Interface Sci. 239, 98-105 (2001).
[26] H.A. Stone, A.D. Stroock, and A. Ajdari, Annu. ‘’Engineering Flows in small devices,’’ Rev. Fluid Mech. 36:381-411 (2004)
[27] R. -C. Chen, Jinn-Liang Liu. ‘’Monotone iterative methods for the adaptive finite element solution of semiconductor equations,’’ J. of Computational and Applied Mathematics 159 (2003) 341–364
[28] Yiming Li, ‘’A parallel monotone iterative method for the numerical solution of multi-dimensional semiconductor Poisson equation,’’ Computer Physics Communications 153 (2003) 359–372
[29] R.J. Hunter Zeta Potential in Colloid Science: Principles and Applications, Academic Press, New York, (1981)
[30] H. J. Keh, and H. C. Tseng, J. Colloid Interface Sci. 242, 450-459, (2001) [31] J. Yang, and D. Y. Kwok, J. colloid Interface Sci., 260, 225-233, (2003) [32] C. C. Chang, and R. J. Yang, J. Micromech. Microeng., 14, 550-558, (2004) [33] Kuo-Hsien Hsu, NCTU MUST ‘’Three-dimensional parallelize Poisson slover’’
[34] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, New York, 1989.
Table
Table3-1: Steps of the adaptive process:
Refinement level
Number of nodes Number of elements Force of interaction(Fs)
0 924 3821 20.9924
1 3045 14305 34.6570
2 18253 96638 40.4853
3 132477 747048 44.0112
4 187732 1049734 48.0722
5 213224 1190741 48.4066
PS: These cases are runed by six parallel processors.
Table3-2: Steps of the adaptive process for previously literature [6]
Refinement level
Number of elements Force of interaction(Fs)
0 946 36.518
1 1992 43.716
2 4466 46.596
3 7642 47.597
4 11314 48.472
5 13525 48.630
6 15187 48.862
7 15477 48.826
8 15541 48.840
9 15555 48.843
10 15578 48.841
11 15588 48.840
Table3-3: The interaction forces of two particles in a cylinder pore at different separation distance.
Separation Distance(h) 0.5 1 3.5 6 8
Force of interaction(Fs) 26 13 -0.7 -4.7 0.9
PS: These cases are runed by six parallel processors.
Figure
Fig 1.1 The illustration of like-charge attractions phenomenon
Fig 1.2 The Electric Double Layer distribution
No
Yes
Read in mesh generation
Start
Com pute the coefficients of shape
function
Setup initial &
boundary conditions
Com pute coefficient m atrix
Com pute loading m atrix
Solve matrix
Converge?
Output
Stop
Fig 2.1 Simplified flow chart of the three-dimensional nonlinear Poisson-Boltzmann solver
Fig 2.2 Distributed memory architecture
Fig 2.3 Shared memory architecture
Fig 2.4 The flow chart of parallel Poisson-Boltzmann solver
Fig 2.5 Isotropic mesh refinement of tetrahedral mesh (T: Tetrahedron)
isotropic ( 1st stage) isotropic ( 2nd stage)
anisotropic ( 2nd stage) initial grid
initial grid
initial grid
2 hanging node anisotropic (2nd stage)
3 hanging node isotropic (2nd stage)
1 hanging node anisotropic (2nd stage)
2 hanging node
3 hanging node 1 hanging node
Fig 2.6 Mesh refinement rules for two-dimensional triangular cell
refined 4 cells
refined 8 cells refined 2 cells
(+1) (+2) (+3) (+1) (+4)
non-coplanar coplanar
6 5 4 3 2 1
coplanar
: n hanging node (+n) : adding n nodes
n
Fig 2.7 Schematic diagram for mesh refinement rules of tetrahedron
Fig 2.8 Schematic diagram of the proposed cell quality control
Fig 2.9 Schematic diagram of typical cell quality control
Fig 2.10 Schematic diagram of simple cell quality control
Fig 2.11 A case that the proposed cell-quality-control would not affect to it
CPU0 gather output data
flag the cells based on refinement criteria
renumber added nodes
synchronize
stop start
add nodes on cell edge refinement required
build neighbor identifier array CPU K (K=0,…,np-1)
N=N+1
distribute the data read grid data and
relative cell data
Fig 2.12 Flow chart of parallel mesh refinement module
Fig 2-13 Flow chart of moduleⅠ
remove hanging nodes
stop start
add nodes on isotropic cells
communicate hanging node information
hanging nodes on IPB
?
yes
no
add nodes on anisotropic cells
Fig 2-14 Coupled PPBS-PAMR method
visual preprocessor
stop PAMR
mesh partition
(weighting from parallel Poisson-Boltzmann solver)
PPB mesh partition
(weighting based on cell volume)
start
(N=0)
N< NMAX
Refinement criteria satisfied ?
no yes
yes
N=N+1
Fig 3.1 Geometry of the potential around a sphere case
Fig 3.2 The comparison of different result at Ψ=1(sphere)
Fig 3.3 The comparison of different result at Ψ =5(sphere)
Fig 3.4 The relationship between iteration and residual with different initial guess (Ψ =5)
Fig 3.5 Geometry of the potential around a cylinder case
Fig 3.6 The comparison of different result at Ψ =1(cylinder)
Fig 3.7 The comparison of different result at Ψ =5(cylinder)
Fig 3.8 The illustration the case of two interacting identical particles
Fig 3.9 Level 0 initial mesh of two interacting identical particles case
Fig 3.10 Level 1 refinement mesh of two interacting identical particles case
Fig 3.11 Level 2 refinement mesh of two interacting identical particles case
Fig 3.12 Level 3 refinement mesh of two interacting identical particles case
Fig 3.13 Level 4 refinement mesh of two interacting identical particles case
Fig 3.14 Level 5 refinement mesh of two interacting identical particles case
Fig 3.15 The potential distribution of two identical particles
Fig 3.16 The potential distribution for the sphere confined in a pore at separation distances h=0.5
Fig 3.17 The potential distribution for the sphere confined in a pore at separation distances h=6
Fig 3.18 Potential along the midplane between two spheres
Fig 3.19 Potential along midplane between two spheres (in local)