3-1 Convergence of the PPBES
Table 3-1 (a) and (b) summarizes the convergence study with a good initial guess and all-zero guess, respectively for a typical simulation (CPU No.=6; 81,354 nodes and 458,064 elements). A good initial guess is from the solution of last level mesh by interpolation. Results show that, with a good initial guess, only three Newton iterations are required to achieve the convergence criterion (ε0=10-7) in the present inexact Newton iterative scheme. On the other hand, if all zeros are used as the initial estimate, five Newton iterations are needed. Compared with all zeros are used as the initial estimate (relative residual=10-7, total time=394.32sec), a good initial guess (relative residual=10-7, total time=301.57sec) is saving about 24% total CPU time.
In addition, approximately 900-1300 iterations are required for the convergence (relative residual of from 10-5 to 10-7) in the parallel CG solver within each Newton’s iteration, since we do not use any preconditioning techniques. In addition, numerical experiments show that relative residual of 10-5 for the CG solver is the optimum choice in terms of the runtime and accuracy. If we choose a low relative residual (10-4), not only increase the number of iterations for the CG solver but also bring the
geometrical additive Schwarz domain decomposition method [Nadeem and Jimack, 2002], could be used. This work is currently in progress in our group and will be reported in the near future.
3-2 Validation 1: The Potential Distribution around a Charged Sphere
In presently papers review, Tuinier [2003] provided simple expressions are presented that give a very accurate description of electrostatic potential around a single sphere and cylinder for arbitrary double-layer thickness and surface potential.
Completed PPBES is used to compute the potential distribution around a charged sphere, and the results are compared with analytical and approximate solutions [Tuinier, 2003] to validate the parallelized code.
Table 3-2 summarizes the previously developed analytical and approximate solutions for this case. Normalized radius of the sphere is kept as 5.0, while two cases of constant value of electrostatic potential (Ψ =1.0 and 5.0) are set at the surface of the 0 charged sphere. Note that since PPBES is a 3D code, only 1/16 of the full domain is used for this simulation by taking advantage of the symmetry of the problem, while the Neumann boundary conditions are used at the symmetric planes. Fig 3-1 shows the surface mesh distribution of the sphere (176,309 elements; 36,396 nodes).
Fig. 3-2a shows that when Ψ =1.0, the computed potential distribution agrees very 0
well with both the analytical and approximate solutions [Tuinier, 2003], because at this lower Ψ , the Poisson-Boltzmann equation can be linearized correctly. However, Fig. 0 3-2b shows that when Ψ =5.0, the computational potential distribution deviates 0
considerably from the analytical solution but stays quite close to the approximate solution. This is because at this higher Ψ , the Poisson-Boltzmann equation cannot be 0 linearized correctly in the derivation of the analytical solution.
3-3 Validation 2: Interaction Between Two Like-charged Spheres
within a Cylindrical Pore
In this section, we will simulate a geometrically confined pair of charged spheres concerning the phenomenon of long-range electrostatic attraction between particles of like charge [Larsen and Grier, 1997]. The long-range electrostatic interaction of two colloidal particles confined in a cylindrical pore was studied in the present paper [Bowen and Sharif, 1998], we use the same values of parameters and geometry, such as the 1:1 electrolyte, the potential on surfaces of the sphere particlesΨs =3.0, the potential on the cylindrical pore surfaceΨp =5.0, the radius of the particles a=1.185, the sphere radius to pore radius ratio 0.13. Taking advantage of the symmetry in this problem, only 1/4 of the whole physical domain is necessary for simulation, resulting in the need for 0.64 million elements (close to 0.11 million nodes).
Fig 3-3 shows the calculated potential distribution for the sphere confined in a pore at separation distances r=6. Fig 3-4 shows the potential along the midplane BC at r=6 for the first-order shape function. We can see the potential along this plane is almost decreasing at all distance, but after Z=3.4 the potential become increasing. By comparing with previous numerical data [Brwen and Sharif, 1998], we also have almost the same results, but still have 1% inaccuracy. Our calculations confirm speculation that the attraction between confined spheres. Besides, if we choose the second-order shape function, the accuracy will be improved.
3-4 Parallel performance of the PPBES
To study the parallel performance of the PPBES, we consider a test case with a charged sphere (radius=5,Ψ =2.0) confined in a cylindrical pore with the sphere radius 0 to pore radius ratio of λ=0.35. Taking advantage of the symmetry in this problem, only 1/16 of the whole physical domain is necessary for simulation, resulting in the need for 0.61 million elements (close to 0.1 million nodes). Fig. 3-5 illustrates the resulting speedup as a function of the number of parallel processors (up to 32), and Table 3-3 summarizes the detailed time breakdown for various components of the parallelized code. Among these, the matrix solver using parallel CG method occupies ~50% of the total runtime and setup the matrix occupies ~8% for the number of processors tested in
the present study. This indicates that further reduction of the runtime can be highly expected if the matrix solver is greatly improved. Total runtime is ~2,486 seconds with the use of a single processor as compared to ~102 seconds with the use of 32 processors, which results in 76.2% in parallel efficiency. The results clearly show that the current parallel implementation of the Poisson-Boltzmann equation using the subdomain-by-subdomain method performs reasonably well for the typical problem size. A smaller problem size was not tested in this study since it is irrelevant in practice for typical three-dimensional applications. Indeed, it is expected that the parallel speedup will be even greater if a larger problem size is considered. Thus, not only can the current parallel implementation help to greatly reduce the runtime required for parametric study to understand the distribution of electrostatic potential but also increase tremendously the size of the problem that the memory-distributed parallel machine can handle.
3-5 The Second-order Shape Function for PPBES
Figure 3-6 shows that the distribution of potential around the charged spheres (radius a=0.325μm) near a like-charged plate (distance h = 2.5μm) at level-0 for solutions of the first-order shape function (Fig 3-6a) and the second-order shape function (Fig 3-6b). For the test cases, Table 3-4 shows that the mesh levels of the
number of nodes and elements. We can find the solution of second-order shape function better than the solution of first-order shape function at level-0 show in Figure 3-7 and Figure 3-8. It is a clear demonstration that when we use the second-order shape function for mesh of level-0, which the solution between the first-order shape function for mesh of level-3 and mesh of level-4. In other words, the solution of the second-order shape function will increase in accuracy over the first-order shape function. The advantage of second-order shape function is that they can define the edge points of elements in order to increase the solution of accuracy. It is not necessary to use very small elements. Therefore, we can use the second-order shape function to reduce the number of elements at refinement level.
Table 3-5 shows the information (case from Section 3-3) that compares the first-order with the second-order shape function for mesh of level 0. For the same order nodes (~110,000 nodes, the second-order shape function is from original 16,280 nodes), we can find the total CPU time is almost closed (~90 second). Approximately 720 iterations are required for the convergence in the parallel CG solver within each Newton’s iteration.
Then, we use the mesh data of Table 3-5 to validate the solution of accuracy. Fig 3-9 shows the case for interaction between two like-charged spheres within a cylindrical pore (the same as case of Section 3-3) but use the second-order shape
function. The result shows that the solution of the second-order shape function will increase in accuracy over the first-order shape function. The inaccuracy reduces near to about 0.2%.