全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Multiwavelet Boundary Element Method for Cavities Admitting Many NURBS Patches

DOI: 10.4236/mnsms.2016.64007, PP. 69-93

Keywords: Wavelet, Patch, NURBS, Connolly, Modeling

Full-Text   Cite this paper   Add to My Lib

Abstract:

We consider the modeling and simulation by means of multiwavelets on many patches. Our focus is on molecular surfaces which are represented in the form of Solvent Excluded Surfaces that are featured by smooth blendings between the constituting atoms. The wavelet bases are constructed on the unit square which maps bijectively onto the patches embedded in the space. The cavity which designates the surface bounding a molecular model is acquired from the nuclei coordinates and the Van-der-Waals radii. We use multi-wavelets for which the wavelet basis functions are organized hierarchically on several levels. Our assembly of the linear system is accomplished by using a hierarchical tree which enables the treatment of large molecules admitting thousands of patches. Along with the patch construction, some wavelet simulation outcomes which are applied to realistic patches are reported.

References

[1]  Hsiao, H., Steinbach, O. and Wendland, W. (2000) Domain Decomposition Methods via Boundary Integral Equations. Journal of Computational and Applied Mathematics, 125, 521-537.
https://doi.org/10.1016/s0377-0427(00)00488-x
[2]  Baker, N., Sept, D., Holst, M. and McCammon, J. (2001) The Adaptive Multilevel Finite Element Solution of the Poisson-Boltzmann Equation on Massively Parallel Computers. IBM Journal of Research and Development, 45, 427-438.
https://doi.org/10.1147/rd.453.0427
[3]  Diedrich, C., Dijkstra, D., Hamaekers, J., Henniger, B. and Randrianarivony, M. (2015) A Finite Element Study on the Effect of Curvature on the Reinforcement of Matrices by Randomly Distributed and Curved Nanotubes. Journal of Computational and Theoretical Nanoscience, 12, 2108-2116.
https://doi.org/10.1166/jctn.2015.3995
[4]  Randrianarivony, M. (2004) Anisotropic Finite Elements for the Stokes Problem: A-Posteriori Error Estimator and Adaptive Mesh. Journal of Computational and Applied Mathematics, 169, 255-275.
https://doi.org/10.1016/j.cam.2003.12.025
[5]  Apel, T. and Randrianarivony, M. (2003) Stability of the Discretizations of the Stokes Problem on Anisotropic Meshes. Mathematics and Computers in Simulation. 61, 437-447.
https://doi.org/10.1016/S0378-4754(02)00098-8
[6]  Tomasi, J., Mennucci, B. and Cammi, R. (2005) Quantum Mechanical Continuum Solvation Models. Chemical Reviews, 105, 2999-3094.
https://doi.org/10.1021/cr9904009
[7]  Weijo, V., Randrianarivony, M., Harbrecht, H. and Frediani, L. (2010) Wavelet Formulation of the Polarizable Continuum Model. Journal of Computational Chemistry, 31, 1469-1477.
[8]  Cohen, A., Daubechies, I. and Feauveau, J. (1992) Biorthogonal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 45, 485-560.
https://doi.org/10.1002/cpa.3160450502
[9]  Beylkin, G. (1992) On the Representation of Operators in Bases of Compactly Supported Wavelets. SIAM Journal on Numerical Analysis, 29, 1716-1740.
https://doi.org/10.1137/0729097
[10]  Lage, C. and Schwab, C. (1999) Wavelet Galerkin Algorithms for Boundary Integral Equations. SIAM Journal on Scientific Computing, 20, 2195-2222.
https://doi.org/10.1137/S1064827597329989
[11]  Petersdorff, T. and Schwab, C. (1996) Wavelet Approximations for First Kind Boundary Integral Equations on Polygons. Numerische Mathematik, 74, 479-519.
https://doi.org/10.1007/s002110050226
[12]  Randrianarivony, M. (2008) Harmonic Variation of Edge Size in Meshing CAD Geometries from IGES Format. In: Bubak, M., van Albada, G.D., Dongarra, J. and Sloot, P.M.A., Eds., Computational Science—ICCS 2008, Springer, Berlin, 56-65.
https://doi.org/10.1007/978-3-540-69387-1_7
[13]  Kleemann, B., Rathsfeld, A. and Schneider, R. (1996) Multiscale Methods for Boundary Integral Equations and Their Application to Boundary Value Problems in Scattering Theory and Geodesy. In: Hackbusch, W. and Wittum, G., Eds., Boundary Elements: Implementation and Analysis of Advanced Algorithms, Vieweg+Teubner Verlag, Wiesbaden, 1-28.
https://doi.org/10.1007/978-3-322-89941-5_1
[14]  Harbrecht, H. and Randrianarivony, M. (2010) From Computer Aided Design to Wavelet BEM. Computing and Visualization in Science, 13, 69-82.
https://doi.org/10.1007/s00791-009-0129-1
[15]  Randrianarivony, M. (2006) Geometric Processing of CAD Data and Meshes as Input of Integral Equation Solvers. Ph.D. Dissertation, Technical University of Chemnitz, Germany.
[16]  Randrianarivony, M. (2009) On Global Continuity of Coons Mappings in Patching CAD Surfaces. Computer-Aided Design, 41, 782-791.
https://doi.org/10.1016/j.cad.2009.04.012
[17]  Randrianarivony, M. (2011) On Transfinite Interpolations with Respect to Convex Domains. Computer Aided Geometric Design, 28, 135-149.
https://doi.org/10.1016/j.cagd.2010.10.003
[18]  Harbrecht, H. and Randrianarivony, M. (2009) Wavelet BEM on Molecular Surfaces: Parametrization and Implementation. Computing, 86, 1-22.
https://doi.org/10.1007/s00607-009-0050-y
[19]  Randrianarivony, M. and Brunnett, G. (2008) Preparation of CAD and Molecular Surfaces for Meshfree Solvers. In: Griebel, M. and Schweitzer, M.A., Eds., Meshfree Methods for Partial Differential Equations IV, Springer, Berlin, 231-245.
https://doi.org/10.1007/978-3-540-79994-8_14
[20]  Hoschek, J. and Lasser, D. (1996) Fundamentals of Computer Aided Geometric Design. A. K. Peters, Taylor & Francis, Natick.
[21]  Randrianarivony, M. (2016) Domain Decomposition for Wavelet Single Layer on Geometries with Patches. Applied Mathematics, 7, 1798-1823.
https://doi.org/10.4236/am.2016.715151
[22]  Genz, A. and Malik, A. (1980) An Adaptive Algorithm for Numeric Integration over an N-Dimensional Rectangular Region. Journal of Computational and Applied Mathematics, 6, 295-302.
https://doi.org/10.1016/0771-050X(80)90039-X
[23]  Johnson, S. (2010).
http://ab-initio.mit.edu/wiki/index.php/Cubature
[24]  Bajaj, C., Xu, G. and Zhang, Q. (2009) A Fast Variational Method for the Construction of Resolution Adaptive C2-Smooth Molecular Surfaces. Computer Methods in Applied Mechanics and Engineering, 198, 1684-1690.
https://doi.org/10.1016/j.cma.2008.12.042
[25]  Lee, B. and Richards, F. (1971) The Interpretation of Protein Structures Estimation of Stratic Accessibility. Journal of Molecular Biology, 55, 379-400.
https://doi.org/10.1016/0022-2836(71)90324-X
[26]  Connolly, M. (1983) Analytical Molecular Surface Calculation. Journal of Applied Crystallography, 16, 548-558.
https://doi.org/10.1107/s0021889883010985
[27]  Schwartz, L. (1966) Théorie des Distribution. Hermann, Paris.
[28]  McLean, W. (2000) Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge.
[29]  Atkinson, K. (1997) The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge.
https://doi.org/10.1017/CBO9780511626340
[30]  Hsiao, G. and Wendland, W. (2008) Boundary Integral Equations. Springer Verlag, Berlin.
https://doi.org/10.1007/978-3-540-68545-6
[31]  DeBoor, C. and Fix, G. (1973) Spline Approximation by Quasi-Interpolants. Journal of Approximation Theory, 8, 19-45.
https://doi.org/10.1016/0021-9045(73)90029-4
[32]  Brunnett, G. (1995) Geometric Design with Trimmed Surfaces. Computing Supplement, 10, 101-115.
https://doi.org/10.1007/978-3-7091-7584-2_7
[33]  Piegl, L. and Tiller, W. (1995) The NURBS Book. Springer, Berlin.
https://doi.org/10.1007/978-3-642-97385-7
[34]  Lyche, T. and Mørken, K. (1992) Spline-Wavelets of Minimal Support. In: Braess, D. and Schumaker, L.L., Eds., Numerical Methods in Approximation Theory, Birkhäuser, Basel, 177-194.
https://doi.org/10.1007/978-3-0348-8619-2_10
[35]  Lyche, T., Mørken, K. and Quak, E. (2001) Theory and Algorithms for Non-Uniform Spline Wavelets, Multivariate Approximation and Applications. Cambridge University Press, Cambridge, 152-187.
[36]  Gallier, J. (2000) Geometric Methods and Applications for Computer Science and Engineering. In: Antman, S.S. and Greengard, L., Eds., Texts in Applied Mathematics, Springer Verlag, New York, 38.
[37]  Silla, E., Pascal-Ahuir, J., Tomasi, J. and Bonaccorsi, R. (1987) Electrostatic Interaction of a Solute with a Continuum. Improved Description of the Cavity and of the Surface Cavity Bound Charge Distribution. Journal of Computational Chemistry, 8, 778-787.
https://doi.org/10.1002/jcc.540080605
[38]  Klamt, A. (2005) COSMO-RS: From Quantum Chemistry to Fluid Phase Thermodynamics and Drug Design. Elsevier Science, Amsterdam.
[39]  Ammar, A. and Chinesta, F. (2008) Circumventing Curse of Dimensionality in the Solution of Highly Multidimensional Models Encountered in Quantum Mechanics Using Meshfree Finite Sums Decompositions. In: Griebel, M. and Schweitzer, M.A., Eds., Meshfree Methods for Partial Differential Equations IV, Springer, Berlin, 1-17.
https://doi.org/10.1007/978-3-540-79994-8_1
[40]  Rao, A. (2005) MPI-Based Parallel Finite Element Approaches for Implicit Dynamic Analysis Employing Sparse PCG Solvers. Advances in Engineering Software, 36, 181-198.
https://doi.org/10.1016/j.advengsoft.2004.10.004
[41]  Zhang, J. (2015) A PETSc-Based Parallel Implementation of Finite Element Method for Elasticity Problems. Mathematical Problems in Engineering, 2015, Article ID: 147286.
https://doi.org/10.1155/2015/147286
[42]  Vega-Gisbert, O., Roman, J. and Squyres, J. (2016) Design and Implementation of Java Binding Open MPI. Parallel Computing, 59, 1-20.
https://doi.org/10.1016/j.parco.2016.08.004

Full-Text

comments powered by Disqus

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133

WeChat 1538708413