Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
Cell Adhesion and Detachment in Shear Flow 223 [PG05] T.-W. Pan and R. Glowinski. Direct simulation of the motion of neutrally buoyant balls in a three-dimensional Poiseuille flow. C. R. Mécanique, 333:884–895, 2005. [SBBR + 02] T. Scott-Burden, J. P. Bosley, D. Rosenstrauch, K. D. Henderson, F. J. Clubb, H. C. Eichstaedt, K. Eya, I. Gregoric, T. J. Myers, B. Radovancevic, and O. H. Frazier. Use of autologous auricular chondrocytes for lining artificial surfaces: a feasibility study. Ann. Thorac. Surg., 73:1528–1533, 2002. [SKE + 99] R. M. Schinagl, M. S. Kurtis, K. D. Ellis, S. Chien, and R. L. Sah. Effect of seeding duration on the strength of chondrocyte adhesion to articular cartilage. J. Orthopaedic Research, 17:121–129, 1999. [SZD03] M. E. Staben, A. Z. Zinchenko, and R. H. Davis. Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Phys. Fluid, 15:1711–1733, 2003. [ZBCAG04] R. Zaidel-Bar, M. Cohen, L. Addadi, and B. Geiger. Hierarchical assembly of cell-matrix adhesion complexes. Biochem. Soc. Trans., 32(3):416–420, 2004.
Computing the Eigenvalues of the Laplace–Beltrami Operator on the Surface of a Torus: A Numerical Approach Roland Glowinski 1 and Danny C. Sorensen 2 1 University of Houston, Department of Mathematics, Houston, TX, 77004, USA roland@math.uh.edu 2 Rice University, Department of Computational and Applied Mathematics, Houston, TX, 77251-1892, USA sorensen@rice.edu Summary. In this chapter, we present a methodology for numerically computing the eigenvalues and eigenfunctions of the Laplace–Beltrami operator on the surface of a torus. Beginning with a variational formulation, we derive an equivalent PDE formulation and then discretize the PDE using finite differences to obtain an algebraic generalized eigenvalue problem. This finite dimensional eigenvalue problem is solved numerically using the eigs function in Matlab which is based upon ARPACK. We show results for problems of order 16K variables where we computed lowest 15 modes. We also show a bifurcation study of eigenvalue trajectories as functions of aspect ration of the major to minor axis of the torus. 1 Introduction A large number of physical phenomena take place on surfaces. Many of these are modeled by partial differential equations, a typical example being provided by elastic shells. It is not surprising, therefore, that many questions have arisen concerning the spectrum of some partial differential operators defined on surfaces. This area of investigation is known as spectral geometry. Among these operators defined on surfaces, a most important one is the so called Beltrami Laplacian, also known as the Laplace–Beltrami operator. The main goal of this chapter is to discuss the computation of the lowest eigenvalues of the Laplace–Beltrami operator associated with the boundary of a torus of R 3 . After a description of our methodology for the computation of these eigenvalues and their corresponding eigenfunctions, we present selected results from our numerical experiments. The methodology consists of obtaining a finite difference discretization of a PDE that is equivalent to a more standard variational formulation; then the resulting finite dimensional generalized
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Cell Adhesion <strong>and</strong> Detachment in Shear Flow 223<br />
[PG05] T.-W. Pan <strong>and</strong> R. Glowinski. Direct simulation of the motion of neutrally<br />
buoyant balls in a three-dimensional Poiseuille flow. C. R.<br />
Mécanique, 333:884–895, 2005.<br />
[SBBR + 02] T. Scott-Burden, J. P. Bosley, D. Rosenstrauch, K. D. Henderson,<br />
F. J. Clubb, H. C. Eichstaedt, K. Eya, I. Gregoric, T. J. Myers,<br />
B. Radovancevic, <strong>and</strong> O. H. Frazier. Use of autologous auricular chondrocytes<br />
for lining artificial surfaces: a feasibility study. Ann. Thorac.<br />
Surg., 73:1528–1533, 2002.<br />
[SKE + 99] R. M. Schinagl, M. S. Kurtis, K. D. Ellis, S. Chien, <strong>and</strong> R. L. Sah.<br />
Effect of seeding duration on the strength of chondrocyte adhesion to<br />
articular cartilage. J. Orthopaedic Research, 17:121–129, 1999.<br />
[SZD03] M. E. Staben, A. Z. Zinchenko, <strong>and</strong> R. H. Davis. Motion of a particle<br />
between two parallel plane walls in low-Reynolds-number Poiseuille<br />
flow. Phys. Fluid, 15:1711–1733, 2003.<br />
[ZBCAG04] R. Zaidel-Bar, M. Cohen, L. Addadi, <strong>and</strong> B. Geiger. Hierarchical<br />
assembly of cell-matrix adhesion complexes. Biochem. Soc. Trans.,<br />
32(3):416–420, 2004.
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