David L. Chopp

MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD

A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (S. Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the extended finite-element method (X-FEM) (N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition, the level set function is used to develop the local enrichment for material interfaces. Numerical examples in two-dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique.

The impact of quorum sensing and swarming motility on Pseudomonas aeruginosa biofilm formation is nutritionally conditional

The role of quorum sensing in Pseudomonas aeruginosa biofilm formation is unclear. Some researchers have shown that quorum sensing is important for biofilm development, while others have indicated it has little or no role. In this study, the contribution of quorum sensing to biofilm development was found to depend upon the nutritional environment. Depending upon the carbon source, quorum‐sensing mutant strains (lasIrhlI and lasRrhlR) either exhibited a pronounced defect early in biofilm formation or formed biofilms identical to the wild‐type strain. Quorum sensing was then shown to exert its nutritionally conditional control of biofilm development through regulation of swarming motility. Examination of pilA and fliM mutant strains further supported the role of swarming motility in biofilm formation. These data led to a model proposing that the prevailing nutritional conditions dictate the contributions of quorum sensing and swarming motility at a key juncture early in biofilm development.

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.

Some Improvements of the Fast Marching Method

The fast marching method published by Sethian [Proc. Natl. Acad. Sci. USA, 93 (1996), pp. 1591--1595] is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic. It has been used in a wide variety of applications such as robotic path planning [R. Kimmel and J. Sethian, Fast Marching Methods for Computing Distance Maps and Shortest Paths, Tech. Report 669, CPAM, University of California, Berkeley, 1996], crack propagation [M. Stolarska et al., Modelling crack growth by level sets in the extended finite element method, Comput. Methods Appl. Mech. Engrg., to appear; Internat. J. Numer. Methods Engrg., 51 (2001), pp. 943--960; N. Sukumar, D. L. Chopp, and B. Moran, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, J. Comput. Phys., submitted], seismology [J. Sethian and A. Popovici, Geophysics, 64 (1999), pp. 516--523], photolithography [J. Sethian, Fast marching level set methods for three-dimensional photolithography development, in Proceedings of the SPIE 1996 International Symposium on Microlithography, Santa Clara, CA, 1996], and medical imaging [R. Malladi and J. Sethian, Proc. Natl. Acad. Sci. USA, 93 (1996), pp. 9389--9392]. It has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and/or an extended velocity function. In this paper, we improve upon the second order fast marching method of Sethian [SIAM Rev., 41 (1999), pp. 199--235] by constructing a second order approximation of the interface generated from local data on the mesh. The data is interpolated on a single box of the mesh using a bicubic approximation. The distance to the front is then calculated by using a variant of Newtons method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node. The result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method.

The extracellular matrix protects Pseudomonas aeruginosa biofilms by limiting the penetration of tobramycin

Biofilm cells are less susceptible to antimicrobials than their planktonic counterparts. While this phenomenon is multifactorial, the ability of the matrix to reduce antibiotic penetration into the biofilm is thought to be of limited importance studies suggest that antibiotics move fairly rapidly through biofilms. In this study, we monitored the transport of two clinically relevant antibiotics, tobramycin and ciprofloxacin, into non-mucoid Pseudomonas aeruginosa biofilms. To our surprise, we found that the positively charged antibiotic tobramycin is sequestered to the biofilm periphery, while the neutral antibiotic ciprofloxacin readily penetrated. We provide evidence that tobramycin in the biofilm periphery both stimulated a localized stress response and killed bacteria in these regions but not in the underlying biofilm. Although it is unclear which matrix component binds tobramycin, its penetration was increased by the addition of cations in a dose-dependent manner, which led to increased biofilm death. These data suggest that ionic interactions of tobramycin with the biofilm matrix limit its penetration. We propose that tobramycin sequestration at the biofilm periphery is an important mechanism in protecting metabolically active cells that lie just below the zone of sequestration.

Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method

A numerical technique for modeling fatigue crack propagation of multiple coplanar cracks is presented. The proposed method couples the extended finite element method (X-FEM) [Int. J. Numer. Meth. Engng. 48 (11) (2000) 1549] to the fast marching method (FMM) [Level Set Methods & Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK, 1999]. The entire crack geometry, including one or more cracks, is represented by a single signed distance (level set) function. Merging of distinct cracks is handled naturally by the FMM with no collision detection or mesh reconstruction required. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity [Comput. Meth. Appl. Mech. Engng. 139 (1996) 289]. This enables the domain to be modeled by a single fixed finite element mesh with no explicit meshing of the crack surfaces. In an earlier study [Engng. Fract. Mech. 70 (1) (2003) 29], the methodology, algorithm, and implementation for three-dimensional crack propagation of single cracks was introduced. In this paper, simulations for multiple planar cracks are presented, with crack merging and fatigue growth carried out without any user-intervention or remeshing.

Flow under curvature: Singularity formation, minimal surfaces, and geodesics

We study hypersurfacesmoving under flow that depends on the mean curvature. The approach is based on a numerical technique that embeds the evolving hypersurface as the zero level set of a family of evolving surfaces. In this setting, the resulting partial differential equation for the motion of the level set function may be solved by using numerical techniques borrowed from hyperbolic conservation laws. This technique is applied to several problems: the evolution of a dumbbell, and related many-armed surfaces, collapsing under mean curvature; the construction of a minimal surface attached to a given one-dimensional wire frame in R3, and, more generally, the construction of surfaces whose mean curvature is a prescribed function of position; the motion of curves on two-manifolds under flow that depends on geodesic curvature. Some experiments involving flow controlled by Gaussian curvature are also included.

A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment

We present a two‐dimensional biofilm growth model in a continuum framework using an Eulerian description. A computational technique based on the eXtended Finite Element Method (XFEM) and the level set method is used to simulate the growth of the biofilm. The model considers fluid flow around the biofilm surface, the advection–diffusion and reaction of substrate, variable biomass volume fraction and erosion due to the interfacial shear stress at the biofilm–fluid interface. The key assumptions of the model and the governing equations of transport, biofilm kinetics and biofilm mechanics are presented. Our 2D biofilm growth results are in good agreement with those obtained by Picioreanu et al. (Biotechnol Bioeng 69(5):504–515, 2000). Detachment due to erosion is modeled using two continuous speed functions based on: (a) interfacial shear stress and (b) biofilm height. A relation between the two detachment models in the case of a 1D biofilm is established and simulated biofilm results with detachment in 2D are presented. The stress in the biofilm due to fluid flow is evaluated and higher stresses are observed close to the substratum where the biofilm is attached. Biotechnol. Bioeng. 2009;103: 92–104.

A projection method for motion of triple junctions by level sets

We develop a projection method to treat the motion of multiple junctions (such as contact lines) in the level set formulation. Multiple junctions are relevant to many fields including fluid dynamics, foams, and semiconductor manufacture. In the level set method an interface is defined as the zero level set of a smooth function. For an N -phase system the location of all interfaces can be specified by N − 1 functions (hence only one level set function is needed for a two-phase system). For N > 2 we describe a symmetric projection of the N level set functions onto an N −1 dimensional manifold. This reduction in phase space eliminates unacceptable values of the level set functions (such as cases where more than one is positive at a given point.) This prevents the formation of vacuums or overlaps at multiple junctions during interface evolution. Further, this method can be applied to any number of phases and spatial dimensions. We present twoand three-dimensional results showing that the method gives correct equilibrium contact angles and produces accurate dynamics in multi-phase fluids.

Influence of the Hydrodynamic Environment on Quorum Sensing in Pseudomonas aeruginosa Biofilms

We provide experimental and modeling evidence that the hydrodynamic environment can impact quorum sensing (QS) in a Pseudomonas aeruginosa biofilm. The amount of biofilm biomass required for full QS induction of the population increased as the flow rate increased.