# David Chopp Engineering Sciences and Applied Mathematics

## Level Set Method

The level set method is a numerical method developed by Osher and Sethian in 1988 that is designed to solve equations involving moving interfaces. In this method, the interface is represented as an isocontour or isosurface of a “level set function” of one higher dimension. The evolution of the interface is then translated into an evolution of the underlying level set function. There are multiple advantages of this method: (1) Interfaces can break and merge naturally without requiring complex rules around topological changes that occur with methods that track the interface. (2) The level set evolution equation can be written as the integral of a hyperbolic conservation law so that the wealth of methods devoted to such equations can be used. (3) The equations and numerical methods are easily adapted to 2, 3 or more dimensions without substantial change.

Chopp and his group have made critical discoveries in the level set method that enabled the method to solve many problems leading to the gain in popularity it now enjoys. It is also a fundamental method that his group uses for much of the research in the group. A list of relevant papers are listed below.

## Level Set Method Articles

Author(s)Title/JournalYear
A. Sadeghirad, D. L. Chopp, X. Ren, E. Fang, J. LuaA Novel Hybrid Approach for Level Set Characterization and Tracking of Non-Planar 3D Cracks in the Extended Finite Element Method, Engineering Fracture Mechanics, 160:1-142016
B. V. Merkey and D. L. ChoppModeling the Impact of Interspecies Competition on Performance of a Microbial Fuel Cell, Bulletin of Mathematical Biology, 76(6):1429-14532014
B. V. Merkey and D. L. ChoppThe Performance of a Microbial Fuel Cell Depends Strongly on Anode Geometry: A Multidimensional Modeling Study, Bulletin of Mathematical Biology, 74:834-8572012
R. Duddu, D. L. Chopp, and P. VoorheesDiffusional Evolution of Precipitates in Elastic Media Using the Extended Finite Element and the Level Set Methods, Journal of Computational Physics, 230(4):1249-12642011
J. Shi, D. L. Chopp, J. Lua, N. Sukumar, and T. BelytschkoAbaqus Implementation of Extended Finite Element Method Using a Level Set Representation for Three-Dimensional Fatigue Crack Growth and LIfe Predictions, Engineering Fracture Mechanics, 77(14):2840-28632010
B. L. Vaughan, B. G. Smith, D.L. ChoppThe Influence of Fluid Flow on Modeling Quorum Sensing in Bacterial Biofilms, Bulletin of Mathematical Biology, 72(5):1143-11652010
D. L. ChoppAnother look at velocity extensions in the level set method. SIAM Journal of Scientific Computing, 31(5):3255-32732009
B. Merkey, B. E. Rittmann, and D.L. ChoppModeling How Soluble Microbial Products (SMP) Support Heterotrophs in Autotroph-Based Biofilms, Journal of Theoretical Biology, 259:670-6832009
R. Duddu, D.L. Chopp, and B. MoranA Two-Dimensional Continuum Model of Biofilm Growth Incorporating Fluid Flow and Shear Stress Based Detachment, Biotechnology and Bioengineering, 103(1):92-1042009
N. Sukumar, D. L. Chopp, E. Béchet, N. MöesThree-dimensional non-planar crack growth by a coupled extended finite element and fast marching method, International Journal of Numerical Methods in Engineering, 76(5):727-7482008
R. Duddu, S. Bordas, D. L. Chopp, and B. MoranA combined extended finite element and level set method for biofilm growth. International Journal of Numerical Methods in Engineering, 74(5):848-8702008
B. G. Smith, B. L. Vaughan, and D. L. ChoppThe extended finite element method for boundary layer problems in biofilm growth, Communications in Applied Mathematics and Computational Science, 2(1):35-562007
M. J. Kirisits, J. Margolis, B. L. Purevdorj-Gage, B. Vaughan, D. L. Chopp, P. Stoodley, and M. R. ParsekThe influence of the hydrodynamic environment on quorum sensing in Pseudomonas aeruginosa biofilms, Journal of Bacteriology, 189(22):8357-83602007
A. Tongen and D. L. ChoppSimulation of multigrain thin film growth. Interfaces and Free Boundaries, 8:1-192006
M. Torres, D. L. Chopp, and T. WalshLevel set methods to compute minimal surfaces in a medium with exclusions (voids). Interfaces and Free Boundaries, 7(2)2005
S. Kodambaka, D. L. Chopp, I. Petrov, and J. E. GreeneCoalescence kinetics of two-dimensional TiN islands on atomically-smooth TiN(001) and TiN(111) terraces. Surface Science, 540(2-3):L611-L6162003
M. Stolarska and D. L. ChoppModeling spiral cracking due to thermal cycling in integrated circuits. International Journal of Numerical Methods in Engineering, 41(20):2381-24102003
D. L. Chopp and J. A. VellingFoliations of hyperbolic space by constant mean curvature surfaces sharing ideal boundary. Journal of Experimental Mathematics, 12(3):339-3502003
D. L. Chopp and N. SukumarFatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. International Journal of Engineering Science, 41(8):845-8692003
N. Sukumar, D. L. Chopp, and B. MoranExtended finite element for three-dimensional fatigue crack propagation, Engineering Fracture Mechanics, 70(1):29-482003
K. A. Smith, F. J. Solis, and D. L. ChoppA projection method for motion of triple junctions by level sets. Interfaces and Free Boundaries, 4(3):263-2762002
H. Ji, D. L. Chopp, and J. E. DolbowA hybrid extended finite element/level set method for modeling phase transformations. International Journal for Numerical Methods in Engineering, 54(8):1209-12332001
D. L. ChoppReplacing iterative algorithms with single-pass algorithms. Proceedings of the National Academy of Sciences, 98(20):10992-109932001
D. L. ChoppSome improvements of the fast marching method, SIAM Journal of Scientific Computing, 23(1):230-2442001
M. Stolarska, D. L. Chopp, N. Möes, and T. BelytschkoModelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering, 51(8):943-9602001
N. Sukumar, D. L. Chopp, N. Möes, and T. BelytschkoModelling holes and inclusions by level sets in the extended finite element method. Computer Methods in Applied Mechanics and Engineering, 190(46-47):6183-62002001
D. L. ChoppA level-set method for simulating island coarsening. Journal of Computational Physics, 162:104-1222000
D. L. Chopp and J. A. SethianMotion by intrinsic Laplacian of curvature. Interfaces and Free Boundaries, 1(1):107-1231999
D. L. Chopp, L. C. Evans, and H. IshiiWaiting time effects for Gauss curvature flows. Indiana University Math Journal, 48(1):311-3341999
S. Angenent, T. Ilmanen, and D.L. ChoppA computed example of nonuniqueness of mean curvature flow in R3. Communications on Partial Differential Equations, 20(11-12):1937-19581995
D. L. ChoppNumerical computation of self-similar solutions for mean curvature flow. Journal of Experimental Mathematics, 3(1):1-151994
D. L. Chopp and J. A. SethianFlow under curvature: Singularity formation, minimal surfaces, and geodesics, Journal of Experimental Mathematics, 2(4):235-2551993
D. L. ChoppComputing minimal surfaces via level set curvature flow, Journal of Computational Physics, 106(1):77-911993
N. Ford and D. L. Chopp