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Optimal control for mass conservative level set methods. (English) Zbl 1321.49047

Summary: This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach.

MSC:

49M30 Other numerical methods in calculus of variations (MSC2010)
49M05 Numerical methods based on necessary conditions
49M25 Discrete approximations in optimal control
49K20 Optimality conditions for problems involving partial differential equations

Software:

Anderson
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Full Text: DOI

References:

[1] Stanley, O.; Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 1, 12-49 (1988) · Zbl 0659.65132
[2] Osher, S.; Fedkiw, R., (Level Set Methods and Dynamic Implicit Surfaces. Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 153 (2003), Springer) · Zbl 1026.76001
[3] Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (1999), Cambridge University Press · Zbl 0973.76003
[4] Ausas, R. F.; Dari, E. A.; Buscaglia, G. C., A geometric mass-preserving redistancing scheme for the level set function, Internat. J. Numer. Methods Fluids, 65, 8, 989-1010 (2011) · Zbl 1444.76084
[6] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077
[7] Basting, Ch.; Kuzmin, D., A minimization-based finite element formulation for interface-preserving level set reinitialization, Computing, 95, 1, 13-25 (2013)
[8] Kuzmin, D., An optimization-based approach to enforcing mass conservation in level set methods, J. Comput. Appl. Math., 258, March, 78-86 (2014) · Zbl 1294.65067
[9] Kees, C. E.; Akkerman, I.; Farthing, M. W.; Bazilevs, Y., A conservative level set method suitable for variable-order approximations and unstructured meshes, J. Comput. Phys., 230, 12, 4536-4558 (2011) · Zbl 1416.76214
[10] Lesage, A.-C.; Dervieux, A., Conservation Correction by Dual Level Set, Research Report 7089, INRIA (2009)
[11] Olsson, E.; Kreiss, G., A conservative level set method for two phase flow, J. Comput. Phys., 210, 1, 225-246 (2005) · Zbl 1154.76368
[12] Olsson, E.; Kreiss, G.; Zahedi, S., A conservative level set method for two phase flow II, J. Comput. Phys., 225, 1, 785-807 (2007) · Zbl 1256.76052
[13] Sussman, M., A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles, J. Comput. Phys., 187, 1, 110-136 (2003) · Zbl 1047.76085
[14] Sussman, M.; Puckett, E. G., A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows, J. Comput. Phys., 162, 2, 301-337 (2000) · Zbl 0977.76071
[15] van der Pijl, S. P.; Segal, A.; Vuik, C.; Wesseling, P., A mass-conserving level-set method for modelling of multi-phase flows, Internat. J. Numer. Methods Fluids, 47, 4, 339-361 (2005) · Zbl 1065.76160
[16] Jung, Y.; Chu, K. T.; Torquato, S., A variational level set approach for surface area minimization of triply-periodic surfaces, J. Comput. Phys., 223, 2, 711-730 (2007) · Zbl 1115.65071
[17] Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034
[18] Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 3, 335-362 (1979) · Zbl 0416.76002
[19] Walker, H. F.; Ni, P., Anderson acceleration for fixed-point iterations, SIAM J. Numer. Anal., 49, 4, 1715-1735 (2011) · Zbl 1254.65067
[20] Fang, H.; Saad, Y., Two classes of multisecant methods for nonlinear acceleration, Numer. Linear Algebra Appl., 16, 3, 197-221 (2009) · Zbl 1224.65134
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