Geodesic
Iterative approach to solving boundary integral equations
Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 4, pp. 54-67.

Voir la notice de l'article provenant de la source Math-Net.Ru

Under consideration are the issues of numerical solution of a boundary integral equation describing the vorticity generation process on the streamlined airfoils in meshless vortex methods. The traditional approach based on the quadrature method leads to the necessity of solving a system of linear algebraic equations with dense matrix. If we consider the system of airfoils moving relative to one another, this procedure has to be performed at each time step of the calculation, and its high computational complexity significantly reduces the efficiency of vortex methods. The transition from the traditional approach expressed by an integral equation of the first kind to an approach with the integral equation of the second kind makes it possible to apply the simple-iteration method for numerical solving the boundary integral equation. By examples of some model problems, we demonstrate that the iterative approach allows reducing the computational complexity of the problem by tens to hundreds times while providing an acceptable accuracy of the approximate solution.
Keywords: vortex method, incompressible flow, vortex sheet, boundary integral equation, singular integral, simple-iteration method. 
@article{SJIM_2019_22_4_a5,
     author = {E. A. Mikhailov and I. K. Marchevskii and K. S. Kuzmina},
     title = {Iterative approach to solving boundary integral equations},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {54--67},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/}
}
TY  - JOUR
AU  - E. A. Mikhailov
AU  - I. K. Marchevskii
AU  - K. S. Kuzmina
TI  - Iterative approach to solving boundary integral equations
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2019
SP  - 54
EP  - 67
VL  - 22
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/
LA  - ru
ID  - SJIM_2019_22_4_a5
ER  - 
%0 Journal Article
%A E. A. Mikhailov
%A I. K. Marchevskii
%A K. S. Kuzmina
%T Iterative approach to solving boundary integral equations
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2019
%P 54-67
%V 22
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/
%G ru
%F SJIM_2019_22_4_a5
E. A. Mikhailov; I. K. Marchevskii; K. S. Kuzmina. Iterative approach to solving boundary integral equations. Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 4, pp. 54-67. http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/

[1] L. Morino, “Helmholtz decomposition revisited: vorticity generation and trailing edge condition”, Comput. Mech., 1:1 (1986), 65–90 | DOI | MR | Zbl

[2] J. C. Wu, J. F. Thompson, “Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation”, Computers Fluids, 1 (1973), 197–215 | DOI | Zbl

[3] G. Ya. Dynnikova, “Analog integralov Bernulli i Koshi Lagranzha dlya nestatsionarnogo vikhrevogo techeniya idealnoi neszhimaemoi zhidkosti”, Izv. RAN. Mekhanika zhidkosti i gaza, 2000, no. 1, 31–41 | MR | Zbl

[4] Vvflow CFD Suite (stable), https://packagecloud.io/vvflow/stable

[5] VM2D: Vortex method for 2D flow simulation, https://github.com/vortexmethods/VM2D

[6] K. Kuzmina, I. Marchevsky, E. Ryatina, “Numerical simulation in 2D strongly coupled FSI problems for incompressible flows by using vortex method”, Proceedings AIP Conf., 2027, 2018, 040045 | DOI

[7] P. R. Andronov, S. V. Guvernyuk, G. Ya. Dynnikova, Vikhrevye metody rascheta nestatsionarnykh gidrodinamicheskikh nagruzok, Izd-vo Mosk. un-ta, M., 2006

[8] G. Ya. Dynnikova, Vikhrevye metody issledovaniya nestatsionarnykh techenii vyazkoi neszhimaemoi zhidkosti, Dis. ... d-ra fiz. mat. nauk: 05.13.18, M., 2011, 269 pp. | Zbl

[9] L. G. Loitsyanskii, Mekhanika zhidkosti i gaza, Drofa, M., 2003

[10] I. K. Lifanov, Metod singulyarnykh integralnykh uravnenii i chislennyi eksperiment (v matematicheskoi fizike, aerodinamike, teorii uprugosti i difraktsii voln), Yanus, M., 1995

[11] S. N. Kempka, M. W. Glass, J. S. Peery, J. H. Strickland, M. S. Ingber, Accuracy considerations for implementing velocity boundary conditions in vorticity formulations, SANDIA report. SAND96-0583, UC-700, 1996 | Zbl

[12] K. S. Kuzmina, I. K. Marchevskii, V. S. Moreva, “Opredelenie intensivnosti vikhrevogo sloya pri modelirovanii vikhrevymi metodami obtekaniya profilya potokom neszhimaemoi sredy”, Mat. modelirovanie, 29:10 (2017), 20–34 | Zbl

[13] K. S. Kuzmina, I. K. Marchevskii, V. S. Moreva, E. P. Ryatina, “Raschetnaya skhema vikhrevykh metodov vtorogo poryadka tochnosti dlya modelirovaniya obtekaniya profilei neszhimaemym potokom”, Izv. vuzov. Aviatsionnaya tekhnika, 2017, no. 3, 73–80

[14] K. S. Kuzmina, I. K. Marchevskii, “O vliyanii vikhrevogo sloya i tochechnykh vikhrei pri priblizhennom reshenii granichnogo integralnogo uravneniya v dvumernykh vikhrevykh metodakh vychislitelnoi gidrodinamiki”, Prikl. matematika i mekhanika, 83:3 (2019), 471–485

[15] K. S. Kuzmina, I. K. Marchevskii, “Obotsenkakh vychislitelnoi slozhnosti i pogreshnosti bystrogo algoritma v metode vikhrevykh elementov”, Tr. In-ta sistemnogo programmirovaniya RAN, 28:1 (2016), 259–274

[16] D. Rossinelli, P. Koumoutsakos, “Vortex methods for incompressible flow simulations on the GPU”, The Visual Computer, 24:7–9 (2008), 699–708 | DOI

[17] G. Ya. Dynnikova, “Ispolzovanie bystrogo metoda resheniya «zadachi N tel» pri vikhrevom modelirovanii techenii”, Zhurn. vychisl. matematiki i mat. fiziki, 49:8 (2009), 1458–1465 | MR | Zbl

[18] G. Morgenthal, J. H. Walther, “An immersed interface method for theVortex-In-Cell algorithm”, Computers and Structures, 85 (2007), 712–726 | DOI | MR