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COMMON PART

Project Number21-11-00039

Project titleMethods of numerical analysis for mathematical models with corner and strong singularity

Project LeadRukavishnikov Viktor

AffiliationComputing Center of the Far Eastern Branch of the Russian Academy of Sciences,

Implementation period | 2021 - 2023 |

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-206 - Computational mathematics

KeywordsWeighted finite element method for boundary value problems with singularity in electrodynamics, hydrodynamics and theory of elasticity

PROJECT CONTENT

Annotation

In the beginning of last century, a Galerkin method was created on the base of notion of generalized (weak) solution. This method allowed developing effective numerical algorithms for determination of approximate solution to the boundary value problems that do not have classic solution, i.e. with discontinued initial data (coefficients and right hands of equation and boundary conditions). Reentrant corners on the boundary do not allow us to calculate with reasonable accuracy both classic and weak solutions using universal classic methods. We suggested to define the solution to the problems with strong and corner singularity as a R-generalized one in weighted spaces. On the base of the notion of R-generalized solution, effective weighted numerical methods were developed. These methods allow us to calculate approximate solution without loss of accuracy and independently of the angle and size of singularity.
The weak solution to the Maxwell equations considered in 2D domain with reentrant angle on the boundary does not belong to the Sobolev space $H^1$. Such problems are called boundary value problems with strong singularity. For the Lame system or linearized Navier-Stokes equations posed in domain with reentrant angle on the boundary it is possible to define weak solution in the space $H^1$, but it does not belong to the space $H^2$. These problems are called problems with weak singularity. According to the principle of coordinated estimates, approximate solution to these problems by the classical finite difference methods and finite element methods converge to the exact one with the rate substantially smaller than 1. Whereas suggested weighted finite element method based on the conception of $R_\nu$-generalized solution has constant convergence rate equal to one and it is not depend on the size of singularity. Basic results were published in leading journals on numerical methods and applied mathematics (quartile Q1 and Q2 Web of Science Core Collection):
V.A. Rukavishnikov, A.O. Mosolapov, E.I. Rukavishnikova. Weighted finite element method for elasticity problem with a crack. Computers and Structures, 2021, Vol. 243 (or 244), Article 106400
V.A. Rukavishnikov, H.I. Rukavishnikova. The Finite Element Method For Boundary Value Problem With Strong Singularity// Journal of Computational and Applied Mathematics. 2010. Vol. 234, № 9, pp.2870-2882.
Rukavishnikov V.A., Mosolapov A.O. New numerical method for solving time-harmonic Maxwell equations with strong singularity // Journal of Computational Physics 231 (2012). - P. 2438-2448.
V. Rukavishnikov and E. Rukavishnikova. On the Existence and Uniqueness of R_ν-Generalized Solution for Dirichlet Problem with Singularity on All Boundary. Abstract and applied analysis. Volume 2014, Article ID 568726, 6 p.
Viktor A. Rukavishnikov, Alexey V. Rukavishnikov. Weighted finite element method for the Stokes problem with corner singularity// Journal of Computation and Applied Mathematics. 341(2018). P. 144-156.
V. A. Rukavishnikov, O. P. Tkachenko. Dynamics of a fluid-filled curvilinear pipeline// Applied Mathematics and Mechanics (English Edition). 39:6 (2018). P. 905–922.
V. A. Rukavishnikov, E. I. Rukavishnikova. Weighted Finite-Element Method for Elasticity Problems with Singularity, Finite Element Method. Simulation, Numerical Analysis and Solution Techniques, eds. Păcurar Răzvan, IntechOpen Limited, London, 2018, P. 295-311.
V.A. Rukavishnikov, A.V. Rukavishnikov. New Numerical Method for the Rotation form of the Oseen Problem with Corner Singularity // Symmetry-Basel, V. 11, Issue: 1 (2019), Article Number: 54.
Rukavishnikov V. A., Tkachenko O. P. Mathematical model of the pipeline with angular joint of elements //Mathematical Methods in the Applied Sciences. 2020. Vol. 43. P. 7550-7568.
Rukavishnikov, V. A., Rukavishnikova, E. I. Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary// Symmetry-Basel, V. 11, Issue: 12 (2019), Article Number: 1455.
Rukavishnikov Viktor A., Rukavishnikova Elena I. On the Dirichlet problem with corner singularity. Mathematics, 2020. V. 8, Issue: 11 (2020), Article Number: 1870.
Suggested approach allows us to compute approximate solution by the weighted FEM for given accuracy $10^{-3}$ in $10^6$ times faster than in the case of classic finite element methods. We also note that for realization of the weighted FEM we take in $10^6$ times less of computing resources and energy consumption. For example, crack problem in the case when mathematical model is represented by the Lame system, cannot be solved with reasonable accuracy by the classic FEM due to singularity caused by reentrant corner of $2\Pi$. At the same time, the weighted final element method allows us to perform computations with high accuracy both inside of the domain and near the crack.
The present project is devoted
to the development of bases of common theory and computational techniques for problems with singularity on the base of R-generalized solution
to the establishing of the algorithm for the determination of the body of optimal computational parameters for the best calculation of the R-generalized solution in dependence of the boundary conditions type, source of singularity and spaces which contain initial data of the equation and boundary conditions
to the development of new mathematical models and evaluation of the theory of numerical methods for boundary value problems with singularity, creation of algorithms and software packages for numerical analysis on high-performance computing systems mathematical models with singularity in electrodynamics, hydrodynamics and elasticity
Realization of stated problems will make significant contribution to the theory of boundary value problems with singularity in the sense of creation of common and effective approach on the base of which analysis of the mathematical models with angular and strong singularity is performed. In most cases this will exclude necessity of particular algorithms for individual classes of problems, such as mesh refinement, singularity separation and so on. Common disadvantage of these methods is worsening of the main matrix in the FEM when the method is leaded to the acceptable accuracy. As a result, this leads to the difficulties in calculations on computers.

Expected results

The present project is devoted to development of common approach for creation of numerical methods without loss of accuracy for analysis of different mathematical models with angular and strong singularity. This will exclude the necessity of particular algorithms development for the individual classes of problems.
The project will result in:
• theoretical basis of common theory and numerical techniques for problems with weak and strong singularity on the base of R-generalized solution will be created;
• algorithm for determination of body of optimal parameters (BOP) in the weighted FEM for the best calculation of numerical solution in dependence of boundary conditions type, source of singularity and functional spaces containing initial data of equations and boundary conditions will be founded;
• BOP for the crack problem, for the problem of elasticity, hydrodynamics and electrodynamics posed in domains with different reentrant angles on the boundary will be derived; this will give opportunities for creation of high-precision industrial codes in different areas of human knowledge;
• mathematical models will be developed and the theory of numerical methods for boundary value problems with singularity will be progressed, algorithms and software packages for numerical analysis on the multiprocessor computer of mathematical models in electrodynamics, hydrodynamics and the theory of elasticity with angular and strong singularity will be created
• creation of industrial software packages for analysis of pipeline with profile fracture under loads close to the emergency conditions, for high-precision calculations of displacements and stresses in building structures with cracks and profile fractures; for determination of liquid flow influence to the shore destruction and different inclusions; for calculation of electromagnetic field in wafer filter in dependence of its cross section deformation and wave power will be prepared.
Creation of high-precision numerical methods and base software packages grounded on these methods will allow us to forecast man-caused accidents in good time and also to conduct high-quality examination of originating complicated technological structures. Moreover, this gives us an opportunity to reduce in millions times both the expenditure of computer resources and computing time.
The main results will be published in leading international journals (Computers and Structures, International Journal of Advances in Engineering Software, Journal of Computational and Applied Mathematics, Applied Mathematics and Computation, etc.), reports on international conferences will be made.

REPORTS

Annotation of the results obtained in 2023

The first boundary value problem for a second-order differential equation with inconsistent degeneracy of the initial data and with a strong singularity in a two-dimensional domain is considered. Based on the known coercive and differential properties of the $R_{\nu}$-generalized solution for its approximate finding, a new finite element weighting method with an increased convergence rate of $O(h^{2})$ is constructed. The basis functions of a finite element space contain a singular component. The convergence analysis of the approximate solution to the exact solution is carried out in the norm of the Sobolev weight space $W^{1}_{2,\nu+\beta/2+2}(\Omega, \delta)$.
For the boundary value problem of the elasticity theory with a singularity, a numerical analysis of the developed finite element weight method is carried out at various re-entrant corner at the boundary of the domain in the range from $\pi$ to $2\pi$. Confirmation of the theoretical estimate of the convergence rate is obtained.
A scheme of the non-conforming Morley finite element method is constructed to find an approximate solution to the Dirichlet problem for a biharmonic equation in an L-shaped domain. The code was created and calculations of model problems were carried out, confirming theoretical estimates of the convergence rate of the approximate solution to the exact solution in the norms of Sobolev spaces and Lebesgue space.
A system of non-stationary non-linear Navier-Stokes equations for the flow of an incompressible viscous fluid in a polygonal domain with a re-entrant corner is considered. Based on the implicit Runge-Kutta methods of the first and second orders, a certain $R_{\nu}$-generalized solution of the problem in weight sets and the constructed finite element weight method for various values of the re-entrant corner, the regions of optimal parameters of the method are experimentally determined. The domains depend on the value $\delta$, which is included in the definition of an $R_{\nu}$-generalized solution, and are constructed in the variables $\nu$ and $\nu^{\ast}$ – indicators of the weight space and the weight function in the basis of the finite element method. They don't depend on the size of the grid and their intersection forms the body of the optimal parameters (BOP) of the method.
Verification of the mathematical model of the stress-strain state of the pipeline segment with an inset has been performed, namely: a computer program has been created that implements an algorithm for numerical analysis of the deformed state of the pipeline by the finite element method; an approach is proposed to suppress the error of the numerical solution, which significantly improved the accuracy of the results; numerical experiments have been carried out to test the proposed algorithms and methods. It is established that the proposed algorithms and methods allow us to find the deformed state of the pipeline with a side inset with high accuracy.
A numerical analysis of the applied problem of calculating stresses on a pipe section containing a singularity point is carried out according to the one-dimensional mathematical model proposed by us, which approximates a three-dimensional model of pipeline deformation. The results of numerical analysis are compared with the results of calculation by the finite element method in CAE ABAQUS and CAE FreeCAD. The analysis of the dynamics of a curved pipe in MATLAB was carried out, its results were compared with the literature data. The practical applicability and high efficiency of the proposed mathematical model and its research method are proved.
In general, 6 reports on the works of 2023 were made at international conferences and 11 articles were published in journals indexed in Web of Science or Scopus, 4 of them in journals included in the Q1Web of Science Core Collection or SJR.
Publications.
1. Rukavishnikov V.A, Rukavishnikova E.I. Weighted finite element method and body of optimal parameters for elasticity problem with singularity. Computers & Mathematics with Applications. 2023, 151: 408-417.
2. Rukavishnikov V.A., Rukavishnikov A.V. Theoretical analysis and construction of numerical method for solving the Navier–Stokes equations in rotation form with corner singularity. Journal of Computational and Applied Mathematics, 2023, 429: 115218.
3. Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P. Numerical Investigation of Pipe Deformation Under Pressure with Branch. International Journal of Applied Mechanics. 2023, 15(07): 2350052.
4. Rukavishnikov V.A., Rukavishnikova E.I. The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity Mathematics. 2023, 11(15): 3272.
5. Rukavishnikov V.A. Weighted Finite Element Method and Body of Optimal Parameters for One Problem of the Fracture Mechanics. Lecture Notes in Computer Science. Numerical Computations: Theory and Algorithms (NUMTA 2023), Parts I & II & III. Springer Nature Switzerland AG. 2023.
6. Rukavishnikov A.V. On the optimal set of parameters for an approximate method for solving stationary nonlinear Navier - Stokes equations with singularity Computational technologies. 2022, 27(6): 70-87.
7. Rukavishnikov A.V. On the optimal set of parameters for an approximate method for solving stationary nonlinear Navier - Stokes equations with singularity arXiv. 2023. https://doi.org/10.48550/arXiv.2309.14589.
8. Rukavishnikov V.A. Body of optimal parameters of the weighted finite element method for the elasticity problem with singularity. AIP Conference Proceedings. 2023, 2849 (1): 450037.
9. Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P. Approximate solution of equations of vibrations propagation in a curved fluid-filled tube. AIP Conference Proceedings, 2023, 2849(1): 450038.
10. Rukavishnikov V.A., Rukavishnikova E.I. Weighted finite element method for one problem of the fracture mechanics. AIP Conference Proceedings. 2024.
11. Rukavishnikov V.A., Rukavishnikov A.V. Unweighted FEM for solving Navier-Stokes Equations in Rotation Form with Reentrant Corner. AIP Conference Proceedings. 2024.
12. Rukavishnikov A.V., Rukavishnikov V.A. On a numerical method for solving a non-stationary problem of hydrodynamics with an angular singularity. Journal of Physics: Conference Proceedings. 2023.
13. Андрианов И.К. Моделирование траектории наклонной трещины в пластине при циклическом растяжении. Информационные технологии и высокопроизводительные вычисления: материалы VII Международной научно-практической конференции, Хабаровск: ХФИЦ ДВО РАН, 2023: 26-28.
14. Рябоконь А.С., Ткаченко О.П. Математическое моделирование напряженно-деформированного состояния тонкостенной трубы с боковой врезкой. Информационные технологии и высокопроизводительные вычисления: материалы VII Международной научно-практической конференции, Хабаровск: ХФИЦ ДВО РАН, 2023: 194-196.
15. Rukavishnikov V.A. Weighted finite element method and body of optimal parameters for one problem of the fracture mechanics. Numerical Computations: Theory and Algorithms. Book of Abstracts of the 4th International Conference and Summer School. 2023: 177. https://www.numta.org/pdf/NUMTA2023_Book.pdf
16. Rukavishnikov A.V., Rukavishnikov V.A. On a Numerical Method for Solving a Non-Stationary Problem of Hydrodynamics with an Angular Singularity. Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences, Book of Abstracts of Fifteenth International Hybrid Conference on Application of Mathematics in Technical and Natural Sciences. 2023: 53. http://2023.eac4amitans.eu/resources/amitansabsbook.pdf

Publications

**1.** *Andrianov I.K.* **Моделирование траектории наклонной трещины в пластине при циклическом растяжении** Информационные технологии и высокопроизводительные вычисления: материалы VII Международной научно-практической конференции, ХФИЦ ДВО РАН, Хабаровск, Материалы VII Международной научно-практической конференции «Информационные технологии и высокопроизводительные вычисления», Хабаровск, 2023, Страницы 26-28 (year - 2023)

**2.** *Rukavishnikov A.V.* **Об оптимальном наборе параметров приближенного метода решения стационарных нелинейных уравнений Навье - Стокса с сингулярностью** Вычислительные технологии, Том 27. Номер 6. Cтраницы 70-87 (year - 2023) https://doi.org/10.25743/ICT.2022.27.6.007

**3.** *Rukavishnikov A.V., Rukavishnikov V.A.* **On a numerical method for solving a non-stationary problem of hydrodynamics with an angular singularity** Journal of Physics: Conference Proceedings, - (year - 2023)

**4.** *Rukavishnikov V. A., Rukavishnikova E. I.* **The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity** MATHEMATICS, Том 11. Выпуск 15. Номер статьи 3272. (year - 2023) https://doi.org/10.3390/math11153272

**5.** *Rukavishnikov V. A., Rukavishnikova E. I.* **Weighted finite element method and body of optimal parameters for elasticity problem with singularity** Computers & Mathematics with Applications, Том 151. Страницы 408-417. (year - 2023) https://doi.org/10.1016/j.camwa.2023.10.021

**6.** *Rukavishnikov V.A.* **Weighted Finite Element Method and Body of Optimal Parameters for One Problem of the Fracture Mechanics** Lecture Notes in Computer Science. Numerical Computations: Theory and Algorithms (NUMTA 2023), Parts I & II & III. Springer Nature Switzerland AG., - (year - 2023)

**7.** *Rukavishnikov V.A.* **Body of optimal parameters of the weighted finite element method for the elasticity problem with singularity** AIP Conference Proceedings, Том 2849. Выпуск 1. Номер статьи 450037. (year - 2023) https://doi.org/10.1063/5.0162089

**8.** *Rukavishnikov V.A., Rukavishnikov A.V.* **Theoretical analysis and construction of numerical method for solving the Navier–Stokes equations in rotation form with corner singularity** Journal of Computational and Applied Mathematics, Том 429. Номер статьи 115218. (year - 2023) https://doi.org/10.1016/j.cam.2023.115218

**9.** *Rukavishnikov V.A., Rukavishnikov A.V.* **Unweighted FEM for solving Navier-Stokes Equations in Rotation Form with Reentrant Corner** AIP Conference Proceedings, - (year - 2023)

**10.** *Rukavishnikov V.A., Rukavishnikova E.I.* **Weighted finite element method for one problem of the fracture** AIP Conference Proceedings, - (year - 2023)

**11.** *Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P.* **Approximate solution of equations of vibrations propagation in a curved fluid-filled tube** AIP Conference Proceedings, Том 2849. Выпуск 1. Номер статьи 450038. (year - 2023) https://doi.org/10.1063/5.0162090

**12.** *Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P.* **Numerical Investigation of Pipe Deformation Under Pressure With Branch** International Journal of Applied Mechanics, Том 15. Номер 07. Номер статьи 2350052. (year - 2023) https://doi.org/10.1142/S1758825123500527

**13.** *Ryabokon A.S., Tkachenko O.P.* **Математическое моделирование напряженно-деформированного состояния тонкостенной трубы с боковой врезкой** Информационные технологии и высокопроизводительные вычисления: материалы VII Международной научно-практической конференции, ХФИЦ ДВО РАН, Хабаровск, Материалы VII Международной научно-практической конференции «Информационные технологии и высокопроизводительные вычисления», Хабаровск, 2023, Страницы 194-196 (year - 2023)

**14.** *Rukavishnikov A.V., Rukavishnikov V.A.* **On a Numerical Method for Solving a Non-Stationary Problem of Hydrodynamics with an Angular Singularity** Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences, Sofia, Book of Abstracts of Fifteenth International Hybrid Conference on Application of Mathematics in Technical and Natural Sciences, страница 53 (year - 2023)

**15.** *Rukavishnikov V.A.* **Weighted finite element method and body of optimal parameters for one problem of the fracture mechanics** Numerical Computations: Theory and Algorithms, Calabria, Italy, Book of Abstracts of the 4th International Conference and Summer School. Страница 177. (year - 2023)

**16.** *Rukavishnikov A.V.* **On the optimal set of parameters for an approximate method for solving stationary nonlinear Navier - Stokes equations with singularity** ArXiv.org, arXiv:2309.14589v1 [math.NA] (year - 2023) https://doi.org/10.1016/j.cam.2023.115218

**17.** *-* **Рукавишников В.А., Чемоданов А.А. «Программа численного решения краевой задачи для бигармонического уравнения с сингулярностью». Программа проходит регистрацию в Роспатенте.** -, - (year - )

Annotation of the results obtained in 2021

The scheme of the finite element method for a boundary value problem with degeneration of the initial data on the entire boundary of the domain and strong singularity is constructed, the estimate of the convergence rate of the approximate solution to the exact one in the norm of the S.L. Sobolev special weighted space is proved, numerical analysis for the calculation of model problems confirming theoretical estimates is carried out on a computing cluster.
The method for determining the body of optimal parameters (BOP) in the weighted finite element method for the best determination of an approximate solution of boundary value problems with a singularity is developed.
On the base of technique for determination of optimal parameters in the weighted finite element method, BOP for optimal computations for the crack problem is constructed, numerical analysis for model problems is carried out, the advantage of the suggested approach over the methods with mesh refinement, singularity extraction and hybrid methods is established.
A technique is developed, a program for the computing cluster is created, and a series of model problems for determination of BOP in the weighted FEM for elasticity problem depending on the value of a reentrant corner on the boundary of the domain is selected.
Based on the concept of the R-generalized solution and weighted finite element method a numerical analysis of the steady-state flow of a viscous incompressible fluid in the convection form of the nonlinear Navier-Stokes equations in the L-shaped domain is carried out. Inf-sup inequality in weighted sets is proved. A comparative analysis with the classical approach is made.
An approximate method with the preconditioning matrix of the system for solving stationary nonlinear Navier-Stokes equations in the convective and rotation forms in a polygonal domain with a reentrant corner with a value more than 180 degrees on its boundary is developed. The optimal set of parameters and its relation to the value (independence from the value) of the reentrant corner for each form are determined.
A 3D mathematical model of a pipeline as a moment shell with an irregular geometry of the median surface is constructed. Correctness of an asymptotic transition to the shell equations for the pipe is proved. Resolving differential equations of the equilibrium of this shell in a formulation of Koiter-Vlasov are obtained; these equations are supplemented with the joint boundary conditions. The analysis of dimensionless parameters of the problem is carried out. A method of an approximate solution of the boundary value problem is proposed; numerical experiments showing high accuracy of this method are performed.
The study of a pipeline with a complex intersected profile is carried out within the framework of the shell theory. An algorithm for reducing the pipe statics problem is developed. Based on this algorithm, a reduced mathematical model of a pipeline with a T-shaped profile is constructed. Areas of main problem parameters, for which the mathematical model is applicable, are found. Numerical experiments to confirm the adequacy of the mathematical model of the pipeline with a T-shaped profile are performed.
To clarify the parameters and verify conformity of the constructed mathematical models, the additional asymptotic and numerical analysis of the dynamics problem of a curved cylindrical shell with a fluid flow oscillations is performed. Numerical solutions of the dynamics problem of a curved fluid-filled tube under fast transients are obtained.
In general, 11 reports were presented at international conferences and 8 papers in editions indexed by Web of Science or Scopus, 2 of them – in journals included in the Q1 Web of Science Core Collection or SJR were published. In addition, three papers are submitted and are under review in journals.
Publications:
1. Rukavishnikov V.A., Rukavishnikova E.I. Error estimate FEM for the Nikol’skij–Lizorkin problem with degeneracy. Journal of Computational and Applied Mathematics. Volume 403. Paper 113841 (year - 2022).
2. Rukavishnikov V.A. Body of Optimal Parameters in the Weighted Finite Element Method for the Crack Problem. Journal of Applied and Computational Mechanics. Volume 7, N 4. Pages 2159-2170 (year - 2021).
3. Tkachenko O.P., Ryabokon A.S. Asymptotic Analysis of the Equations of Hydroelastic Oscillations in Thin-Walled Elastic Pipeline. Materials Physics and Mechanics. Volume 47 (5) (year - 2021).
4. Rukavishnikov V.A. Weighted Finite Element Method with Set of Optimal Parameters For Crack Problem. AIP Conference Proceedings. (year - 2021).
5. Rukavishnikov A.V., Rukavishnikov V.A. Numerical simulation for the one stationary nonlinear hydrodynamics problem in non-convex domain. AIP Conference Proceedings. (year - 2021).
6. Rukavishnikov V.A., Mosolapov A.O. Numerical Solution of the Crack Problem by the Weighted FEM. CEUR Workshop Proceedings. Volume 2930. Pages142-147. http://ceur-ws.org/Vol-2930/paper19.pdf (year - 2021).
7. Rukavishnikov V.A., Ткаченко О.П. Mathematical models of pipelines alternative stress states. CEUR Workshop Proceedings. Volume 2930. Pages 173-179. http://ceur-ws.org/Vol-2930/paper24.pdf (year - 2021).
8. Ryabokon A.S., Tkachenko O.P., Rukavishnikov V.A. Two-dimensional mathematical model of pipelines with a complex intersected profile. CEUR Workshop Proceedings. Volume 2930. Pages 200-205. http://ceur-ws.org/Vol-2930/paper28.pdf (year - 2021).
9. Rukavishnikov A.V. New approach for solving stationary nonlinear Navier-Stokes equations in non-convex domain. In arxiv.org. Url: https://arxiv.org/abs/2110.15649v1. [math.NA] (year - 2021).

Publications

**1.** *Rukavishnikov A.V. , Rukavishnikov V.A.* **Numerical simulation for the one stationary nonlinear hydrodynamics problem in non-convex domain** AIP Conference Proceedings, - (year - 2021)

**2.** *Rukavishnikov V.A.* **Weighted finite element method with set of optimal parameters for crack problem** AIP Conference Proceedings, - (year - 2021)

**3.** *Rukavishnikov V.A.* **Body of optimal parameters in the weighted finite element method for the crack problem** Journal of Applied and Computational Mechanics, Том 7. Выпуск 4. Страницы 2159–2170. (year - 2021) https://doi.org/10.22055/JACM.2021.38041.3142

**4.** *Rukavishnikov V.A., Mosolapov A.O.* **Numerical Solution of the Crack Problem by the Weighted FEM** CEUR Workshop Proceedings, Номер 0074-2930-8. Том 2930. Страницы 142-147. (year - 2021)

**5.** *Rukavishnikov V.A., Rukavishnikova E.I.* **Error estimate FEM for the Nikol’skij–Lizorkin problem with degeneracy** Journal of Computational and Applied Mathematics, Том 403. Номер статьи 113841. (year - 2022) https://doi.org/10.1016/j.cam.2021.113841

**6.** *Rukavishnikov V.A., Tkachenko O.P.* **Mathematical Models of Pipelines Alternative Stress States** CEUR Workshop Proceedings, Номер 0074-2930-8. Том 2930. Страницы 173-179. (year - 2021)

**7.** *Ryabokon A.S., Tkachenko O.P., Rukavishnikov V.A.* **Two-dimensional mathematical model of pipelines with a complex intersected profile** CEUR Workshop Proceedings, Номер 0074-2930-8. Том 2930. Страницы 200-205. (year - 2021)

**8.** *Tkachenko O.P., Ryabokon A.S.* **Asymptotic analysis of the equations of hydroelastic oscillations in thin-walled elastic pipeline** Materials Physics and Mechanics, - (year - 2021)

**9.** *Rukavishnikov A.V.* **New approach for solving stationary nonlinear Navier-Stokes equations in non-convex domain** ArXiv.org, Номер 2110.15649. Страницы 1-13. (year - 2021)

Annotation of the results obtained in 2022

A new method for the boundary value problem of the elasticity theory in a domain with a reentrant corner at the boundary is developed. On its basis, we prove an estimate for the convergence rate of an approximate solution by the weighted finite element method (WFEM) to an exact one in the norm of a weighted Sobolev space. For the effective use of the WFEM, it is necessary to correctly set the control parameters for performing calculations. An algorithm for determining the optimal parameters of the WFEM for finding an approximate solution of the Lamé system without loss of accuracy in domains with a boundary containing reentrant corners α in the range from π to 2π is constructed. Bodies of optimal parameters for the weighted finite element method are determined. This bodies do not depend on the corner α for a set of grids of various dimensions. The dependence of the method error on the change in the values of the input parameters on the stability is investigated. This opens up possibilities for creating industrial codes based on WFEM.
An algorithm for processing the calculation results of the problem of elasticity theory with a singularity is structured. This made it possible to determine the body of optimal parameters in the weighted finite element method. A program for visualizing the results of numerical analysis is created.
For a boundary value problem with degeneracy of the initial data on the entire boundary of the domain, an estimate for the convergence rate of an approximate solution by the finite element method to an exact one in the norm of a weighted Lebesgue space was proved. A numerical analysis of the calculation of model problems, confirming theoretical estimates, was carried out on a computing cluster.
The properties of the operators of the Stokes problem with a corner singularity in a nonsymmetric variational formulation are established. Estimates of the norms are obtained, and a connection with the asymmetric bilinear form a( . , .) of the problem is determined. The inf-sup inequalities of the bilinear forms b1( . , . ) and b2( . , . ) in weighted sets are proved.
An existence and uniqueness theorem for an $R_{\nu}$-generalized solution of the Stokes problem with a corner singularity in sets of weighted Sobolev spaces is proved.
A system of non-stationary nonlinear Navier-Stokes equations for the flow of an incompressible viscous fluid in a rotation form in a domain with a reentrant corner is considered.
Discretization of the problem by time was carried out using implicit Runge-Kutta methods of the first and second orders. For both cases, an $R_{\nu}$-generalized solution of the problem in weighted sets is defined. In the first of them, the Crank-Nicholson scheme was used. An estimate related to the conservation of the energy balance of the approximation velocity field is proved. In the second, a scheme with L-stability is applied. A method for finding an approximate solution is implemented for each scheme. Sets of parameters are experimentally found. When using them, at each moment of time the convergence order of the approximate solution to the exact one does not depend on the value of the reentrant corner.
An analysis of the coefficient set and right-hand sides for the equations of the mathematical model of a curved pipeline is performed. The basic dimensionless parameters are determined, on the basis of which this mathematical model can be set. An algorithm for the numerical solution of the boundary value problem of the equilibrium of a complexly curved cylindrical shell has been developed. It is considered as a section of the pipeline with the found right-hand sides of the equations. Using the results of numerical experiments, the boundaries of parameters are determined, at which the mathematical model is applicable to the calculation of pipelines.
An algorithm for finding a numerical solution to a model problem of the statics of a pipe with a fractured profile fluid-filled is developed. Numerical estimates of the adequacy of the proposed model and the method of its research are obtained.
An extended mathematical model of a pipeline with an inset is constructed based on the theory of elastic shells. The formulation of the model is reduced to a two-dimensional boundary value problem in a flat domain. The accuracy of the description of the stress-strain state of the pipe was checked by solving this problem. The computational experiment showed good agreement between the exact and numerical solution of the problem.
In general, for the works of 2022: 7 reports were made at international conferences and 8 papers were published in journals indexed in Web of Science or Scopus, 4 of them in journals included in the Q1 Web of Science Core Collection or SJR. In addition, 3 papers have been submitted and are under peer review in journals.
Publications.
1. Rukavishnikov V.A., Tkachenko O.P. Approximate resolving equations of mathematical model of a curved thin-walled cylinder. Applied Mathematics and Computation. Volume 422. Paper 126961. (year - 2022).
2. Rukavishnikov V.A., Rukavishnikov A.V. On the properties of operators of the Stokes problem with corner singularity in nonsymmetric variational formulation. Mathematics. Volume 10, N 6. Paper 889. (year - 2022).
3. Rukavishnikov A.V., Rukavishnikov V.A. New numerical approach for the steady-state Navier–Stokes equations with corner singularity. International Journal of Computational Methods. Volume 19. Paper 2250012. (year - 2022).
4. Rukavishnikov V.A., Rukavishnikov A.V. On the existence and uniqueness of an Rν-generalized solution to the Stokes problem with corner singularity. Mathematics. Volume 10, N 10. Paper 1752. (year - 2022).
5. Rukavishnikov V.A, Rukavishnikova E.I. On the error estimation of the FEM for the Nikol’skij-Lizorkin problem with degeneracy in the Lebesgue space. Symmetry. Volume 14, N 6. Paper 1276. (year - 2022).
6. Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P. Mathematical modeling of the stress–strain state of T‐shaped connection of cylindrical pipes. Mathematical Methods in the Applied Sciences. (year - 2022).
7. Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P. Finite element analysis of mathematical model for pipe system. Informatika Sistemy Upravlenaya № 2 (72). P. 3-12. (year - 2022).
8. Rukavishnikov V.A., Seleznev D.S., Guseinov A.A. Algorithm for processing the results of calculations for determining the body of optimal parameters in the weighted finite element method. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software. (year - 2022).
9. Tkachenko O.P. Principal stress-strain states of thin-walled complexly bent pipelines. // Materials Physics and Mechanics. (year - 2022).
10. Rukavishnikov V.A., Seleznev D.S., Guseinov A.A. The program for processing the results of calculations of the problem of the theory of elasticity with a singularity for determining the body of optimal parameters in the weighted finite element method. Computer programs. Database. Topologies of integrated circuits. 2021668932, 11/22/2021. Application No. 2021667919 dated 11/12/2021. (year - 2021).

Publications

**1.** *Rukavishnikov A.V., Rukavishnikov V.A.* **New numerical approach for the steady-state Navier–Stokes equations with corner singularity** International Journal of Computational Methods, Том 19. Номер статьи 2250012. (year - 2022) https://doi.org/10.1142/S0219876222500128

**2.** *Rukavishnikov V.A., Rukavishnikov A.V.* **On the existence and uniqueness of an Rν-generalized solution to the Stokes problem with corner singularity** Mathematics, Том 10. Номер 10. Номер статьи 1752. (year - 2022) https://doi.org/10.3390/math10101752

**3.** *Rukavishnikov V.A., Rukavishnikov A.V.* **On the properties of operators of the Stokes problem with corner singularity in nonsymmetric variational formulation** Mathematics, Том 10. Номер 6. Номер статьи 889. (year - 2022) https://doi.org/10.3390/math10060889

**4.** *Rukavishnikov V.A., Rukavishnikova E.I.* **On the error estimation of the FEM for the Nikol’skij-Lizorkin problem with degeneracy in the Lebesgue space** Symmetry, Том 14. Номер 6. Номер статьи 1276. (year - 2022) https://doi.org/10.3390/sym14061276

**5.** *Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P.* **Mathematical modeling of the stress–strain state of T‐shaped connection of cylindrical pipes** Mathematical Methods in the Applied Sciences, 30 June 2022 First publication (year - 2022) https://doi.org/10.1002/mma.8536

**6.** *Rukavishnikov V.A., Ryabokon A.S., Tkachenko O.P.* **Конечно-элементный анализ математической модели системы труб** Информатика и системы управления, Том 72. Выпуск 2. Страницы 3-12. (year - 2022) https://doi.org/10.1002/mma.8536

**7.** *Rukavishnikov V.A., Seleznev D.S., Guseinov A.A.* **Algorithm for processing the results of calculations for determining the body of optimal parameters in the weighted finite element method** Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software, - (year - 2022) https://doi.org/10.14529/mmp220407

**8.** *Rukavishnikov V.A., Tkachenko O.P.* **Approximate resolving equations of mathematical model of a curved thin-walled cylinder** Applied Mathematics and Computation, Том 422. Номер статьи 126961 (year - 2022) https://doi.org/10.1016/j.amc.2022.126961

**9.** *Tkachenko O.P.* **Principal stress-strain states of thin-walled complexly bent pipelines** Materials Physics and Mechanics, - (year - 2022)

**10.** *Rukavishnikov V.A., Seleznev D. S., Guseinov A.A.* **Программа обработки результатов вычислений задачи теории упругости с сингулярностью для определения тела оптимальных параметров в весовом методе конечных элементов** Программы для ЭВМ. Базы данных. Топологии интегральных микросхем, 2021668932, Бюл. № 12 (year - 2021)

**11.** *Rukavishnikov V.A., Seleznev D. S., Guseinov A.A.* **Программа обработки результатов вычислений задачи теории упругости с сингулярностью для определения тела оптимальных параметров в весовом методе конечных элементов** Федеральная служба по интеллектуальной собственности. Государственная регистрация программы для ЭВМ, Номер регистрации (свидетельства): 2021668932. (year - 2021)