Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters. The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters. The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions. Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions. Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.

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Thermal Analysis on a Satellite Box during Launch Stage by Analytical Solution

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.277-279.765 ◽

2005 ◽

Vol 277-279 ◽

pp. 765-770 ◽

Cited By ~ 1

Author(s):

Hui Kyung Kim ◽

Joon Min Choi ◽

Bum Seok Hyun

Keyword(s):

Thermal Analysis ◽

Analytical Solution ◽

Numerical Solutions ◽

Thermal Environment ◽

Time Dependent ◽

Space Environment ◽

Numerical Techniques ◽

Order Ordinary Differential Equation ◽

Time Line ◽

Time Period

Simple methods are developed to predict temperatures of a satellite box during the launch stage. The box is mounted on the outer surface of a satellite and directly exposed to a space thermal environment for the time period from fairing jettison to separation. These simple methods involve solving a 1st order ordinary differential equation (ODE), simplified from the full governing equation when several assumptions are made. The existence of an analytical solution for the 1st order ODE is determined depending on the treatment of the time-dependent molecular heating term. Even for the case that the analytical solution is not available due to the time dependent term, the 1st order ODE can be solved by relatively simple numerical techniques. The temperature difference between two different approaches (analytical and numerical solutions) is relatively small (less than 1°C along the time line) when they are applied to the STSAT-I launch scenario. The present methods can be generally used as tools to quickly check whether a satellite box is safe against the space environment during the launch stage for the case that the detailed thermal analysis is not available.

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Analytical Solution for Three-Dimensional Hyperbolic Heat Conduction Equation with Time-Dependent and Distributed Heat Source

Journal of Mechanics ◽

10.1017/jmech.2016.42 ◽

2016 ◽

Vol 33 (1) ◽

pp. 65-75 ◽

Cited By ~ 2

Author(s):

M. R. Talaee ◽

V. Sarafrazi

Keyword(s):

Heat Conduction ◽

Analytical Solution ◽

Heat Source ◽

Heat Conduction Equation ◽

Numerical Solutions ◽

Three Dimensional ◽

Closed Form Solution ◽

Time Dependent ◽

Conduction Equation ◽

Hyperbolic Heat Conduction

AbstractThis paper is devoted to the analytical solution of three-dimensional hyperbolic heat conduction equation in a finite solid medium with rectangular cross-section under time dependent and non-uniform internal heat source. The closed form solution of both Fourier and non-Fourier profiles are introduced with Eigen function expansion method. The solution is applied for simple simulation of absorption of a continues laser in biological tissue. The results show the depth of laser absorption in tissue and considerable difference between the Fourier and Non-Fourier temperature profiles. In addition the solution can be applied as a verification branch for other numerical solutions.

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Generalized analytical solution for advection-dispersion equation in finite spatial domain with arbitrary time-dependent inlet boundary condition

Hydrology and Earth System Sciences ◽

10.5194/hess-15-2471-2011 ◽

2011 ◽

Vol 15 (8) ◽

pp. 2471-2479 ◽

Cited By ~ 38

Author(s):

J.-S. Chen ◽

C.-W. Liu

Keyword(s):

Boundary Condition ◽

Analytical Solution ◽

Numerical Solutions ◽

Generalized Solution ◽

Spatial Domain ◽

Time Dependent ◽

Hydrodynamic Dispersion ◽

Arbitrary Time ◽

Special Cases ◽

Inlet Boundary Condition

Abstract. This study presents a generalized analytical solution for one-dimensional solute transport in finite spatial domain subject to arbitrary time-dependent inlet boundary condition. The governing equation includes terms accounting for advection, hydrodynamic dispersion, linear equilibrium sorption, and first order decay processes. The generalized analytical solution is derived by using the Laplace transform with respect to time and the generalized integral transform technique with respect to the spatial coordinate. Some special cases are presented and compared to illustrate the robustness of the derived generalized analytical solution. Result shows an excellent agreement between the analytical and numerical solutions. The analytical solutions of the special cases derived in this study have practical applications. Moreover, the derived generalized solution which consists an integral representation is evaluated by the numerical integration to extend its usage. The developed generalized solution offers a convenient tool for further development of analytical solution of specified time-dependent inlet boundary conditions or numerical evaluation of the concentration field for arbitrary time-dependent inlet boundary problem.

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Numerical Simulations for Chemotaxis Models

Biomath Communications ◽

10.11145/bmc.2019.04.277 ◽

2019 ◽

Vol 6 (1) ◽

pp. 16

Author(s):

Gurgen Dallakyan

Keyword(s):

Numerical Methods ◽

Mathematical Models ◽

Numerical Simulations ◽

Numerical Solutions ◽

Chemotaxis Models

In the paper, we study the usage of numerical methods in solution of mathematical models of biological problems. More specifically, Keller-Segel type chemotaxis models are discussed, their numerical solutions by sweep and Lax-Friedrichs methods are obtained and interpreted biologically.

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Numerical comparison between simplified mathematical models for rock-ice avalanches

10.5194/egusphere-egu21-7090 ◽

2021 ◽

Author(s):

Stefania Sansone ◽

Giorgio Rosatti ◽

Daniel Zugliani

Keyword(s):

Numerical Methods ◽

Mathematical Models ◽

Numerical Solutions ◽

Numerical Comparison ◽

Mathematical Framework ◽

Numerical Schemes ◽

Geophysical Research ◽

Simplified Models ◽

Three Phase ◽

Numerical Fluxes

Rock-ice avalanches correspond to three-phase mixtures composed of a liquid and of particles of rock and ice. The presence of ice inside the mixture plays a key role in the mobility of rock-ice avalanches, since the heat produced by basal friction and particle collisions induces its transformation into water. Due to this continuous supply of liquid to the mixture, rock-ice avalanches can threaten populations living in cold mountainous areas. Thus, for a good hazard assessment and management, there is the need to construct mathematical models able to predict the flow of rock-ice avalanches. In the literature, there exist only few models that deal with this type of mass flows (Pudasaini & Krautblatter 2014, Bartelt et al. 2018, Sansone et al. 2021). As proposed in Sansone et al. (2021), a framework of different simplified rock-ice avalanche models can be derived by starting from a complete three-phase approach and by imposing two specific assumptions, namely the isokinetic and incompressibility hypotheses. In this way, five classes of simplified approaches can be detected, and these mathematical models are characterized by different levels of approximations of the physics of rock-ice avalanches.In this work, we provide some numerical solutions for the depth-integrated one-dimensional versions of all the simplified mathematical models detected in Sansone et al. (2021). These numerical solutions are constructed using three different numerical schemes that distinguish themselves from the way the numerical fluxes are evaluated. While one of the three chosen numerical methods evaluates the numerical fluxes without considering the eigenstructure of the systems of equations, the other two schemes take partially or entirely account of the eigenstructure of the equation systems. Due to the possible loss of hyperbolicity detectable in some simplified models, we consider as test cases the problems of the small perturbations of the flow depth and of the concentrations.The first result of the analysis computed corresponds to the comparison between the numerical solutions derived from the three numerical schemes for each class of models. In this way, the responses of the different numerical methods to each equation system can be investigated. The second result consists in comparing numerically the different classes of simplified models detected by Sansone et al. (2021), thus allowing us to quantify the effects of the assumptions of each class of models on the flow dynamics.&#160;References:Bartelt P., Christen M., B&#252;hler Y., Buser O. (2018), Thermomechanical modelling of rock avalanches with debris, ice and snow entrainment. In 9th European Conference on Numerical Methods in Geotechnical Engineering (NUMGE), University of Porto, Porto, PORTUGAL.Pudasaini S., Krautblatter M. (2014), A two-phase mechanical model for rock-ice avalanches. Journal of Geophysical Research: Earth Surface 119 (10), 2272-2290.Sansone S., Rosatti G., Zugliani D. (2021), A mathematical framework for modelling rock-ice avalanches. Paper under review.

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Plastic failure zone characteristics and stability control technology of roadway in the fault area under non-uniformly high geostress: A case study from Yuandian Coal Mine in Northern Anhui Province, China

Open Geosciences ◽

10.1515/geo-2020-0154 ◽

2020 ◽

Vol 12 (1) ◽

pp. 406-424 ◽

Cited By ~ 1

Author(s):

Yaoguang Huang ◽

Aining Zhao ◽

Tianjun Zhang ◽

Weibin Guo

Keyword(s):

Numerical Methods ◽

Numerical Solutions ◽

Stress Test ◽

Pressure Coefficient ◽

Stability Control ◽

Failure Zone ◽

Plastic Failure ◽

Comparison Results ◽

High Geostress ◽

The Difference

AbstractIn order to explore the effective support method for deep broken roadway, based on the in situ stress test results, the analytical and numerical solutions of the stress and the range of plastic failure zone (PFZ) in a circular roadway subjected to non-uniform loads were obtained using analytical and finite difference numerical methods based on the elastoplastic theory, respectively. Their comparison results show that the analytical and numerical methods are correct and reasonable. Furthermore, the high geostress causes the stress and range of PFZ in roadway roof and floor to increase sharply while those in roadway ribs decrease. Moreover, the greater the difference of horizontal geostress in different horizontal directions is, the larger the range of PFZ in roadway roof and floor is. The shape of PFZ in roadway varies with the ratio of horizontal lateral pressure coefficient in x-direction and y-direction. Finally, according to the distribution characteristics of PFZ and range of PFZ under the non-uniformly high geostress, this paper has proposed a combined support scheme, and refined and optimized supporting parameters. The field monitoring results prove that the roadway deformation and fracture have been effectively controlled. The research results of this paper can provide theoretical foundation as well as technical reference for the stability control of deep broken roadway under non-uniformly high geostress.

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RESIDUAL DISPLACEMENTS‘ PROGRESIVE ANALYSIS OF THE MULTISUPPORTED BEAM / DAUGIAATRAMĖS SIJOS LIEKAMŲJŲ POSLINKIŲ PROGRESINĖ ANALIZĖ

Mokslas - Lietuvos ateitis ◽

10.3846/mla.2014.686 ◽

2014 ◽

Vol 6 (5) ◽

pp. 461-467 ◽

Cited By ~ 1

Author(s):

Liudas Liepa ◽

Agnė Gervytė ◽

Ela Jarmolajeva ◽

Juozas Atkočiūnas

Keyword(s):

Numerical Methods ◽

Mathematical Models ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Analysis Problem ◽

Residual Displacements ◽

Repeated Load ◽

Energy Principles

This paper focuses on a shakedown behaviour of the ideally elasto-plastic beams system under variable repeated load. The mathematical models of the analysis problems are created using numerical methods, extremum energy principles and mathematic programming. It is shown that during the shakedown process the residual displacements vary non-monotonically. By solving analysis problem, where the load locus is being progressively expanded, it is possible to determine the upper and lower bounds of residual displacements. Suggested methods are ilustrated by solving multisupported beam example problem. The results are obtained considering principle of the small displacements. Nagrinėjamas idealiai tampriai plastinės lenkiamos strypinės sistemos prisitaikomumo būvis, veikiant kartotinei kintamajai apkrovai. Analizės uždavinių matematiniai modeliai sudaromi, pasitelkus skaitinius metodus, ekstreminius energinius principus ir matematinį programavimą. Parodoma, kad prisitaikant konstrukcijai jos liekamieji poslinkiai gali kisti nemonotoniškai. Išsprendus analizės uždavinį, kuriame progresyviai plečiama apkrovos veikimo sritis, galima nustatyti viršutines ir apatines liekamųjų poslinkių kitimo ribas. Siūloma metodika iliustruota daugiaatramės sijos liekamųjų poslinkių skaičiavimo pavyzdžiu. Rezultatai gauti, esant mažų poslinkių prielaidai.

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