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1

Force analysis and treatments for Lidong tunnel of Renxin expressway crossing karst caves

Zhu Kai, Zhang Kui, Wu Xiang-Dong, Chen Xiang-Ge

Archives of Civil Engineering

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2024

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Vol. 70, nr 1

693--705

EN

There are several large karst caves at haunch part of the Lidong Tunnel during construction, together with inrush water due to high pressure within these caves. In light of it, this paper takes YK342+113 section as an example and adopts finite difference software FLAC 3D, so as to analyze tunnel deformation when crossing karst caves under six different working conditions, including with or without karst cave, before and after karst treatment, along with support locations. According to analysis results: First, the wall rock mainly had deformation at tunnel vault when evacuating at the third bench, which is a critical monitoring focus for tunnel construction; Second, karst cave treatment contributed to better conduct forces on both sides of wall rock, thus reducing vault settlement, while not affecting horizontal convergence and upturn of vaults; Third, treatment measures were proved to be effective in minimizing wall rock deformation by comparing deformation curves under different conditions; Fourth, after treatment measures, the angular points within the cave’s chamber had stress concentration, which might cause secondary collapse. Field monitoring data revealed that the final settlement of the tunnel vault was relatively consistent with the numerical analysis results, with a distinct change in daily settlement after initial support construction. By integrating numerical analysis and field monitoring, the rationality of the karst treatment plan was fully verified, providing a valuable reference for similar projects.

2

Simulation of seismic wave propagation in poroelastic media using vectorized Biot’s equations: an application to a CO2 sequestration monitoring case

Anthony Emmanuel, Vedanti Nimisha

Acta Geophysica

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2020

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Vol. 68, no. 2

435--444

EN

Wave propagation through porous media allows us to understand the response and interaction that occur between the elastic rock matrix and the fuid. This interaction has been described by Biot in his theory of poroelasticity. Seismic wave simulation using Biot’s formulations is computationally expensive when compared with the acoustic and elastic cases. This computational burden can be reduced by reformulating the numerical derivative operators to improve the efciency. To achieve this, we used a staggered-grid fnite diference operator to discretize 2D velocity stress equations as given by Biot’s theory. A vectorized derivative is applied on the staggered grid by shifting the coordinates. The reformulated equations were applied to compute the seismic response of a reservoir, where CO2 is being injected and the efect of injected CO2 in the formation is clearly seen in the synthetic data generated. The algorithm was coded in Python and to test its efciency, the simulation run-time was compared for both serial and vectorized equations, and the speed-up ratio was calculated. Our results show a decrease in the simulation run-time for the vectorized execution with over a factor of a hundred percent (100%). We further observed that the amplitudes of the events increase with an increase in CO2 saturation in the formation. This matches well with the real data.

3

Analiza stanów krytycznych mostu w czasoprzestrzeni

Obrębski J.

Mechanik

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2018

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R. 91, nr 7

591--593

PL

W pracy zbadano możliwość oceny stanu krytycznego mostu pod wpływem obciążeń ruchomych, z zastosowaniem jednolitego kryterium oceny geometrycznej zmienności i utraty stateczności konstrukcji oraz czasoprzestrzeni modelowanej różnicami skończonymi. Oba ujęcia numeryczne zostały zaproponowane przez autora. Wspomnianym kryterium jest zerowa wartość głównego wyznacznika dynamicznej macierzy sztywności dla mostu, a nawet zadania – gdy obciążenie porusza się dalej poza przęsłem. Wyniki przykładów liczbowych pokazują skuteczność metody i wpływ różnych parametrów na stanykrytyczne.

EN

In the paper was pointed possibility to evaluate critical state of bridge under travelling loading, applying uniform criterion for geometrical changeability and instability of structure and 3D-time space method modelled by finite differences. The both numerical methods are formulated by present author. In the above criterion is used value of main determinant of dynamical stiffness matrix for bridge or even for task, when loading is travelling beyond span. Results shows efficacy of the method and influence of some parameters.

4

Czasoprzestrzeń w analizie dynamicznej mostów

Obrębski J. B.

Problemy Mechatroniki : uzbrojenie, lotnictwo, inżynieria bezpieczeństwa

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2017

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Vol. 8, Nr 3 (29)

109--126

PL

W pracy zaprezentowano kolejne zastosowania i możliwości analizy dynamicznej obiektów inżynierskich za pomocą czasoprzestrzeni z wykorzystaniem różnic skończonych, w ujęciu numerycznym zaproponowanym przez autora. Obiekty te mogą być poddane różnego rodzaju obciążeniom. Tym razem jest to dźwigar mostowy z pojazdami poruszającymi się po nim z określoną prędkością. Może to być jeden pojazd lub cała ich grupa. Rozważania przeprowadzono od ogólnych różniczkowych równań ruchu przez ich uproszczenia wynikające z przyjętych założeń dla mostu, do układu dwóch równań opisujących jego giętno-skrętne zachowanie się. Pokazano przykłady konkretnych zastosowań tej metody.

EN

The paper presents next application and possibilities of dynamical analysis for engineering structures by means of 3D-Time Space Method in numerical edition proposed by the author with application of finite differences. These structures can be charged by different kinds of loadings. This time, it is a bridge girder with a car or the whole group of different carriages moving with certain velocity. The considerations were taken, starting from general differential equations of motion, through their simplified versions - resulting from taken assumptions for a bridge, until a set of two equations describing its bending-torsion behaviour. There are given examples of particular applications of the method.

5

The discontinuous Galerkin method with higher degree finite difference compatibility conditions and arbitrary local and global basis functions

Jaśkowiec J.

Computer Assisted Methods in Engineering and Science

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2016

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Vol. 23, no. 2/3

109--132

EN

This paper focuses on the discontinuous Galerkin (DG) method in which the compatibility condition on the mesh skeleton and Dirichlet boundary condition on the outer boundary are enforced with the help of one-dimensional finite difference (FD) rules, while in the standard approach those conditions are satisfied by the penalty constraints. The FD rules can be of arbitrary degree and in this paper the rules are applied up to fourth degree. It is shown that the method presented in this paper gives better results in comparison to the standard version of the DG method. The method is based on discontinuous approximation, which means that it can be constructed using arbitrary local basis functions in each finite element. It is quite easy to incorporate some global basis functions in the approximation field and this is also shown in the paper. The paper is illustrated with a couple of two-dimensional examples.

6

Implementation of Elastic Prestack Reverse-Time Migration Using an Efficient Finite-Difference Scheme

Yan H., Yang L., Dai H., Li X. Y.

Acta Geophysica

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2016

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Vol. 64, no. 5

1605--1625

EN

Elastic reverse-time migration (RTM) can reflect the underground elastic information more comprehensively than single-component Pwave migration. One of the most important requirements of elastic RTM is to solve wave equations. The imaging accuracy and efficiency of RTM depends heavily on the algorithms used for solving wave equations. In this paper, we propose an efficient staggered-grid finite-difference (SFD) scheme based on a sampling approximation method with adaptive variable difference operator lengths to implement elastic prestack RTM. Numerical dispersion analysis and wavefield extrapolation results show that the sampling approximation SFD scheme has greater accuracy than the conventional Taylor-series expansion SFD scheme. We also test the elastic RTM algorithm on theoretical models and a field data set, respectively. Experiments presented demonstrate that elastic RTM using the proposed SFD scheme can generate better images than that using the Taylor-series expansion SFD scheme, particularly for PS images. Furthermore, the application of adaptive variable difference operator lengths can effectively improve the computational efficiency of elastic RTM.

7

Mathematical modeling of air duct heater using the finite difference method

Sarafraz M. M. M. M., Peyghambarzadeh S. M. S. M., Marahel A. A.

Polish Journal of Chemical Technology

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2011

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Vol. 13, nr 4

47-52

EN

In this research, mathematical modeling of a duct heater has been performed using energy conservation law, Stefan-Boltzman law in thermal radiation, Fourier’s law in conduction heat transfer, and Newton’s law of cooling in convection heat transfer. The duct was divided to some elements with equal length. Each element has been studied separately and air physical properties in each element have been used based on its temperature. The derived equations have been solved using the finite difference method and consequently air temperature, internal and external temperatures of the wall, internal and external convection heat transfer coeffi cients, and the quantity of heat transferred have been calculated in each element and effects of the variation of heat transfer parameters have been surveyed. The results of modelling presented in this paper can be used for the design and optimization of heat exchangers.

8

Unsteady free convection flow along a vertical cone in a stable thermally stratified medium

Bapuji P., Ekambavanan K., Pop I.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 4

1073-1092

EN

Unsteady natural convection flow of a viscous and incompressible fluid flow over a vertical cone immersed in a stable thermally stratified medium is theoretically studied in this paper. The dimensionless coupled partial differential boundary layer equations are solved numerically using an efficient and unconditionally stable finite-difference scheme of Crank-Nicolson type. The effects of the Prandtl number and stratification parameter on the velocity and temperature profiles as well as the local and average skin friction and Nusselt numbers on the flow and heat transfer characteristics have been determined and discussed in detail. The present results are compared with available results from the open literature and are found to be in good agreement.

9

A time-step insensitive recurrent approach to analyze non-stationary random responses

Chen K. F., Zhang Q., Zhang S. W.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 1

1-9

EN

The recurrent approach constructed via the stochastic central difference (SCD) is a very fast method for analyzing non-stationary random responses. However, the computational results depend to a great extent upon the discrete time-step size. A new recurrent approach is proposed in this report. It is based on the theory of linear differential equations. Theoretical analysis shows that this algorithm is unconditionally stable for all damped systems. Two examples show that the proposed approach is not sensitive to the time-step.

10

Modelowanie numeryczne transportu substancji aktywnych z uwzględnieniem sorpcji i reakcji chemicznych w oparciu o równania różniczkowe adwekcji i kinetyki procesów

Sławomirski M. R.

Prace Instytutu Mechaniki Górotworu PAN

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2004

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Vol. 6, no. 3-4

243--248

EN

The problem of transfer of active substances dissolved from underground depository of industrial waste materials by subsurface water has been considered. It is assumed that the advection of dissolved active substance is modified by diffusion, sorption, and chemical reactions occurring in the solution. The problem has been described by the system of non-linear partial differential equations. The system of equations has been solved applying the finite difference technique. The original semi-implicite finite difference algorithm has been implemented within the computer program for the Digital AlphaStation RISC type computer. The solution of the problem is represented by the single concentration wave which moves from the boundary of depository with the advection velocity (cf. Fig. 1). The magnitude of the concentration wave is progressively decreased in distance from the boundary of the depository owing to sorption, ion exchange, and chemical disintegration of the active substance. The sharp edge of the wave is gradually `flattened’ as the result of diffusion and dispersion effects.

11

Blasius flow of viscoelastic fluids: a numerical approach

Sadeghy K., Sharifi M.

International Journal of Applied Mechanics and Engineering

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2004

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Vol. 9, no 2

399-411

EN

The effects of a fluid elasticity on the characteristics of a boundary layer in a Blasius flow are investigated for a second-grade fluid, and also for a Maxwell fluid. Boundary layer approximations are used to simplify the equations of motion which are finally reduced to a single ODE using the concept of similarity solution. For the second-grade fluid, it is found that the number of boundary conditions should be augmented to match the order of the governing equation. A combination of finite difference and shooting methods are used to solve the governing equations. Results are presented for velocity profiles, boundary layer thickness, and skin friction coefficient in terms of the local Deborah number. An overshoot in velocity profiles is predicted for a second-grade fluid but not for a Maxwell fluid. The boundary layer is predicted to become thinner for the second-grade fluid but thicker for the Maxwell fluid, the higher the Deborah number. By an increase in the level of fluid elasticity, a drop in wall skin friction is predicted for the second-order fluid but not for the Maxwell fluid.

12

Transient free convection flow of viscous dissipative fluid past a semi-infinite inclined plate

Ganesan P., Palani G.

International Journal of Applied Mechanics and Engineering

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2003

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Vol. 8, no 3

395-402

EN

An analysis is performed to study the transient free convection flow of a viscous incompressible fluid past a semi-infinite inclined plate with viscous dissipation. The dimensionless governing equations are unsteady, coupled and non-linear integro partial differential equations. An analytical method fails to give a solution. Hence an implicit finite difference scheme of Crank-Nicolson method is employed. The effect of the dissipation parameter on the velocity, temperature, skin friction and Nusselt number are studied in detail. It is observed that greater viscous dissipative heat causes a rise in the temperature irrespective of the Prandtl number.

13

Discrete approximations of the Hamiltonian-Jacobi equation for an optimal control problem of a differential-algebraic system

Bonnans J. F., Chartier P., Zidani H.

Control and Cybernetics

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2003

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Vol. 32, no 1

33-56

EN

This paper discusses the numerical resolution of the Hamilton-Jacobi-Bellman equation associated with optimal control problem when the state equation is of algebraic differential type. We discuss two numerical schemes. The first reduces to the standard framework, while the second does not suppose any knowledge of the Jacobian of the data. We obtain some error estimates, and display numerical results obtained on a simple test problem.

PL

Artykuł rozpatruje rozwiązanie numeryczne równania Hamiltona-Jacobiego-Bellmana, związanego z zagadnieniem sterowania optymalnego w przypadku, gdy równanie stanu jest algebraiczno-różniczkowe. Rozważane są dwie procedury numeryczne. Pierwsza z nich sprowadza się do postępowania standardowego, podczas gdy druga nie zakłada znajomości Jakobianu danych. Otrzymano pewne oceny błędu, a na końcu artykułu pokazano wyniki numeryczne dla prostego zadania testowego.

14

Set-valued iterative square roots of bijections

Powierża T.

Bulletin of the Polish Academy of Sciences. Mathematics

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1999

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Vol. 47, no 4

377-383

EN

There are different ideas how to generalize the notion of an iterative root, especially when a function does not have a "real" root. We consider a multifunction as a substitute for this notion.