Search results for: compact finite difference

Authors: Mohd Agos Salim Nasir, Ahmad Izani Md. Ismail

Abstract:

Several numerical schemes utilizing central difference approximations have been developed to solve the Goursat problem. However, in a recent years compact discretization methods which leads to high-order finite difference schemes have been used since it is capable of achieving better accuracy as well as preserving certain features of the equation e.g. linearity. The basic idea of the new scheme is to find the compact approximations to the derivative terms by differentiating centrally the governing equations. Our primary interest is to study the performance of the new scheme when applied to two Goursat partial differential equations against the traditional finite difference scheme.

Keywords: Goursat problem, partial differential equation, finite difference scheme, compact finite difference

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Authors: Sam Teek Ling, Norhashidah Hj. Mohd. Ali

Abstract:

We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization. The methods are compared with the existing second order standard five point formula to show the dramatic improvement in computed accuracy. Numerical experiments are presented to illustrate the effectiveness of the proposed methods.

Keywords: Explicit group iterative method, finite difference, fourth order compact, Poisson equation.

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Authors: R. M. Barron, B. Zogheib

Abstract:

A new numerical method for solving the twodimensional, steady, incompressible, viscous flow equations on a Curvilinear staggered grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well-established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity-pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation in a manner similar to the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results obtained from the present method are based on the first-order upwind scheme for the convective terms, but the methodology can easily be modified to accommodate higher order differencing schemes.

Keywords: Curvilinear, finite difference, finite volume, SIMPLE.

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Authors: Shigeyuki Haruyama, Hiroyuki Tanaka, Dai-Heng Chen, Aidil Khaidir Bin Muhamad

Abstract:

In this paper, the deformation modes of a compact impact absorption member subjected to axial compression are investigated using finite element method and experiments. A multiple combination compact impact absorption member, referred to as a 'compress-expand member', is proposed to substitute the conventional thin-walled circular tube. This study found that the proposed compact impact absorption member has stable load increase characteristics and a wider range of high load efficiency (Pave/Pmax) than the thin-walled circular tube. Moreover, the proposed compact impact absorption member can absorb larger loads in a smaller radius than the thin-walled cylindrical tube, as it can maintain its stable deformation in increased wall thicknesses.

Keywords: axial collapse, compact impact absorption member, finite element method, thin-walled cylindrical tube.

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Authors: Na Cheng, Zi-li Chen

Abstract:

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. We present a necessary and sufficient condition under which each positive semi-compact operator is b-weakly compact.

Keywords: b-weakly compact, Dunford-Pettis operator, M-weakly compact operator, L-weakly compact operator, semi-compact operator, weakly sequentially continuous lattice operations, order continuous norm, positive Schur property.

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Authors: Reza Mollapourasl, Majid Haghi

Abstract:

In this study, we develop local meshfree methods known as radial basis function-generated finite difference (RBF-FD) method and Hermite finite difference (RBF-HFD) method to design stencil weights and spatial discretization for Helmholtz equation. The convergence and stability of schemes are investigated numerically in three dimensions with irregular shaped domain. These localized meshless methods incorporate the advantages of the RBF method, finite difference and Hermite finite difference methods to handle the ill-conditioning issue that often destroys the convergence rate of global RBF methods. Moreover, numerical illustrations show that the proposed localized RBF type methods are efficient and applicable for problems with complex geometries. The convergence and accuracy of both schemes are compared by solving a test problem.

Keywords: Radial basis functions, Hermite finite difference, Helmholtz equation, stability.

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Authors: Aso Rahimzadegan, Seyyed Poorya Hosseini, Kamran Qaderi

Abstract:

In this paper, we have proposed two novel plasmonic demultiplexing structures based on metal-insulator-metal surfaces which, beside their compact size, have a very good transmission spectrum. The impact of the key internal parameters on the transmission spectrum is numerically analyzed by using the twodimensional (2D) finite difference time domain (FDTD) method. The proposed structures could be used to develop ultra-compact photonic wavelength demultiplexing devices for large-scale photonic integration.

Keywords: Photonic integrated devices, Plasmonics, Metalinsulator- metal (MIM) waveguide, Demultiplexers.

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Authors: Norhashidah Hj. Mohd Ali, Teng Wai Ping

Abstract:

In this paper, the formulation of a new group explicit method with a fourth order accuracy is described in solving the two dimensional Helmholtz equation. The formulation is based on the nine-point fourth order compact finite difference approximation formula. The complexity analysis of the developed scheme is also presented. Several numerical experiments were conducted to test the feasibility of the developed scheme. Comparisons with other existing schemes will be reported and discussed. Preliminary results indicate that this method is a viable alternative high accuracy solver to the Helmholtz equation.

Keywords: Explicit group method, finite difference, Helmholtz equation, five-point formula, nine-point formula.

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Authors: Nopparat Pochai, Rujira Deepana

Abstract:

Water pollution assessment problems arise frequently in environmental science. In this research, a finite difference method for solving the one-dimensional steady convection-diffusion equation with variable coefficients is proposed; it is then used to optimize water treatment costs.

Keywords: Finite difference, One-dimensional, Steady state, Waterpollution control, Optimization, Convection-diffusion equation.

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Authors: Khaled Moaddy

Abstract:

In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: Standard finite difference schemes, non–standard schemes, Laplace equation, Dirichlet boundary conditions.

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Authors: Akram G., Jasmy Y.

Abstract:

The finite-difference time-domain (FDTD) method is one of the most widely used computational methods in electromagnetic. This paper describes the design of two-dimensional (2D) FDTD simulation software for transverse magnetic (TM) polarization using Berenger's split-field perfectly matched layer (PML) formulation. The software is developed using Matlab programming language. Numerical examples validate the software.

Keywords: Finite difference time domain (FDTD) method, perfectly matched layer (PML), split-filed formulation, transverse magnetic (TM) polarization.

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Authors: M. M. Shokrieh, A. Karamnejad

Abstract:

This paper deals with a numerical analysis of the transient response of composite beams with strain rate dependent mechanical properties by use of a finite difference method. The equations of motion based on Timoshenko beam theory are derived. The geometric nonlinearity effects are taken into account with von Kármán large deflection theory. The finite difference method in conjunction with Newmark average acceleration method is applied to solve the differential equations. A modified progressive damage model which accounts for strain rate effects is developed based on the material property degradation rules and modified Hashin-type failure criteria and added to the finite difference model. The components of the model are implemented into a computer code in Mathematica 6. Glass/epoxy laminated composite beams with constant and strain rate dependent mechanical properties under dynamic load are analyzed. Effects of strain rate on dynamic response of the beam for various stacking sequences, load and boundary conditions are investigated.

Keywords: Composite beam, Finite difference method, Progressive damage modeling, Strain rate.

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Authors: N. Fusun Oyman Serteller

Abstract:

In this paper, the techniques to solve time dependent electromagnetic wave propagation equations based on the Finite Difference Method (FDM) are proposed by comparing the results with Finite Element Method (FEM) in 2D while discussing some special simulation examples.  Here, 2D dynamical wave equations for lossy media, even with a constant source, are discussed for establishing symbolic manipulation of wave propagation problems. The main objective of this contribution is to introduce a comparative study of two suitable numerical methods and to show that both methods can be applied effectively and efficiently to all types of wave propagation problems, both linear and nonlinear cases, by using symbolic computation. However, the results show that the FDM is more appropriate for solving the nonlinear cases in the symbolic solution. Furthermore, some specific complex domain examples of the comparison of electromagnetic waves equations are considered. Calculations are performed through Mathematica software by making some useful contribution to the programme and leveraging symbolic evaluations of FEM and FDM.

Keywords: Finite difference method, finite element method, linear-nonlinear PDEs, symbolic computation, wave propagation equations.

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Authors: Vineet K. Srivastava, Mukesh K. Awasthi, Mohammad Tamsir

Abstract:

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.

Keywords: Burgers’ equation, Implicit Finite-difference method, Newton’s method, Gauss elimination with partial pivoting.

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Authors: Jaipong Kasemsuwan

Abstract:

A numerical solution of the initial boundary value problem of the suspended string vibrating equation with the particular nonlinear damping term based on the finite difference scheme is presented in this paper. The investigation of how the second and third power terms of the nonlinear term affect the vibration characteristic. We compare the vibration amplitude as a result of the third power nonlinear damping with the second power obtained from previous report provided that the same initial shape and initial velocities are assumed. The comparison results show that the vibration amplitude is inversely proportional to the coefficient of the damping term for the third power nonlinear damping case, while the vibration amplitude is proportional to the coefficient of the damping term in the second power nonlinear damping case.

Keywords: Finite-difference method, the nonlinear damped equation, the numerical simulation, the suspended string equation

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Authors: Mahdi Sharifian, Mohammad Ali Fanaei

Abstract:

Saccharomyces cerevisiae (baker-s yeast) can exhibit sustained oscillations during the operation in a continuous bioreactor that adversely affects its stability and productivity. Because of heterogeneous nature of cell populations, the cell population balance models can be used to capture the dynamic behavior of such cultures. In this paper an unstructured, segregated model is used which is based on population balance equation(PBE) and then in order to simulation, the 4th order Rung-Kutta is used for time dimension and three methods, finite difference, orthogonal collocation on finite elements and Galerkin finite element are used for discretization of the cell mass domain. The results indicate that the orthogonal collocation on finite element not only is able to predict the oscillating behavior of the cell culture but also needs much little time for calculations. Therefore this method is preferred in comparison with other methods. In the next step two controllers, a globally linearizing control (GLC) and a conventional proportional-integral (PI) controller are designed for controlling the total cell mass per unit volume, and performances of these controllers are compared through simulation. The results show that although the PI controller has simpler structure, the GLC has better performance.

Keywords: Bioreactor, cell population balance, finite difference, orthogonal collocation on finite elements, Galerkin finite element, feedback linearization, PI controller.

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Authors: Guang Zeng, Jin Huang, Zicai Li

Abstract:

In this paper, we research the standard 13-point difference schemes for solving the biharmonic equation. Heuristic method is applied to judging the stability of multi-level difference schemes of the biharmonic equation. It is showed that the standard 13-point difference schemes are stable.

Keywords: Finite-difference equation, computational stability, hirt method.

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Authors: Jaipong Kasemsuwan

Abstract:

This paper presents the finite difference scheme and the numerical simulation of suspended string. The vibration solutions when the various external forces are taken into account are obtained and compared with the solutions without external force. In addition, we also investigate how the external forces and their powers and coefficients affect the amplitude of vibration.

Keywords: Nonlinear external forces, Numerical simulation, Suspended string equation.

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Authors: Jyothi Digge, B. U. Rindhe, S. K. Narayankhedkar

Abstract:

A compact 1x3 power splitter based on Photonic Crystal Waveguides (PCW) with flexible power splitting ratio is presented in this paper. Multimode interference coupler (MMI) is integrated with PCW. The device size reduction compared with the conventional MMI power splitter is attributed to the large dispersion of the PCW. Band Solve tool is used to calculate the band structure of PCW. Finite Difference Time Domain (FDTD) method is adopted to simulate the relevant structure at 1550nm wavelength. The device is polarization insensitive and allows the control of output (o/p) powers within certain percentage points for both polarizations.

Keywords: Dispersion, MMI Coupler, Photonic Bandgap, Power Splitter.

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Authors: Kelong Zheng, Jinsong Hu,

Abstract:

In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable.

Keywords: Generalized Rosenau-Burgers equation, difference scheme, stability, convergence.

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Authors: Davod Khojasteh Salkuyeh

Abstract:

An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.

Keywords: Ordinary differential equations, optimal stepsize, error, stochastic arithmetic, CESTAC, CADNA.

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Authors: H. Taheri Shahraiyni, B. Ataie Ashtiani

Abstract:

Flow movement in unsaturated soil can be expressed by a partial differential equation, named Richards equation. The objective of this study is the finding of an appropriate implicit numerical solution for head based Richards equation. Some of the well known finite difference schemes (fully implicit, Crank Nicolson and Runge-Kutta) have been utilized in this study. In addition, the effects of different approximations of moisture capacity function, convergence criteria and time stepping methods were evaluated. Two different infiltration problems were solved to investigate the performance of different schemes. These problems include of vertical water flow in a wet and very dry soils. The numerical solutions of two problems were compared using four evaluation criteria and the results of comparisons showed that fully implicit scheme is better than the other schemes. In addition, utilizing of standard chord slope method for approximation of moisture capacity function, automatic time stepping method and difference between two successive iterations as convergence criterion in the fully implicit scheme can lead to better and more reliable results for simulation of fluid movement in different unsaturated soils.

Keywords: Finite Difference methods, Richards equation, fullyimplicit, Crank-Nicolson, Runge-Kutta.

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Authors: Vijay Kumar Kukreja, Ravneet Kaur

Abstract:

In this paper, one dimensional advection diffusion model is analyzed using finite difference method based on Crank-Nicolson scheme. A practical problem of filter cake washing of chemical engineering is analyzed. The model is converted into dimensionless form. For the grid Ω × ω = [0, 1] × [0, T], the Crank-Nicolson spatial derivative scheme is used in space domain and forward difference scheme is used in time domain. The scheme is found to be unconditionally convergent, stable, first order accurate in time and second order accurate in space domain. For a test problem, numerical results are compared with the analytical ones for different values of parameter.

Keywords: Consistency, Crank-Nicolson scheme, Gerschgorin circle, Lax-Richtmyer theorem, Peclet number, stability.

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Authors: Musthafah b. Mohd.Tahir, Noboru Yamada, Tetsuya Hoshino

Abstract:

This paper describes the experimental efficiency of a compact organic Rankine cycle (ORC) system with a compact rotary-vane-type expander. The compact ORC system can be used for power generation from low-temperature heat sources such as waste heat from various small-scale heat engines, fuel cells, electric devices, and solar thermal energy. The purpose of this study is to develop an ORC system with a low power output of less than 1 kW with a hot temperature source ranging from 60°C to 100°C and a cold temperature source ranging from 10°C to 30°C. The power output of the system is rather less due to limited heat efficiency. Therefore, the system should have an economically optimal efficiency. In order to realize such a system, an efficient and low-cost expander is indispensable. An experimental ORC system was developed using the rotary-vane-type expander which is one of possible candidates of the expander. The experimental results revealed the expander performance for various rotation speeds, expander efficiencies, and thermal efficiencies. Approximately 30 W of expander power output with 48% expander efficiency and 4% thermal efficiency with a temperature difference between the hot and cold sources of 80°C was achieved.

Keywords: Organic Rankine cycle, Thermodynamic cycle, Thermal efficiency, Turbine efficiency, Waste heat recovery, Powergeneration, Low temperature heat engine.

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Authors: Serifenur Cebesoy, Yelda Aygar, Elgiz Bairamov

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In this study, we examine some spectral properties of non-selfadjoint matrix-valued difference equations consisting of a polynomial-type Jost solution. The aim of this study is to investigate the eigenvalues and spectral singularities of the difference operator L which is expressed by the above-mentioned difference equation. Firstly, thanks to the representation of polynomial type Jost solution of this equation, we obtain asymptotics and some analytical properties. Then, using the uniqueness theorems of analytic functions, we guarantee that the operator L has a finite number of eigenvalues and spectral singularities.

Keywords: Difference Equations, Jost Functions, Asymptotics, Eigenvalues, Continuous Spectrum, Spectral Singularities.

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Authors: Atilla Akpinar

Abstract:

In this paper, we study on finite projective Hjelmslev planes M(Zq) coordinatized by Hjelmslev ring Zq (where prime power q = pk). We obtain finite hyperbolic Klingenberg planes from these planes under certain conditions. Also, we give a combinatorical result on M(Zq), related by deleting a line from lines in same neighbour.

Keywords: Finite Klingenberg plane, finite hyperbolic Klingenberg plane.

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Authors: Zone-Ching Lin, Meng-Hua Lin, Ying-Chih Hsu

Abstract:

This paper uses quasi-steady molecular statics model and diamond tool to carry out simulation temperature rise of nanoscale orthogonal cutting single-crystal silicon. It further qualitatively analyzes temperature field of silicon workpiece without considering heat transfer and considering heat transfer. This paper supposes that the temperature rise of workpiece is mainly caused by two heat sources: plastic deformation heat and friction heat. Then, this paper develops a theoretical model about production of the plastic deformation heat and friction heat during nanoscale orthogonal cutting. After the increased temperature produced by these two heat sources are added up, the acquired total temperature rise at each atom of the workpiece is substituted in heat transfer finite difference equation to carry out heat transfer and calculates the temperature field in each step and makes related analysis.

Keywords: Quasi-steady molecular statics, Nanoscale orthogonal cutting, Finite difference, Temperature.

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Authors: S. M. Abdur Razzak, M. A. Rashid, Y. Namihira, A. Sayeem

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A simple microstructure optical fiber design based on an octagonal cladding structure is presented for simultaneously controlling dispersion and leakage properties. The finite difference method with anisotropic perfectly matched boundary layer is used to investigate the guiding properties. It is demonstrated that octagonal photonic crystal fibers with four rings can assume negative ultra-flattened dispersion of -19 + 0.23 ps/nm/km in the wavelength range of 1.275 μm to 1.68 μm, nearly zero ultra-flattened dispersion of 0 ± 0.40 ps/nm/km in a 1.38 to 1.64 μm, and low confinement losses less than 10-3 dB/km in the entire band of interest.

Keywords: Finite difference modeling, group velocity dispersion, optical fiber design, photonic crystal fiber.

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Authors: Vijay Janyani, Neetu Joshi, Jigyasa Pagaria, Parul Pathak

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Photonic Crystal (PhC) based devices are being increasingly used in multifunctional, compact devices in integrated optical communication systems. They provide excellent controllability of light, yet maintaining the small size required for miniaturization. In this paper, the band gap properties of PhCs and their typical applications in optical waveguiding are considered. Novel PhC based applications such as nonlinear switching and tapers are considered and simulation results are shown using the accurate time-domain numerical method based on Finite Difference Time Domain (FDTD) scheme. The suitability of these devices for novel applications is discussed and evaluated.

Keywords: Band gap engineering, Nonlinear switching, Photonic crystals, PhC tapers, waveguides.

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Authors: Sourabh Agrawal, Ashok K. Jain

Abstract:

Stick models are widely used in studying the behaviour of straight as well as skew bridges and viaducts subjected to earthquakes while carrying out preliminary studies. The application of such models to highly curved bridges continues to pose challenging problems. A viaduct proposed in the foothills of the Himalayas in Northern India is chosen for the study. It is having 8 simply supported spans @ 30 m c/c. It is doubly curved in horizontal plane with 20 m radius. It is inclined in vertical plane as well. The superstructure consists of a box section. Three models have been used: a conventional stick model, an improved stick model and a 3D finite element model. The improved stick model is employed by making use of body constraints in order to study its capabilities. The first 8 frequencies are about 9.71% away in the latter two models. Later the difference increases to 80% in 50th mode. The viaduct was subjected to all three components of the El Centro earthquake of May 1940. The numerical integration was carried out using the Hilber- Hughes-Taylor method as implemented in SAP2000. Axial forces and moments in the bridge piers as well as lateral displacements at the bearing levels are compared for the three models. The maximum difference in the axial forces and bending moments and displacements vary by 25% between the improved and finite element model. Whereas, the maximum difference in the axial forces, moments, and displacements in various sections vary by 35% between the improved stick model and equivalent straight stick model. The difference for torsional moment was as high as 75%. It is concluded that the stick model with body constraints to model the bearings and expansion joints is not desirable in very sharp S curved viaducts even for preliminary analysis. This model can be used only to determine first 10 frequency and mode shapes but not for member forces. A 3D finite element analysis must be carried out for meaningful results.

Keywords: Bearing, body constraint, box girder, curved viaduct, expansion joint, finite element, link element, seismic, stick model, time history analysis.

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