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M. T. Capilla, A. Balaguer-Beser

Then the cell average of the solution in (5) is the sum of the four quarter-cell averages defined in (11):

Using the reconstruction polynomials Ri,j (x, y, tn), we also approximate the point-values uˆni,j at time tn on the non-staggered grid.

3.2Source term integration

Following the procedure described in [6], an integration by parts with respect to the variable x is performed, in order to involve the spatial derivative of the free surface elevation instead of the bed elevation. This was suggested by Cale et al. in [5] for the one dimensional

where ψ(x, y) = g · Zb(x, y) · the free surface elevation which at cell-centers and to prevent Equation (12) in the y variable

In a similar way we evaluate the third component of the averaged source term in (6), obtaining:

A new high-order well-balanced central scheme for 2D shallow water equations

where the interpolating three-degree polynomial j,x( ; ˆ) approximates point values start-

P y φ

ing from point values (see [6]).

Again, we apply a centered quadrature rule for the integral with respect to the x variable:

The source term time integrals in (4) are also evaluated using a Gaussian quadrature rule with two nodes:

3.3Reconstruction of Runge-Kutta fluxes

In order to evaluate the time integrals of the source term (17) and the time flux integrals in

(7) we have to predict the point-values of the solution at two intermediate states: uˆni,j+βk ≡ u(xi, yj , tn +βk t), k = 0, 1. The prediction of these intermediate values at times tn+β0 and tn+β1 is obtained by means of a Runge-Kutta scheme coupled with the natural continuous extension (NCE) [4]:

l=1

where the constants bl(βk) are given in [6] and ki,j(l), 1 ≤ l ≤ 4, are the Runge-Kutta fluxes, which coincide with a numerical evaluation of (−fx − gy + s) in the shallow water system

(3). We use the point values of the solution {uˆ(i,jl)} to calculate the functions Fij and Gij , defined by

M. T. Capilla, A. Balaguer-Beser

where for simplicity we have omitted the index (l) from uˆ(i,jl). Equation (20) is a twodimensional extension of the definition presented in [5, 6]. This definition, (20), guarantees that the numerical scheme maintains the exact C-property. The evaluation of the RungeKutta fluxes is approximated using the interpolating polynomials that approximate the functions Fij and Gij , by means of:

For more details about these polynomials we refer the reader to [1].

4C-property verification

In this Section we will prove that our numerical scheme satisfies the exact C-property. It can

be verified through straightforward calculations that in case of quiescent flow, starting from ηi,jn = η = constant and qk;ni,j = 0, where qk = vkh, for k = 1, 2, gives ηi,jn+1 = ηˆi,jn+1 = η and

qk;ni,j+1 = qˆk;ni,j+1 = 0, n. This means that, starting from the mentioned initial conditions, the scheme maintains the steadiness of the point values and the cell averages of the solution.

4.1C-property for cell averages

The steadiness of the cell averages of the solution, i.e., ui,jn+1 = ui,jn , in the case of quiescent flow, must be verified. To achieve this result, it is su cient to show that in the following

equation:

A new high-order well-balanced central scheme for 2D shallow water equations

the vector valued term in square brackets is zero, for water at rest.

It can be seen from Equation (2), that the first component of the term in square brackets in (23) is identically zero. We have to prove that the second and third components of this term are zero. Remembering Equation (13) and assuming ηˆi(y) = ηˆi+1(y) = η = constant,

where ψi+ 12 (y) is the cell average of the function ψ(x, y) = g Zb(x, y) ∂x∂η on the cell [xi, xi+1]. In a similar way, it can be proved that the third component of the term in square brack-

the averaged values ψi+ 12 (y) and φj+ 12 (x) are zero, and the C-property verification for cell averages is proved. In an analogous form we can prove the C-property for point-values.

5Applications

In this Section we examine the behavior and accuracy of the numerical scheme, in several numerical tests. We show a numerical verification of the exact C-property and we present results for several standard tests proposed in the literature. In all cases, we consider x = y for the spatial steps. In the case of the shallow water system with fixed bottom topography, the numerical stability of the scheme is assured by selecting a time step satisfying the CFL condition which is related to the numerical stability of the Runge-Kutta scheme used in the time integration (see [8]). We focus our attention on the behavior of the reconstruction method described in the previous Section. To assure that our numerical scheme obtains accurate results with di erent CFL numbers, we have taken a fixed time step in all simulations: t = 0.0025 x = 0.0025 y.

M. T. Capilla, A. Balaguer-Beser

5.1Test for the exact C-property

This test, presented in [10], is used to verify numerically that our scheme maintains the exact C-property over a non flat bottom. The bottom topography is given by a two-dimensional

hump:

Zb(x, y) = 0.8 e−50((x−0.5)2 +(y−0.5)2) , x, y [0, 1] .

As initial condition for the water depth we set h(x, y, 0) = 1−Zb(x, y) and the initial velocity is set to be zero: q1(x, y, 0) = q2(x, y, 0) = 0. This surface should remain flat. We consider a rectangular mesh with x = y = 0.01 which corresponds to a NX=NY=100 uniform mesh. Table 1 contains the L1 and L∞ errors for the water height h and the discharges q1 and q2, at time t = 0.05 s.

Table 1: L1 and L∞ errors for the C-property analysis

From these results, we can conclude that the exact C-property is verified.

5.2Circular dam-break problem

We consider a test problem described in [7] and [9]. The domain is the square [0, 2] × [0, 2] and the bottom topography is given by the function:

and q1(x, y, 0) = q2(x, y, 0) = 0. We compute the numerical results using (200 × 200) grid points (Δx = y = 0.01). Figure 1 (left) shows the numerical solution for the free surface level at time t = 0.15 s. The analytical solution is not known for this problem, and therefore we have used our numerical scheme to obtain a reference solution in a mesh composed by (800 × 800) cells. In Figure 1 (right) we compare the numerical and the reference solutions for the free surface level at time t = 0.15 s, along the line y = 1.

These results can be compared with those in [7] and [9].

A new high-order well-balanced central scheme for 2D shallow water equations

Figure 1: The circular dam-break problem: solution at t = 0.15 s. Left: free surface and topography. Right: a longitudinal section at y = 1.

5.3Perturbation of a lake at rest in 2D

We show numerical results for a classical example of a small perturbation of a two-dimensional steady-state flow [10, 11, 9]. The perturbation occupies a small portion of the computational domain. The domain is the rectangle [0, 2] × [0, 1] and the topography is an isolated elliptical shaped hump:

Zb(x, y) = 0.8 e−5((x−0.9)2 −50(y−0.5)2) .

The initial conditions are q1(x, y, 0) = q2(x, y, 0) = 0 and

!

So the surface is almost flat except for 0.05 ≤ x ≤ 0.15, where h is perturbed upward by 0.01. Figure 2 displays the results obtained with our numerical scheme on a uniform mesh with (200 × 100) nodes. The Figure shows the contours of the surface level h + Zb and a longitudinal section at y = 0.5, at two di erent ending times: t = 0.24 s and t = 0.36 s. These results can be directly compared with those in [10, 11, 9]. We observe that our scheme can resolve consistently this problem and no oscillations are observed.

6Conclusions

In this paper we have extended the well-balanced central scheme described in [6] to solve the shallow water system in two space dimensions. The resulting scheme satisfies the exact

M. T. Capilla, A. Balaguer-Beser

Figure 2: Perturbation of a lake at rest: Left: contours of the surface level h + Zb with 30 uniformly spaced contour lines. Right: solution along the line y = 0.5 (Zb = Zb/80 + 0.98).

C-property and it has been designed to have high-resolution and a non-oscillatory behavior. A circular dam-break problem and a test with small perturbation of a lake at rest are solved to demonstrate these theoretical properties. The capability of the numerical scheme in reproducing the space-time evolution of the variables has been proved, and the agreement between our results and what are reported in literature has been confirmed.

Acknowledgements

This work was partially funded by the “Programa de Apoyo a la Investigaci´on y Desarrollo” (PAID-06-10) of the Universidad Polit´ecnica de Valencia. Angel Balaguer-Beser thanks the support of the Spanish Ministry of Education and Science in the framework of the Projects

A new high-order well-balanced central scheme for 2D shallow water equations

CGL2009-14220-C02-01 and CGL2010-19591.

References

[1]A. Balaguer and C. Conde, Fourth-Order Non-oscillatory Upwind and Central Schemes for Hyperbolic Conservation Laws, SIAM J. Numer. Anal. 43(2) (2005) 455– 473.

[2]A. Balaguer-Beser, A new reconstruction procedure in central schemes for hyperbolic conservation laws, Int. J. Numer. Meth. Engng. 86 (2011) 1481–1506.

[3]A. Bermudez´ and M. E. Vazquez´ , Upwind methods for hyperbolic conservation laws with source terms, Computers and Fluids 23 (1994) 1049–1071.

[4]F. Bianco, G. Puppo, and G. Russo, High-order central schemes for hyperbolic systems of conservation laws, SIAM J. Sci. Comput. 21(1) (1999), 294–322.

[5]V. Caleffi, A. Valiani, and A. Bernini, Fourth-order balanced source term treatment in central WENO schemes for shallow water equations, J. Comput. Phys. 218

(2006) 228–245.

[6]M. T. Capilla and A. Balaguer-Beser, A well-balanced high-resolution shapepreserving central scheme to solve one-dimensional sediment transport equations, Adv. Eng. Softw. (2012) http://dx.doi.org/10.1016/j.advengsoft.2012.04.003.

[7] M. J. Castro, E. D. Fernandez´ -Nieto, A. M. Ferreiro, J. A. Garc´ıaRodr´ıguez and C. Par´es, High order extensions of Roe schemes for two dimensional nonconservative fyperbolic systems, J. Sci. Comput. 39 (2009) 67–114.

[8]D. Levy, G. Puppo, and G. Russo, A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput. 24 (2002) 480–506.

[9]A. Mart´ınez-Gavara and R. Donat, A hybrid second order scheme for shallow water flows, J. Sci. Comput. 48 (2011) 241–257.

[10]Y. Xing and C. W. Shu, High order finite di erence WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys. 208 (2005) 206–227.

[11]Y. Xing and C. W. Shu, A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, Commun. Comput. Phys. 1(1) (2006) 100–134.

Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

A faster than real-time simulator of motion platforms

Sergio Casas1, Ricardo Olanda1, Marcos Fern´andez1 and Jos´e V. Riera1

1 Robotics Institute and Information Technology and Communications (IRTIC),

University of Valencia

emails: Sergio.Casas@uv.es, Ricardo.Olanda@uv.es, Marcos.Fernandez@uv.es,

J.Vicente.Riera@uv.es

Abstract

Motion platforms have been used for several decades in real-time simulations for motion cueing generation. Although many motion cueing algorithms (MCA) have been proposed to generate appropriate signals for motion platforms, this software always needs to be set-up for the specific application and motion platform used. This process is both expensive and time consuming, as the tests are performed on expensive motion platforms and the parameters are often adjusted by successive approximations, which may take plenty of time. Moreover, these tests can perform harmful sequences than can damage the hardware and even hurt human testers in case of failure. In order to solve these problems, we present a generic real-time virtual simulator of motion platforms. A motion platform simulator emulates the behavior of a real counterpart without the need to actually build it, and unlike a real one, it can run faster than real time, allowing fast tests and parameterizations of MCA. This simulator is assessed with a comparative study with a 3-DoF (Degrees of Freedom) and a 6-DoF real motion platforms.

Key words: Motion platform, virtual prototyping, real-time simulation, MCA.

1Introduction

A motion platform [11] is a powered, mechanical and self-contained motion generator system. Motion platforms have been used for motion cueing in di erent kind of applications, like real-time flight [21] and driving simulators [18], industrial equipment training [8] or medical rehabilitation [4]. Despite their widespread use, it is not easy to generate the appropriate signals for these manipulators in order to obtain the desired behavior [17]. A large set of tests for each motion-platform and for each specific use is necessary, requiring a

A faster than real-time simulator of motion platforms

successive approximation adjustment of the parameters [14]. This process is both expensive and time consuming, because the motion platform has to be built and then, tested for a long time. Moreover, some tests perform severe and potentially harmful movements that can damage the motion platform, and more importantly, hurt human testers in case of software or hardware failure.

In order to solve these problems, a motion platform simulator can be used. This simulator performs a computer-based emulation of a real motion platform, receiving the same inputs than the real motion platform does, and generating the same (simulated) outputs. This way, multiple tests do not wear or hurt the platform hardware, ensuring the safety of human testers. In addition, depending on the hardware used to run the simulator, tasks performed by the virtual platform can be executed faster than the real ones in the real motion platform, saving time. For all these reasons, we think that the use of a motion platform simulator is justified.

Hence, in this paper, we present a faster than real-time generic physically-based motion platform simulator, which allows us to simulate di erent kinds of motion platforms. The rest of the paper is organized as follows: section 2 reviews previous works on motion platform simulation. In section 3 the simulator itself is discussed. Section 4 shows the simulation validation tests. Finally, section 5 shows the conclusions and outlines the future work.

2Related Work

Motion platforms are widely used in order to add inertial cues to virtual reality applications, but the simulation of motion platforms is less frequent. Nevertheless, we can find some works that compare a virtual motion platform behavior to analytical results, like in Selvakumar et al. work [19] where a 3-DoF parallel manipulator simulated using ADAMS [13] is compared using MATLAB [10]. In Hajimirzaalian et al. work [6], a Stewart platform [20] is simulated using also ADAMS, and a comparison between the direct dynamics, provided by ADAMS, and the inverse solution, based on a Lagrangian formulation, is presented. In Gosselin et al. work [5], a simulator that allows interactive kinematic analysis of spherical parallel mechanisms is presented. In Lee et al. work [9], ADAMS is used to perform a kinematic and a dynamic simulation of a 3-DoF manipulator, in order to compare it to an inverse dynamic formulation based on a Newton-Euler approach.

However, all these works use analytical solutions in order to study the behavior of a specific motion platform. Our goal is to test MCA, and thus, we want to build a virtual motion platform that is able to simulate a generic motion platform physically and numerically, not kinematically nor analytically. A solution similar to our needs is presented in Hulme and Pancotti work[7], where a 6-DoF Moog 2000E platform [12] simulator is presented. However, the simulation is restricted to this model, not allowing a generic specification. This simulator uses a detailed CAD model to provide visual information, and allows “in the