Computation of Two-Dimensional Disturbances in an Elastic Body
A.A. Aganin* , Khismatullina N.A.**
Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, 420111 Russia
E-mail: *aganin@kfti.knc.ru, **nailya hism@mail.ru
Received March 15, 2017
Abstract
The classical Godunov method is widely used for the numerical study of waves in continuous media. If the Courant condition is satisfied, the Godunov scheme is stable and monotonous. However, due to its first order of accuracy it can lead to rather large smearing of jumps, contact discontinuities, and other features of the solution in the domains where the solution gradients are great.
In this paper, the possibility of increasing the efficiency of computation of linear waves in an elastic body by applying one of the second-order accurate UNO modifications of the classical Godunov method has been studied. In that UNO modification, the monotonicity requirement is replaced by the TVD condition, and all the parameters inside each grid cell are assumed linear rather than constant as they are in the Godunov method. The TVD condition is met just on the level of approximation. To derive the second order of accuracy in time, the time derivatives of the unknown functions have been expressed in terms of their spatial derivatives. Those expressions allow to calculate the next half-time-layer values of the unknown functions at the center of the grid cells and on both sides of the cell boundaries. The values of the unknown functions on the boundaries themselves have been found by solving the corresponding Riemann problems. Subsequently, numerical flows across the cell boundaries have been computed. Computation of the values at the next time layer has been carried out by an explicit scheme. Some limiters for the values of the spatial derivatives have been introduced. The present UNO scheme limiters use approximations to both the first and second derivatives.
The efficiency of the proposed UNO modification has been estimated by computing a number of one- and two-dimensional problems on propagation of linear waves in an elastic body and their interaction with each other and with the surface of the body. The results of those computations have been compared with the exact solutions and the results of applying the Godunov method. It has been shown that the UNO scheme considered allows one to reduce computational costs by more than a factor of ten.
Keywords: UNO scheme, Godunov scheme, efficiency of difference schemes, linear wave, elastic body
Figure Captions
Fig. 1. Distributions of the uniform pressure P in the body at two moments of time t1 and t2 upon changes in the velocity (a) and pressure (b) at the body surface y = 0 following the laws on the insertions. Black lines - analytical solution, red lines - UNO scheme, green lines - classical Godunov method.
Fig. 2. The interaction of the wave with the rigid wall and with the wave, which is formed at this surface: (a) - initial distribution of the uniform pressure P in the body and the velocity defect law v on the rigid wall; (b); (c) - distributions of P at the moments of time t1 and t2 , respectively, t2 > t1 > 0. Arrows show the direction in which the waves move. The curves are designated in the same way as in Fig. 1.
Fig. 3. The interaction of the wave with the mobile rigid wall and the wave which is formed in this wall: (a) -
at the body surface; (b); (c) - distributions u at the moments of time t1 and t2 , respectively, t2 > t1 > 0. Arrows show the direction in which the waves move. The curves are designated in the same way as in Fig. 1.
Fig. 5. The dynamics of impulses inside the body and their interaction with the non-mobile rigid wall: initial distributions of the velocities u and v in the body (a) and their distributions at one of the moments before their interaction with the wall (b) and at one of the moments after the onset of interaction with (c) . Arrows show the direction in which the waves move. The curves are designated in the same way as in Fig. 1.
Fig. 6. Impact on the part of the body surface (the area of the impact is colored in black): (a) - isolines σi , computation on the grid 100 × 100 by the Godunov (black curves) and UNO (red curves) methods; (b) - isolines σi, computed by the Godunov method on the grid 800 × 800 (black curves) and by the UNO method on the grid 100 × 100 (red curves); (c) - distributions σi along the symmetry plane x = 0 calculated by the UNO method on the grids 100 × 100, 200 × 200, 400 × 400,
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For citation: Aganin A.A., Khismatullina N.A. Computation of two-dimensional disturbances in an elastic body. Uchenye Zapiski Kazanskogo Universiteta. Seriya FizikoMatematicheskie Nauki, 2017, vol. 159, no. 2, pp. 143–160. (In Russian)
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