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Partial differential Equations, Numerics and Control

Asymmetric propagation of a wave

Vincent Lescarret

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In mechanical engineering (see for example [1]), an important issue is to construct multi-structured materials such as nets of beams, glass or walls connected to each other through links insuring the continuity of displacement of the whole structure.
We consider here such a structure with elastic property. In realistic situations, the connections/interfaces between the different elements of the structure often have a higher mass density than in the elements themselves. So, one should also take the interface as an element of the structure (of lower dimension).

So, the easiest structure which has the features just described is made of two n-d (n=1,2,3...) vibrating elements connected through a (n-1)-d weighted interface. The corresponding mathematical model is made of three equations: two n-d wave equations describing the evolution of the displacement of the n-d elements and a (n-1)-d wave equation describing the motion of the interface. This last equation is coupled with the other two.

In [2], H. Koch and E. Zuazua have studied the property of regularity of such a system and have got that it is well posed in asymmetric Sobolev spaces. More precisely, the solution of the system on each n-d element is in a Sobolev space of the considered element and the degree of regularity differs by one between the two spaces as long as it is true at the initial time.

This result depends on the dimension of the physical setting. In 1-d, one already knows from Hansen-Zuazua [3] that the problem is well-posed in asymmetric spaces. In 2-d (or higher dimension), the main result of Koch-Zuazua expresses that such a result holds if the speed of propagation on the interface is slower than that in the medium where the initial data are less regular.

In the present work, we have considered the space semi-discretization of the continuous mathematical model in view of performing numerical computations. In a first part we have shown that the translation of Koch-Zuazua’s result in this discrete setting applies. More precisely, we have shown the existence of solution in some discrete asymmetric space for the semi-discrete finite difference and semi-discrete mixed finite element discretization.

Then, we have highlighted those results by some numerical computations on the full finite difference discretization of the mathematical model for which the previous results also apply. In the video from above there is a simulation in 2D with a fast interface speed.

[1] J. Lagnese, G. Leugering, E.J.P.G. Schmidt, “Modelling, Analysis and Control of Multi-Link Flexible Structures”, Basel, Birkhuser, 1994, 398 pages
[2] H. Koch and E. Zuazua, “A hybrid system of PDE’s arising in multi-structure interaction: coupling of wave equations in n and n-1 space dimensions”. Recent trends in partial differential equations, Contemp. Math., vol. 409 (2006), pp. 55–77.
[3] S. Hansen and E. Zuazua. “Controllability and stabilization of strings with point masses”. SIAM J. Cont. Optim., vol. 33 (5) (1995), pp. 1357–1391.

This code allows to obtain these results:

[Code.zip] (10 KB)

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Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del PaÃŒs Vasco (UPV/EHU)