| The inverse problem in fluid mechanics, because of its wide application, such as en-gineering, medical, military and so on, made people do a lot of research on it.There are inverse problem of parameter control, inverse problem of source item control, bound-ary condition control inverse problems, initial condition control inverse problems and inverse problem of shape control in the inverse problem in fluid mechanics[19].We re-searched about the measurement of obstacles in the ideal fluid in this article.And the reason of people interesting in this problem is from the submarine campaign. Water is a liquid, its compressibility is smaller.So it can be regarded as ideal fluid.Obviously, sub-marine detection is a practical and full of meaning problems.This paper mainly dis-cussed the mathematical problems about measuring in the ideal fluid.Because of the in-teresting of submarine detection in underwater, I want to try to find the methods through the measurement of fluid nearby the obstacle to detect the moving obstacle in fluid, and then began to study the relationship between the flow field conditions in somewhere in the ideal and the moving obstacle in fluid.Through reference, the equations of the system solid and fluid as follows: div v=0,(x, t)∈(Ω\S (t))×R (13) v·n=(h+r(x-h))·n,(x, t)∈dS(t)×R (14) u·n=g,(x, t)∈dΩ×R (15)In these equations, v=v(x, t), P=P{x, t) is the velocity and the pressure of the fluid, g is the flow through the boundary Ω, r is the angular velocity of the solid, n is the outward unit normal vector, f(t), T(t) stands for the external force and external torque applied to the solid. J is moment of inertia.Because our problem is to get rid of the boundary, we want to plus infinity conditions.Through the deliberate physical meaning and mathematical assumptions simplify the equations as follows:For the direct problem, we want to get u by the given(h,l).But according to the expression of the Laplace equationwe known, value u of the boundary on B is unknown, so we firstly got value u of the boundary on B, then got u interior.Firstly, change (7)-(9) to the boundary integral equations as follows:Then through the method using slicing constant generating space, picking up the collocation method to determine the parameters, we can get the system (2.25).The so-lution of this equations is the value u of the boundary on B.Pluging this into the (10), we also use the method which use slicing constant generating space, and pick up the collo-cation method to determine the parameters, then getting the system (2.30). Getting the solution is the solution of the direct problem. Through the numerical experiments, obvi-ously, despite there is error by this numerical method, there is still the correct property.The inverse problem has the same precondition with the direct, but the different given condition.Here we want to get the data (h, l) that can describe the moving ob-stacle, though:u=φonΓm, Γm is near obstacles in fluid internal, the equations as follows: According to the above equations, the calculation is the same with the direct prob-lem.First, we get the boundary integral equation, and pick up the collocation method to get the equations.Then, according to the iterative method, we got the expression of solution (3.4), the solution can be got by the Landweber iteration on MATLAB.In a word, there are correct points in both direct and inverse problems, but it still needs for improving and further test in future. |
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