Aikido Shugyo: Harmony in Confrontation by Gozo Shioda, Jacques Payet, Christopher Johnston, Kyoichi

By Gozo Shioda, Jacques Payet, Christopher Johnston, Kyoichi Inoue

Trans. Jacques Payet, Christopher Johnston

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The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. Roy. Soc. London, Ser. A, Vol. 323, pp. 201-210, 1971. ; Numerical solution of scalar diffraction problems, Numerical Solution of In­ tegral Equations, Ch. 21, Eds. M. , Clarendon Press, Oxford, pp. 275-290, 1974. ; Solutions for sound-radiation problems by integral equations at the critical wavenumbers,/. Acoust. Soc. , Vol. 47, pp. 387-388, 1970. ; Integral equation method for radiation from vibrating bodies, /.

7) is easily verified in the following way. Let ί b e a solution of the adjoint homogeneous equation of eq. 2. Thus there is a solution in eq. L problem. Furthermore, this solution of eq. 7) is also the solution of eq. 8). Let U\ be any solution of eq. 7). Then, Green's interior displacement formula with this Ui and the given boundary traction g as densities has the following form v(X) = (Sg)(X)-(DUl)(X) leö_. 10) Interior limit of this expression yields v(x) = (Sg)(x)+[[-jI-K*)ul](x) = 0 x^fl, where eq.

6 should be used for the exterior problem. Also, as shown in Fig. 1, displacements ( f) and tractions (g) are given on dDi and dD2 respectively. For example, we formulate the integral equations for the third interior problem with the help of the method ii). On JC G dD\, from the interior displacement formula, we have (St)dDl-(K*u)dD2=(±f + K*f)dDl-(Sg)dD2. 1) On x e dD2, from the interior traction formula, we have (Kt)dDl-(Dnu)dD2 = (Dnf)dDl+(±g-Kg)dD2. 2) In the above expressions, (-)aDl and (· )aD% stand for the integration on dDi and dD2 , respectively.

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