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The boundary element method with programming for engineers and scientists - phần 9

K ma trận cho tất cả các vùng này sau đó được lắp ráp tại cùng một cách như với FEM. Phương pháp thứ hai là hiệu quả hơn và nhiều hơn nữa tuân theo thực hiện trên máy tính song song. | 396 The Boundary Element Method with Programming We discretise the total time into arbitrary small steps of size At then we have N VN u Q t YNn t un Q q Q t Y Nn t qn Q 14.34 n 1 n 1 where Nn t are shape functions in time and un and qn are the pressure and pressure gradient at time step n at time tn n t . If we assume the variation of u and q to be constant within one time step At then the convolution integrals may be evaluated analytically. In this case the shape functions are Nn t H t - t -1 -H t - tn 14.35 where H is the Heaviside function. The time interpolation is shown in Figure 14.7. Substituting 14.34 into 14.33 we obtain the integral equation discretised in time and written for the time tN time step N cuN P U P T Q tN q Q T - T P t Q tN u g.r dS S 14.36 The convolution integrals are approximated by N U P T Q tN q Q 0 qn Q Un 14.37 n-1 and N T P t Q tN uQ t Un Q - Nn n 1 14.38 where Un u P t Q tNd KTNn Jt P r Q tN dT 14.39 t -1 t -1 This means that only the fundamental solutions are inside the integrals and these may be integrated analytically3. The time discretised integral equation now becomes N N Cun P J EAUNn qn Q dS Q -J TNn u.n Q dS Q 14.40 S n 1 S n 1 DYNAMICS 397 or taking the sum outside the integral .Ar -Ar . cuN P JaUn qn Q dS Q - JaTn u Q dS Q 14.41 n 1 S n 1 S For each time step N we get an integral equation. In a well posed boundary value problem either u or q is specified on the boundary and the values of u and q are known at the beginning of the analysis t 0 . Furthermore the integral equation 11.41 must be satisfied for any source point P. If we ensure the satisfaction at a discrete number of points Pi then we can get for each time step N as many equations that are necessary to compute the unknowns. Similar to static problems we specify the points Pi to be the node points of the boundary element mesh point collocation . To solve the integral equation we introduce the discretisation in space of Chapter 3 J J u Q x Nj u q Q Y Nj qenj 14.42 j