Tamtam Proceedings - lamsin

552 AyadiFor all 0 ≤ p ≤ i, P p ([1, i]) denotes the set of all subsets, containing p elements chosenin the set {1, 2, ..., i}. It is convenient to recall that P p ([1, i]) is finite and its cardinal isequal to C p i . Setting N = i + j, we have the linear problem:Theorem 3.1 The pair (λ h , u ∗ h) is solution to the unidentified linear eigenvalue problem:λ h =min R(A h (u h ), B h (u h ))(v). (4)v∈R N−p(u h ) −{0}Remark 3.2 In order to solve problem (3), according to Theorem 3.1, we can solve afinite number of identified linear problems of type (4). This number is at most equal to2 i . This means, at least, that the convergence of the algorithm is guaranteed. Then, thereis a direct relation between the number of mesh nodes in the contact region ω c and thatof linear eigenvalue problems to be solved. Indeed, the smaller the former is, the smallerthe later will be. Moreover, the computing of these problems may be done in parallel ifwe dispose of parallel machines.3.2. The Practical Algorithmλ = ∞, q = 0For p = 0 to iFor m = 1 to C p iλ(p, m) = R(A h (p, m), B h (p, m))(w ∗ )If w ∈ K h then λ = min(λ, λ(p, m))If (λ = λ(p, m)) then q = wElse q = qElse λ = λ and q = qEndEnd4. Numerical ResultsIn order to test our algorithm and validate our numerical unilateral buckling model,we consider a unit length beam simply supported of its boundary and fixed of its middle.The obstacle is located just above the central part of the beam (see figure 1 below).TAMTAM –Tunis– 2005