The Preconditioning SRQCG Methods For Structural Modal Reanalysis

With the development of science and technology, many large-complicated structures need to be designed. In structural optimization, the solution is iterative and consists of repeated analyses followed by redesign steps. The high computational cost involved in repeated analyses of large-scale problems is one of the main obstacles in the solution process. In many problems the analysis part will require most of the computational effort, therefore only methods that do not involve numerous time-consuming implicit analyses might prove useful. Reanalysis methods, intended to reduce the computational cost, have been motivated by some typical difficulties involved in the solution process. The object of reanalysis is to evaluate the structural response for successive modifications efficiently and precisely without solving the set of the modified implicit equations so that the computational cost is significantly reduced.In this thesis, the Simultaneous Rayleigh quotient modified conjugate gradient method which combine the preconditioned technique and iterate method is developed for structural modal reanalysis problems.Suppose the stiffness matrix and mass matrix of initial design is and with dimension of . The frequencies and corresponding modesobtained from solution of the initial eigenproblemAssume a change in the design, the corresponding stiffness and mass matrices can be expressed The modified analysis equations to be solved areThe object is to evaluate the modified structural responseλi, using the initial information from without solving the set of the modified implicit equations so that the computational cost is significantly reduced.The SRQMCG2 scheme for structural modal reanalysis consists of the following steps:Step 1. Use initial stiffness matrix as preconditioning matrix, that is K0Step 2. Using the eigenvectors obtained from the modal analysis for initial structure as starting matrix X ( 0), a tolerance value TOLL, the allowed maximum number of iterations NITMAX and a"restart"value NREST. Compute the initial residual matrix as where is the diagonal matrix whose entries are the diagonal coefficients of the Rayleigh matrix , then compute the average relative residual (i teration index). ( k)As long as k is smaller than NITMAX and ra is greater than TOLL, execute Steps 3, 4, and 5; otherwise go to Step 6. 3.2.1. Programing a Ritz projection step, which is equivalent to solving the problem: Where y is a p-dimensional vector and . An orthogonal matrix is found such that Step 4. Evaluate the new matrix 4.1. The vector is evaluated by M-orthogonalizing p(j k) with respect to 4.2. The coefficientsαj are obtained by minimizing the Rayleigh quotient is evaluated by M-orthonormalizing xj with respect to Vj. Step 5. Compute the residual matrix M r X together with the value ra( k+1), Increment the iteration counter k and go to the loop start.Step 6. If ra ( k) is smaller than TOLL, R x j and x(j k), ( j =1,,p),are the smallest p eigenvalues of (3), and the corresponding eigenvectors, respectively. Here, ra ( k) is the average relative residual, it is defined as follows:Consider now the preconditioning matrix . In mathematics, two preconditioning matrices can be chosen. The cheapest selection for is provided by where D is the diagonal matrix formed by the diagonal entries of K ,named SRQMCG2(D). Another choice iswhere L being the pointwise incomplete Cholesky factor of K, named SRQMCG2(LLT).Numerical examples shows that, compared with the algorithms SRQMCG2(D) and SRQMGC2(LLT), the proposed algorithm SRQMCG2(K0) is an efficient method for solving structural modal reanalysis.