FAST ALGORITHMS FOR DETERMINING POSITIVE DEFINITENESS OF LARGE SCALE MATRICES IN SOLID STRUCTURE ANALYSIS

In order to determine the positive definiteness of the super-large-scale matrix in the structural stability analysis, the parallel computing method must be adopted. However, the traditional direct methods such as the eigenvalue’s analysis, the master subordinate determinant’s computation and LDLT decomposition are difficult to realize in parallel computing. In this paper, some iterative algorithms which are suitable for parallel computing are proposed. A new approach is developed that determining the positive definiteness of a matrix can be transformed into an optimization problem, which is solved by various optimization algorithms. The algorithms based on the steepest descent method and the conjugate gradient method are proposed. Considering the sparseness of the stiffness matrix of the mechanical system and the weakly non-positive definite property of at the critical buckling state, we propose a method via calculating the stationary point on a cutting plane to determine the non-positive definite matrices. In order to ensure the accuracy of the determination of strong non-positive definite matrices, a hybrid method combing the cutting plane method and the conjugate gradient method is developed. The numerical results show that the proposed algorithms are accurate and efficient. The conjugate gradient algorithm is more efficient for strongly non-positive definite matrices while the hybrid method is more efficient for the weak non-positive definite matrices. These algorithms are easy to realize in parallel computing on the cluster and can quickly determine the positive definiteness of large-scale matrices.