Onnonlineardynamicanalysis Usingsubstructuringandmode Superposition

The solution of nonlinear dynamic equilibrium equations using mode superposition and substructuring is studied. The objective is to design schemes that in some analyses can significantly decrease the computational effort involved when compared to a complete direct integration solution. Specific schemes for mode superposition analysis and substructuring are proposed. These techniques have been implemented in ADINA. The results of a few sample analyses are presented and recommendations are given on the use of these procedures in practical analysis. INTRODUC’I’ION The analysis of the nonlinear dynamic response of large finite element systems has become of much interest during recent years. The problems to be considered can quite generally be divided into wave propagation problems and structural vibration problems. Both types of problems are solved using an incremental step-by-step solution of the governing equations of motion. In this paper we consider the solution of structural vibration problems only. These problems can be solved effectively using an implicit time integration scheme with a. modified Newton iteration in each time step. The solution requires that for each of the time steps At, 2At, 3At,. . . , nAt, where n is the total number of time steps considered, the incremental dynamic equilibrium equations be established and then solved using equilibrium iterations. The evaluation of the incremental equilibrium equations involves as a significant computational expense the calculation and the triangular factorization of the effective tangent stiffness matrix, and the equilibrium iterations require the calculation of out-of-balance nodal point force vectors and the forward reduction and backsubstitution of these vectors, until the corrections to the incremental quantities are sufficiently small. The complete solution process is summarized, for example in Table 1 of Refs. [1,2]. Since a large part of the cost in this incremental analysis lies in the calculation of the tangent stiffness matrices and the equilibrium iterations, it is desirable that, without loss of solution stability and accuracy, a new tangent stiffness matrix is only very infrequently formed and equilibrium iterations be only performed when really necessary. Mathematical analysis and practical experience show that because of stability and accuracy considerations, equilibrium iterations are best performed in each time step. However, depending on the problem considered, a new tangent stiffness matrix need not be calculated in each time step, and indeed in many analyses, the original stiffness matrix can be employed throughout the complete response calculation. Also, in many nonlinear analyses, we only deal with local nonlinearities. The objective in this paper is to consider the nonlinear dynamic analysis of finite element systems for which the original stiffness matrix can be employed throughout the incremental solution, or which only contain local nonlinearities, and demonstrate how the response can be calculated effectively using the basic principles of mode superposition and substructuring. The possible use of mode superposition in nonlinear analysis is quite natural, because mathematically only a change of basis to a computationally more effective system of equations is performed [ I, 31. The method of mode superposition has already been applied in nonlinear analysisj4-61, but the effectiveness of the schemes employed in actual practical analyses is still questionable. The possible use of substructuring on the linear degrees of freedom is equally natural, if we recognize that the solution of the incremental equations involves an “effective stiffness matrix” on which we operate as in static ana!ysis[lJ]. However, there are some important questions with regard to the actual implementation of a substructuring scheme, and these must be addressed in detail in order to arrive at an effective solution strategy. In the paper we first present the genera! equations that we employ in an incremental nonlinear dynamic anhlysis. We then discuss how we use mode superposition and substructuring procedures to reduce the computational effort of solution. The proposed techniques are only effective in the solution of certain nonlinear systems, but then they may yield a significantly more effective solution than the use of direct integration of the complete system of equations. The techniques are implemented in ADINA, and we present the results of some sample analyses using the methods. Finally, we give specific recommendations on the practical usage of the techniques. 2. INCREMENTAL EQCJATtONS OF EQUILIBRtUM The incremental nodal point equilibrium equations for an assemblage of nonlinear finite elements considered here have been discussed in[l, 31. Using an implicit time integration scheme and the modified Newton iteration to establish dynamic equilibrium at time t + At, the governing finite element equations are M I+A1@ + c t+Ar&l) + ‘K d"(i) where =f+A*R-f+ArF(i-l)/i= 1,2,3,_ ] (1) I+Atu(i) = I+Al”Ct--1) + AU”’ (2) and M = constant mass matrix, C = constant damping matrix, ‘K = tangent stiffness matrix at time t, ‘+“R = external nodal point load vector due to body forces, surface loads and concentrated loads, ‘+A’F”-‘) = nodal point force vector equivalent to the element stresses that