مرزهای دقیق برای تجزیه و تحلیل حساسیت سازه با پارامترهای نامشخص ولی محدود شده

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.

The purpose of sensitivity analysis is to work out the structure response or the variety of performance through the transformation of parameters or designing variables [1]: equation(1) u=u(b1,b2,…,bn),u=u(b1,b2,…,bn), Turn MathJax on where b1,b2,…,bnb1,b2,…,bn are structure parameters or designing variables. Thus, via partial differential operation and bring b0=(b10,b20,…,bn0)Tb0=(b10,b20,…,bn0)T into Eq. (1), we have equation(2) View the MathML source∂u∂bb0=∂u(b10,b20,…,bn0)∂b. Turn MathJax on The absolute value of Eq. (2) denotes the sensitivity degree of structure responses or capability to structure parameters. On condition that View the MathML source∂u∂bb0>0 the structure response or performance u is monotone increased around the parameter b0=(b10,b20,…,bn0)Tb0=(b10,b20,…,bn0)T. If View the MathML source∂u∂bb0<0 the structure response or performance u is monotone degressive around the parameter b0=(b10,b20,…,bn0)Tb0=(b10,b20,…,bn0)T. Bear in mind Eq. (2), we also gain the transformation of the structure response or performance as equation(3) View the MathML sourceδu=∂u∂bδb. Turn MathJax on In engineering practice, very often difference operation methods are used instead of differential operation methods, but the results are often unreliable. However, the above-mentioned normal sensitivity analysis method has many problems: Case I. The mathematical foundation of the normal sensitivity analysis method is the differential calculus of real analysis. In terms of differential calculus principle, based on partial differential, the sensitivity analysis result is only local information. Namely, in this local bound the structure response or performance is most sensitive to this parameter; but in another local bound, the structure response or performance is likely least sensitive to this parameter. In the same way, the structure response or performance is monotone increased to this parameter in this local bound; but in another local bound, the structure response or performance is likely monotone degressive to this parameter. However, in practice analysis and designing, people were concerned with the sensitivity information of the structure response or performance in a certain large bound. The sensitivity analysis which based on differential calculus cannot satisfy the requirement of such global information. Although we can process normal sensitivity analysis many times and receive the sensitivity information in a certain large bound, the efficiency of the calculation will decrease seriously. Case II. For most practical engineering, it is impossible to present the parse expressions of structure response or performance in virtue of complexity and legion dimensions. So, usually we use difference or perturbation analysis instead of differential calculus. However, in engineering practice, the variety of parameters or designing variables oversize or undersize will all impact the precision of the sensitivity analysis and present complete incorrect information. As shown in Fig. 1, we have equation(4) View the MathML sourceδu1δb1=u(b1)-u(b0)b1-b0=u1-u0b1-b0>0 Turn MathJax on and equation(5) View the MathML sourceδu2δb2=u(b2)-u(b0)b2-b0=u2-u0b2-b0<0. Turn MathJax on It presents an opposite sensitivity information. So, sometimes the results of the difference sensitivity analysis method are without confidence to the distinct nonlinearity structures. Full-size image (7 K) Fig. 1. Distinct nonlinearity function. Figure options Case III . Having found the results of the sensitivity analysis, how can we work out the variation of structure responses or performances through the variation of parameters or designing variables? The normal sensitivity analysis calculate the variation by Eq. (3), but it is only present the local information and δb should not be too large, otherwise the result will be without confidence. For instance, as shown in Fig. 1, we process a calculation according to δu=(∂u/∂b)δbδu=(∂u/∂b)δb as following: equation(6) View the MathML sourceu1=u0+δu1δb1δb1. Turn MathJax on From Eq. (6) we get that u 1 is increased when compared with u 0. Also, we process another calculation according to δu=(∂u/∂b)δbδu=(∂u/∂b)δb as following: equation(7) View the MathML sourceu2=u0+δu2δb2δb2. Turn MathJax on From Eq. (7) we work out that u2 is decreased when compared with u0. Case IV. In the difference calculation of sensitivity, the result that calculated by δu is the distance u(b)-u(b0)u(b)-u(b0) of structure responses or parameters between b and b 0 rather than the distance umax-uminumax-umin of structure responses or parameters between b and b 0. Where u(b)≠umaxu(b)≠umax and u(b0)≠uminu(b0)≠umin will be correct with distinct nonlinearity problem. Therefore, in engineering practice the variation of parameters or design variables δb=b-b0δb=b-b0 oversize or undersize will all impact the precision of sensitivity analysis and present complete incorrect information indeed. Case V. If the structure parameters in the expressions of structure responses, performances or mathematical calculations are uncertain, especially in the large-scale structure, the result of the sensitivity analysis is often without confidence due to the cumulation of the uncertainty. If we use interval mathematics to process sensitivity analysis, the above problem of normal sensitivity analysis will be completely solved.

In this paper, considering the properties of the stiffness matrix and the mass matrix in structural engineering, making use of the structural parameters and the non-negative decomposition of a matrix, we present a new method to determine the lower and upper bounds on the sensitivity due to uncertain-but-bounded parameters for the interval sensitivity analysis problem. Without any partial differential operations, the interval analysis method obtained the information on the response of the structure with the structural parameters (or design variables) changing. The effectiveness and correctness of the algorithm was illustrated by two numerical examples. For large-scale structures with interval parameters, a fast computing technique for obtaining the approximate sensitivity and the corresponding errors is desirable.