Interval Perturbation Finite Element Method For The Response Analysis Of The Acoustic Filed With Uncertain Parameters

In order to control the noise of the products in its development process, theacoustic behavior of the products has to be predicted previously. In the traditionalacoustic field prediction, the geometry, materials and environment related to thenumerical models are treated as deterministic factors, and the response of acousticfield are evaluated latter by using numerical analysis techniques such as finiteelement method. However, for most practical engineering cases, uncertainties ingeometric dimensions, applied loads, density, speed and other parameters of theacoustic field are unavoidable. These uncertainties are very small values in most cases,but the coupling of all uncertain parameters may lead to large uncertain levels of theresponse. To analyze the response of system more accurately, uncertain analysismethods need to be introduced. The main approaches to solve uncertain problemsinclude probabilistic methods, fuzzy methods and interval methods. The intervalanalysis method only needs the upper and lower bound information of uncertainparameters when it is used for uncertain problems. Thus, the interval analysis isperfectly appropriate for the numerical analysis of non-deterministic models withuncertain parameters whose information about the probability density function orfuzzy membership is missing. The interval analysis method has been becoming animportant complementary method of probability method.Up to now, the application of interval approaches in the acoustic domain isrelatively few. This thesis makes an intensive and systemic research of the acousticalfirst-order interval perturbation finite element method and the modified first-orderinterval perturbation finite element method. To further improve the computationalefficiency and accuracy of these existing acoustical interval methods, new approachesfor reducing the truncation error of the perturbation expansion of system response areproposed.The main research work and innovative achievements in this thesis are as follows(1) Aiming at the problem that the accuracy of the first-order intervalperturbation finite element method(FIPFEM) is not satisfactory when it is used for theresponse analysis of the acoustic filed, the acoustical second-order intervalperturbation finite element method(SIPFEM) is proposed in this thesis based on thesecond-order Taylor series expansion and the acoustic finite element method. In the acoustical SIPFEM, the non-deterministic sound pressure vector of the acoustic filedwith interval parameters is expanded to the second order Taylor series. The upper andlower bounds of the sound pressure response are evaluated latter in the inner feasibledomain of the interval parameters based on the extreme value theorem. Numericalresults on a2D acoustic tube with interval parameters show that the computationalefficiency of SIPFEM is slightly lower than that of FIPFEM; compared with theimprovement in accuracy, the additional computational burden of SIPFEM isacceptable. Therefore, SIPFEM is more suitable to the uncertain acoustic fieldprediction with small uncertain-but-bounded parameters.(2) In order to reduce the large computational cost for executing matrix inversesin uncertain acoustic analysis, an efficient method using the epsilon-algorithm forresponse analysis of the uncertain acoustic field is presented. In the epsilon-algorithmprocedures, the initial iterative matrix sequence of the inverse perturbation matrix isgenerated based on the Neumann series expansion. Numerical results on a2D acoustictube with uncertain parameters verify that the epsilon-algorithm achieves desirableaccuracy and can be applied to evaluate the inverse of the perturbation matrix withlarge uncertainty levels.(3) To further improve the computational accuracy and efficiency of the modifiedinterval perturbation finite element method (MIPFEM), a decomposed interval matrixperturbation finite element method (DIMPFEM) is proposed. In the proposed method,the dynamic stiffness matrix is decomposed into a sum of several sub-matrixes whoseperturbation matrix can be decomposed to the products of perturbation factors anddetermine matrices. To achieve a higher computational efficiency, the inverseperturbation matrix, approximated by the modified Neumann series expansion, iscalculated by the epsilon-algorithm. Numerical examples on a2D acoustic tube and a2D acoustic cavity of a MPV with interval parameters verify that the computationalaccuracy and efficiency of DIMPFEM are higher than those of MIPFEM.A detail study for the acoustic interval perturbation finite element method isimplemented in this thesis. Some acoustic numerical methods, such as the acousticalsecond-order interval perturbation finite element method, response analysis of theuncertain acoustic field based on the epsilon-algorithm and decomposed intervalmatrix perturbation finite element method, are proposed for the acoustic problems.The research findings can be applied to solving the uncertain acoustic problems andhas more engineering application foreground.