Research On Automatic Differentiation Method Of Stochastic Finite Element

The reliability and safe of mechanical structures is one of the major objectives of mechanical structures design. Parameter uncertainty of mechanical structures is inherent in most engineering problems, and loads that are applied to the mechanical structures are same random. Thus, the random structure systems with random parameters subjected to random loads are appeared. The research on reliability of the uncertain systems is very important for design purposes. The reliability analysis of mechanical structures can help the designer to establish acceptable tolerances on mechanical structures and govern the fluctuations of the system parameters for safe operations. In the end, the reasonable mechanical structures could be designed with enough reliability and little cost.The common method to study reliability of rendom mechanical structure is Stochastic finite element which calculates partial derivative with critical condition function respects to rendom structure parameters. Only few not complex question which can get analytic result is calculated the derivative manually. Most of industrial programs can't get analytic result but the derivative is replaced by limited differential. Numerical defference mothod with disadvantages of big tolerance of cut off and lower precise of compute result will have very big differences in further calculation which is to bring on very inaccurate result. So, the numerical difference method limits enormously the application of stochastic finite element to put into practice. Automitic differentiation is a kind of pragrame to calculate derivative of function by compiling system working out automatically according to the programe of calculating function. It doesn't have cutting off errors and avoid tedious analytic deriving to derivative. Its accuracy of the result derivative is the same as the analytic derivative that can be match to the accuracy of computer floating decimal. Automitic diffenrentiation supply a kind of forceful tool to calculate partial derivative of stochastic response.This work proposes automatic differentiation method with structure analysis of stochastic finite element. It applys to calculate reliability of complex mechanical structure. It is including as follow:Charpter 1: Introduction.Introduce the commmon method to calculate reliability of stochastic mechanical structure. Review the history of stochastic finite element and its application in industy. Introduce the status of automatic differentiation. Clearify the mechnical background of this work study. Define the direction and target as well as meaning of project.Charpter 2: Mathematics foundation.Introduce briefly all mentioned before mathematic theory and mothod in this work study. These theories which including in Kronecker algebra theory, technology of second moment and fourth moment, Edgeworth theory and its application are strong methematic foundation of this work study. Then introduce to calculate the function of high order derivative.Charpter 3: Basic theory of automatic differentiation and achieve to calculate it by programe.Introduce basic theory of automatic differentiation. Introduce that how to derive derivative of monadic function and derivative of multi-monadic function by Taylor parameter formula. Then developed one mould MADF for calculating automatic differentiation. This mould defined a new data type ADF which is for stocking function and partial derivative of function then reloading common calculating code to this new type. Example showed calculating process of automatic differentiation.Charpter 4: Automatic differentiation method of stochastic finite element to analysize structure random response.Introduce the theory and formula of current and common method of stochastic finite element. Then developed Automatic differentiation method of stochastic finite element based on scatter theory of stochastic space and stochastic perturbation finite element and combine with the technology of automatic defferentiation. Described the process of its application. One plane frame as an example to show that how to calculate all bars stochastic responses by using automatic differentiation method of stochastic finite element under which the length of bar, material character E and the load are random various.Charpter 5: Automatic differentiation method of stochastic finite element on analysing reliability.Introduce common strenth theory in industry and its limited status function. Then introduced theory to calculate reliability separately under which the limited status function is in distribution and in non-distribution. In final, described the process to calculate the reliability by automatic differentiation method of stochastic finite element method. Calculated the reliability of a plane frame as example.Charpter 6: Application on Automatic differentiation method of stochastic finite element.Calculated and analysized reliability of long cylindrical high pressure container by using automatic differentiation method of stochastic finite element. Worked out the failure occurrence and reliability in two kinds of failure modes. Then compared the result to the Monte-Carlo analysized result. The analysized result on long cylindrical high pressure container by using automatic differentiation method of stochastic finite element match well with Monte-Carlo analysized result. So, the conclusion is that automatic differentiation method of stochastic finite element not only precise but also be applicated easily.Charpter 7: Conclusion and expectation.Sum-up all conclusions in this work study. Expecte to apply automatic differentiation method of stochastic finite element in industry and other fields in order to optimize calculating, imitate numerous data, analysize sensibility to system, work out result by differentiation formula and so on.