Topology Optimization Of Continuum Structures Considering Uncertainties In Loading Magnitude And Direction

Topology optimization has become an important method of maximum stiffness of the structure design which makes the compliance as the objective function. External factors such as the loading uncertainties always have an important impact for the optimal configuration of structure. Topology optimization needs to include the design goal and the constraint of the statistical indicators such as mean and variance. Accurate and fast calculation of the mean and variance of the uncertainties of structural topology optimization have become a difficulty in the design and an effective analysis method needs to be learn. Robust design has become an important method for configuration design under loading uncertainty. Robust design of structure requires average (mean) optimal and also the uncertain parameters of structure properties (standard deviation) could not be too large. Therefore, how to optimize these parameters considering the interaction model, how to define a suitable description of the index is another important issue. Robust design at present, uses the mean and standard deviation to describe the structure of the robustness, and their weighted forms is a good idea, but the combined weight coefficients of different structure will make different results. Therefore, it is important to study the rules of weight coefficient and how to put forward appropriate robustness indices. According to this requirement, this paper study the analytical method to calculate the mean and variance of structural uncertainty and the weighted forms of the robustness of the rules in describing the weight coefficient, and then proposes a reasonable description, thus a robust topology optimization model can be established.The research work of this paper are as follows:(1) Analytical formula of the compliance expectation and variance considering the loading magnitude and direction uncertainties. When the loading magnitude and direction are all obey the normal distribution assumption, the analytical method for solving the expectation and variance of compliance with uncertain loads have been studied, the structure compliance is expressed as the form of reciprocal compliance and the sum of loading magnitude and direction. We make the calculation of mean and variance into the mean value problems of trigonometric function. By solving the Gauss integral with the trigonometric function, a series of those load uncertainties’ description called "mean function" are proposed, and then we give the accurate formulas of expectation and variance for compliance. Numerical examples demonstrate the accuracy of the method. The analytical formula are used to get the configuration design of typical examples with the robust topology optimization which makes compliance expectations as the goal. The design results show that the magnitude and direction of considering load uncertainties have a significant impact on the result of topology optimization, also verified the correctness and effectiveness of the analytical formulas for the topological optimization model.(2) The robustness descriptive index and robust topology optimization model. Robust design requires both the average performance of structure and the uncertain parameters’ balances. In order to describe the robust requirements and structure optimization model, we study the objective function of the weighted form of structural compliance expectations and standard deviations, find the rules of the weighted form, then a new coefficient is proposed, called "coefficient of variation for compliance". The index has a clear physical meaning and it is more reasonable. Secondly, we study the robustness of topology optimization model. based on the minimum coefficient of variation. Based on the load of uncertain magnitude and direction, we have compared some numerical examples to verify the correctness of the robustness topology optimization model in this paper, and the results show that it can effectively reduce the coefficient of variation for compliance and also improve the structural robustness.(3) Minimization expected compliance design based on the stochastic finite element method. In this paper, we make the method which based on the stochastic finite element method of Taylor expansion into the topology optimization, considering load uncertainties, and derive different analytical formulas compared with the previous research. The corresponding numerical examples verify the validity of this method, and we make the basic research in order to better use the stochastic finite element method into topology optimization in the future.