Nondeterministic Linear Static Finite Element Analysis: An Interval Approach

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Zhang, Hao
Muhanna, Rafi L.
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This thesis presents a nontraditional treatment for uncertainties in the material, geometry, and load parameters in linear static finite element analysis (FEA) for mechanics problems. Uncertainties are introduced as bounded possible values (intervals). FEA with interval parameters (interval FEA, IFEA) calculates the bounds on the system response based on the ranges of the system parameters. The obtained results should be accurate and efficiently computed. Toward this end, a rigorous interval FEA is developed and implemented. In this study, interval arithmetic is used in the formulation to guarantee an enclosure for the response range. The main difficulty associated with interval computation is the dependence problem, which results in severe overestimation of the system response ranges. Particular attention in the development of the present method is given to control the dependence problem for sharp results. The developed method is based on an Element-By-Element (EBE) technique. By using the EBE technique, the interval parameters can be handled more efficiently to control the dependence problem. The penalty method and Lagrange multiplier method are used to impose the necessary constraints for compatibility and equilibrium. The resulting structure equations are a system of parametric linear interval equations. The standard fixed point iteration is modified, enhanced, and used to solve the interval equations accurately and efficiently. The newly developed dependence control algorithm ensures the convergence of the fixed point iteration even for problems with relatively large uncertainties. Further, special algorithms have been developed to calculate sharp results for stress and element nodal force. The present method is generally applicable to linear static interval FEA, regardless of element type. Numerical examples are presented to demonstrate the capabilities of the developed method. It is illustrated that the present method yields rigorous and accurate results which are guaranteed to enclose the true response ranges in all the problems considered, including those with a large number of interval variables (e.g., more than 250). The scalability of the present method is also illustrated. In addition to its accuracy, rigorousness and scalability, the efficiency of the present method is also significantly superior to conventional methods such as the combinatorial, the sensitivity analysis, and the Monte Carlo sampling method.
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