A fundamental mathematical modeling and computational tool in engineering and applied sciences is the finite element method (FEM). The formulation of every such problem begins with the building of a mathematical model with carefully chosen simplifying assumptions that allow an engineer or scientist to obtain an approximate FEM-based solution without sacrificing the essential features of the physics of the problem. Consequently, one needs to deal with many types of uncertainty inherent in such an approximate solution process. Using a public-domain statistical analysis software package named DATAPLOT, we present a metrological approach to the verification and uncertainty estimation of FEM-based simulations by treating each simulation as a "virtual experiment." Similar to a physical experiment, the uncertainty of a virtual experiment is addressed by accounting for its physical (modeling), mathematical (discretization), and computational (implementation) errors through the use of a rigorous statistical method known as the design of experiments (DOE). An introduction of the methodology is presented in the form of five specific topics: (a) the fundamentals of DOE, (b) the assumptions of model building, (c) setting objectives for an experiment, (d) selecting process input variables (factors) and output responses, and (e) weighing the objectives of the virtual experiment versus the number of factors identified in order to arrive at a choice of an experimental design. The method is then specialized for FEM applications by choosing a specific objective and a subclass of experimental designs known as the fractional factorial design. Two examples of this FEM-specific approach are included: (1) The free vibration of an isotropic elastic cantilever beam with a known theoretical solution, and (2) The calculation of the first resonance frequency of the elastic bending of a single-crystal silicon cantilever beam without known solutions. In each example, the FEM-simulated result is accompanied by a 95% confidence interval. Significance and limitations of this metrological approach to advancing FEM as a precision simulation tool for improving engineering design appear at the end of this paper.
Proceedings Title: Proceedings of the 2008 Pittsburgh Conference of the American Society of Engineering Education
Conference Dates: June 22-25, 2008
Conference Location: Pittsburgh, PA
Conference Title: 2008 Pittsburgh Conference of the American Society of Engineering Education
Pub Type: Conferences
anisotropic elasticity, design of experiments, engineering statistics, finite element method, mathematical modeling, metrology, silicon cantilever beam, simulation, statistical data analysis, uncertainty estimation, vibration engineering, virtual measurement systems