Finite Element Method Solution Uncertainty, Asymptotic Solution, and a New Approach to Accuracy Assessment

Errors and uncertainties in finite element method (FEM) computing can come from the following eight sources, the first four being FEM-method-specific, and the second four, model-specific: (1) Computing platform such as ABAQUS, ANSYS, COMSOL, LS-DYNA, etc.; (2) choice of element types in designing a mesh; (3) choice of mean element density or degrees of freedom (d.o.f.) in the same mesh design; (4) choice of a percent relative error (PRE) or the Rate of PRE per d.o.f. on a log-log plot to assure solution convergence; (5) uncertainty in geometric parameters of the model; (6) uncertainty in physical and material property parameters of the model; (7) uncertainty in loading parameters of the model, and (8) uncertainty in the choice of the model. By considering every FEM solution as the result of a numerical experiment for a fixed model, a purely mathematical problem, i.e., solution verification, can be addressed by first quantifying the errors and uncertainties due to the first four of the eight sources listed above, and then developing numerical algorithms and easy-to-use metrics to assess the solution accuracy of all candidate solutions. In this paper, we present a new approach to FEM verification by applying three mathematical methods and formulating three metrics for solution accuracy assessment. The three methods are: (1) A 4-parameter logistic function to find an asymptotic solution of FEM simulations; (2) the nonlinear least squares method in combination with the logistic function to find an estimate of the 95% confidence bounds of the asymptotic solution; and (3) the definition of the Jacobian of a single finite element in order to compute the Jacobians of all elements in a FEM mesh. Using those three methods, we develop numerical tools to estimate (a) the uncertainty of a FEM solution at one billion d.o.f., (b) the gain in the rate of PRE per d.o.f. as the asymptotic solution approaches very large d.o.f.’s, and (c) the estimated mean of the Jacobian distribution (mJ) of a given mesh design. Those three quantities are shown to be useful metrics to assess the accuracy of candidate solutions in order to arrive at a so-called “best” estimate with uncertainty quantification. Our results include calibration of those three metrics using problems of known analytical solutions and the application of the metrics to sample problems, of which no theoretical solution is known to exist.