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

Published

### Author(s)

Jeffrey T. Fong, Pedro V. Marcal, Robert B. Rainsberger, Li Ma, Nathanael A. Heckert, James J. Filliben

### Abstract

Errors and uncertainties in finite element method (FEM) computing can come from the following eight sources: (1) Computing platform; (2) choice of element types; (3) choice of mean element density or degrees of freedom (d.o.f.); (4) choice of a percent relative error (PRE) or the Rate of PRE per d.o.f. to assure solution convergence; (5) uncertainty in geometric parameters; (6) uncertainty in physical and material property parameters; (7) uncertainty in loading parameters, and (8) uncertainty in the model. By considering every FEM solution as the result of a numerical experiment, 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 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 node in a finite element mesh in order to compute the Jacobians of all nodes in a given mesh. Using these 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. 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.
Proceedings Title
Proceedings of ASME Verification and Validation Symposium, May 16-18, 2018, Minneapolis, MN,
U.S.A.
Conference Dates
May 16-18, 2018
Conference Location
Minneapolis, MN

### Keywords

ABAQUS, Accuracy assessment metric (AAM), ANSYS, barrel vault, cantilever beam, computational modeling, COMSOL, DATAPLOT, element type, FEM, finite element method, hexahedron element, holed composite laminate strength test, Jacobian, logistic function, LS-DYNA, mesh density, mesh quality, MPACT, nonlinear least squares method, relative error convergence rate, resonance frequency, statistical analysis, super-parametric method, tetrahedron element, TrueGrid, uncertainty quantification.

## Citation

Fong, J. , Marcal, P. , Rainsberger, R. , Ma, L. , Heckert, N. and Filliben, J. (2018), Finite Element Method Solution Uncertainty, Asymptotic Solution, and a New Approach to Accuracy Assessment, Proceedings of ASME Verification and Validation Symposium, May 16-18, 2018, Minneapolis, MN, U.S.A., Minneapolis, MN, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=925462 (Accessed September 15, 2024)