Abstract

Computational modeling and simulation is a central tool in science and engineering, directed at solving partial differential equations for which analytical solutions are unavailable. The continuous equations are generally discretized in time, space, energy, etc., to obtain approximate solutions using a numerical method. The aspiration is for the numerical solutions to asymptotically converge to the exact-but-unknown solution as the discretization size approaches zero. A generally applicable procedure to assure convergence is unavailable. The Richardson extrapolation is the main method for dealing with this challenge, but its assumptions introduce uncertainty to the resulting approximation. We use info-gap decision theory to model and manage its main uncertainty, namely, in the rate of convergence of numerical solutions. The theory is illustrated with a numerical application to Hertz contact in solid mechanics.

References

1.
Oden
,
J. T.
,
Belytschko
,
T.
,
Fish
,
J.
,
Hughes
,
T. J. R.
,
Johnson
,
C.
,
Keyes
,
D.
,
Laub
,
A.
,
Petzold
,
L.
,
Srolovitz
,
D.
, and
Yip
,
S.
,
2006
, “
Simulation-Based Engineering Science: Revolutionizing Engineering Science Through Simulation
,” Report of the National Science Foundation Blue Ribbon Panel on Simulation-based Engineering Science, Alexandria, VA,
Report
.https://www.nsf.gov/attachments/106803/public/TO_SBES_Debrief_050306.pdf
2.
Adam
,
J. M.
,
Parisi
,
F.
,
Sagaseta
,
J.
, and
Lu
,
X.
,
2018
, “
Research and Practice on Progressive Collapse and Robustness of Building Structures in the 21st Century
,”
Eng. Struct.
,
173
, pp.
122
149
.10.1016/j.engstruct.2018.06.082
3.
Chen
,
C.-J.
, and
Jaw
,
S.-Y.
,
1997
,
Fundamentals of Turbulence Modeling
,
Taylor and Francis
,
Bristol, PA
.
4.
Durbin
,
P. A.
,
2018
, “
Some Recent Developments in Turbulence Closure Modeling
,”
Annu. Rev. Fluid Mech.
,
50
(
1
), pp.
77
103
.10.1146/annurev-fluid-122316-045020
5.
Blevins
,
R. D.
,
1993
,
Formulas for Natural Frequency and Mode Shape
,
Krieger Publishing Company
, Malabar, FL.
6.
Lax
,
P. D.
, and
Richtmyer
,
R. D.
,
1956
, “
Survey of the Stability of Linear Finite Difference Equations
,”
Commun. Pure Appl. Math.
,
9
(
2
), pp.
267
293
.10.1002/cpa.3160090206
7.
Lax
,
P. D.
, and
Milgram
,
A. N.
,
1954
,
Parabolic Equations, Annals of Mathematics Studies
, Vol.
33
,
Princeton University Press
,
Princeton, NJ
, pp.
167
190
.
8.
Lions
,
J.-L.
, and
Margenes
,
E.
,
1972
,
Non-Homogeneous Boundary Value Problems and Applications
,
Springer-Verlag
,
New York
.
9.
Leveque
,
R. J.
,
2002
,
Finite Volume Methods for Hyperbolic Problems
(Cambridge Texts in Applied Mathematics),
Cambridge University Press
,
New York
.
10.
Grinstein
,
F. F.
,
Margolin
,
L. G.
, and
Rider
,
W. J.
,
2007
,
Implicit Large Eddy Simulation
,
Cambridge University Press
,
New York
.
11.
Ciarlet
,
P. G.
,
1991
, “
Basic Error Estimates for Elliptic Problems
,”
Finite Element Methods: Handbook of Numerical Analysis
,
North-Holland
,
New York
.
12.
Rivière
,
R.
,
2008
, “
Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation
,”
Frontiers in Applied Mathematics
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
13.
Babuška
,
I.
,
1971
, “
Error-Bounds for the Finite Element Method
,”
Numerische Mathematik
, Vol.
16
,
Springer-Verlag
,
Berlin, Germany
, pp.
322
333
.
14.
Girault
,
V.
, and
Raviart
,
P.-A.
,
1986
,
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Series in Computational Mathematics
,
Springer-Verlag
,
Berlin, Germany
.
15.
Zienkiewicz
,
O. C.
, and
Zhu
,
J. Z.
,
1992
, “
The Super-Convergent Patch Recovery and a Posteriori Error Estimates, Part 1: The Recovery Technique
,”
Int. J. Numer. Methods Eng.
,
33
(
7
), pp.
1331
1364
.10.1002/nme.1620330702
16.
Hirt
,
C. W.
,
1968
, “
Heuristic Stability Theory for Finite Difference Equations
,”
J. Comput. Phys.
,
2
(
4
), pp.
339
355
.10.1016/0021-9991(68)90041-7
17.
Warming
,
R. F.
, and
Hyett
,
B. J.
,
1974
, “
The Modified Equation Approach to the Stability and Accuracy Analysis of Finite Difference Methods
,”
J. Comput. Phys.
,
14
(
2
), pp.
159
179
.10.1016/0021-9991(74)90011-4
18.
Richardson
,
L. F.
,
1910
, “
The Approximate Arithmetical Solution by Finite Differences of Physical Problems Including Differential Equations, With an Application to the Stresses in a Masonry Dam
,”
Philos. Trans. R. Soc. A
,
210
(
459–470
), pp.
307
357
.10.1098/rspa.1910.0020
19.
Richardson
,
L. F.
, and
Gaunt
,
J. A.
,
1927
, “
The Deferred Approach to the Limit
,”
Philos. Trans. R. Soc. A
,
226
(
636–646
), pp.
299
349
.10.1098/rsta.1927.0008
20.
Roache
,
P. J.
,
2009
,
Fundamentals of Verification and Validation
,
Hermosa Publishers
,
Albuquerque, NM
.
21.
Roache
,
P. J.
,
1994
, “
Perspective: A Method for Uniform Reporting of Grid Refinement Studies
,”
ASME J. Fluids Eng.
,
116
(
3
), pp.
405
413
.10.1115/1.2910291
22.
Stern
,
F.
,
Wilson
,
R. V.
,
Coleman
,
H. W.
, and
Paterson
,
E. G.
,
2001
, “
Comprehensive Approach to Verification and Validation of Computational Fluid Dynamics Simulations—Part 1: Methodology and Procedures
,”
ASME J. Fluids Eng.
,
123
(
4
), pp.
793
802
.10.1115/1.1412235
23.
Wilson
,
R. V.
,
Stern
,
F.
,
Coleman
,
H. W.
, and
Paterson
,
E. G.
,
2001
, “
Comprehensive Approach to Verification and Validation of Computational Fluid Dynamics Simulations—Part 2: Application to RANS Simulation of a Cargo/Container Ship
,”
ASME J. Fluids Eng.
,
123
(
4
), pp.
803
810
.10.1115/1.1412236
24.
Roy
,
C. J.
,
2005
, “
Review of Code and Solution Verification Procedures for Computational Simulation
,”
J. Comput. Phys.
,
205
(
1
), pp.
131
156
.10.1016/j.jcp.2004.10.036
25.
Eça
,
L.
, and
Hoekstra
,
M.
,
2014
, “
A Procedure for the Estimation of the Numerical Uncertainty of Computational Fluid Dynamics (CFD) Calculations Based on Grid Refinement Studies
,”
J. Comput. Phys.
,
262
, pp.
104
130
.10.1016/j.jcp.2014.01.006
26.
Rider
,
W.
,
Witkowski
,
W.
,
Kamm
,
J. R.
, and
Wildey
,
T.
,
2016
, “
Robust Verification Analysis
,”
J. Comput. Phys.
,
307
(
C
), pp.
146
163
.10.1016/j.jcp.2015.11.054
27.
Kamm
,
J. R.
,
Rider
,
W. J.
, and
Brock
,
J. S.
,
2003
, “
Combined Space and Time Convergence Analyses of a Compressible Flow Algorithm
,”
AIAA
Paper No. 2003–4241.10.2514/6.2003-4241
28.
Hemez
,
F. M.
,
Brock
,
J. S.
, and
Kamm
,
J. R.
,
2006
, “
Nonlinear Error Ansatz Models in Space and Time for Solution Verification
,”
47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
,
American Institute of Aeronautics and Astronautics
, Newport, RI, May 1–4, Paper No. AIAA 2006–1995.10.2514/6.2006-1995
29.
Roy
,
C. J.
,
2003
, “
Grid Convergence Error Analysis for Mixed-Order Numerical Schemes
,”
AIAA J.
,
41
(
4
), pp.
595
604
.10.2514/2.2013
30.
Banks
,
J. W.
,
Aslam
,
T.
, and
Rider
,
W. J.
,
2008
, “
On Sub-Linear Convergence for Linearly Degenerate Waves in Capturing Schemes
,”
J. Comput. Phys.
,
227
(
14
), pp.
6985
7002
.10.1016/j.jcp.2008.04.002
31.
Ben-Haim
,
Y.
,
2006
,
Info-Gap Decision Theory: Decisions Under Severe Uncertainty
, 2nd ed.,
Academic Press
, Amsterdam, The Netherlands.
32.
Ben-Haim
,
Y.
,
2018
,
Dilemmas of Wonderland: Decisions in the Age of Innovation
,
Oxford University Press
, Oxford, UK.
33.
Phillips
,
T. S.
, and
Roy
,
C. J.
,
2014
, “
Richardson Extrapolation-Based Discretization Uncertainty Estimation for Computational Fluid Dynamics
,”
ASME J. Fluids Eng.
,
136
(
12
), p.
121401
.10.1115/1.4027353
34.
Roache
,
P. J.
,
2003
, “
Conservatism of the Grid Convergence Index (GCI) in Finite Volume Computations on Steady State Fluid Flow and Heat Transfer
,”
ASME J. Fluids Eng.
,
125
(
4
), pp.
731
732
.10.1115/1.1588692
35.
Stern
,
F.
,
Wilson
,
R.
, and
Shao
,
J.
,
2006
, “
Quantitative Verification and Validation (V&V) of Computational Fluid Dynamics (CFD) Simulations and Certification of CFD Codes
,”
Int. J. Numer. Methods Fluids
,
50
(
11
), pp.
1335
1355
.10.1002/fld.1090
36.
Freitas
,
C. J.
,
Celik
,
I. B.
,
Ghia
,
U.
,
Roache
,
P. J.
,
Freitas
,
C. J.
,
Coleman
,
H.
, and
Raad
,
P. E.
,
2008
, “
Procedure for Estimation and Reporting of Uncertainty Due to Discretization in Computational Fluid Dynamics (CFD) Applications
,”
ASME J. Fluids Eng.
,
130
(
7
), p.
078001
.10.1115/1.2960953
37.
Eça
,
L.
,
Hoekstra
,
M.
, and
Roache
,
P. J.
, “
Verification of Calculations: An Overview of the 2nd Lisbon Workshop
,”
AIAA
Paper No. 2007-4089.10.2514/6.2007-4089
You do not currently have access to this content.