The NAFEMS Simulation Governance & Management Working Group (SGMWG) would like to commission a "Code Verification Exemplar" in the field of solid mechanics. The exemplar would demonstrate the methodology for performing rigorous code verification. Although this exemplar would be in the field of solid mechanics, the methodology demonstrated would apply to any field of computational simulation, for example, fluid dynamics, electromagnetics, and structural mechanics.
Rigorous code verification requires that a series of numerical solutions is computed on continually refined meshes so that the observed order of accuracy of the numerical solution can be determined. The most refined meshes must be in the asymptotic convergence region of the numerical solution to the mathematical model of the system of interest. The observed order of accuracy, also referred to as the observed order of convergence, is determined by examining the rate of change of the numerical error of the solution as a function of the mesh size. Mathematically, the observed order of accuracy is the value of the exponent p of the first term in the power series expansion of the numerical error in terms of the mesh size h, i.e., Chp. The observed order of accuracy can be found by plotting the logarithm of the numerical error of each solution as a function of the logarithm of the representative mesh size of each solution. The slope of the straight line in the asymptotic region on this log-log plot is the observed order of accuracy of the numerical solution. An exact solution of the mathematical model must be available so that the numerical error in quantities of interest (QOI), either the local error in the solution domain or the global error over the entire solution domain, can be computed exactly. Typical QOI are the local displacement or strain energy and the total strain energy in the structure. The numerical solution error is a combination of the discretization error within the solution domain of the original mathematical model, the numerical error due to the imposition of boundary conditions and external loading of the original mathematical model, and the numerical error due to integration of the numerical solution.
This procedure for rigorous code verification goes beyond the traditional technique of assessing numerical accuracy used in NAFEMS benchmarks. The traditional technique compares the numerical solution for QOI on a moderately refined mesh with a “target” or “reference” solution. There are two technical difficulties with this traditional approach. First, the finest mesh may not be refined sufficiently to be in the asymptotic convergence region. Second, the target solution may not be sufficiently accurate to assesses the accuracy of the numerical solution. Most importantly, the traditional technique does not determine the observed order of accuracy of the numerical method. The observed order of accuracy is much more sensitive to any shortcoming in any aspect of the numerical solution procedure, e.g., mesh size, numerical error in imposition of the boundary conditions given in the original mathematical model, and numerical error in the imposition of the structural loading given in the original mathematical model. Because of the effectiveness of the observed order of accuracy method for testing software, the SGMWG is seeking the development and documentation of an exemplar using this method. For a detailed discussion of the methodology for determining the observed order of accuracy, see Knupp, P. and K. Salari (2002), Verification of Computer Codes in Computational Science and Engineering, Chapman & Hall/CRC, Boca Raton, Florida, and Oberkampf, W. L. and C. J. Roy (2010), Verification and Validation in Scientific Computing, Cambridge University Press, Cambridge, UK.
The following briefly describes the steps required for formulating and conducting the rigorous code verification exemplar:
The proposal should identify which competencies from the Verification and Validation Technical of the NAFEMS Professional Simulation Engineer scheme will be addressed by the exemplar problem.
The main body of the documentation of the exemplar should generally follow the steps outlined above. Note that some of these steps can be combined into major sections of the main body of the documentation, as appropriate. The appendix should give information and procedures that are excessively detailed for the main body of the text, for example, the procedure used to generate the series of uniformly refined meshes and the input and output data files for the software package used to solve the numerical model. The information should be adequately explained and commented so that it is easily understood and reproducible by other computational analysts.
The total cost is not expected to exceed £2K (GBP), and it is expected that the document will be completed within 6 months from NAFEMS approval. The length of the document is not anticipated to exceed 20 A4 pages.
Each proposal should consist of:
All proposals should be sent to NAFEMS at sgmwg@nafems.org to arrive no later than 1 July 2022. Further details are available from theSGMWG chairman who are contactable using sgmwg@nafems.org.
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