The Method of Manufactured Solutions

by N. Tardieu, EDF R&D / AMA

The Method of Manufactured Solutions is a method of verification of scientific software. This is a systematic way for obtaining analytical solutions for problems that can be very complex and nonlinear.

It is interesting to note that the "American Institute of Aeronautics and Astronautics" (AIAA) has included in its software quality assurance standards in 1998 [1]. This practice is far less common in solid mechanics, and the "American Society of Mechanical Engineers" (ASME) has only integrated into its standard in 2006 [2]. This is not related to a specific problem, extensive studies in solid mechanics have been conducted using this method [3].

The introduction of this method in Code_Aster is another step forward in our approach to software quality. It is described in the Code_Aster document "Using the method of manufactured solutions for testing software" and U2.08.08 used in cases following tests:

• v4.04.107 : Method of manufactured solutions in 2D thermics
• v6.03.150 : Method of manufactured solutions in 2D contact and large deformations
• v6.04.219 : Method of manufactured solutions in 3D contact and large deformations

Elastic cube under large deformations in contact with a rigid paraboloid surface

Optimal convergence rate in L2 norm of the computed solution towards the analytical solution

References :

[1] American Institute for Aeronautics and Astronautics, Guide for the Verification and Validation of Computational Fluid Dynamics Simulations, 1998.

[2] American Society of Mechanics Engineers, Guide for Verification and Validation in Computational Solid Mechanics, 2006.

[3] Eric Chamberland, André Fortin, Michel Fortin, Comparison of the performance of some finite element discretizations for large deformation elasticity problems, Computers & Structures, Vol. 88, no. 11-12, pp. 664-673, 2010