r4.10.03 Residual spatial error indicator for transient thermal

Summary

During numerical simulations using finite elements, obtaining a raw result is no longer sufficient. The user is increasingly requesting calculation of spatial error in relation to his mesh. He needs methodological support and advanced « numerical » -« computer » tools to assess the quality of his studies and improve them.

For this purpose, the spatial error indicators a posteriori make it possible to locate, on each element, an error map on which the remeshing tools will be able to rely: a first calculation on a coarse mesh makes it possible to exhume the error map based on the discretized data and solution (hence the term « a posteriori »), the refinement is then carried out locally by prioritizing this information.

The new a posteriori indicator (called « pure residue ») that has just been implemented for post » -« to treat the thermal solvers of the Code_Aster is based on their local residues extracted from the semi » -« discretizations in time. Via the option “ERTH_ELEM” in CALC_ERREUR, it uses the thermal fields (EVOL_THER) emanating from THER_LINEAIRE and THER_NON_LINE. It thus complements the code’s offer in terms of advanced tools to improve the quality of studies, their pooling and their comparisons.

The purpose of this note is to detail the theoretical, numerical and computer work that led to its implementation. As far as the theoretical study is concerned, we initially limited ourselves to the linear thermics of an immobile structure discretized by standard isoparametric finite elements. But, in practice , the scope of use of this option has been partially extended to non-linear thermal.

The reader is given the theoretical and practical properties and limitations of the unearthed indicator, while linking these considerations, which can sometimes seem a bit « ethereal », to the precise configuration of the offending operators and to the choices of code modeling. We tried to constantly link the various items discussed, while detailing, at a minimum, slightly technical demonstrations that are rarely explained in specialized literature.

• 1. Problem — Document description
• 2. The boundary problem
  • 2.1. background
  • 2.2. From strong to weak formulations
• 3. Discretization and controllability
  • 3.1. Controllability of the ongoing problem
  • 3.2. Semi-discretization in time
  • 3.3. Temporal discretization error
  • 3.4. Total discretization in time and space
• 4. Pure residue indicator
  • 4.1. Ratings
  • 4.2. Increase in the global spatial error
  • 4.3. Different types of indicators possible
  • 4.4. Reduction of the local spatial error
  • 4.5. Complements
• 5. Summary of the theoretical study
• 6. Implementation in Code_Aster
  • 6.1. Specific difficulties
  • 6.2. Environment required/configuration
  • 6.3. Presentation/exploitation of error calculation results
  • 6.4. Scope of use
  • 6.5. Example of use
• 7. Conclusion — Perspective
• 8. Bibliography
• 9. Description of document versions