r3.06.07 Diagonalization of the thermal mass matrix

Summary:

To improve the consistency of the solution in transient thermal problems, one of the solutions consists in « lumping » (i.e.: condensing on the diagonal) the thermal mass matrix (capacity matrix).

This possibility is accessible by the models PLAN_DIAG, AXIS_DIAG and 3D_ DIAG for the phenomenon THERMIQUE. It is activated when the thermal calculation commands THER_LINEAIRE and THER_NON_LINE are called.

When using these models, only linear finite elements (2D and 3D) have a lumped mass matrix. In fact, direct diagonalization does not give satisfactory results for quadratic finite elements, so it is not allowed.

The theoretical results are illustrated by the thermo-mechanical calculation of a cylinder subjected to thermal shock.

• 1. Introduction
• 2. Principle of the maximum
  • 2.1. Statement of principle for the continuous case
  • 2.2. Respect for the principle of maximum discretion
  • 2.3. Sufficient conditions for compliance with the principle of maximum at the discrete level
• 3. Diagonalization method used and types of elements
  • 3.1. Boundary elements and conditions such as the extra-diagonal terms in \(K\) are negative
  • 3.2. Diagonalization method: Integration into element nodes
• 4. Implemented in Code_Aster
  • 4.1. 2D modellings
  • 4.2. 3D modeling
• 5. Thermal calculation of a cylinder subjected to a cold shock
  • 5.1. Data
  • 5.2. Results
• 6. Conclusion
• 7. Bibliography
• 8. Description of document versions