The solution of multiphysical problems by weak coupling originates from the combined use of disjoint codes solving, for example, each a physical problem whose inputs themselves originate from the output of the other code. It’s obviously iterative. The codes can thus be used in a black box by a driver controlling the input-output files. Of course, the opening of the codes will allow for a finer transfer of information. At the extreme, it is possible to search for the strong coupling which will solve at each iteration in one pass the unknowns coming respectively from the two codes. This means that it is necessary to construct a larger linear system mixing the two (or more) unknown variables.
How, for example, to assemble, in the supervector (second member) and the Jacobian supermatrix, the elementary vectors and matrices of the two coupled equations?
A multiphysical model is still mathematical
Naupacte uses its unique tensor representation for the construction of such a coupled system. Diagonal or crossed blocks are placed there, as well as isolated coefficients coming from imposed values or multiple value constraints to be processed by Lagrange multipliers (Navpactos provides functions avoiding the pain of numbering). Stacking the equations does not add structural complexity.
The physics are embedded in the mathematical formulas that model them and that appear in the construction of the residual vector and the matrix in particular at this stage of the algorithm. Navpactos is above all mathematical, algebraic precisely. This is the basis of calculations and the commonplace of physical models.
If you know the physics, do express it as a model that Navpactos calculates.
The advantage of formal representation
The object used to define the matrix (the same for the vector but with fewer properties) gets a lot of information before it is calculated.
- It can deduce the possible symmetry of the matrix (except in special cases such as the symmetry resulting from the potential from which the pressure force internal to a volume derives, because the symmetry is obtained only on the total assembly and not on the elementary level).
- It can also optimize matrix construction by identifying common patterns.
- The automatic renumbering of equations can be envisaged to obtain triangular systems when they occur (strong coupling but unilateral dependence resolving to a weak coupling with the right order of resolution between the unknowns).
- If the matrix is the Jacobian of a vector, it can be obtained by automatic derivation… It becomes unnecessary to set it manually!
This is an extension of tensor use in Navpactos. The tensor formulas are not only used to calculate the tensor value but to construct a certain object, here the sparse matrix.
In the end, the quintessence of Naupacte’s know-how in formal computation is perfectly expressed in the construction of a sparse matrix derived from the finite element method, whatever the equations.