In this methodology, the Bayesian network is pursued in two directions: (1) probabilistic surrogate modeling to estimate the output uncertainty given values of the design variables, and (2) probabilistic multidisciplinary analysis (MDA) to infer the distributions of the coupling and output variables that satisfy interdisciplinary compatibility conditions.
The inversion capability of the Bayesian network is exploited in a novel manner to enforce the interdisciplinary compatibility condition in feedback-coupled MDA, by setting the difference between the values of the same coupling variable from two successive iterations to zero.
Therefore, the joint PDF of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given the compatibility condition is:
It turns out that in fact additional compatibility conditions are necessary.
Just as continuity and Kirchhoff-type conditions are natural for a manifold of partial differential equations on networks, and not only for the (BBMG) system, the compatibility conditions (3.
Thus, using in particular the compatibility conditions
As emphasized in the theoretical section, the connection between a single variant of martensite and the parent austenitic phase is provided by a twinned martensitic structure, which is able to satisfy the compatibility conditions
both with the austenite (over the habit plane) and with the single variant.
0]), which are specified through the compatibility conditions of (2.
x] and are determined through compatibility conditions of the overdetermined system composed of (3.
where the functions [phi], [psi] and [lambda] may be determined through compatibility conditions.
The compatibility conditions
(continuity of the body in the deformed state), in conjunction with the mentioned symmetry conditions, mean that in the deformed state the basic cell must keep its rectangular shape.
Then the compatibility condition
(4) holds and u is Lipschitz continuous with the Lipschitz constant [L.