The discrete Painleveequations are nonlinear difference equations that arise as compatibility conditions
of linear systems, says Joshi, then warns that the deceptive simplicity of this statement hides deep layers of mathematical properties, which she outlines here.
The inclusion of smart materials layers will modify material properties between neighboring segments, producing discontinuities and fulfilling compatibility conditions
for the continuity of the displacement and rotation of the beam and for the equilibrium of bending moment and shear at the discontinuity points.
The basic concept in the study originated from the extension of constraints of the compatibility conditions
at free interfaces between structures and substructures.
One way in which these auxiliary equations arise is through compatibility conditions
. Moreover, an interesting result is presented by showing that there is an important link between a nonhomogeneous Schrodinger equation and a combination of surface variables which are relevant to the more physical case of space-like isothermic surfaces.
The characteristic parameters of cyclically stabilized stress-strain curves can be derived from strain-life curves using the compatibility conditions
of Equations (3) and (4):
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
. A copula-based sampling technique is employed for efficient sampling from the joint and conditional distributions.
It turns out that in fact additional compatibility conditions
Newly derived compatibility conditions
for residual deformations [[theta].sub.r] improve the quality of the mathematical model of the optimization problem and avoid a direct use of the abovementioned influence matrices.
Next, the sixth-order governing differential compatibility equations, as well as the associated end-boundary compatibility conditions
, are derived based on the virtual force principle.
The bar and elastic substrate compatibility conditions
are both enforced in the strong sense.
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.
satisfy the compatibility conditions
. These conditions apply to the equations (2.17) are (3ii).