When in a non-linear system the non-linear function is treated as a piecewise constant force in a small enough time interval, the resulting response agrees with the solution of the original non-linear equation.
Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect.
It is easy to check that (18) together with (23) agrees with the solution derived in [14].
Hence if the generator Q ( x ) = ( q 11 ( x ) q 12 ( x ) q 21 ( x ) q 22 ( x ) ) satisfies the q-property, then it follows that q ˜ ( x, i ) = 0 for all x, so the solution reduces to the form: u ( x, i ) = E x, i φ ( X , which agrees with the solution to the Dirichlet problem given by: { L u ( x, i ) = 0 in O × { 1, 2 }, u ( x, i ) = φ ( x, i ) on ∂ O × { 1, 2 }.
But I do not agree with the solutions Mr. Egan suggests.
These quadratic FCPM solutions of simply supported, multilayered composite cylinders and sandwiched FGM ones obtained in this way are in excellent agreement with the exact 3D solutions available in the literature, and those solutions for the cylinders with combinations of clamped and simply-supported edge conditions closely agree with the solutions obtained using the ANSYS commercial software.
These estimated results are agreed with the exact solution even in the irregular shape fin.
The indistinguishable cytoplasmic signals agreed with the indistinguishable solution fluorescence results.
Numerical simulations of a multiple parallel fracture reservoir using the TOUGH2 code agreed with the derived analytical solutions over a range of flow rates and interfracture spacings with only small deviation due to multidimensional effects.
We found that the simple potential of Novello et al. leads to the solution which agrees with the Schwarzschild solution of GR and small corrections to the potential are permitted and the generic potential leads to the 'black hole solution' with a modified mass-horizon radius relation (i.e., (r_gne 2GM)), of which asymptotic behavior describes weak gravity up to the post-Newtonian order.
The solution for the first order isothermal case agreed with the pseudo steady state solution within the conditions defined by Bischoff.
