The phrase "achieved for the case" is not correct and may be unclear in written English.
It could be used in a context where you are discussing a specific situation or example, but it lacks clarity without additional context.
Example: "The results achieved for the case of the missing documents were conclusive."
Alternatives: "obtained for the situation" or "accomplished for the instance".
Firstly, it is evaluated the percentage of simulations in which the global optimal point is achieved for the case with square solid wall.
For the three times optimized geometry, the presence of thermal contact resistance changed the optimal length and width ratios in nearly 16%and12%2% in comparison with optimal ratios achieved for the case with perfect thermal contact.
The maximum thermal efficiency of 49.16% could be achieved for the case of fuel and air inlet temperature of 873 K and the fuel mass flow rate of 3.202 × 10−7 kg s−1.
Excellent correlation of the present numerical solutions is achieved for the case of free convection micropolar flow neglecting Sort/Dufour effects, with the earlier study by Nazar et al. (2002) [12].
And the highest ionic conductivity of 3.3 × 10−2 S cm−1 was achieved for the case of Mg2+-GPE with 200 wt.% of liquid electrolyte contents at 20 °C.
The prestress strain corresponding to the optimum ductility index was found to be 2830 με (the strain value can be determined using an accepted arbitrary test such as monotonic test) while no optimum ductility was achieved for the case of the un-fatigued beams.
For gender, correct classification rates (CCR) of 80% and 94% were achieved for the Vancouver and Oakland case studies, respectively.
Detailed characterizations of these modes from their derived mode shapes have been achieved for the typical case of triple-layered GSs, as well as general MLGSs.
Detailed characterizations of these modes from their derived mode shapes have been achieved for the typical case of an embedded triple-layered GSs, as well as general embedded MLGSs.
The obtained results are similar to those achieved for the MIMO case.
In previous work we have shown that this can be achieved for the special case that the weight kernel is constructed from a linear combination of zeroth order modified Bessel functions of the second kind [21].
