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Exact(2)
is evaluated exactly as in the case of DWE, already described in Section 3.1.
Since the integrand is a polynomial of degree 2 s − 1, it is evaluated exactly by the s-stage Gaussian quadrature formula.
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The influences of many neighboring groups to the central group are evaluated exactly.
Therefore, the problem can be modeled as a simulation optimization problem, where the objective function cannot be evaluated exactly.
However, the partial derivatives can now be evaluated exactly by solving only one mesoscale problem per Gauß point.
Through-the-thickness integrals are evaluated exactly, and those over an element in the midplane of the plate are evaluated by using the 2×2 Gauss quadrature rule.
A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants.
The integrals along the Mach cone and along the cell interfaces are evaluated exactly, as well as by means of numerical quadratures.
Some of the resultant integrals are evaluated by using the Gauss Chebyshev integration rules after moving the series coefficients to the outside of the integral sign; others are evaluated exactly, including the modified hypersingular integral.
For inclusions with a circular cross-section, the lower bound of the band gap can be evaluated exactly by means of the derived analytical solution, which is also valid for compressible coatings and can therefore be used to determine lower bounds of higher band gaps as well.
When fitness can be evaluated exactly, the different kinds of stability can be evaluated.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com