Characteristics of four-dimensional normal distribution of properties of composite in stochastic compromise optimization

Abstract

In a numerical example of compromise optimization by computerized mathematical modeling (2000 realizations) for a known deterministic solution, in the case of an isotropic spatially reinforced porous composite, certain scatter characteristics of the optimal solution have been established, namely four standard deviations and six coefficients of linear correlation for four properties—density, modulus of elasticity, thermal conductivity, and linear thermal expansion coefficient. Of the 17 input data (parameters of the composite components), 10 are stochastic, the others deterministic. An equation is presented for the four-dimensional hyperellipsoid of normal distribution with numerical values of the coefficients, as well as all invariants and roots of the characteristic equation, the matrix of direction cosines of the principal axes of the hyperellipsoid, and the lengths of the principal semiaxes, depending on the dimensionality of the scattering region and the assigned probability P. The four-dimensional hyperellipsoid has been projected onto three-dimensional space and then onto a plane. A section of the scattering region has been constructed.