Sensitivity Analysis for Some Composites 191 whereK is the eth element stiffness matrix,h is the d th component of the D-dimensional design variable vector h,e represents a small perturbation and the D-dimensional vector 1 is equal to 1 at the d th position and zeroes elsewhere. Such a scheme is known as forward finite difference rule,however backward and central differences can be applied too.Backward differentiation uses the values of a function in actual (h)and previous point (h-E),while central difference is returned from arithmetic averaging of equations containing forward and backward differences. 4.1.2 Sensitivity of Homogenised Heat Conductivity As is known,it is possible to obtain the effective heat conductivity tensor components by the application of some algebraic approximations for particular types of composite materials.However,numerical procedure is not very general in this case.The effective heat conductivity for a periodic fibre-reinforced composite in a 2D problem where the fibre has the round cross-section and the total composite volume is relatively large in comparison to the single inclusion can be approximated using the Cylinder Assemblage Model(CAM)for a fibre-reinforced plane structure.The Spherical Inclusion Model (SIM)[65]for spherical inclusions distributed periodically (3D composite).The heat conductivity coefficients of composite components k,k2 are such that k>k2(the same results hold true for electrical conductivity,magnetic permeability and the dielectric constant for composites,for instance). A concept of the first test is to compare the effective heat conductivities obtained for the 1D,2D(fibre)and 3D(particle-reinforced)composites in terms of various reinforcement volume ratios and the interrelation between heat conductivity coefficients for both components.The following equations are used: 。1D composite L(et) rdΩ ak(y) ·2 D composite -学》 3D composite 学Sensitivity Analysis for Some Composites 191 where (e) Kαβ is the eth element stiffness matrix, d h is the d th component of the D-dimensional design variable vector h, ε represents a small perturbation and the D-dimensional vector ( ) 1 d is equal to 1 at the d th position and zeroes elsewhere. Such a scheme is known as forward finite difference rule, however backward and central differences can be applied too. Backward differentiation uses the values of a function in actual (h) and previous point (h-ε), while central difference is returned from arithmetic averaging of equations containing forward and backward differences. 4.1.2 Sensitivity of Homogenised Heat Conductivity As is known, it is possible to obtain the effective heat conductivity tensor components by the application of some algebraic approximations for particular types of composite materials. However, numerical procedure is not very general in this case. The effective heat conductivity for a periodic fibre-reinforced composite in a 2D problem where the fibre has the round cross-section and the total composite volume is relatively large in comparison to the single inclusion can be approximated using the Cylinder Assemblage Model (CAM) for a fibre-reinforced plane structure. The Spherical Inclusion Model (SIM) [65] for spherical inclusions distributed periodically (3D composite). The heat conductivity coefficients of composite components k1, k2 are such that k1>k2 (the same results hold true for electrical conductivity, magnetic permeability and the dielectric constant for composites, for instance). A concept of the first test is to compare the effective heat conductivities obtained for the 1D, 2D (fibre) and 3D (particle-reinforced) composites in terms of various reinforcement volume ratios and the interrelation between heat conductivity coefficients for both components. The following equations are used: • 1D composite ∫ Ω Ω Ω = ( ) ( ) k y d k eff • 2D composite ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − = + −1 1 2 2 2 ( ) 2 2 1 1 k k v k k k v f f eff D • 3D composite ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − = + −1 1 2 2 2 ( ) 3 3 1 1 k k v k k k v f f eff D