190 Computational Mechanics of Composite Materials 34=G4+G8KaQ-Kaqy) (4.5) The alternative AVM strategy begins with the introduction of an adjoint variable vector =1,2,...,N such that Aa =G.pKap (4.6) It yields the adjoint equations forA in the form KaplB =Ga (4.7) and then,the sensitivity gradient coefficients may be obtained as 34=G4+.0a-K9B】 (4.8) having solved the above equation for the adjoint variables A.The main ideas of the DDM and AVM seem to be identical but in realistic engineering design problems their computer performance is considerably different.Since most of the functions are given explicitly in the problems considered,the DDM technique has found its application below. The matrices of derivatives of practically any order of the global stiffness matrix with respect to design variables are obtained simply by adding derivatives of element stiffness expressed in the global coordinate system.It is done quite similarly to the assembling procedure for the global stiffness matrix.This process is usually essentially simplified,because almost all entries in the matrices of their derivatives with respect to the particular design variables are equal to 0 and then all arithmetic operations can be carried out at the element level. Effective computation of stiffness derivatives with respect to design variables for finite elements is another issue to be taken into account in developments of any sensitivity-oriented software.Most up-to-date finite element codes engage numerical integration instead of using the closed form expressions in terms of design variables to generate the element stiffness matrices.For such numerically generated element matrices a differentiation process with respect to design variables can be performed through a sequence of computations (at least two solution for initial and for a slightly perturbed design parameter)used to generate these matrices,leading to implicit design derivative procedures. The element matrices of the design derivatives can also be obtained by using a finite difference scheme,which is demonstrated for the eth element of the stiffness matrix 器k6+l小=2。 (4.9)
190 Computational Mechanics of Composite Materials ( ) G GβKαβ Qα Kαγ qγ d d 1 .d .d . . . ℑ = + − − (4.5) The alternative AVM strategy begins with the introduction of an adjoint variable vector λα , α=1,2,…,N such that 1 . − λα = GβKαβ (4.6) It yields the adjoint equations for λα in the form Kαβλβ = G.α (4.7) and then, the sensitivity gradient coefficients may be obtained as ( ) G λα Qα Kαβqβ .d .d .d .d ℑ = + − (4.8) having solved the above equation for the adjoint variables λβ . The main ideas of the DDM and AVM seem to be identical but in realistic engineering design problems their computer performance is considerably different. Since most of the functions are given explicitly in the problems considered, the DDM technique has found its application below. The matrices of derivatives of practically any order of the global stiffness matrix with respect to design variables are obtained simply by adding derivatives of element stiffness expressed in the global coordinate system. It is done quite similarly to the assembling procedure for the global stiffness matrix. This process is usually essentially simplified, because almost all entries in the matrices of their derivatives with respect to the particular design variables are equal to 0 and then all arithmetic operations can be carried out at the element level. Effective computation of stiffness derivatives with respect to design variables for finite elements is another issue to be taken into account in developments of any sensitivity-oriented software. Most up-to-date finite element codes engage numerical integration instead of using the closed form expressions in terms of design variables to generate the element stiffness matrices. For such numerically generated element matrices a differentiation process with respect to design variables can be performed through a sequence of computations (at least two solution for initial and for a slightly perturbed design parameter) used to generate these matrices, leading to implicit design derivative procedures. The element matrices of the design derivatives can also be obtained by using a finite difference scheme, which is demonstrated for the eth element of the stiffness matrix [ ] K ( ) h K ( ) h h K e d e d e ( ) ( ) ( ) ( ) 1 1 αβ αβ αβ ε ε ≅ + − ∂ ∂ , d =1,2,...,D , (4.9)
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
192 Computational Mechanics of Composite Materials where vr is the reinforcement volume fraction,while ki,k2 are heat conductivity coefficients of composite components such that ki>k2. Furthermore,the sensitivities of effective heat conductivity with respect to those characterising original composite components are determined:the computations are performed using the mathematical package MAPLE.All the results of the numerical experiments are presented in Figures 4.1-4.9:the effective heat conductivities for the 1D,2D and 3D composites are plotted in Figures 4.1- 4.3,their material sensitivities with respect to design variable ki in Figures 4.4- 4.6,while sensitivity studies with respect to the parameter k2 are presented in Figures 4.7-4.9. 1- 16 14 12 04 03020 Figure 4.1.Effective heat conductivity for 1D composite 05 04时 03 02 01 05 04 k15 03 02 100 01 Figure 4.2.Material sensitivity of in 1D problem to k 05 04 03时 02 01 05 0102w0304 10 Figure 4.3.Sensitivity of in ID problem to
192 Computational Mechanics of Composite Materials where vf is the reinforcement volume fraction, while k1, k2 are heat conductivity coefficients of composite components such that k1>k2. Furthermore, the sensitivities of effective heat conductivity with respect to those characterising original composite components are determined: the computations are performed using the mathematical package MAPLE. All the results of the numerical experiments are presented in Figures 4.1-4.9: the effective heat conductivities for the 1D, 2D and 3D composites are plotted in Figures 4.1- 4.3, their material sensitivities with respect to design variable k1 in Figures 4.4- 4.6, while sensitivity studies with respect to the parameter k2 are presented in Figures 4.7-4.9. Figure 4.1. Effective heat conductivity for 1D composite Figure 4.2. Material sensitivity of k(eff) in 1D problem to k1 Figure 4.3. Sensitivity of k (eff) in 1D problem to vf
Sensitivity Analysis for Some Composites 193 24 1 124 11 05 03 40201 Figure 4.4.Effective heat conductivity for 2D composite 044 03 02 014 0.5 02w030 01 100 Figure 4.5.Material sensitivity of in 2D problem to k 0.50.4 030201 8 10 0 24k6 Figure 4.6.Sensitivity of in 2D problem to 2.8 2.6 2.4 2 1.8 1.6 1.4 1.21 0.5 0.40.30.201024k6 10 Figure 4.7.Effective heat conductivity for 3D composite
Sensitivity Analysis for Some Composites 193 Figure 4.4. Effective heat conductivity for 2D composite Figure 4.5. Material sensitivity of k (eff) in 2D problem to k1 Figure 4.6. Sensitivity of k (eff) in 2D problem to vf Figure 4.7. Effective heat conductivity for 3D composite
194 Computational Mechanics of Composite Materials 04时 03 02 0.1 0 0.5 04 0.1 0203 100 Figure 4.8.Material sensitivity of in 3D problem to k 2 1 0.5 04 0.3 02 0.1 0 Figure 4.9.Sensitivity of in 3D problem to Analysing numerical results it can be observed that the effective heat conductivity surface has an analogous shapes for 1D,2D and 3D composites. However the values of this coefficient obtained for the same reinforcement ratio are largest for 3D composite with spherical inclusion,next largest for 2D fibre- reinforced composite,and smallest for the 1D case.Therefore,3D composites seem to be most optimal-using the same volume of reinforcement,the highest value of the effective material property is obtained.According to engineering intuition,it is found that increasing both k and vr an increasing of final value of is obtained.The results of sensitivity studies presented in Figures 4.3.4.6 and 4.9 make it possible to observe the greatest sensitivity of composite effective characteristics with respect to both design parameters(k and vr)for extremely small values of the coefficient ki and the largest value of the reinforcement ratio. The sensitivity gradients ofkm with respect to vr have almost constant value, while with respect to k are efficiently nonlinear and reach the maximum forv=0.5 (cf.Figure 4.2 and 4.3,for instance).This result means that the effective conductivity value is most sensitive to the changes of ki,if the reinforcement volume ratio is maximal,which is predictable result and it positively validates this homogenisation method. The smallest sensitivity of to the parameter k can be noted for r tending to 0,while the inverse relation is observed with respect to the reinforcement volume fraction.The variability of the sensitivity surface for with respect to the heat
194 Computational Mechanics of Composite Materials Figure 4.8. Material sensitivity of k(eff) in 3D problem to k1 Figure 4.9. Sensitivity of k(eff) in 3D problem to vf Analysing numerical results it can be observed that the effective heat conductivity surface has an analogous shapes for 1D, 2D and 3D composites. However the values of this coefficient obtained for the same reinforcement ratio are largest for 3D composite with spherical inclusion, next largest for 2D fibrereinforced composite, and smallest for the 1D case. Therefore, 3D composites seem to be most optimal - using the same volume of reinforcement, the highest value of the effective material property is obtained. According to engineering intuition, it is found that increasing both k1 and vf an increasing of final value of k (eff) is obtained. The results of sensitivity studies presented in Figures 4.3, 4.6 and 4.9 make it possible to observe the greatest sensitivity of composite effective characteristics with respect to both design parameters (k1 and vf) for extremely small values of the coefficient k1 and the largest value of the reinforcement ratio. The sensitivity gradients of k (eff) with respect to vf have almost constant value, while with respect to k1 are efficiently nonlinear and reach the maximum for vf=0.5 (cf. Figure 4.2 and 4.3, for instance). This result means that the effective conductivity value is most sensitive to the changes of k1, if the reinforcement volume ratio is maximal, which is predictable result and it positively validates this homogenisation method. The smallest sensitivity of k (eff) to the parameter k1 can be noted for vf tending to 0, while the inverse relation is observed with respect to the reinforcement volume fraction. The variability of the sensitivity surface for k(eff) with respect to the heat