4 Sensitivity Analysis for Some Composites 4.1 Deterministic Problems As is known,the sensitivity analysis in engineering systems is employed to verify how input parameters of a specific engineering problem influence the analysed state functions (displacements,stresses and temperatures,for instance). The sensitivity coefficients [269],being the purpose of such an analysis,are computed using partial derivatives of the considered state function with respect to the particular input parameter(s).These derivatives can be obtained numerically starting from the fundamental algebraic equations system of the problem,for instance,or alternatively,by a simple derivation if only a closed form solution exists;some combined analytical-numerical methods are also known [99].It is important to underline that this methodology is common for all discrete numerical techniques:Boundary Element Method (BEM)[51,206],Finite Difference Method (FDM)[90,206],FEM [7,21,387]as well as hybrid and meshless strategies [81] From the computational point of view,there are the following numerical methods in structural design sensitivity analysis [75,76,103,134,207]:the Direct Differentiation Method (DDM),the Adjoint Variable Method (AVM)applied together with the Material Derivative Approach (MDA)or the Domain Parametrisation Approach (DPA)suitable for shape sensitivity studies. Considering these capabilities and,on the other hand,a very complex structure of composite materials,sensitivity analysis should be applied especially in design studies for such structures.Instead of a single (or two)parameters characterising the elastic response of a homogeneous structure,the total number of design parameters is obtained as a product of component numbers in a composite and the number of material and geometrical parameters for a single component.Even some extra state variables should be analysed to define interfacial behaviour,general interaction of the constituents and/or the lack of periodicity.Usually,to reduce the complexity of the original composite,the so-called effective homogenisation medium having the same strain (or complementary)energy is analysed. This chapter is devoted to general computational sensitivity studies of the homogenisation method for some periodic composite materials with linear elastic and transversely isotropic constituents.The composite is first homogenised,the effective material tensor components are computed using the FEM-based additional computer program.Further,material parameters of the composite most decisive for its effective material properties are determined numerically.It should be underlined that the homogenisation method is generally an intermediate numerical tool applied to exclude the necessity of composite micro-scale discretisation and,in the same time,to reduce the total number of degrees of freedom of the entire model.On the other hand,there are many numerical
4 Sensitivity Analysis for Some Composites 4.1 Deterministic Problems As is known, the sensitivity analysis in engineering systems is employed to verify how input parameters of a specific engineering problem influence the analysed state functions (displacements, stresses and temperatures, for instance). The sensitivity coefficients [269], being the purpose of such an analysis, are computed using partial derivatives of the considered state function with respect to the particular input parameter(s). These derivatives can be obtained numerically starting from the fundamental algebraic equations system of the problem, for instance, or alternatively, by a simple derivation if only a closed form solution exists; some combined analytical-numerical methods are also known [99]. It is important to underline that this methodology is common for all discrete numerical techniques: Boundary Element Method (BEM) [51,206], Finite Difference Method (FDM) [90,206], FEM [7,21,387] as well as hybrid and meshless strategies [81]. From the computational point of view, there are the following numerical methods in structural design sensitivity analysis [75,76,103,134,207]: the Direct Differentiation Method (DDM), the Adjoint Variable Method (AVM) applied together with the Material Derivative Approach (MDA) or the Domain Parametrisation Approach (DPA) suitable for shape sensitivity studies. Considering these capabilities and, on the other hand, a very complex structure of composite materials, sensitivity analysis should be applied especially in design studies for such structures. Instead of a single (or two) parameters characterising the elastic response of a homogeneous structure, the total number of design parameters is obtained as a product of component numbers in a composite and the number of material and geometrical parameters for a single component. Even some extra state variables should be analysed to define interfacial behaviour, general interaction of the constituents and/or the lack of periodicity. Usually, to reduce the complexity of the original composite, the so-called effective homogenisation medium having the same strain (or complementary) energy is analysed. This chapter is devoted to general computational sensitivity studies of the homogenisation method for some periodic composite materials with linear elastic and transversely isotropic constituents. The composite is first homogenised, the effective material tensor components are computed using the FEM-based additional computer program. Further, material parameters of the composite most decisive for its effective material properties are determined numerically. It should be underlined that the homogenisation method is generally an intermediate numerical tool applied to exclude the necessity of composite micro-scale discretisation and, in the same time, to reduce the total number of degrees of freedom of the entire model. On the other hand, there are many numerical
186 Computational Mechanics of Composite Materials homogenisation techniques.They can be divided generally into two essentially different approaches:stress averaging (the boundary stresses are introduced between the composite constituents plus displacement-type periodicity conditions) and strain approach (uniform extensions of the RVE boundaries in various directions plus periodicity conditions on the remaining cell edges).Considering this,different results of the homogenisation method in terms of the effective material tensors are obtained(as a result quite different sensitivity gradients must be computed in these two approaches).The sensitivity analysis introduces a new aspect of the homogenisation technique-it can be verified if the homogenised and original structures have the same or even analogous (in terms of their signs) sensitivity gradients.The composites can be optimised then by manipulating its material or geometrical design parameters [310]as well as by choosing various constituent materials with computationally determined shape for the new designed composite structure. The sensitivity gradients are computed here by application of a homogenisation-oriented computer program MCCEFF according to the DDM implementation approach and presented as functions of the composite design parameters -Young moduli and Poisson ratios of the constituents.Since a finite difference scheme is used for the sensitivity gradient computations,numerical sensitivity of the final results to the increase of an arbitrarily introduced parameter must be verified.This numerical phenomenon makes it necessary to determine the most suitable interval of parameter increments for the particular effective elasticity tensor components. The entire computational methodology is illustrated with two examples-1D and 2D two component periodic composites.The closed form effective Young modulus is used in the first example,while the homogenisation function is to be computed in the second case.Both illustrations show that different components of the effective elasticity tensor show different sensitivities to particular mechanical properties of the original composite and,further,the illustrations make it possible to determine the most decisive elastic parameters for the homogenisation-based computational design studies.Quite similar sensitivity studies are carried out in the case of heat conductivity coefficient for 1D,2D and 3D two component composites. It should be noticed that sensitivity analysis can be used for validation of various homogenisation methods.In most cases an increase in Young moduli of composite components should result in a corresponding increase of the effective material tensor components;an opposite phenomenon can be observed for some specific cases,but usually in an extremely small range only.Therefore,if the sensitivity analysis shows that most of the gradients are negative,the homogenisation theory should be essentially corrected. The applied effective modulus method is verified below using the examples of 1D distributed heterogeneities in the periodic two-component bar structure and of the fibre-reinforced periodic composite.As is demonstrated for plane composite structure,the sensitivity gradients of a homogenised elasticity tensor show some instabilities observed for an extremely small value of the perturbation parameter
186 Computational Mechanics of Composite Materials homogenisation techniques. They can be divided generally into two essentially different approaches: stress averaging (the boundary stresses are introduced between the composite constituents plus displacement-type periodicity conditions) and strain approach (uniform extensions of the RVE boundaries in various directions plus periodicity conditions on the remaining cell edges). Considering this, different results of the homogenisation method in terms of the effective material tensors are obtained (as a result quite different sensitivity gradients must be computed in these two approaches). The sensitivity analysis introduces a new aspect of the homogenisation technique – it can be verified if the homogenised and original structures have the same or even analogous (in terms of their signs) sensitivity gradients. The composites can be optimised then by manipulating its material or geometrical design parameters [310] as well as by choosing various constituent materials with computationally determined shape for the new designed composite structure. The sensitivity gradients are computed here by application of a homogenisation-oriented computer program MCCEFF according to the DDM implementation approach and presented as functions of the composite design parameters - Young moduli and Poisson ratios of the constituents. Since a finite difference scheme is used for the sensitivity gradient computations, numerical sensitivity of the final results to the increase of an arbitrarily introduced parameter must be verified. This numerical phenomenon makes it necessary to determine the most suitable interval of parameter increments for the particular effective elasticity tensor components. The entire computational methodology is illustrated with two examples – 1D and 2D two component periodic composites. The closed form effective Young modulus is used in the first example, while the homogenisation function is to be computed in the second case. Both illustrations show that different components of the effective elasticity tensor show different sensitivities to particular mechanical properties of the original composite and, further, the illustrations make it possible to determine the most decisive elastic parameters for the homogenisation-based computational design studies. Quite similar sensitivity studies are carried out in the case of heat conductivity coefficient for 1D, 2D and 3D two component composites. It should be noticed that sensitivity analysis can be used for validation of various homogenisation methods. In most cases an increase in Young moduli of composite components should result in a corresponding increase of the effective material tensor components; an opposite phenomenon can be observed for some specific cases, but usually in an extremely small range only. Therefore, if the sensitivity analysis shows that most of the gradients are negative, the homogenisation theory should be essentially corrected. The applied effective modulus method is verified below using the examples of 1D distributed heterogeneities in the periodic two-component bar structure and of the fibre-reinforced periodic composite. As is demonstrated for plane composite structure, the sensitivity gradients of a homogenised elasticity tensor show some instabilities observed for an extremely small value of the perturbation parameter
Sensitivity Analysis for Some Composites 187 At the same time,for Poisson ratios values tending to their physical bounds,an uncontrolled increase of all sensitivity gradients is observed.That is why a continuation of this study is necessary in the context of computational error,to extend constitutive models of composite components as well as to evaluate geometrical and material sensitivity gradients for more complex heterogeneous structures,especially in the probabilistic context. Another important topic studied here is the application of the parameter finite difference analysis to the sensitivity analysis of the uniform plane strain problem of the real composite.This is done under the assumption that the RVE of plane cross- section is uniformly extended in two perpendicular directions and the unit shear strain is applied on the RVE.Therefore,the sensitivity functional is proposed as the elastic strain energy stored in the cell,which is treated as some type of representative strain state of the composite under real conditions.To reflect the real conditions of the composite service more accurately,the particular strain component can be scaled over some multipliers to illustrate pure horizontal and/or vertical extension of the composite specimen.The sensitivity of this functional is taken as a measure of influence of various material parameters on the overall behaviour of the composite.According to the previous results,we observe the Poisson ratio of the matrix as a dominating material parameter for the fibre- reinforced periodic composites with the RVE specified below. Finally,it should be mentioned that this sensitivity analysis is introduced and performed to validate the homogenisation theory itself.In the case when the external boundary conditions are known together with the micromorphology of a certain composite,the homogenisation theory makes it possible to determine the effective characteristics of this structure and,according to the sensitivity analysis the sensitivity gradients of both real and homogenised structures are computed.If these gradients have consistent signs and comparable values,the homogenisation algorithm proposed is useful in computational modelling;otherwise another method should be proposed.It can happen that some homogenisation theories(or even closed form equations)are valid for some specific boundary value problems and it can be verified in this way.Another promising field of application of such an analysis is optimization and/or identification of composite materials and structures. Sensitivity gradients cannot be obtained analytically if the homogenisation function components are determined numerically in some cell problem solutions. Hence,two separate ways can be followed,the first one being purely computational finite difference based studies,where the gradients are obtained as differences of some slightly modified homogenisation tests.Alternatively,a semi- analytical method can be implemented where the spatial averages of the constitutive tensor components (independent from homogenisation functions)are differentiated symbolically and the remaining part resulting from homogenisation FEM tests is analysed using the finite differences;analogous opportunities are available for probabilistic (and next stochastic)analyses.Taking into account the consistency of the Monte Carlo simulation application and the computational time savings,full numerical differentiation is implemented.A semi-analytical approach can be implemented partially in some mathematical symbolic computation
Sensitivity Analysis for Some Composites 187 At the same time, for Poisson ratios values tending to their physical bounds, an uncontrolled increase of all sensitivity gradients is observed. That is why a continuation of this study is necessary in the context of computational error, to extend constitutive models of composite components as well as to evaluate geometrical and material sensitivity gradients for more complex heterogeneous structures, especially in the probabilistic context. Another important topic studied here is the application of the parameter finite difference analysis to the sensitivity analysis of the uniform plane strain problem of the real composite. This is done under the assumption that the RVE of plane crosssection is uniformly extended in two perpendicular directions and the unit shear strain is applied on the RVE. Therefore, the sensitivity functional is proposed as the elastic strain energy stored in the cell, which is treated as some type of representative strain state of the composite under real conditions. To reflect the real conditions of the composite service more accurately, the particular strain component can be scaled over some multipliers to illustrate pure horizontal and/or vertical extension of the composite specimen. The sensitivity of this functional is taken as a measure of influence of various material parameters on the overall behaviour of the composite. According to the previous results, we observe the Poisson ratio of the matrix as a dominating material parameter for the fibrereinforced periodic composites with the RVE specified below. Finally, it should be mentioned that this sensitivity analysis is introduced and performed to validate the homogenisation theory itself. In the case when the external boundary conditions are known together with the micromorphology of a certain composite, the homogenisation theory makes it possible to determine the effective characteristics of this structure and, according to the sensitivity analysis the sensitivity gradients of both real and homogenised structures are computed. If these gradients have consistent signs and comparable values, the homogenisation algorithm proposed is useful in computational modelling; otherwise another method should be proposed. It can happen that some homogenisation theories (or even closed form equations) are valid for some specific boundary value problems and it can be verified in this way. Another promising field of application of such an analysis is optimization and/or identification of composite materials and structures. Sensitivity gradients cannot be obtained analytically if the homogenisation function components are determined numerically in some cell problem solutions. Hence, two separate ways can be followed, the first one being purely computational finite difference based studies, where the gradients are obtained as differences of some slightly modified homogenisation tests. Alternatively, a semianalytical method can be implemented where the spatial averages of the constitutive tensor components (independent from homogenisation functions) are differentiated symbolically and the remaining part resulting from homogenisation FEM tests is analysed using the finite differences; analogous opportunities are available for probabilistic (and next stochastic) analyses. Taking into account the consistency of the Monte Carlo simulation application and the computational time savings, full numerical differentiation is implemented. A semi-analytical approach can be implemented partially in some mathematical symbolic computation
188 Computational Mechanics of Composite Materials packages,where probabilistic moments can be derived according to the classical integral definitions,while the random fields of homogenising stresses averaged over the RVE are treated using the numerical differentiation approach. The results of computations in the form of deterministic derivatives or their probabilistic equivalents can next be implemented in deterministic and/or probabilistic optimisation problems based on the gradient techniques.Such an analysis will enable us to optimise various composites [84,240,264,281,320]using their homogenised models -without the necessity of complicated multiscale problem discretisation and their further solution.The main benefits of the integrated computational approach to the composites are (a)the most effective choice of composite components (sensitivity to the expected values of material parameters),(b)selection of the best processing technology from the necessary accuracy point of view (standard deviation levels),(c)efficient durability control and analysis(sensitivity to the interface and structural defects parameters),etc.The proposed method is significantly more complicated than the previous approaches. However it makes the computational model of composite materials and their behaviour more realistic and focused on the engineering analyses. 4.1.1 Sensitivity Analysis Methods The main aim of the structural design sensitivity analysis is to study the interrelation between the response (or state variables)of a structure determined from a solution for the boundary-value problem and design variables begin the input data for the solution process.Displacements,stresses,temperatures or velocities can be taken as the structural response measures,whereas such parameters as truss and beam cross-sectional areas,plate and shell thicknesses and material characteristics are usually chosen as design variables.Let us note that even for linear elastic problems the equilibrium equations may generally contain some nonlinear expressions for the state and design variables-this is the case of plate/shell thickness and/or truss lengths and,especially,material parameters in composites. The sensitivity gradients are the main numerical tool to evaluate the design sensitivity of a structure with respect to some design parameter.For engineers a more interesting issue is the overall sensitivity of the structure examined under general loading conditions than particular state function gradients.The gradients of the structural response functionals with respect to design variables give a useful measure of structural response variation together with the change of a given design input. The sensitivity analysis is especially applicable with common implementation with one of the well-established numerical methods of structural analysis,i.e.with the finite element formulation.To illustrate the main ideas let us consider the static structural response of a linear elastic system with N degrees of freedom defined by the functional [208]
188 Computational Mechanics of Composite Materials packages, where probabilistic moments can be derived according to the classical integral definitions, while the random fields of homogenising stresses averaged over the RVE are treated using the numerical differentiation approach. The results of computations in the form of deterministic derivatives or their probabilistic equivalents can next be implemented in deterministic and/or probabilistic optimisation problems based on the gradient techniques. Such an analysis will enable us to optimise various composites [84,240,264,281,320] using their homogenised models – without the necessity of complicated multiscale problem discretisation and their further solution. The main benefits of the integrated computational approach to the composites are (a) the most effective choice of composite components (sensitivity to the expected values of material parameters), (b) selection of the best processing technology from the necessary accuracy point of view (standard deviation levels), (c) efficient durability control and analysis (sensitivity to the interface and structural defects parameters), etc. The proposed method is significantly more complicated than the previous approaches. However it makes the computational model of composite materials and their behaviour more realistic and focused on the engineering analyses. 4.1.1 Sensitivity Analysis Methods The main aim of the structural design sensitivity analysis is to study the interrelation between the response (or state variables) of a structure determined from a solution for the boundary-value problem and design variables begin the input data for the solution process. Displacements, stresses, temperatures or velocities can be taken as the structural response measures, whereas such parameters as truss and beam cross-sectional areas, plate and shell thicknesses and material characteristics are usually chosen as design variables. Let us note that even for linear elastic problems the equilibrium equations may generally contain some nonlinear expressions for the state and design variables – this is the case of plate/shell thickness and/or truss lengths and, especially, material parameters in composites. The sensitivity gradients are the main numerical tool to evaluate the design sensitivity of a structure with respect to some design parameter. For engineers a more interesting issue is the overall sensitivity of the structure examined under general loading conditions than particular state function gradients. The gradients of the structural response functionals with respect to design variables give a useful measure of structural response variation together with the change of a given design input. The sensitivity analysis is especially applicable with common implementation with one of the well-established numerical methods of structural analysis, i.e. with the finite element formulation. To illustrate the main ideas let us consider the static structural response of a linear elastic system with N degrees of freedom defined by the functional [208]
Sensitivity Analysis for Some Composites 189 (hd)=Gga(hd),hd.d=1.2.....D:a=1.2....N (4.1) where G is a given function of structural displacements vector (g)and design variables,h represents a D-dimensional vector of design variables;the displacement vector satisfies classical equilibrium equations,i.e. Keke)galh）=Q.e） (4.2) The displacement vector is assumed to be an implicit function of design variables,because the stiffness matrix K and the load vector o are some functions of these variables. Now,the SDS analysis is employed to determine the changes of the structural response functional with variations in design parameters,so the so-called sensitivity gradient 03/oh"is to be determined.The chain rule of differentiation applied to(4.1)returns here 34=G4+G.9a (4.3) where (.)and (.denote first partial derivatives with respect to the dth design variable and the ath nodal displacement,respectively.The design variables hare introduced as the only arguments in the functions K and,therefore, partial derivatives of these functions with respect to h are in fact equal to the corresponding total derivatives.Nevertheless,there holds 0G/oh=in case of G.Since it is an explicitly given function of h and the derivatives G and G may be computed directly,while is to be determined numerically. The first technique for computing of the sensitivity gradients known as the direct differentiation method (DDM)extensively employed in structural optimisation reflects the following algorithm.Let us assume that K and are continuously differentiable with respect to the design variablesh; then,the vector is also continuously differentiable.Differentiation of both sides of(4.2)with respect to ha gives Ko898 Qa -KoB9B (4.4) Since the stiffness matrix K is assumed to be nonsingular,(4.4)can be solved for it yields
Sensitivity Analysis for Some Composites 189 ( ) [ ( ) ] d d d ℑ h = G qα h , h , d=1,2,…,D; α = 1,2,...,N (4.1) where G is a given function of structural displacements vector ( α q ) and design variables, d h represents a D-dimensional vector of design variables; the displacement vector satisfies classical equilibrium equations, i.e. ( ) ( ) ( ) d d d Kαβ h qβ h = Qα h (4.2) The displacement vector is assumed to be an implicit function of design variables, because the stiffness matrix Kαβ and the load vector Qα are some functions of these variables. Now, the SDS analysis is employed to determine the changes of the structural response functional with variations in design parameters, so the so-called sensitivity gradient d ∂ℑ ∂h is to be determined. The chain rule of differentiation applied to (4.1) returns here d d d G G q. . . . ℑ = + α α (4.3) where .d (.) and .α (.) denote first partial derivatives with respect to the dth design variable and the αth nodal displacement, respectively. The design variables d h are introduced as the only arguments in the functions ℑ, Kαβ , qα , Qα and, therefore, partial derivatives of these functions with respect to d h are in fact equal to the corresponding total derivatives. Nevertheless, there holds d d G h . ∂ ∂ = ℑ in case of G. Since it is an explicitly given function of d h and qα , the derivatives d G. and G.α may be computed directly, while d q. α is to be determined numerically. The first technique for computing of the sensitivity gradients known as the direct differentiation method (DDM) extensively employed in structural optimisation reflects the following algorithm. Let us assume that ( ) d Kαβ h and ( ) d Q h α are continuously differentiable with respect to the design variables d h ; then, the vector ( ) d qβ h is also continuously differentiable. Differentiation of both sides of (4.2) with respect to d h gives Kαβ qβ Qα Kαβ qβ .d .d .d = − (4.4) Since the stiffness matrix Kαβ is assumed to be nonsingular, (4.4) can be solved for d q. β ; it yields