efﬁcient solution for composites reinforced by...

Efficient Solution for CompositesReinforced by Particles

Vladimı́r Kompiš, Mário Štiavnický and Qing-Hua Qin

Abstract A very effective method for some kind of problems is the Method ofFundamental Solutions (MFS). It is a boundary meshless method which does notneed any mesh and in linear problems only nodes (collocation points) on the domainboundaries and a set of source functions (fundamental solutions, i.e. Kelvin func-tions) in points outside the domain are necessary to satisfy the boundary conditions.Another kind of source functions can be obtained from derivatives of the Kelvinsource functions. Dipoles are the derivatives of Kelvin function in direction of actingforce.

Putting the dipoles into the particle, i.e. outside the domain of the matrix ofcomposite material, they can very efficiently simulate the interaction of the particlewith the matrix and with the other particles. A dipole satisfies both the force andmoment equilibrium and so the models do not require any additional condition forthe simulation of interactions. If the particles of reinforcing material are in the formof spheres, or ellipsoids, a triple dipole (i.e. dipoles in the direction of main axes ofthe particle) can give satisfactory accuracy for practical problems. It is given howthe models can be used for homogenization of composite material.

A novel method is used for evaluation of material properties of homogenizedmaterial in order to increase the numerical efficiency of the composite. The numeri-cal results are compared with Mori-Tanaka analytic models with all rigid inclusionsas well as for problems with elastic inclusions when the differences in stiffness ofmatrices and particle are small.

1 Introduction

Efficiency of a numerical method depends on how accurately it approximates thereal conditions. It is closely related to the number of interpolation functions and

V. Kompiš (B)Department of Mechanical Engineering, Academy of Armed Forces of General M. R. Štefánik,Demänovská 393, Liptovský Mikuláš, 03119, Slovakiae-mail: [email protected]

A chapter in honor of Dimitri Beskos’ 65th birthday

G.D. Manolis, D. Polyzos (eds.), Recent Advances in Boundary Element Methods,DOI 10.1007/978-1-4020-9710-2 18, C© Springer Science+Business Media B.V. 2009

277

278 V. Kompiš et al.

complexity of evaluated expressions necessary to satisfy the governing equationsand boundary conditions of a given problem. Usually geometric concentrators,holes, inclusions, material inhomogeneities, contact with other bodies and otherirregularities lead to local effects and large gradients in all mechanical, thermal,electromagnetic and coupled fields. In mechanics they lead to the effects known asstress concentration. Moreover, if such peculiarities are close to each other, or to theboundary, they can very strongly interact and further influence the fields.

Accurate approximation of the local effects is necessary to correct evaluationof such parameters as local and global behaviour of the structure, local and globaldamage conditions and other important properties.

The approximate function used to simulate the real structure will be moreefficient, which will best approximate both governing equations and boundaryconditions. The unknown functions are mostly approximated by polynomials. Cor-rect approximation of large gradients requires very fine subregions, submeshing,remeshing during the solution when solution errors are controlled, etc. This reducesthe efficiency of the model. If a structure can be defined as a continuum, then thelocal effects decay with the distance from the concentrator. Boundary type formula-tions, which keep the decaying effect, need much less parameters to obtain similaraccuracy than the domain methods.

A very effective method for some kind of problems is the Method of FundamentalSolutions (MFS) (Golberg and Chen 1998; Karageorghis and Fairweather 1989).It is a boundary meshless method which does not need any mesh and in linearproblems only nodes (collocation points) on the domain boundaries and a set ofsource functions (fundamental solutions) in points outside the domain are necessaryto satisfy the boundary conditions. The MFS uses discrete source functions, thefundamental solutions, to approximate the fields in the domain. In domains withsmooth boundary, if the boundary conditions are smooth it is necessary to use verymany source points acting in some distances from the boundary in order to simu-late smooth functions along the boundaries. Using polynomial Trefftz (T-)functions(Jirousek and Wroblewski 1997; Kompiš and Štiavnický 2006) in combination toMFS can improve both accuracy and numerical stability of the model. Oppositely,if the boundary conditions are discontinuous like contact problems of bodies withcurved boundaries of different curvature, it is necessary to use continuous distribu-tion of source functions placed along the boundary with corresponding discontinuity(Kompiš and Dekýš 2003) for the best approximation.

Problems with inhomogeneous micro/nano-structure and especially compositematerials with stiff or weak inclusions are especially important (Shonaike andAdvani 2003; Atieh et al. 2005; Schwarz et al. 2004). In structures like these, theboundary conditions are usually smooth, if the inhomogeneity surface is smooth,but the inhomogeneities introduce large gradients in local fields and many inhomo-geneities would require to solve millions or billions equations after discretizationby classical methods like FEM or BEM. It will be shown here how the use ofdiscrete source functions, dipoles, in problems with small aspect ratio can leadto very effective solution for homogenization of Representative Volume Elements(RVE). The method has all attractive features of MFS, moreover, the inclusions

Efficient Solution for Composites Reinforced by Particles 279

in composite material do not contain any load for static isothermic problems andthe loads by dipoles simulating the interaction are in force and moment equilib-rium and do not need any additional conditions. Also the rigid body modes ofparticles are correctly introduced. The presented method can use all features of theFast Multipole Solution (Greengard and Rokhlin 2002; Nishimura 2002; Nishimuraet al. 1999) and of the Boundary Point Method (Ma and Qin, 2007; Ma et al. 2008;Zhang et al. 2004), i.e. reduced computations for far field interaction, moreover, alsothe near field interaction is considerably reduced in our formulations. Comparingto MFS usually only few collocation points are necessary to obtain results withrequired accuracy as the source points do not introduce large gradients along theinterdomain (particle-matrix) surface.

A cube which contains specified distribution of reinforcing particles is chosenfor RVE and auxiliary particles placed outside the RVE are used in order to increasethe efficiency of the model for evaluation of homogenized material constants of thecomposite. The numerical results are compared with Mori-Tanaka analytic modelswith all rigid inclusions as well as for problems with elastic inclusions when thedifferences in stiffness of matrices and particle are small.

2 Description of the Model

Under discrete source functions we will understand all unit forces (the Kelvin solu-tion (Beskos 1991; Cheng and Cheng 2005)), point dislocations, dipoles, couples,or some other type of Radial Basis Functions (RBF) satisfying homogeneous gov-erning equations in infinite space or half space except for the source point forceitself, i.e. this type of source functions are Trefftz functions, if the source points arelocated outside the discretized domain. A very attractive concept is dipoles havingmechanical meaning of two collinear forces acting in the same point in oppositedirection. Mathematically it is a derivative of the Kelvin function in direction ofacting force. The displacement field of a dipole is

U (D)pi = U (F)pi,p = −1

16πG (1 − ν)1

r2[3r,i r

2,p − r,i + 2 (1 − ν) r,pδi p

](1)

The upper index (D) denotes corresponding dipole field and (F) a force (Kelvin)field. The lower indices belong to components of the field; the first index corre-sponds to the component of the source quantity and the other indices the componentsof the field quantity. The index after comma denotes partial derivative, i.e.

r,i = �r/

�xi (t) = ri/r (2)

where r is the distance between the source point s, where the dipole or force areacting and the field point t , where the displacement (field variable) is expressed, i.e.

r = √riri , ri = xi (t) − xi (s) (3)


and xi are coordinates of the points. G, υ and δi j are material shear modulus, Pois-son ratio and Kronecker delta function, respectively. Summation convection overrepeated indices will be used, but it will not act over the repeatedindex p.

Corresponding strain and stress fields (see Kompiš et al. (2008) for more details)are

E (D)pi j =1

2

(U (D)pi, j + U (D)pj,i

)= − 1

16πG (1 − ν)1

r3[−15r,i r, j r,p + 3r,i r, j+

+ 2 (1 − 2ν) δi pδ j p + 6ν(δi pr, j r,p + δ j pr,i r,p

) + δi j(3r2,p − 1

)](4)

S(D)pi j = 2G E (D)pi j +2Gν

1 − 2νδi j E(D)pkk

= − 18π (1 − ν)

1

r3[(1 − 2ν) (2δi pδ j p + 3r2,pδi j − δi j

)+

+ 6νr,p(r,iδ j p + r, jδi p

) + 3 (1 − 5r2,p)

r,i r, j]

(5)

If unit forces acting in source points are located in discrete points outside thesolution domain for computational models and also the collocation points (i.e. thepoints in which the boundary conditions have to be satisfied) are chosen in somediscrete points of the domain boundary, then the method of solution is known as theMethod of Fundamental Solutions (MFS) (Golberg and Chen 1998; Karageorghisand Fairweather 1989). The method is very simple one, it does not need any ele-ments and any integration and thus, it is a fully meshless method. These functionsare Trefftz functions and they serve as interpolators in the whole domain. Also anyother Trefftz functions can be used for this purpose (Kompiš and Štiavnický 2006).They are very convenient to modeling of inhomogeneous materials with spherical,ellipsoidal, or other smooth inclusions or holes, especially when the density of par-ticles is small. Note that the domain which is approximated by source functionsis the matrix and thus, the domains of particles are outside of the domain. A dipolelocated inside the particle, i.e. outside of the domain represented by the matrix, givesboth zero resulting force and moment along any closed surface and thus the globalequilibrium is not destroyed by local errors as it can be by using MFS (Kompiš andŠtiavnický 2006), however, the location of the source points is important for the bestsimulation of continuity and equilibrium along the interdomain boundaries.

In micromechanics (MM) the material properties are homogenized over the RVE.Corresponding average homogeneous material properties are obtained by integralsover the RVE (Qu and Cherkaoui 2006). The boundary conditions can influencethe results and the inhomogeneities closed to boundaries give large gradients inintegration surface.

In the present model the strains and stresses are split into a constant part (verysmall RVE is considered for MM) and a local part (corresponding to so calledeigenstrain (Ma et al. 2008; Qu and Cherkaoui 2006)).


εtoti j = ε∞i j + εloci jσ toti j = σ∞i j + σ loci jutoti j = u∞i j + uloci j (6)

where the upper indices tot, ∞ and loc denote the total component, componentcorresponding to constant strains and stresses acting in infinity on the continuousmatrix material without any particles in it and local components acting in the matrixwith eigenstrains from dipoles (including the eigenstrains in auxiliary points outsidethe RVE) with zero loads in infinity, respectively.

Corresponding displacements vary linearly in the RVE. In order to reduce thegradients we located auxiliary source points outside the RVE so that the local trac-tions are equal to zero (see Fig. 1) in the control points on the surface of the RVE.The problem in Fig. 1 is presented in the plane x1x3 for better understanding themodel.

On each interface of matrix-inclusion six pairs of control points are chosen toprescribe the local displacements as shown in Fig. 2. A triple dipole (i.e. dipole ineach coordinate direction) is placed into the centre of the inclusion and into auxiliarysource points and there intensities are computed so that boundary displacementson inclusions and tractions on the RVE surface are satisfied. Note that we do notwork with displacements in the model but with the differences of displacements indirection of vectors connecting corresponding points on opposite boundary of theparticle.

x3

x1

RVE

auxiliary dipoles

control points

inhomogeneity

L R midpoints

Fig. 1 RVE computational model


Fig. 2 Spherical inclusionand control points

x3

x1

x2

control points

In order to reduce the computations an elimination-iterative procedure is devel-oped. In the first step the intensities of dipoles inside the inclusions (internal dipoles)are chosen to be equal those without influence the interaction and intensities in theauxiliary dipoles outside the RVE are computed. In the second step the intensities ininternal dipoles are calculated from interaction of all dipoles and averaged multiplieris chosen for internal dipoles. Then intensities in auxiliary dipoles are recalculated.In the next steps the intensities of auxiliary dipoles are computed from intensities ofinternal dipoles by solving system of linear equations and intensity of each internaldipole is found from interaction with all dipoles (i.e. no averaging is used again).

The elastic constants of homogenized material are evaluated from the elasticenergy. For this purpose both average strains and stresses are to be evaluated inthe RVE. The total strains and stresses consist of the constant components (as givenfor material without inhomogeneities) and local strains (eigenstrains as defined inQu and Cherkaoui (2006)) and the local stresses. The local averaged strains areevaluated from the total deformation of RVE by action of the dipoles. For this pur-pose the displacements of control points on the boundaries of the RVE are used anddifference between the displacements on opposite boundaries are evaluated. In thisway the strains are evaluated as the averaged deformation of the RVE. This defor-mation has to introduce the integral value of the whole RVE. As the displacementfunction of the RVE surface has complicated form given by the interaction of alldipoles (see Fig. 3), two procedures were used as follows. The control points wereregularly distributed along the RVE surface. The midpoint and trapezoidal ruleswere used in the procedures. In regularly distributed particles the number of controlpoints for midpoint rule integration was identical to the number of particles closestto corresponding side of the RVE. For trapezoidal rule points between these pointsand along the edges of the RVE were included. In following examples only axialreinforcement was evaluated from axial strains as

εlocii =1

Fi

ˆui d Fi (7)

where Fi is crossectional area of the RVE with normal in direction of xi coordinateaxis. The integral is evaluated numerically.

The local averaged surface tractions, identical to corresponding normal stresscomponent, are evaluated from the sum of dipole intensities inside the RVE in each


a b

Fig. 3 Deformation of the matrix reinforced with a patch of particles

coordinate direction. The component sum is substituted by a “finite dipole” definedas constant traction acting in normal direction to corresponding side of the RVEand the distance between opposite sides. This is identical to division of the sum ofcorresponding component of the dipoles by the total volume of the RVE.

σ locii =∑

DDi /VRV E (8)

where Di is intensity of a dipole in direction of xi coordinate axis.Elastic constants for the homogenized material are evaluated from the elastic

energy

Ee = 12σ toti j ε

toti j =

1

2

(σ∞i j ε

∞i j + σ∞i j εloci j + σ loci j ε∞i j + σ loci j εloci j

)(9)

and the stiffening is defined by relative increase of corresponding component of theelastic modulus.

The procedure will be explained on following examples.

3 Examples and Discussion

Let the matrix is an elastic material with modulus of elasticity E = 1000 andPoisson’s ratio ν = 0.3. Further let rigid spherical inclusions with radius R areregularly distributed in the RVE (Fig. 1) and there are 8 × 8 × 8 inclusions insidethe RVE. The hydrostatic stress state is chosen in this example with hydrostatictension p = σ11 = σ22 = σ33 = 25, which corresponds to constant strain ofthe RVE without inclusion ε11 = ε22 = ε33 = 0.01. There are 512 triple-dipoles


inside the RVE and 384 auxiliary triple-dipoles outside the RVE. If the problemwould be solved taking into account all interactions, it has to be solved a system of5760 × 5760 (= 3.32 × 107 coefficients!) linear equations. Instead 1152 × 1152equations was solved in 3–4 iterations (relative errors less than 0.5 %) for differentpercentage of the inclusions in the RVE, solution time in MATLAB in a notebookwas less than 5 min. The stiffening effect was evaluated as an increase of stiffnessin direction of coordinate axis x1.

Note that if the elastic inclusions are considered then the distance of the controlpoints on the interface changes according to corresponding tractions. For the stressstate in inclusion it is sufficient to use constant terms. They correspond to linearterms in displacement fields. Recall that if Trefftz polynomials are used for approx-imation of homogeneous equilibrium equations then all constant and linear termsare Trefftz polynomials, i.e. they satisfy the homogeneous equilibrium equations(Kompiš and Štiavnický 2006) and there are 12 independent displacement modescorresponding to the constant and linear terms. Six modes correspond to rigid bodydisplacements and another 6 terms to independent constant stress states. One neednot keep for rigid body terms of the inclusions in the present formulation and theyare correctly simulated in all inclusions contained in the RVE model and only 3d.o.f. (3 dipoles) for simulation of each inclusion are necessary for approximatesolution of the problem in this model.

Table 1 gives the stiffening of the RVE for different relative volume of stiffen-ers (1st column), L/R (where is the radius of particles and L is their distance) andpresent results are compared with Mori-Tanaka (Qu and Cherkaoui 2006) modelsas well. The last two columns give the results obtained by M.-T. and numericalmodels when the ratio of modules of elasticity of particle to the one of matrix is2 : 1. As the Mori-Tanaka’s (M.-T.) model is also an approximate model obtainedanalytically by simplified assumptions. The M.-T. model does not enable to studyall fields by different configurations (probabilistic distribution of particles, proba-bilistic dimensions of particles, influence of the domain boundaries, etc. as it is inthe present model.

Figure 3 shows deformation of the matrix reinforced with a patch of particles(only regularly distributed spherical particles with constant radius are shown in thispaper). It is possible to see that the displacements are not constant between theparticles because of the boundary effect of the patch. Recall that we have chosen theboundary conditions prescribed by constant (zero) local tractions in control points.Instead constant displacements can be chosen in such points.

Table 1 Stiffening effect

% Stiffener L/R M.-T., rigid Numerical, rigid M.-T., elastic Numerical, elastic

0.0155 30 1.000316 1.000340 1.000104 1.0001281 7.4882 1.0205 1.0226 1.0067 1.00833.35 5 1.0684 1.0744 1.0227 1.027915.5 3 1.3218 1.3746 1.1095 1.1341


4 Conclusions

We have shown that discrete source functions, which can be defined as Trefftz radialbasis functions, are convenient approximation or interpolation functions for compu-tational simulation (Oden et al. 2006) of material behaviour with local fields containlarge gradients. They can very accurately simulate the decaying of the large gradi-ents and thus, they are very useful tool for multiscale modeling and for simulationof large subdomains containing many large gradients inside. Problems with materialinhomogeneities are important examples for practical applications. The hard inclu-sions are only one problem of the application. Also weak or mixed hard and weakinhomogeneities or microvoids can be solved very effectively by this technique.Very important applications come for materials with micro- or nano-structure, whereinteraction of micro/nano-particles with considerably larger particles comes intoeffect (Perez 2005). Important simulation of materials reinforced with hard particlesis also the investigation of the interaction such material with free boundaries andthe behavior of thin surface layers reinforced with the particles. Simulation of suchproblems is considered in the next future.

Presented model can be used for reinforcing particles with the aspect ratio closeto one. If the aspect ratio is large, continuous source functions are to be used forbetter accuracy of the model as described in Kompiš, et al. (2008).

The interaction of fibre like inclusions is very complicated and very large gra-dients can rise in different part of the fibre. Discrete source functions are not ableto simulate correctly the continuity conditions between fibre and matrix and contin-uous distribution of source functions is much more effective for this purpose thanother methods.

Much research has to be done in all these simulations, as many effects notincluded into present models are important for correct simulation. We can mentionthe inter-layers in very small (nano-structures), curved fibers, nonlinear materialbehavior, interaction between micro/nanoparticles and polymeric and other matrixmaterials with fiber like structure, etc.

Acknowledgments The support of NATO (grant 001-AVT-SVK) and Slovak Agency APVV(grant APVT-20-035404) is gratefully acknowledged.

References

M. A. Atieh, et al., Multi-wall carbon nanotubes/natural rubber nanocomposites, AzoNano –Online Journal of Nanotechnology, 1, (2005), pp. 1–11.

D. E. Beskos, ed. Boundary Element Analysis of Plates and Shells. Springer-Verlag, Berlin (1991).A. H. D. Cheng, D. T. Cheng, Heritage and early history of the boundary element method,

Engineering Analysis with Boundary Elements, 29, (2005), pp. 268–302.M. A. Golberg, C. S. Chen, The method of fundamental solutions for potential, Helmholtz and

diffusion problems, in M. A. Golberg ed. Boundary Integral Methods – Numerical and Mathe-matical Aspect, Computational Mechanics Publications, Southampton (1998), pp. 103–176.

F. L. Greengard, V. Rokhlin, A fast algorithm for particle simulations, Journal of ComputationalPhysics, 73, (1987), pp. 325–348.


J. Jirousek, A. Wroblewski, T-elements: State of the art and future trends, Archive of AppliedMechanics, 3, (1997), pp. 323–434.

A. Karageorghis, G. Fairweather, The method of fundamental solutions for the solution ofnonlinear plane potential problems, IMA Journal Numerical Analysis, 9, (1989), pp. 231–242.

V. Kompiš, M. Štiavnický, M. Kompiš, Z. Murčinková, Q.H. Qin, Method of continuous sourcefunctions for modelling of matrix reinforced with finite fibres, in V. Kompiš, ed. Compositeswith Micro- and Nano-Structure, Chapter 3, (2008), pp. 27–46. Springer, New York.

V. Kompiš, V. Dekýš, Effective evaluation of local contact fields, in Proceedings of the 4th Inter-national Congress of Croatian Society of Mechanics, Bizovac, 2003, Croatia, (2003), CDROM.

V. Kompiš, M. Štiavnický, Trefftz functions in FEM, BEM and meshless methods, ComputerAssisted mechanics and Engineering Sciences, 13, (2006), pp. 417–426.

H. Ma, Q.-H. Qin, Solving potential problems by a boundary-type meshless method–the bound-ary point method based on BIE, Engineering Analysis with Boundary Elements, 31, (2007),pp. 749–761.

H. Ma, Q.-H. Qin, V. Kompiš, Computational models and solution procedure for inhomoge-neous materials with eigen-strain formulation of boundary integral equations, in V. Kompiš,ed., Composites with Micro- and Nano-Structure, Chapter 13, (2008), pp. 239–256. Springer,New York.

N. Nishimura, Fast multipole accelerated boundary integral equations, Applied MechanicsReviews, 55, (2002), pp. 299–324.

N. Nishimura, K. Yoshida, S. Kobayashi, A fast multipole boundary integral equation method forcrack problems in 3D, Engineering Analysis with Boundary Elements, 23, (1999), pp. 97–105.

J. T. Oden et al. Simulation – Based Engineering Science: Revolutionazing Engineering Sciencethrough Simulation. Report NSF Blue Ribbon Panel, (2006).

I. Perez, Polymer Nanotube Composites, RTO-MP-AVT-122 (2005), pp. 13-1–13-11.J. Qu, M. Cherkaoui, Fundamentals of Micromechanics of Solids, John Wiley &Sons, Hoboken,

New Jersey (2006).J. A. Schwarz, C. I. Contescu, K. Putyera, eds., Nanoscience and Nanotechnology, Vols. 1–5,

Marcel Dekker, New York (2004).G. O. Shonaike, S.G. Advani, eds., Advanced Polymeric Materials, Structure Property Relation-

ships, CRC Press, Boca Raton (2003).J. M. Zhang, M. Tanaka, T. Matsumoto, Meshless analysis of potential problems in threee dimen-

sions with the hybrid boundary node method, International Journal for Numerical Methods inEngineering, 59, (2004). pp. 1147–1160.

ContentsStability Analysis of PlatesMulti-Level Fast Multipole BEM for 3-D ElastodynamicsA Semi-Analytical Approach for Boundary Value Problems with Circular BoundariesThe Singular Function Boundary Integral Method for Elliptic Problems with Boundary SingularitiesFast Multipole BEM and Genetic Algorithms for the Design of Foams with Functional-Graded Thermal ConductivityAn Integral Equation Formulation of Three-Dimensional Inhomogeneity ProblemsEnergy Flux Across a Corrugated Interface of a Basin Subjected to a Plane Harmonic SH WaveBoundary Integral Equations and Fluid-Structure Interaction at the Micro ScaleA 2D Time-Domain BEM for Dynamic Crack Problems in Anisotropic SolidsSimulation of Elastic Scattering with a Coupled FMBE-FE ApproachAn Application of the BEM Numerical Green's Function Procedure to Study Cracks in Reissner's PlatesGeneral Approaches on Formulating Weakly-Singular BIES for PDESDynamic Inelastic Analysis with BEM: Results and NeedsQuantifier-Free Formulae for Inequality Constraints Inside Boundary ElementsMatrix Decomposition Algorithms Related to the MFS for Axisymmetric ProblemsBoundary Element Analysis of Gradient Elastic ProblemsThe Fractional Diffusion-Wave Equation in Bounded Inhomogeneous Anisotropic Media. An AEM SolutionEfficient Solution for Composites Reinforced by ParticlesDevelopment of the Fast Multipole Boundary Element Method for Acoustic Wave ProblemsSome Issues on Formulations for Inhomogeneous Poroelastic MediaAxisymmetric Acoustic Modelling by Time-Domain Boundary Element TechniquesFluid-Structure Interaction by a Duhamel-BEM / FEM CouplingBEM Solution of Creep Fracture Problems Using Strain Energy Density Rate ConceptMFS with RBF for Thin Plate Bending Problems on Elastic FoundationTime Domain B-Spline BEM Methods for Wave Propagation in 3-D Solids and Fluids Including Dynamic Interaction Effects of Coupled MediaA BEM Solution to the Nonlinear Inelastic Uniform Torsion Problem of Composite BarsTime Domain BEM: Numerical Aspects of Collocation and Galerkin FormulationsSome Investigations of Fast Multipole BEM in Solid MechanicsThermomechanical Interfacial Crack Closure: A BEM ApproachAuthor IndexABCDFGHIJKLMPQRSTWXYZ

Subject Index

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