NUMERICAL SOLUTION OF DYNAMIC PROBLEMS IN COUPLE ...

INTERNATIONAL JOURNAL OF c© 2011 Institute for ScientificNUMERICAL ANALYSIS AND MODELING, SERIES B Computing and InformationVolume 2011, Number 2-3, Pages 215–230

NUMERICAL SOLUTION OF DYNAMIC PROBLEMS

IN COUPLE-STRESSED CONTINUUM

ON MULTIPROCESSOR COMPUTER SYSTEMS

VLADIMIR SADOVSKII, OXANA SADOVSKAYA, AND MARIYA VARYGINA

Abstract. Computational algorithm for the solution of 2D and 3D dynamic problems of theCosserat elasticity theory using multiprocessor computer systems is worked out. The algorithm isbased on the two-cyclic splitting method with respect to the spatial variables in combination withthe explicit monotone finite-difference ENO–scheme for solving 1D problems adapted for the cal-

culation of discontinuities. Boundary conditions of symmetry, permitting to reduce computationsmany times, are formulated. Computations of 3D Lamb’s problem on the action of concentratedload on the surface of a homogeneous elastic half-space and the problem on the action of con-centrated impulsive periodic load are performed. The shock waves of four types (longitudinal,transverse, torsional and rotational waves), characteristic of a couple-stressed elastic medium, arerecognized numerically. The oscillations of rotational motion of particles on the wave fronts arealso found.

Key words. Cosserat continuum, dynamics, elasticity, shock-capturing method, finite-differencescheme, and parallel computational algorithm.

Introduction

One hundred years celebrated last year from the date of appearance of the firsttheory of a continuum with internal rotations (the Cosserat continuum), whichinitiated the development of mechanics of continua with microstructure. The firstwork by brothers Eugene and Francois Cosserat [1], dedicated to the constructionof the couple-stressed elasticity theory, was published in 1909. Further developmentof this theory refers to the sixties of XX century [2, 3, 4, 5].

Mathematical model of the Cosserat elasticity theory, which takes into accountthe microstructure of materials, is used to describe the stressed-strained state ofcomposites, powdered, granular, micropolar and liquid-crystal materials. Simplifi-cation of this model is a reduced Cosserat model where additional rotational degreesof freedom are taken into account, but the couple stresses are not considered, atfirst it was suggested in [6]. Current researches on modeling of nanoscale structuresshow that the Cosserat model results from a limit transition in discrete molecular-dynamic models with an unbounded increase in a number of particles [7, 8, 9].Therefore, in the near future this model will find wide application.

One of the main factors constraining the analysis of models of non-classical mediais a lack of information about the material constants. This, in turn, opposes theadaptation of such models into practice of calculating dynamic characteristics ofgranular materials. First publications about experimental determination of theelasticity parameters in media with microstructure appeared in 1970 – 1980 years.Parameters for some materials such as bone tissue, highly porous polymer materialsand composites were obtained in [10, 11, 12].

Received by the editors October 7, 2010 and, in revised form, May 19, 2011.2000 Mathematics Subject Classification. 35L50, 65Y05, 74S20.

215

216 V. SADOVSKII, O. SADOVSKAYA, AND M. VARYGINA

A fundamental difference between the model of the couple-stressed elasticitytheory and the classical one is that the former implicitly includes the small para-meter that characterizes the size of particles in the microstructure. As a result, inorder to obtain correct numerical solutions, computations must be performed ona grid whose meshes are smaller than the characteristic size of particles. To solve2D and 3D dynamic problems, parallel algorithms can be efficiently used becausethey make it possible to distribute the computational load between multiple nodesof a cluster. The use of distributed computing allows to do much more fine grids,thereby increasing the accuracy of numerical solutions.

A part of the results of the present paper was reported at the Fifth Conferenceon Finite Difference Methods: Theory and Applications (June 28 – July 2, 2010,Lozenetz, Bulgaria).

1. Mathematical model

In the model of the Cosserat continuum, along with translational motion which isdefined by the velocity vector v, independent rotations of particles with the angularvelocity vector ω are considered and, along with the stress tensor σ with nonsym-metric components, the nonsymmetric tensor m of couple stresses is introduced.Complete system of equations of the model consists of the motion equations, thekinematic equations, and the generalized law of the linear elasticity theory:

(1)

ρ v = ∇ · σ + ρ g, j ω = ∇ ·m− 2 σa + j q,

Λ = ∇v + ω, M = ∇ω,

σ = λ (δ : Λs) δ + 2µΛs + 2αΛa,

m = β (δ : M s) δ + 2 γM s + 2 εMa.

Here ρ is the material density, g and q are the vectors of body forces and moments,Λ and M are the strain and curvature tensors, j is the product of the momentof inertia of a particle about to the axis passing through its center of gravity bythe number of particles in a unit volume, δ is the metric tensor, and λ, µ, α,β, γ, ε are the phenomenological elasticity coefficients for an isotropic medium.The conventional notations and operations of the tensor analysis are used: a colondenotes double convolution of tensors, a dot over a symbol denotes a derivative withrespect to time, and superscripts s and a denote the symmetric and antisymmetrictensor components. Where needed, the antisymmetric component is identified withthe corresponding vector. In particular, the vector of the tensor σa = (σ − σ∗)/2appears in the equations of motion (an asterisk denotes a tensor transposition).

The formula r =√

5 j/(2 ρ) is valid to estimate the linear parameter of micro-structure of a material. It is based on the model approximation about a mediumas dense packing of an ensemble of ball-shaped particles of identical radius r.

For 3D case the system of equations (1) in a Cartesian coordinate system hasthe next expanded form:

(2)

ρ vi = σ1i,1 + σ2i,2 + σ3i,3 + ρ gi,

a1 σii + a2 (σkk + σll) = vi,i,

a3 σik + a4 σki = vk,i − ωl, a4 σik + a3 σki = vi,k + ωl,

j ωi = m1i,1 +m2i,2 +m3i,3 + σkl − σlk + j qi,

b1 mii + b2 (mkk + mll) = ωi,i,

b3 mik + b4 mki = ωk,i, b4 mik + b3 mki = ωi,k.

NUMERICAL SOLUTION OF SOME DYNAMIC PROBLEMS 217

System (2) includes 24 equations with respect to 24 unknown functions. The sub-scripts after a comma denote the partial derivatives with respect to the correspon-ding spatial coordinates. For brevity, we use the notations:

i, k, l = 1, 2, 3, i 6= k 6= l, k = i+ 1 mod 3, l = k + 1 mod 3,

a1 =λ+ µ

µ (3λ+ 2µ), a2 = − λ

2µ (3λ+ 2µ), a3 =

µ+ α

4µα, a4 =

α− µ

4µα,

b1 =β + γ

γ (3 β + 2 γ), b2 = − β

2 γ (3 β + 2 γ), b3 =

γ + ε

4 γ ε, b4 =

ε− γ

4 γ ε.

The system of equations (1) can be written in matrix form, convenient for nu-merical realization, [13]:

(3) A U =

n∑

i=1

Bi U,i +QU +G,

where n is the spatial dimension of a problem, U is the vector-function consisting ofthe components of the velocity and angular velocity vectors and the nonsymmetricstress and couple stress tensors. In two-dimensional problems

U =(

v1, v2, σ11, σ22, σ33, σ12, σ21, ω3, m23, m32, m31, m13

)

,

in three-dimensional problems

U =(

v1, v2, v3, σ11, σ22, σ33, σ23, σ32, σ31, σ13, σ12, σ21,

ω1, ω2, ω3, m11, m22, m33, m23, m32, m31, m13, m12, m21

)

.

The matrix coefficients A and Bi, containing the elasticity parameters of a mate-rial, are symmetric, Q is antisymmetric, G is the given vector of body forces andmoments. The matrix A is positive definite if its diagonal blocks

a1 a2 a2a2 a1 a2a2 a2 a1

,

(

a3 a4a4 a3

)

,

b1 b2 b2b2 b1 b2b2 b2 b1

,

(

b3 b4b4 b3

)

are positive definite. According to the Sylvester criterion, this condition restrictsthe admissible values of the material parameters:

(4)3λ+ 2µ > 0, µ > 0, α > 0;

3 β + 2 γ > 0, γ > 0, ε > 0.

If the inequalities (4) are fulfilled, the potential energy of elastic deformation isa positive definite quadratic form and the system (3) is hyperbolic in the senseof Friedrichs. The characteristic properties of this system are described by theequation

det(

cA+

n∑

i=1

νiBi)

= 0,

n∑

i=1

ν2i = 1.

The positive roots of this equation are velocities of the longitudinal waves cp, ofthe transverse waves cs, of the torsional waves cm, and of the rotational waves cω.These roots are

(5) cp =

√

λ+ 2µ

ρ, cs =

√

µ+ α

ρ, cm =

√

β + 2 γ

j, cω =

√

γ + ε

j.

The complete system of left eigenvectors in 2D case contains 6 vectors correspondingto the nonzero eigenvalues ±cp, ±cs, and ±cω and 6 vectors for c = 0; and in 3Dcase it contains 12 vectors corresponding to the nonzero eigenvalues ±cp, ±cs, ±cm,and ±cω (among them, cs and cω are double roots) and 12 vectors for c = 0.


In the reduced Cosserat model additional rotational degrees of freedom are takeninto account but couple stresses are absent. Three out of eight material constantsof the linear Cosserat model are zero here. The system of constitutive equationscan also be written in form (3) with the vector-function

U =(

v1, v2, v3, σ11, σ22, σ33, σ23, σ32, σ31, σ13, σ12, σ21, ω1, ω2, ω3

)

.

For the hyperbolic system (3) the boundary-value problem subject to the initialconditions U(0, x) = U0(x) and the dissipative boundary conditions is well-posed.

The fulfillment of dissipative conditions for any two vector-functions U and U en-sures that the inequality

(U − U)

n∑

i=1

νi Bi(U − U) ≤ 0

holds at the points of the boundary. In expanded form this inequality looks asfollows (summation is performed over the repeated indexes):

(vl − vl)(σkl − σkl) νl + (ωl − ωl)(mkl −mkl) νl ≤ 0.

In particular, among the dissipative conditions are the conditions in terms of veloci-ties and in terms of stresses that are formulated in the model of a couple-stressedmedium in the next form:

v = v, ω = ω; σkl νk = pl, mkl νk = ql.

Here, ν is the outward normal vector, p and q are the given vectors of the externalforces and moments specified on the boundary. If the matching conditions for theboundary and initial values of the given functions are not fulfilled on the boundaryof the domain or if these functions are discontinuous, discontinuous solutions (shockwaves) appear. Such solutions can be found from the strong discontinuity equations

(6)(

cA+

n∑

i=1

νi Bi)

[U ] = 0,

where [U ] is the jump of the solution on the discontinuity front, c is the veloci-ty of front in the direction of normal vector. It follows from (6) that the shockwaves of small amplitude in a couple-stressed medium can propagate only with thevelocities (5), and scalar products of the left eigenvectors and the vector U remaincontinuous on the wave fronts.

Under numerical analysis of problems an important role is played by artificialboundary conditions that appear due to the symmetry of the stressed-strainedstate. They enable one to reduce considerably the computational domain. Theseconditions follow from the invariance of the system of equations relative to simpletransformations that take the artificial boundaries (for example, symmetry planes)to themselves. Note that the angular velocity vector ω is actually a second-ranktensor, while the moment tensorm is a third-rank tensor. The formulas for reducingthe rank of these tensors have the form

ω1 = ω32, ω2 = ω13, ω3 = ω21,

m11 = m132, m22 = m213, m33 = m321,

m23 = m221, m32 = m313, m31 = m332,

m13 = m121, m12 = m113, m21 = m232.

Therefore, for example, under transformation of the mirror reflection x2 7→ −x2

the sign of ω1, ω3, m11, m22, m33, m31, and m13 changes because the index 2


appears only ones in their tensor components. These values are equal to zero onthe plane x2.

In order to solve 3D problems concerning elastic waves originated as a result ofthe action of a concentrated impulsive source, the boundary conditions of symmetryare presented in [14] for various types of loading. If on the surface of the half-spacex1 > 0 the normal force σ11 = −p1 δ(x) δ(t), concentrated at the origin, acts, thenthe planes x2 = 0 and x3 = 0 are the symmetry planes (here, δ is the Dirac deltafunction). The system of equations (2) is invariant under the mirror reflectionx2 7→ −x2, therefore, the components of the tensors that change their sign underthis transformation vanish on the plane x2 = 0:

(7) v2 = 0, σ21 = σ23 = 0, ω1 = ω3 = 0, m22 = 0.

The plane x3 = 0 goes to itself under the mirror reflection x3 7→ −x3, therefore, onthis plane

(8) v3 = 0, σ31 = σ32 = 0, ω1 = ω2 = 0, m33 = 0.

If the concentrated tangential force σ12 = −p2 δ(x) δ(t) is applied at the pointx = 0, then x3 = 0 is a symmetry plane and x2 = 0 is an antisymmetry plane. Thetransformation x2 7→ −x2, under which the plane x2 = 0 goes to itself, changes thesign of the solution: U 7→ −U . Hence, on this plane

(9) v1 = v3 = 0, σ22 = 0, ω2 = 0, m21 = m23 = 0.

On the plane x3 = 0 the conditions (8) are fulfilled. Under the action of theconcentrated torsional moment m11 = −q1 δ(x) δ(t), which torques the particleabout the axis x1, both mirror reflections x2 7→ −x2 and x3 7→ −x3 change thesign of the solution: U 7→ −U . Therefore, the conditions (9) are fulfilled on theantisymmetry plane x2 = 0, and the similar conditions

(10) v1 = v2 = 0, σ33 = 0, ω3 = 0, m31 = m32 = 0

are fulfilled on the plane x3 = 0. If the concentrated rotational moment m12 =−q2 δ(x) δ(t) acts on the boundary, the mirror reflection x3 7→ −x3 changes thesign of the solution and the transformation x2 7→ −x2 does not change this sign.Hence, the conditions (7) are fulfilled on the symmetry plane x2 = 0, and theconditions (10) are fulfilled on the antisymmetry plane x3 = 0. It can be shownthat boundary conditions (7)–(10) are dissipative; so, their statement provides thecorrectness of a problem.

A more general mathematical model of a material with microstructure, mecha-nical properties of which vary depending on the sign of stresses and which takesinto account the elastic connections between the particles and the plastic strains ofparticles, can be results in the variational inequality, [13],

(11)(

V − V)

(

A U −n∑

i=1

Bi V,i −QV −G)

≥ 0(

V, V ∈ F)

.

Here the vector-function V includes the components of the velocity and angularvelocity vectors, of the stress and couple stress tensors, and the vector-function Uinstead of the actual stresses includes the components of the conditional stress andconditional couple stress tensors. Conditional stress tensors are calculated by thelinear Hooke law from the system of equations (1). The vector-functions V and Uare related by the equations

(12) V = ξ U + (1− ξ)Uπ, U =1

ξV − 1− ξ

ξV π,


where ξ is the ratio of the elastic compliance moduli in tension and compression,Uπ is the projection of U onto the cone K with respect to the Euclidean norm|U |A =

√UAU , K is a convex cone of stresses permitted by the strength criterion,

F is a convex and closed set defined by the criterion of plasticity.

2. Computational algorithm

The computational algorithm is worked out for a model of the Cosserat continu-um and for its nonlinear generalization to media with different resistances to ten-sion and compression, [13, 14]. The algorithm of the solution of linear system (3)is based on the two-cyclic splitting method with respect to the spatial variablesand time. For the solution of one-dimensional systems in spatial directions, the ex-plicit monotone finite-difference ENO (Essentially Non-Oscillatory) scheme of the“predictor–corrector” type is used. The nonlinear system (11)–(12) is realized bymeans of the splitting method with respect to physical processes. In this case itis necessary to solve the variational inequality, describing the plastic deformationof a material, at each mesh of a spatial grid with using special algorithms for thecorrection of stresses.

For 3D case the two-cyclic splitting method on the time interval (t, t+∆t) consistsof seven stages: the solution of a one-dimensional problem in the x1 direction onthe interval (t, t + ∆t/2), similar stages in the x2 and x3 directions, the stage ofsolution of a system of linear ordinary differential equations with the matrix Q,the stage of repeated recalculation of a problem in the x3 direction on the interval(t+∆t/2, t+∆t), and stages of repeated recalculations in the x2 and x1 directions.As applied to the system (3), the splitting procedure gives the next one-dimensionalsystems:

A U1 = B1 U1,1 +G1, U1(t, x) = U(t, x),

A U2 = B2 U2,2 +G2, U2(t, x) = U1(t+∆t/2, x),

A U3 = B3 U3,3 +G3, U3(t, x) = U2(t+∆t/2, x),

A U4 = QU4, U4(t, x) = U3(t+∆t/2, x),

A U5 = B3 U5,3 +G3, U5(t+∆t/2, x) = U4(t+∆t, x),

A U6 = B2 U6,2 +G2, U6(t+∆t/2, x) = U5(t+∆t, x),

A U7 = B1 U7,1 +G1, U7(t+∆t/2, x) = U6(t+∆t, x).

Here G1 + G2 + G3 = G. Desired value U(t + ∆t, x) is U7(t + ∆t, x). Underthe calculation of 2D problem there are absent third and fifth stages, related tothe x3 direction. This method of two-cyclic splitting has second order of accuracyprovided that at its stages second-order schemes are used. Besides, it ensuresthe stability of a numerical solution in multi-dimensional case provided that theCourant–Friedrichs–Levy stability condition for one-dimensional systems is fulfilled.

At fourth stage of the method the implicit Crank–Nickolson finite-differencescheme with full time step is used:

AUk+1 − Uk

∆t= Q

Uk+1 + Uk

2

(k is the number of time step). This scheme has good computational properties: itis conservative in the sense of consistency with the corresponding energy balanceequation and, hence, it is stable independently of the value of a time step.


Each of six remaining one-dimensional problems is solved by means of an explicitmonotone ENO–scheme of the “predictor–corrector” type (see [15]). This schemeis a generalization of the Godunov scheme with piecewise-linear distributions ofvelocities and stresses over meshes. We consider a procedure of the construction ofthe scheme for 1D problem, using the system of equations

(13) A U = Bi U,i +Gi,

which is of a common form for stages of the splitting. In the case of constantmatrix–coefficients, at the “corrector” step the relationships

Uk+1/2 = Uk+1/2 +∆t

2A−1

(

Bi Uk+1 − Uk

∆xi k+1/2+Gi

)

are used. Here index k+1/2 is related to the center of a mesh of a spatial differencegrid, a superscript corresponds to an actual time level and a subscript correspondsto a previous level. If the matrices are variable then the corresponding terms ofthe conservative approximation are taken as a difference derivative with respectto xi. To close the scheme, we have to complement it with the relationships of the“predictor” step which are obtained with the use of the idea of grid-characteristicmethods. Let Yl and cl be a complete system of left eigenvectors and eigenvaluesof the matrix BiA−1, respectively:

Yl Bi = cl Yl A, Yl AYh = δlh.

This system exists since the matrices A, Bi are symmetric and, in addition, A ispositive definite. In the case of an inhomogeneous material, the eigenvalues involvedin this system are functions of xi of constant signs. Multiplying the equality (13)by the vector Yl from the left, we arrive at a system of differential equations, whichare equations on characteristics:

Yl A U = cl Yl AU,i + Yl Gi.

After approximation we obtain

(14)(

Ik+1/2l

)±= Il k+1/2 ± αl k+1/2

∆xi k+1/2

2+(

cl αl + Yl Gi)

k+1/2

∆t

4,

where αl k+1/2 are derivatives of coefficients of decomposition of the vector–func-tion U in the basis Yl: Il = (Yl A)k+1/2 U , received with the help of the iterativeprocedure of limit reconstruction. Indices “–” and “+” in (14) mark the values ofthese coefficients on the left and on the right boundaries of a mesh, respectively. Theprocedure of limit reconstruction enables one to improve an accuracy of a numericalsolution and consists in the construction of monotone piecewise-linear splines whichapproximate Il with minimal discontinuities on boundaries of neighbouring meshesof a grid. First each of these coefficients is approximated with the help of piecewise-constant splines which take the values Il k+1/2 on the mesh (xi k, xi k+1). Thenjumps are determined by the formulae

∆I0l k = Il k+1/2 − Il k−1/2.

When solving boundary-value problems, jumps related to boundaries of a compu-tational domain are assumed to be zero. After this, the functions

Il = Il k+1/2 + α0l k+1/2(xi − xi k+1/2),

linear on a mesh, with angular coefficients

α0l k+1/2 =

1

∆xi k+1/2sgn∆I0l k min

{

∣

∣∆I0l k∣

∣,∣

∣∆I0l k+1

∣

∣

}

for ∆I0l k ∆I0l k+1 > 0


or α0l k+1/2 = 0 for ∆I0l k ∆I0l k+1 ≤ 0 become approximations. Each of these func-

tions halves a minimal jump of a corresponding piecewise-constant spline with re-spect to the nodes xi = xi k and xi = xi k+1. As a result, piecewise-linear disconti-nuous splines being monotone in domains of monotone change of piecewise-constantsplines are obtained. Then new jumps

∆I1l k = ∆I0l k −(

α0l k+1/2 ∆xi k+1/2 + α0

l k−1/2 ∆xi k−1/2

)

/2

are calculated and the procedure of decreasing jumps is repeated with preservationof monotonicity. To this end, to the old values of angular coefficients α0

l k+1/2 their

increments α1l k+1/2 calculated by the above formula in terms of ∆I1l k are added.

The process is repeated four or five times and then is terminated since, as shownby practical calculations, further iterations do not improve a solution. At internalnodes of a computational domain, at the “predictor” step the quantities Uk aredetermined by the averaging formula

Uk =1

2(U−

k + U−k )

via the quantities U±k relating to different sides of a boundary between meshes and

satisfying the system of equations

(YlA)k+1/2 U+k = I+l k+1/2 for cl ≥ 0,

(YlA)k−1/2 U−k = I+l k−1/2 for cl ≤ 0.

Used variant of a splitting method with respect to spatial variables allows to re-alized the unreflecting boundary conditions rather simply. These conditions simu-late an unimpeded waves passing outside the computational domain. So, for examp-le, at the right boundary the invariants, which transfer the perturbations from theright to the left, are supposed equal to zero. Such equations together with equations,setting the values of invariants on outgoing characteristics, are used under therealization of boundary conditions for 1D problems of the splitting method.

3. Parallel program system

This computational algorithm is realized as a parallel program system for nu-merical solution of 2D and 3D dynamic problems of couple-stressed media on mul-tiprocessor computers (see [13, 16]). The programming is fulfilled in Fortran bymeans of the SPMD (Single Program – Multiple Data) technology with the useof the MPI (Message Passing Interface) library. Program system consists of thepreprocessor, the main program, the subprograms for the realization of boundaryconditions, and the postprocessor. The universality of programs is achieved by aspecial packing of the variables, used at each node of the cluster, into large one-dimensional arrays. The parallelization of computations is performed at the stageof splitting the problem with respect to the spatial variables. Under the solution ofone-dimensional systems of equations, the exchange of data between computationalnodes takes place at the step “predictor” of the difference scheme. At first eachprocessor exchanges with neighboring processors the boundary values of their data,and then calculates the required quantities in accordance with the finite-differencescheme.

Computational domain is distributed between the cluster nodes by means of 1D,2D or 3D decomposition so as to load the nodes uniformly and to minimize thenumber of passing data. Examples of 1D, 2D and 3D decompositions of computa-tional domain in the form of a cube between 7, 9 and 12 processors, respectively,are represented in Fig. 1.


Figure 1. 1D, 2D and 3D decompositions of a computational domain.

The preprocessor program is intended for the generation of finite-difference grid,the initial data preparation in a packed form and for their uniform distributionbetween parallel computational nodes. Each processor of cluster packs its part ofdata into binary files of direct access – a file of real numbers, in which parametersof a material, a part of grid and initial values of a solution for this part of a gridare written, and a file of integers with corresponding them addresses (pointers), i.e.serial numbers of the first elements. Further, for the organization of the controlpoints each processor generates similar files of real numbers, containing the solutionat some time steps.

The main program on each node of cluster makes a similar computations consis-ting of mutually coordinated step-by-step realization of the space-variable splittingmethod (on each time step). Data exchange between the processors is carried outat the level of coefficients of the solution decomposition on the basis from lefteigenvectors at the stage of limit reconstruction of the solution of one-dimensionalsystems. Standard technology of the contour meshes is used, data exchange betweenthe contour meshes is realized by means of the function MPI SendRecv. Results ofcomputations are conserved in the control points.

The postprocessor program performs special compressing of files, containing theresults of computations in the control points. Such data compression is necessary,because the total size of files on the same time step can considerably exceed thesize of RAM of a single processor. Besides, for the transportation of files alongnetwork the significant resources are required. One of the methods for file compres-sion is proposed in [16]. Graphical representation of results is carried out by specialprograms for personal computer. The program for representation of the results ofcomputations of wave problems, received on clusters, in SEG-Y format of Interna-tional Geophysical Society is worked out with the purpose of further processing ofdata in the system SeisView.

4. Numerical results

Testing of the proposed algorithm and software package is performed by compa-rison the results of computations with the exact solutions describing the 1D motionsof a medium with plane waves, as well as with the exact solutions of 2D prob-lems on the propagation of the surface Rayleigh waves in the Cosserat mediumand in reduced Cosserat medium. Analytical solutions for 2D problems are takenfrom [17, 18].

The results of 2D computations of the elastic waves propagation in a rectangularbody are presented in Figs. 2–4. All sides of the body, excepting the left one, arenonreflecting boundaries. On the left side distributed periodic load of Λ-shaped


impulses of tangential stress σ12 is given. The area of application of impulses takeshalf a side in its central part. As a result of impulsive action, a sequence of load-ing and unloading waves propagates over a material. These waves are domains ofsmooth variation of a solution with clearly defined fronts. This is seen in Fig. 2where level curves of tangential stress σ12 are shown for t = 26 and t = 78µs. In thefirst case (Fig. 2a) a single wave induced by the first loading impulse on the bound-ary is observed, in the second case (Fig. 2b) we have three waves caused by threeimpulses. Notice that contrary to the one-dimensional solution, where tangentialstress is positive everywhere, in the plane problem areas of negative value arise be-cause of the lateral unloading. In Fig. 3 one can see the oscillations of level curvesof angular velocity ω3 for the same instants of time. Characteristic scale of theseoscillations can be estimated with the help of a solution of the problem on simpleshear again. In Fig. 4 similar results for linear velocity v1 are presented. Waves aregenerated at the points which belong to the boundary of the area of application ofload on the left side of the body. Analysis shows that these waves are related tointensive rotational motion of particles in a neighbourhood of points at which loadchange sharply. They are severely damped waves moving inwards a body and in

(a) (b)

Figure 2. The action of Λ-shaped impulses of tangential stress(2D case): level curves of σ12.

(a) (b)

Figure 3. The action of Λ-shaped impulses of tangential stress(2D case): level curves of ω3.


(a) (t)

Figure 4. The action of Λ-shaped impulses of tangential stress(2D case): level curves of v1.

the vertical direction with velocity close to that of longitudinal waves and gradu-ally leaving the domain of solution of the problem through horizontal sides. Thecalculations were performed on a square of side 0.1 m for synthetic polyurethane.Characteristic scale of the microstructure of a material is r = 0.15 mm. Velocitiesof elastic waves are cp = 2687, cs = 1395, and cω = 893 m/s. The uniform diffe-rence grid, used in computations, consists of 1000× 1000 meshes with a mesh sizeof 0.1 mm (this value is less than r). On coarser grids calculations with satisfactoryaccuracy may not be performed. For example, on a grid consisting of 500 × 500meshes an understated number of oscillations of angular velocity within any waveis obtained.

Figure 5. Lamb’s problem for normal load (2D case). Levelcurves of angular velocity ω3 and of stresses σ11, σ12, and σ33

(from left to right).


Figure 6. Lamb’s problem for tangential load (2D case). Levelcurves of angular velocity ω3 and of stresses σ11, σ12, and σ33 (fromleft to right).

In Figs. 5 and 6 the results of numerical solution of Lamb’s problem on theaction of concentrated force on a surface of a half-space for plane strain are pre-sented. These result are obtained on a grid consisting of 1000 × 500 meshes. Inthe first case impulsive load is applied at the center of the upper boundary ofcomputational domain in the normal direction and in the second case it is appliedat a tangent to the boundary. Level curves of angular velocity ω3 and stressesσ11, σ12, and σ33 are shown from left to right, respectively. All waves characteris-tic for the solution of Lamb’s problem in the framework of the classical elasticitytheory (incident longitudinal and transverse waves with circular fronts, two conicaltransverse waves in the form of symmetric straight-line segments tangent to thesemicircle of smaller radius, the Rayleigh surface waves rapidly damped with depthshown by bright points on the boundary following an incident transverse wave) areclearly distinguished on level curves. The distinction is that in the Cosserat modelof an elastic medium a solution has clearly pronounced oscillatory nature. Stressoscillations with respect to moderate amplitude are superimposed on significantlyinhomogeneous fields which vary rapidly near wave fronts.

The similar computations were performed for 3D case. With regard to the sym-metry conditions, the computation domain is a quarter of the half-space. In fact,the computations were performed on the cube with the side 0.01 m for the syntheticpolyurethane with the microstructure particles of the size r = 0.15 mm. Velocitiesof elastic waves (5) are cp = 2687, cs = 1395, cm = 1050, and cω = 893 m/s.The uniform difference grid is used which consists of 200× 200× 200 meshes witha mesh size of 0.05 mm. The computations cannot be performed with adequateaccuracy using a coarser grid because the size of the meshes becomes comparablewith the size of the medium particles. On the artificially defined faces of the cube,the symmetry conditions (7)–(10) and the nonreflective boundary conditions thatmodel the free wave passage were imposed. The computations were performed on64 processors of the cluster MVS–100k of JSCC RAS (Moscow), every processorsolved a part of the problem on the subgrid of the size 50× 50× 50 meshes.


Figure 7. Lamb’s problem for normal load (3D case): level sur-faces of stresses σ11, σ12 and of moment m23 (from left to right).

Fig. 7 shows the level surfaces of the normal stress σ11, of the tangential stressσ12 and of the moment m23 for Lamb’s problem on the action of a concentratednormal force σ11 = −p1 δ(x) δ(t), applied at one of the vertices of the computationdomain. In this figure, we can see the incident longitudinal and transverse waves,conical transverse waves, and the Raleigh surface waves. The loading schemes andthe seismograms of incident waves for different types of loading are represented inFigs. 8 and 9. More precisely, the displacement in the direction of the x1 axis and theangular displacement of the particles in the plane x1x2 along a trace passing throughthe point of application of the force on the x2 axis for action of the normal stress σ11

(a)

(b)

Figure 8. Lamb’s problem for the action of normal (a) and tan-gential (b) stresses (3D case): loading schemes and seismograms ofincident waves.


(a)

(b)

Figure 9. Lamb’s problem for the action of torsional (a) and ro-tational (b) moments (3D case): loading schemes and seismogramsof incident waves.

on the boundary are shown in Fig. 8a; the displacement in the direction of the x2

axis and the angular displacement in the plane x1x2 along the same trace for actionof the tangential stress σ12 = −p2 δ(x) δ(t) are shown in Fig. 8b; the displacement inthe direction of the x3 axis and the angular displacement in the plane x2x3 for actionof the torsional moment m11 = −q1 δ(x) δ(t) are shown in Fig. 9a; the displacementin the direction of the x3 axis and the angular displacement in the plane x1x3 foraction of the rotational moment m12 = −q2 δ(x) δ(t) are shown in Fig. 9b. Onthe seismograms one can see four waves propagating at the velocities cp, cs, cm,and cω (the points where these waves appear at the domain boundary are markedby triangles). We also see the oscillations, which depend on the characteristic sizeof the particles. Note that in Figs. 9a and 9b the longitudinal wave is almost absentand the front of the transverse wave is strongly smeared.

Numerical results for 3D problem about the action of a concentrated rotationalmoment m12 = −q2 δ(x)δ(2 π ν t), periodic by time, on the surface of homogeneouselastic half-space are represented in Fig. 10. Here one can see the loading schemeand the level surfaces of the angular velocity ω2 for the nonresonance frequencyof external action ν = 1.5 ν∗ and for the resonance frequency ν = ν∗ (from left toright) at different points of time; ν∗ = 1/T is the frequency of natural oscillations

of rotational motion of particles; T = π√

j/α is the oscillation period. The levelsurfaces correspond to the same range of values ω2. The maximum amplitude ofoscillations of the angular velocity is achieved at the point of load application, andthe wavelength depends essentially on the frequency. Comparison of the graphsshows that for the frequency ν∗ of external action, equal to the natural frequency


(a) (b) (c)

Figure 10. Periodic action of a concentrated rotational momenton the surface of elastic half-space. Loading scheme (a) and levelsurfaces of angular velocity ω2 for nonresonance frequency (b) andresonance frequency (c) at different points of time (t = 6.5µsabove, t = 13µs below).

of the rotational motion of particles, the growth of amplitude with time occurs anda more smooth decay of oscillations with increasing the distance from the point ofload application, characteristic of the acoustic resonance, takes place (Fig. 10c).

Thus, computations of 3D problems have confirmed the main qualitative diffe-rence of the wave field in the Cosserat continuum as compared with the classicalelasticity theory, which consists in the appearance of oscillations of the rotationalmotion of particles on the wave fronts. Comparative calculations with differentvalues of scale of the microstructure of a material were performed, in which a directproportional dependence of the period of natural oscillations from this scale wasfound. The results of the analysis of the oscillation processes show that the Cosseratmedium possesses the eigenfrequency of acoustic resonance, which appears undercertain conditions of perturbation and depends only on the inertial properties ofthe microstructure particles and the elasticity parameters of the material.

The efficiency of parallelization is calculated as acceleration of the problem onn processors (n varies from 2 to 150). The efficiency was calculated at the clusterMVS–100k of Joint Supercomputer Center of the Russian Academy of Sciences.Computations show that the efficiency of the parallel program system is higherthan 80 %.


Acknowledgments

This work was supported by the Russian Foundation for Basic Research (grantno. 11-01-00053), the Complex Fundamental Research Program no. 2 of the Pre-sidium of the Russian Academy of Sciences, and the Interdisciplinary IntegrationProject no. 40 of the Siberian Branch of the Russian Academy of Sciences.

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[17] M.A. Kulesh, V.P. Matveenko and I.N. Shardakov, Construction and analysis of an analyticalsolution for the surface Rayleigh wave within the framework of the Cosserat continuum,J. Appl. Mech. Techn. Phys., 46, No. 4 (2005), 556–563.

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Institute of Computational Modeling SB RAS, Siberian Federal University,Krasnoyarsk, Russia

E-mail : [email protected], o [email protected], and [email protected]

URL: http://icm.krasn.ru/

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