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Proceedings Papers

*Proc. ASME*. IMECE2018, Volume 1: Advances in Aerospace Technology, V001T03A025, November 9–15, 2018

Paper No: IMECE2018-86515

Abstract

One considers linear thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. non-ellipsoidal) shape. The representations of the effective properties (effective moduli, thermal expansion, and stored energy) are expressed through the statistical averages of the interface polarization tensors (generalizing the initial concepts, see e.g. [1] and [2]) introduced apparently for the first time. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals estimated by the method of fundamental solution for a single inclusion inside the infinite matrix. This enables one to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. The results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

Proceedings Papers

*Proc. ASME*. IMECE2017, Volume 1: Advances in Aerospace Technology, V001T03A034, November 3–9, 2017

Paper No: IMECE2017-70776

Abstract

One considers linearly elastic composite material (CM), which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. nonellipsoidal) shape. The new general integral equations (GIE) connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. In particular, we used a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity. As a particular problem allowing us to get an exact result, we consider a linear elastic CM, which consists of a homogeneous matrix containing a statistically homogeneous random set of non-canonical inclusions. The elastic properties of the matrix and the inclusions are the same, but the stress-free strains are different. Increasing of volume fraction of inclusions can lead to change of a sign of local residual stresses estimated by either the new approach or the classical one. The main properties of the method are analyzed and illustrated with several numerical simulations in 2D infinite domains containing statistically homogeneous random field of inclusions.

Proceedings Papers

*Proc. ASME*. IMECE2017, Volume 9: Mechanics of Solids, Structures and Fluids; NDE, Structural Health Monitoring and Prognosis, V009T12A054, November 3–9, 2017

Paper No: IMECE2017-70777

Abstract

One considers a linear elastic random structure composite material (CM) with a homogeneous matrix. The idea of the effective field hypothesis (EFH, H1) dates back to Faraday, Poisson, Mossotti, Clausius, and Maxwell (1830–1870, see for references and details [1], [2]) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). New GIEs essentially define the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields). Estimates of the Hashin-Shtrikman (H-S) type are developed by extremizing of the classical variational functional involving either a classical GIE [1] or a new one. In the classical approach by Willis (1977), the H-S functional is extremized in the class of trial functions with a piece-wise constant polarisation tensors while in the current work we consider more general class of trial functions with a piece-wise constant effective fields. One demonstrates a better quality of proposed bounds, that is assessed from the difference between the upper and lower bounds for the concrete numerical examples.

Proceedings Papers

*Proc. ASME*. IMECE2016, Volume 9: Mechanics of Solids, Structures and Fluids; NDE, Diagnosis, and Prognosis, V009T12A078, November 11–17, 2016

Paper No: IMECE2016-65840

Abstract

One considers a linear heterogeneous media (e.g. filtration in porous media, composite materials, CMs, nanocomposites, peristatic CMs, and rough contacted surfaces). The idea of the effective field hypothesis (EFH, H1, see for references and details [1], [2]) dates back to Mossotti (1850) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). An operator form of the general integral equations (GIEs) is obtained which connects the driving fields and fluxes in a point being considered. Either the volume integral equations or boundary ones are used for these GIEs, new concept of the interface polarisation tensors are introduced. New GIEs present in fact the new (second) background (which does not use the EFH) of multi-scale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

Proceedings Papers

*Proc. ASME*. IMECE2016, Volume 9: Mechanics of Solids, Structures and Fluids; NDE, Diagnosis, and Prognosis, V009T12A079, November 11–17, 2016

Paper No: IMECE2016-65046

Abstract

This paper concerns with the quasi static linear theory of thermoelasticity for triple porosity materials. The system of governing equations based on the equilibrium equations, conservation of fluid mass, the constitutive equations, Darcy’s law for materials with triple porosity and Fourier’s law of heat conduction. The cross-coupled terms are included in the equations of conservation of mass for the fluids of the three levels of porosity (macro-, meso- and micropores) and in the Darcy’s law for materials with triple porosity. The system of general governing equations is expressed in terms of the displacement vector field, the pressures in the three pore systems and the temperature. The basic internal and external boundary value problems (BVPs) are formulated and on the basis of Green’s identities the uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method and the theory of singular integral equations.

Proceedings Papers

*Proc. ASME*. IMECE2016, Volume 12: Transportation Systems, V012T16A008, November 11–17, 2016

Paper No: IMECE2016-65876

Abstract

This work investigates the elasto-plastic response of platelets-like inclusions reinforced polymer composites showing an imperfect interface. The solution of the heterogeneous material problem is solved through a kinematic integral equation. To account for the interfacial behaviour, a linear spring model LSM is adopted, leading to an expression of the modified Eshelby’s tensor. As a consequence, the interfacial contributions with respect to the strain concentration tensor within each phase as well as in the average strain field are described by a modified version of the Mori-Tanaka scheme for the overall response. The non-linear response is established in the framework of the J 2 flow rule. An expression of the algorithmic tangent operator for each phase can be obtained and used as uniform modulus for homogenisation purpose. Numerical results are conducted on graphene platelets GPL-reinforced polymer PA6 composite for several design parameters such as GPL volume fraction, aspect ratio and the interfacial compliance. These results clearly highlight the impact of the aspect ratio as well as the volume fraction by a softening in the overall response when imperfection is considered at the interface. Present developments are analytical-based solutions. They constitute a theoretical framework for further multi-scale applications in automotive. The crashworthiness simulation incorporating an influence of the interfacial behaviour on the strain energy absorption SEA is of interest.

Proceedings Papers

*Proc. ASME*. IMECE2016, Volume 1: Advances in Aerospace Technology, V001T03A044, November 11–17, 2016

Paper No: IMECE2016-65841

Abstract

In contrast to the classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J. Mech. Phys. Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. The new general integral equations (GIE) connecting the displacement fields in the point being considered and the surrounding points of random structure composite materials (CMs) is proposed. For statistically homogeneous thermoperistatic media subjected to homogeneous volumetric boundary loading, one proved that the effective behaviour of this media is governing by conventional effective constitutive equation which is intrinsic to the local thermoelasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional thermoelasticity of CMs and adapted to thermoperistatics. A generalization of the Hills equality to peri-static composites is proved. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CMs are generalized to the case of peristatics, and the energetic definition of effective elastic moduli is proposed. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of the decomposition of local fields into load and residual fields. Effective properties of thermoperistatic CM are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase. This similarity opens a way for straightforward expansion of analytical micromechanics tools for locally elastic CMs to the new area of random structure peri-dynamic CMs. Detailed numerical examples for 1D case are considered.

Proceedings Papers

*Proc. ASME*. IMECE2016, Volume 1: Advances in Aerospace Technology, V001T03A004, November 11–17, 2016

Paper No: IMECE2016-65842

Abstract

One considers a linear elastic composite material (CM, [1]), which consists of a homogeneous matrix containing the random set of heterogeneities. An operator form of the general integral equation (GIE, [2–6]) connecting the stress and strain fields in the point being considered and the surrounding points are obtained for the random fields of inclusions in the infinite media. The new GIE is presented in a general form of perturbations introduced by the heterogeneities and defined at the inclusion interface by the unknown fields of both the displacement and traction. The method of obtaining of the GIE is based on a centering procedure of subtraction from both sides of a new initial integral equation their statistical averages obtained without any auxiliary assumptions such as the effective field hypothesis (EFH), which is implicitly exploited in the known centering methods. One proves the absolute convergence of the proposed GIEs, and some particular cases, asymptotic representations, and simplifications of proposed GIEs are presented for the particular constitutive equations of linear thermoelasticity. In particular, we use a meshfree method [7] based on fundamental solutions basis functions for a transmission problem in linear elasticity. Numerical results were obtained for 2D CMs reinforced by noncanonical inclusions.

Proceedings Papers

*Proc. ASME*. IMECE2015, Volume 1: Advances in Aerospace Technology, V001T01A034, November 13–19, 2015

Paper No: IMECE2015-51161

Abstract

In contrast to the classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J. Mech. Phys. Solids 2000; 48 : 175–209) is free of any spatial derivatives of displacement. The new general integral equations (GIE) connecting the displacement fields in the point being considered and the surrounding points of random structure composite materials (CMs) is proposed. For statistically homogemneous thermoperistatic media subjected to homogeneous volumetric boundary loading, one proved that the effective behaviour of this media is governing by conventional effective constitutive equation which is intrinsic to the local thermoelasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional thermoelasticity of CMs and adapted to thermoperistatics. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of decomposition of local fields into the load and residual fields similarly to the locally elastic CMs. This similarity opens a way for straightforward expansion of analytical micromechanics tools for locally elastic CMs to the new area of random structure peridynamic CMs. Detailed numerical examples for 1D case are considered.

Proceedings Papers

*Proc. ASME*. IMECE2015, Volume 10: Micro- and Nano-Systems Engineering and Packaging, V010T13A008, November 13–19, 2015

Paper No: IMECE2015-50560

Abstract

Micro-cantilever beams are currently employed as sensor in various fields. Of main applications, is using such beams in wind speed sensors. For this purpose, curved out of plane micro-cantilever beams are used. Uniform pressure on such beams causes a large deflection of beam. General mechanics of material theory deals with small deflection and thus cannot be used for explaining this deflection. Although there are a body of works on analysing of large deflection [1], nonlinear deflection, of curved beams [2], yet there is no research on large deflection of curved beam under horizontal uniform distributed force. Theoretically, the wind force is applying horizontally on curved micro-cantilever beam. Here, we neglect the effect of moving weather from beam sides. We first aim how to drive the governed equation. A curved beam does not have a calculable centroid. Also large deflection of beam changes its curvature which would change the centroid of beam consciously. The variation of centroid makes very though calculating the bending moment of each cross section in the beam. To address this issue, an integral equation will be used. The total force will be considered as a single force applied at the centroid. The second challenge is solving the governed nonlinear ordinary differential equation (ODE). Although there are several methods to solve analytically nonlinear ODE, Lie symmetry method, with all its complication, is a general method for this kind of equations. This approach covers all current methods in analytical solving nonlinear ODEs. In this method, an infinitesimal transformation should be calculated. All transformations under one parameter creates a group that called Lie group. A value of parameter which transfers the equation onto itself is called invariant of ODE. One can calculate canonical coordinates ODEs by the invariant. Solving the canonical coordinates ODEs yields to calculating the canonical coordinates. Canonical coordinate are used to reduce the order of nonlinear ODE [3]. By repeating this method one can solve high order ODEs. Our last question is how to do numerical solution of ODE. The possible answer will help to explain the phenomena of deflection clearly and compare the analytical solution with numerical results. Small dimensions of beam, small values of applied force from one side and Young modules value from the other side, will create a stiff ODE. Authors experience in this area shows that the best method to sole these kind of equations is LSODE. This method can be used in Maple. Here, primary calculations show that the governed equation is second order nonlinear ODE and we propose two possible invariants to solve ODE. Overall, the primary numerical solution has shown perfect match with the exact solution.

Proceedings Papers

*Proc. ASME*. IMECE2015, Volume 9: Mechanics of Solids, Structures and Fluids, V009T12A060, November 13–19, 2015

Paper No: IMECE2015-51159

Abstract

One considers a linear heterogeneous media (e.g. filtration in porous media, composite materials, CMs, nanocomposites, peristatic CMs, and rough contacted surfaces). The idea of the effective field hypothesis (EFH, H1, see for references and details [1], [2]) dates back to Mossotti (1850) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). An operator form of the general integral equations (GIEs) is obtained which connects the driving fields and fluxes in a point being considered. New GIEs present in fact the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

Proceedings Papers

*Proc. ASME*. IMECE2014, Volume 9: Mechanics of Solids, Structures and Fluids, V009T12A055, November 14–20, 2014

Paper No: IMECE2014-37917

Abstract

In this paper, the sliding contact of a rigid sinusoid over a viscoelastic halfplane is studied by means of an analytical procedure that reduced the original viscoelastic system to an elastic equivalent one, which has been already solved in [1]. In such a way, the solution of the original viscoelastic contact problem requires just to numerically solve a set of two integral equations. Results show the viscoelasticity influence on the solution by means of a detailed analysis of contact area, pressure and displacement distribution. A particular attention is paid to the transition from full contact to partial contact conditions.

Proceedings Papers

*Proc. ASME*. IMECE2011, Volume 8: Mechanics of Solids, Structures and Fluids; Vibration, Acoustics and Wave Propagation, 299-302, November 11–17, 2011

Paper No: IMECE2011-65787

Abstract

This study is concerned with extending the use of boundary algebraic equations (BAEs) to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as BAEs provide seamless bridging between discrete and continuum models. BAEs share many features with boundary integral equations and are particularly effective for solving problems involving infinite domains. However, it is shown that, BAEs for irregular lattices containing certain terms may require the same amount of computational effort as the original problem for which the equations are formulated. In this paper, we formulate a BAE for a model problem and expose the fundamental obstacle that prevents us from using that BAE as an effective tool. It is shown that, in contrast to regular lattices, BAEs for irregular lattices require a statistical rather than deterministic treatment. This is a very interesting direction for future research.

Proceedings Papers

*Proc. ASME*. IMECE2010, Volume 9: Mechanics of Solids, Structures and Fluids, 581-588, November 12–18, 2010

Paper No: IMECE2010-37048

Abstract

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing statistically inhomogeneous random set of noncanonical inclusions. The elastic properties of the matrix and the inclusions are the same, but the stress-free strains are different. The new general volume integral equation (VIE) proposed by Buryachenko (2010a, 2010b) is implemented. This equation is obtained by a centering procedure without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. The results of this abandonment are quantitatively estimated for some modeled composite with homogeneous fibers of nonellipsoidal shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

Proceedings Papers

*Proc. ASME*. IMECE2010, Volume 9: Mechanics of Solids, Structures and Fluids, 209-217, November 12–18, 2010

Paper No: IMECE2010-40621

Abstract

Peridynamics is a nonlocal extension of classical solid mechanics that allows for the modeling of bodies in which discontinuities occur spontaneously. Because the peridynamic expression for the balance of linear momentum does not contain spatial derivatives and is instead based on an integral equation, it is well suited for modeling phenomena involving spatial discontinuities such as crack formation and fracture. In this study, both peridynamics and classical finite element analysis are applied to simulate material response under dynamic blast loading conditions. A combined approach is utilized in which the portion of the simulation modeled with peridynamics interacts with the finite element portion of the model via a contact algorithm. The peridynamic portion of the analysis utilizes an elastic-plastic constitutive model with linear hardening. The peridynamic interface to the constitutive model is based on the calculation of an approximate deformation gradient, requiring the suppression of possible zero-energy modes. The classical finite element portion of the model utilizes a Johnson-Cook constitutive model. Simulation results are validated by direct comparison to expanding tube experiments. The coupled modeling approach successfully captures material response at the surface of the tube and the emerging fracture pattern.

Proceedings Papers

*Proc. ASME*. IMECE2009, Volume 2: Biomedical and Biotechnology Engineering, 179-185, November 13–19, 2009

Paper No: IMECE2009-11626

Abstract

Mathematical modeling of biological tissue ablation performed using a short pulse laser and the corresponding experimental analysis is of fundamental importance to the understanding and predicting the temperature distribution and heat affected zone for advancing surgical application of lasers. The objective of this paper is to use mathematical models to predict the thermal ablated zones during irradiation of freshly excised mouse skin tissue samples by a novel approach using a focused laser beam from a short pulse laser source. Suggested mathematical model is Stefan kind free boundary problem for the heat equation in unknown region. Temperature of the skin satisfies the classical heat equation subjected to Neumann boundary condition on the known boundary, while along the time-dependent unknown boundary, which characterizes the ablation depth, two conditions are met: (1) temperature is equal to the ablation temperature and (2) classical Stefan condition is satisfied. The latter expresses the conservation of energy at the ablation moment. A method of integral equations is used to reduce the Stefan problem to a system of two Volterra kind integral equations for temperature and ablation depth. MATLAB is used subsequently for the numerical solution. Experiments are performed using two lasers—a diode laser having a wavelength of 1552 nm and pulsewidth of 1.3 ps. The surface temperature distribution is measured using an imaging camera. After irradiation, histological studies of laser irradiated tissues are performed using frozen sectioning technique to determine the extent of thermal damage caused by the laser beam. The ablation depth and width is calculated based on the interpolated polygon technique using image processing software. The surface temperature distribution and the ablation depth obtained from the mathematical models are compared with the experimental measurements and are in very good agreement. A parametric study of various laser parameters such as time-average power, pulse repetition rate, pulse energy, and irradiation time is performed to determine the necessary ablation threshold parameters.

Proceedings Papers

*Proc. ASME*. IMECE2009, Volume 15: Sound, Vibration and Design, 513-518, November 13–19, 2009

Paper No: IMECE2009-10163

Abstract

Some recent development of the fast multipole boundary element method (BEM) for modeling acoustic wave problems in both 2-D and 3-D domains are presented in this paper. First, the fast multipole BEM formulation for 2-D acoustic wave problems based on a dual boundary integral equation (BIE) formulation is presented. Second, some improvements on the adaptive fast multipole BEM for 3-D acoustic wave problems based on the earlier work are introduced. The improvements include adaptive tree structures, error estimates for determining the numbers of expansion terms, refined interaction lists, and others in the fast multipole BEM. Examples involving 2-D and 3-D radiation and scattering problems solved by the developed 2-D and 3-D fast multipole BEM codes, respectively, will be presented. The accuracy and efficiency of the fast multipole BEM results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale acoustic wave problems that are of practical significance.

Proceedings Papers

*Proc. ASME*. IMECE2009, Volume 15: Sound, Vibration and Design, 65-79, November 13–19, 2009

Paper No: IMECE2009-13344

Abstract

This work, the first of two parts, presents the development of a new analytic solution of acoustic scattering and/or radiation by arbitrary bodies of revolution under heavy fluid loading. The approach followed is the construction of a three-dimensional Wiener-Hopf technique with Fourier transforms that operate on the finite object’s arclength variable (the object’s practical finiteness comes about, in a Wiener-Hopf sense, by formally bringing to zero the radius of its semi-infinite generator curve for points beyond a prescribed station). Unlike in the classical case of a planar semi-infinite geometry, the kernel of the integral equation is non-translational and therefore with independent wavenumber spectra for its receiver and source arclengths. The solution procedure begins by applying a symmetrizing spatial operator that reconciles the regions of (+) and (−) analyticity of the kernel’s two-wavenumber transform with those of the virtual sources. The spatially symmetrized integral equation is of the Fredholm 2nd kind and thus with a strong unit “diagonal” — a feature that makes possible the Wiener-Hopf factorization of its transcendental doubly-transformed kernel via secondary spectral manipulations. The companion paper [1] will present a numerical demonstration of the new analysis for canonical problems of fluid-structure interaction for finite bodies of revolution.

Proceedings Papers

*Proc. ASME*. IMECE2007, Volume 8: Heat Transfer, Fluid Flows, and Thermal Systems, Parts A and B, 1371-1380, November 11–15, 2007

Paper No: IMECE2007-41417

Abstract

This work presents hybrid numerical-analytical solutions for transient laminar forced convection over flat plates of non-negligible thickness, subjected to arbitrary time variations of applied wall heat flux at the interface fluid-solid wall. This conjugated conduction-convection problem is first simplified through the employment of the Coupled Integral Equations Approach (CIEA) to reformulate the heat conduction problem on the plate by averaging the related energy equation in the transversal direction. As a result, a partial differential formulation for the average wall temperature is obtained, while a third kind boundary condition is achieved for the fluid in the heat balance at the solid-fluid interface. From the available velocity distributions, the solution method is then proposed for the coupled partial differential equations, based on the Generalized Integral Transform Technique (GITT) under its partial transformation mode, combined with the method of lines implemented in the Mathematica 5.2 routine NDSolve .

Proceedings Papers

*Proc. ASME*. IMECE2007, Volume 9: Mechanical Systems and Control, Parts A, B, and C, 511-519, November 11–15, 2007

Paper No: IMECE2007-43046

Abstract

This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a Fractional Optimal Control Problem (FOCP) is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The scheme presented rely on reducing the equations for distributed system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. This involves discretizing the space domain into several segments, and writing the spatial derivatives in terms of variables at space node points. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] for scalar case is extended for the vector case. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is also descretized into several segments. For the linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for various order of fractional derivatives and various order of space and time discretizations. Numerical results show that for the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.