Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Exact Numerical Processing

Paper submitted to Euromicro Symposium on Digital Systems Design (DSD), Belek-Antalya, Turkey, 2003.

A compact difference scheme for numerical solutions of second order dual-phase-lagging models of microscale heat transfer

Dual-phase-lagging (DPL) models constitute a family of non-Fourier models of heat conduction that allow for the presence of time lags in the heat flux and the temperature gradient. These lags may need to be considered when modeling microscale heat transfer, and thus DPL models have found application in the last years in a wide range of theoretical and technical heat transfer problems. Consequently, analytical solutions and methods for computing numerical approximations have been proposed for particular DPL models in different settings. In this work, a compact difference scheme for second order DPL models is developed, providing higher order precision than a previously proposed method. The scheme is shown to be unconditionally stable and convergent, and its accuracy is illustrated with numerical examples.

3D numerical modelling of acoustic horns using the method of fundamental solutions

In the present work, a three-dimensional (3D) formulation based on the method of fundamental solutions (MFS) is applied to the study of acoustic horns. The implemented model follows and extends previous works that only considered two-dimensional and axisymmetric horn configurations. The more realistic case of 3D acoustic horns with symmetry regarding two orthogonal planes is addressed. The use of the domain decomposition technique with two interconnected sub-regions along a continuity boundary is proposed, allowing for the computation of the sound pressure generated by an acoustic horn installed on a rigid screen. In order to reduce the model discretization requirements for these cases, Green’s functions derived with the image source methodology are adopted, automatically accounting for the presence of symmetry conditions. A strategy for the calculation of an optimal position of the virtual sources used by the MFS to define the solution is also used, leading to improved reliability and flexibility of the proposed method. The responses obtained by the developed model are compared to reference solutions, computed by well-established models based on the boundary element method. Additionally, numerically calculated acoustic parameters, such as directivity and beamwidth, are compared with those evaluated experimentally.

A Finite Element Numerical Algorithm for Modelling and Data Fitting in Complex Systems

Numerical modelling methodologies are important by their application to engineering and scientific problems, because there are processes where analytical mathematical expressions cannot be obtained to model them. When the only available information is a set of experimental values for the variables that determine the state of the system, the modelling problem is equivalent to determining the hyper-surface that best fits the data. This paper presents a methodology based on the Galerkin formulation of the finite elements method to obtain representations of relationships that are defined a priori, between a set of variables: y = z(x1, x2,...., xd). These representations are generated from the values of the variables in the experimental data. The approximation, piecewise, is an element of a Sobolev space and has derivatives defined in a general sense into this space. The using of this approach results in the need of inverting a linear system with a structure that allows a fast solver algorithm. The algorithm can be used in a variety of fields, being a multidisciplinary tool. The validity of the methodology is studied considering two real applications: a problem in hydrodynamics and a problem of engineering related to fluids, heat and transport in an energy generation plant. Also a test of the predictive capacity of the methodology is performed using a cross-validation method.

Numerical and experimental investigation of structural stiffness influence on ratcheting convection cell in granular soils under cyclic loading

Lateral cyclic loaded structures in granular soils can lead to an accumulation of irreversible strains by changing their mechanical response (densification) and forming a closed convective cell in the upper layer of the bedding. In the present thesis the convective cell dimension, formation and grain migration inside this closed volume have been studied and presented in relation to structural stiffness and different loads. This relation was experimentally investigated by applying a cyclic lateral force to a scaled flexible vertical element embedded in dry granular soil. The model was monitored with a camera in order to derive the displacement field by means of the PIV technique. Modelling large soil deformation turns out to be difficult, using mesh-based methods. Consequently, a mesh-free approach (DEM) was chosen in order to investigate the granular flow with the aim of extracting interesting micromechanical information. In both the numerical and experimental analyses the effect of different loading magnitudes and different dimensions of the vertical element were considered. The main results regarded the different development, shape and dimensions of the convection cell and the surface settlements. Moreover, the Discrete Element Method has proven to give satisfactory results in the modelling of large deformation phenomena such as the ratcheting convective cell.

Physical characterization and numerical modeling of a WEST-like tokamak divertor plasma

Il presente elaborato è incentrato sulla modellizzazione del plasma di bordo nei dispositivi per la produzione di energia da fusione nucleare noti come tokamak. La tecnologia che nel corso di tutta la seconda metà del XX secolo fino ad oggi è stata sviluppata a questo fine deve necessariamente scontrarsi con alcuni limiti. Nei tokamak il confinamento del plasma è di tipo magnetico e vincola le particelle a muoversi di moto elicoidale all'interno del vessel, tuttavia il confinamento non risulta perfetto e parte dell'energia si scarica sulle pareti della camera, rischiando pertanto di fondere i materiali. Alcune strategie possono essere messe in atto per limitare questo problema, per esempio agendo sulla geometria del tokamak, oppure sulla fisica, inducendo nel plasma una data concentrazione di impurezze che ionizzino irraggiando parte dell'energia di plasma. Proprio tale meccanismo di perdita è stato simulato in un modello monodimensionale di plasma monofluido di bordo. I risultati del codice numerico relativo al modello dimostrano che per concentrazioni di impurezze crescenti è possibile diminuire in modo significativo flusso di calore e temperatura al divertore. Per di più risulta possibile controllare la posizione del fronte di irraggiamento per mezzo di parametri di controllo del plasma quali la pressione. Si osserva inoltre l'insorgere del cosiddetto fenomeno di biforcazione alle basse temperature di divertore, fenomeno in cui il plasma si comporta in modo instabile a causa di fenomeni fisici tipici delle basse energie ("detachment") e a seguito del quale può improvvisamente spegnersi (disruzione). Infine lo stesso modello è stato migliorato inserendo l'ipotesi di plasma bifluido. Anche per gli ioni viene osservato il fenomeno di biforcazione. I risultati numerici evidenziano le dinamiche dello scambio energetico fra le specie gettando le basi di una progettazione efficiente della chimica del plasma finalizzata al raffreddamento del divertore.

On the initiation of boudinage and folding instabilities in layered elasto-visco-plastic solids - Insights from natural microstructures and numerical simulations

The present understanding of the initiation of boudinage and folding structures is based on viscosity contrasts and stress exponents, considering an intrinsically unstable state of the layer. The criterion of localization is believed to be prescribed by geometry-material interactions, which are often encountered in natural structures. An alternative localization phenomenon has been established for ductile materials, in which instability emerges for critical material parameters and loading rates from homogeneous conditions. In this thesis, conditions are sought under which this type of instability prevails and whether localization in geological materials necessarily requires a trigger by geometric imperfections. The relevance of critical deformation conditions, material parameters and the spatial configuration of instabilities are discussed in a geological context. In order to analyze boudinage geometries, a numerical eigenmode analysis is introduced. This method allows determining natural frequencies and wavelengths of a structure and inducing perturbations on these frequencies. In the subsequent coupled thermo-mechanical simulations, using a grain size evolution and end-member flow laws, localization emerges when material softening through grain size sensitive viscous creep sets in. Pinch-and-swell structures evolve along slip lines through a positive feedback between the matrix response and material bifurcations inside the layer, independent from the mesh-discretization length scale. Since boudinage and folding are considered to express the same general instability, both structures should arise independently of the sign of the loading conditions and for identical material parameters. To this end, the link between material to energy instabilities is approached by means of bifurcation analyses of the field equations and finite element simulations of the coupled system of equations. Boudinage and folding structures develop at the same critical energy threshold, where dissipative work by temperature-sensitive creep overcomes the diffusive capacity of the layer. This finding provides basis for a unified theory for strain localization in layered ductile materials. The numerical simulations are compared to natural pinch-and-swell microstructures, tracing the adaption of grain sizes, textures and creep mechanisms in calcite veins. The switch from dislocation to diffusion creep relates to strain-rate weakening, which is induced by dissipated heat from grain size reduction, and marks the onset of continuous necking. The time-dependent sequence uncovers multiple steady states at different time intervals. Microstructurally and mechanically stable conditions are finally expressed in the pinch-and-swell end members. The major outcome of this study is that boudinage and folding can be described as the same coupled energy-mechanical bifurcation, or as one critical energy attractor. This finding allows the derivation of critical deformation conditions and fundamental material parameters directly from localized structures in the field.

Benchmarking numerical models of brittle thrust wedges

We report quantitative results from three brittle thrust wedge experiments, comparing numerical results directly with each other and with corresponding analogue results. We first test whether the participating codes reproduce predictions from analytical critical taper theory. Eleven codes pass the stable wedge test, showing negligible internal deformation and maintaining the initial surface slope upon horizontal translation over a frictional interface. Eight codes participated in the unstable wedge test that examines the evolution of a wedge by thrust formation from a subcritical state to the critical taper geometry. The critical taper is recovered, but the models show two deformation modes characterised by either mainly forward dipping thrusts or a series of thrust pop-ups. We speculate that the two modes are caused by differences in effective basal boundary friction related to different algorithms for modelling boundary friction. The third experiment examines stacking of forward thrusts that are translated upward along a backward thrust. The results of the seven codes that run this experiment show variability in deformation style, number of thrusts, thrust dip angles and surface slope. Overall, our experiments show that numerical models run with different numerical techniques can successfully simulate laboratory brittle thrust wedge models at the cm-scale. In more detail, however, we find that it is challenging to reproduce sandbox-type setups numerically, because of frictional boundary conditions and velocity discontinuities. We recommend that future numerical-analogue comparisons use simple boundary conditions and that the numerical Earth Science community defines a plasticity test to resolve the variability in model shear zones.

Comparisons between analogue and numerical models of thrust wedge development

Analogue and finite element numerical models with frictional and viscous properties are used to model thrust wedge development. Comparison between model types yields valuable information about analogue model evolution, scaling laws and the relative strengths and limitations of the techniques. Both model types show a marked contrast in structural style between ‘frictional-viscous domains’ underlain by a thin viscous layer and purely ‘frictional domains’. Closely spaced thrusts form a narrow and highly asymmetric fold-and-thrust belt in the frictional domain, characterized by in-sequence propagation of forward thrusts. In contrast, the frictional-viscous domain shows a wide and low taper wedge and a thrust belt with a more symmetrical vergence, with both forward and back thrusts. The frictional-viscous domain numerical models show that the viscous layer initially simple shears as deformation propagates along it, while localized deformation resulting in the formation of a pop-up structure occurs in the overlying frictional layers. In both domains, thrust shear zones in the numerical model are generally steeper than the equivalent faults in the analogue model, because the finite element code uses a non-associated plasticity flow law. Nevertheless, the qualitative agreement between analogue and numerical models is encouraging. It shows that the continuum approximation used in numerical models can be used to model frictional materials, such as sand, provided caution is taken to properly scale the experiments, and some of the limitations are taken into account.

Application of numerical classification in ecological investigations of water pollution /

Mode of access: Internet.

Comparison of numerical and physical hydraulic models, Masonboro Inlet, North Carolina /

"June 1977."

Comparison of numerical and physical hydraulic models, Masonboro Inlet, North Carolina : Appendix 1, fixed-bed hydraulic model results /

"June 1977."