Search

Dynamic mechanical spectroscopy is a common analysis for polymers. Rectangular specimens are usually used for measurement in the solid state due to their easy designing. However, in reason of the non‐symmetric shape of sample, rectangular specimens do not experience linear stress on their all body, leading to overestimation of shear modulus. Corrections are required to determine the right shear modulus. In this present work, straight shear modulus determination is carried out using specimens with a cylindrical shape. The reliability of the technique is described on a biosourced polymer materials made of thermoplastic starch, plasticized by glycerol, choline chloride/urea mixture (ChCl/U), 1‐ethyl‐3‐methylimidazolium chloride ([EMIM]Cl) and 1‐ethyl‐3‐imidazolium acetate ([EMIM]Ac). The technique is shown to be particularly interesting to avoid any additional shaping of the specimens prior to measurements, that can be detrimental to their properties (evaporation, degradation). A comparison between cylinder and rectangular specimen has also been made to illustrate the interest of cylinder clamping on biosourced polymer material. [ABSTRACT FROM AUTHOR]

Abstract This study provides analytic solutions for the magnetic field of coils and magnets that have a non‐axisymmetric cylindrical geometry with a rectangular cross‐section. New analytic solutions are provided for radially magnetized permanent magnet arcs, thin coil disc sectors, and thick coil sectors. If components of the 3D field are not representable in closed‐form or as canonical Legendre elliptic integrals, the exact solution is given in terms of a series of regularized beta functions. The limit and hence spatial convergence is found to these series, giving a well‐defined and fast solving algorithm for computation. The equations can be readily applied to find the magnetostatic field in linear or non‐linear systems that contain a large set of elements. Example applications are provided to demonstrate how the field can be used to calculate forces and benchmark computational efficiency of the equations. A thorough review of the preceding literature and background theory is provided before a detailed methodology obtaining the analytic solutions contained in this compendium, and further related geometries in cylindrical or spherical coordinates. This is the first study to comprehensively solve the field equations for this collection of electromagnetic geometries.

Plant roots are characterized by a cylindrical nature. The worldwide bi-dimensionality in suites of traits for cylindrical-shaped roots reflects essential plant strategies to cope with different environments. Of the two dimensions, one is the root economic spectrum, representing a trade-off between nutrient acquisition and conservation, while the second is the gradient of nutrient foraging from mycorrhizal dependence to reliance on roots themselves. However, the origin of this bi-dimensionality remains unknown. We propose that this bi-dimensionality arises from the cylindrical geometry of roots, the allometric relationship of root anatomical structures, and the independence between root cell wall thickness and cell number. As such, the cylindrical shape of roots, together with mycorrhizal collaborations, may account for the origin of the bi-dimensionality observed in root traits. Plant roots show extraordinary diversity in form and function in heterogeneous environments. Mounting evidence has shown global bi-dimensionality in root traits, the root economics spectrum (RES), and an orthogonal dimension describing mycorrhizal collaboration; however, the origin of the bi-dimensionality remains unresolved. Here, we propose that bi-dimensionality arises from the cylindrical geometry of roots, allometry between root cortex and stele, and independence between root cell wall thickness and cell number. Root geometry and mycorrhizal collaboration may both underlie the bi-dimensionality. Further, we emphasize why plant roots should be cylindrical rather than flat. Finally, we highlight the need to integrate organ-, cellular-, and molecular-level processes driving the bi-dimensionality in plant roots to fully understand plant diversity and functions. [ABSTRACT FROM AUTHOR]

A numerical study in a two-dimensional, axisymmetric cylindrical geometry is performed to evaluate the effect of Marangoni convection in shallow liquid layers of silicone oil. The study aims to understand Marangoni convection for a range of temperature gradients (9.0 K, 11.5 K, and 14.0 K), layer depths (4.0 mm, 4.5 mm, and 5.0 mm) under earth gravity (1.0 g), and microgravity (0.5 × 10 −3 g) conditions. The finite element method is used to solve the present problem using COMSOL V5.6 commercial tool. The microgravity studies ensure Marangoni convection dominates natural convection. In the present simulation, it is found that Marangoni convection is influenced more by temperature difference than layer depths as the change in temperature gradient gives a higher velocity deviation than a change in layer depths. In the presence of fluid meniscus, the thermo-capillary force is observed to be less significant in driving the flow as the peak velocity magnitude is observed to get reduced by 60%. From the microgravity study, it is clear that for velocity magnitude at z = 2.5 mm the Marangoni convection contributes to almost 90% of the total convection.

Phase change material (PCM)-based thermal energy storage (TES) systems are widely used for repeated intermittent heating and cooling applications. However, such systems typically face some challenges due to the low thermal conductivity and expensive encapsulation process of PCMs. The present study overcomes these challenges by proposing a lightweight, low-cost, and low thermal resistance TES system that realizes a fluid-to-PCM additively manufactured metal-polymer composite heat exchanger (HX), based on our previously developed cross-media approach. A robust and simplified, analytical-based, 1D reduced-order model (ROM) was developed to compute the TES system performance, saving computational time compared to modeling the entire TES system using PCM-related transient CFD modeling. The TES model was reduced to a segment-level model comprising a single PCM-wire cylindrical domain based on the tube-bank geometry formed by the metal fin-wires. A detailed study on the geometric behavior of the cylindrical domain and the effect of overlapped areas, where the overlapped areas represent a deviation from 1D assumption on the TES performance, was conducted. An optimum geometric range of wire-spacings and size was identified. The 1D ROM assumes 1D radial conduction inside the PCM and analytically computes latent energy stored in the single PCM-wire cylindrical domain using thermal resistance and energy conservation principles. The latent energy is then time-integrated for the entire TES, making the 1D ROM computationally efficient. The 1D ROM neglects sensible thermal capacity and is thus applicable for the low Stefan number applications in the present study. The performance parameters of the 1D ROM were then validated with a 2D axisymmetric model, typically used in the literature, using commercially available CFD tools. For validation, a parametric study of a wide range of non-dimensionalized parameters, depending on applications ranging from pulsed-power cooling to peak-load shifting for building cooling application, is included in this paper. The 1D ROM appears to correlate well with the 2D axisymmetric model to within 10%, except at some extreme ranges of a few of the non-dimensional parameters, which lead to the condition of axial conduction inside the PCM, deviating from the 1D ROM. [ABSTRACT FROM AUTHOR]

We propose a spectral Particle-In-Cell (PIC) algorithm that is based on the combination of a Hankel transform and a Fourier transform. For physical problems that have close-to-cylindrical symmetry, this algorithm can be much faster than full 3D PIC algorithms. In addition, unlike standard finite-difference PIC codes, the proposed algorithm is free of spurious numerical dispersion, in vacuum. This algorithm is benchmarked in several situations that are of interest for laser-plasma interactions. These benchmarks show that it avoids a number of numerical artifacts, that would otherwise affect the physics in a standard PIC algorithm - including the zero-order numerical Cherenkov effect.

We propose a spectral Particle-In-Cell (PIC) algorithm that is based on the combination of a Hankel transform and a Fourier transform. For physical problems that have close-to-cylindrical symmetry, this algorithm can be much faster than full 3D PIC algorithms. In addition, unlike standard finite-difference PIC codes, the proposed algorithm is free of spurious numerical dispersion, in vacuum. This algorithm is benchmarked in several situations that are of interest for laser-plasma interactions. These benchmarks show that it avoids a number of numerical artifacts, that would otherwise affect the physics in a standard PIC algorithm - including the zero-order numerical Cherenkov effect.

Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green's functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal–normal component of the stress–energy tensor) as a suitable second-order differential operator acting on the corresponding Green's functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force. [ABSTRACT FROM AUTHOR]

On the basis of Landau–de Gennes theory and the finite-difference iterative method, spontaneous chiral structures in cylinders with degenerate planar boundary conditions are investigated. A double-twist director configuration can be achieved by the saddle–splay contribution, corresponding to the L24 term in Landau–de Gennes theory. The twist angle increases as the radius R of the cylinder is reduced because the curvature of the cylindrical surface becomes larger. Moreover, we find fine structures of the domain walls and point defects between opposite-handed domains. An energy comparison shows that domain walls are the stable state for K24/K> 0.6 (K11 = K33 = K, K11, K33 and K24 are the splay, bend, saddle–splay elastic constants in Frank theory), whereas point defects are the stable state for K24/K< 0.6. [ABSTRACT FROM AUTHOR]

This work presents a study of the relationship between the dynamic characteristics of a full-size structure and its scaled version. By understanding the similitude laws and suitably scaling material properties, one can utilize an efficient scale model to obtain realistic structural performance. The main goal is to identify whether the natural frequencies of a coupled fluid-structure system follow the same similitude law as the full-size structure and its scaled prototype. The coupled fluid-structure systems are those where the structural behavior affects the fluid response and the fluid behavior affects the structural response so both must be solved, i.e., coupled together, to obtain the interacting solution. Scaling of the structure in specific numerical analysis defines two models that are similar in their geometry and material properties and is considered only in relation to the need for laboratory testing of engineering models. The finite element representation of a laboratory tested physical model is a bridge between this small-scale prototype and the computational model of a real structure. A thin-walled water storage tank with a simple cylindrical geometry is adopted for the analysis as representative of a coupled fluid-structure system whose dynamic behavior is strongly influenced by the presence of fluid and neither the structure nor the fluid alone governs the response. [ABSTRACT FROM AUTHOR]

Using the support operator technique for two-dimensional problems of the elasticity theory we constructed integrally consistent approximations of the components of the strain tensor and the elastic energy of the medium for the equations of the elasticity theory in terms of displacements. Approximations are constructed for the case of irregular difference grids in the R–Z plane of a cylindrical coordinate system. We use the limiting process assuming that the azimuthal angle tends to zero for passing from the full three-dimensional approximations to the two-dimensional approximations in the R–Z plane. The used technique preserves the divergent form, self-adjointness, and sign-definiteness of the two-dimensional approximations. These properties are inherent in their 3D predecessors corresponding to the operators in the governing differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the influence of thermocapillary convection in a shallow annular pool using different cylinder configurations is numerically investigated. An axisymmetric, two-dimensional, cylindrical geometry with two concentric cylinders kept at two different temperatures constitutes the model. The main cylinder has a radius of 40 mm and the secondary cylinder has a radius of 20 mm. The effects of Prandtl numbers (Pr = 6.7, 16.1, and 25.2) and layer depths (d = 3.0 mm and 11.0 mm) on the streamlines, isotherms, velocity, and temperature profiles at various locations of the flow are investigated. Depending on the position of the secondary cylinder, three different configurations: 1) an internal cylinder 2) an external cylinder, and a 3) central cylinder is analysed. The flow is characterized by a rapid inward flow along the hot outer wall towards the colder inner wall. In deeper pool at lower Prandtl numbers, flow patterns are characterized by a single rolling cell structure. Conversely, secondary cell formations appear in shallow pool regardless of the Prandtl number. Based on the comparative analysis of various cylinder configurations, it is clear that the external cylinder generates greater velocities and temperatures along its surface. The maximum disparity in surface velocity between an external cylinder and other configurations is 7%, 12%, and 13% for Prandtl numbers of 6.7, 16.1, and 25.2 respectively. Therefore, positioning the cylinder externally promotes swifter convection and more effective mixing of the fluid. [ABSTRACT FROM AUTHOR]

A theoretical approach based on a numerical simulation using experimental data is proposed as a contribution to the study of laser-based additive manufacturing by direct energy deposition (DED). The aim is to simulate the stages of layer-by-layer build-ups by considering the induced generated liquid metal in a cylindrical geometry. For normal atmospheric conditions, a 3D model is developed using the finite volumes method to solve the governing equations of the occurring fluid flows and heat transfers. The model is based on the experimental work performed on the LENS system at the Council for Scientific and Industrial Research’s (CSIR) National Laser Centre (NLC), where Ti-6Al-4V cylindrical coupons are laser printed and characterised.

• A 2-D model was investigated for the FT reaction for solid and hollow cylindrical Co catalyst pellets. • Conditions considered where liquid hydrocarbons accumulate in the pores. • Effectiveness factor exceeded that of spherical and slab geometry for Thiele moduli range 0.75–1.15. • FT growth parameter, α , in this range was lowest with cylinders and so gave greatest CH 4 selectivity. Previous modelling of fixed-bed, Fischer-Tropsch (FT) reactors has demonstrated the advantages relative to spherical pellets of using cylindrical and shaped pellets to provide improved transport attributes under conditions relevant to industrial operation. However, mass transport models have focussed on the investigation of transport within pellets with spherical symmetry, whilst detailed investigations of more complex shapes have not been undertaken. Here, a pseudo-isothermal, steady-state, two-dimensional model was investigated for catalyst pellets of cylindrical form, both solid and hollow. A cobalt-based catalyst was considered at conditions where the rate of condensable hydrocarbon generation is large enough to result in the accumulation of liquid hydrocarbons in the pores of a catalyst. It was found that effectiveness factors were bounded by those of sphere and slab above and below Thiele moduli of ∼0.75 and ∼1.15, respectively, for the conditions examined, with the effectiveness factors exceeding those of both sphere and slab models between these moduli. Here, comparisons were made on the basis of the characteristic diffusion length, the catalyst particle's volume divided by its external surface area. However, values of the FT chain growth parameter, α , between these values of Thiele modulus were lower than those of both sphere and slab geometry, and thus under these conditions hollow cylinders gave the greatest methane selectivity. [ABSTRACT FROM AUTHOR]

In this paper we implement the stretched-coordinate Perfectly Matched Layer (PML) technique in [28] to emulate full power absorption outside the simulation domain for time-harmonic electromagnetic wave propagation in presence of gyrotropic dielectric tensor and curved geometry relevant for magnetized plasma devices. We recall the PML formulation as an artificial inhomogeneous lossy medium, following the stretching into the complex plane of a general system of three orthogonal curvilinear coordinates. We apply the general method in cylindrical and toroidal geometries. We then assess this technique in a simple case combining gyrotropy and coordinate curvature. Our test problem analytically quantifies the reflection of Transverse Electric (TE) cylindrical eigenmodes in a gyrotropic medium by a radial PML in cylindrical geometry. The obtained reflection coefficient involves wave, PML and geometric parameters at the PML location. The new coefficient generalizes the one obtained earlier with Cartesian coordinates, and becomes equivalent when the effects of the local cylindrical curvature at the PML (stretched) location can be neglected. These curvature effects are outlined and the limitations they impose on the properties of the PML are quantified as a function of the relevant parameters. Peculiarities related to the gyrotropy are also highlighted. Finite element calculations of the test problem in two-dimensional cylindrical geometry are exploited to verify these properties numerically. Indications are finally given on how to choose the PML parameters in order to obtain a minimal wave reflection at given numerical cost, taking into account errors associated with the numerical scheme. • Stretched-coordinate PMLs were implemented for EM waves in curved gyrotropic media. • Analytic reflection coefficient was derived for cylindrical TE waves in radial PML. • Formula was assessed against 2D Finite Element (FE) simulations in cylinder. • Peculiarities associated with gyrotropy and cylindrical geometry were highlighted. • We also tested the role of FE discretization and unbounded stretching functions. [ABSTRACT FROM AUTHOR]

Expressions for X‐ray absorption and secondary scattering are developed for cylindrical sample geometries. The incident‐beam size is assumed to be smaller than the sample and in general directed off‐axis onto the cylindrical sample. It is shown that an offset beam has a non‐negligible effect on both the absorption and multiple scattering terms, resulting in an asymmetric correction that must be applied to the measured scattering intensities. The integral forms of the corrections are first presented. A small‐beam limit is then developed for easier computation. Absorption and secondary scattering corrections for a cylindrical sample geometry are developed for the case when the incident beam is smaller than the sample and, in general, off‐center from the cylinder. [ABSTRACT FROM AUTHOR]

Highlights • The solution of neutron diffusion equation is important to describe the behaviour of the neutrons in the nuclear reactors. • The essential cylindrical reactor geometry will be studied in this work, where the reactor reflector part is added to the core part to minimize its critical radius, and the neutrons diffuse in two different velocities. • To assure our results, a comparison with classical results and transport theory data has been achieved which made after needed simplification to one velocity case. • The necessary C++ codes using GSL library are accomplished to attain this comparison. • The application of new developed approaching technique of homotopy perturbation method improves the solution of two-group neutron diffusion problem in reflected cylindrical reactor symmetry which represents its flexibility and suitability to deal with different nuclear reactor boundary conditions. • This usage gives results closer than classical calculations, as well as HPM without new approaching technique, with respect to the transport theory data (which considered as our benchmark) after applied in three computational examples. Abstract The solution of neutron diffusion equation is important to describe the behavior of the neutrons in the nuclear reactors. The essential cylindrical reactor geometry will be studied in this work, where the reactor reflector part is added to the core part to minimize its critical radius, and the neutrons diffuse in two different velocities. The massive results when diversification of the appropriate new approaching technique of homotopy perturbation method (HPM) represent its flexibility and suitability to deal with different nuclear reactor boundary conditions. To assure our results, a comparison with classical results and transport theory data has been achieved which made after needed simplification to one velocity case. The necessary C ++ codes using GSL library are accomplished to attain this comparison. [ABSTRACT FROM AUTHOR]

Nodal expansion method (NEM) is a typical nodal method in solving neutron diffusion equation in coarse mesh spatial discretization. NEM has been extended to cylindrical geometry in previous studies. Cylindrical NEM can be realized by different forms, where the variables in the nodal coupling equations are classified as net current, flux or partial current at the nodal surface, respectively. The three types of coupling forms of NEM have been implemented in PANGU code for the high temperature gas-cooled reactor (HTGR) physics analysis. This paper derives the three types of coupling forms of NEM, and analyzes their performance in solving the HTGR model containing void region.

ABSTRACT This paper aims at describing the osmotic dehydration of radish cut into cylindrical pieces, using one- and two-dimensional analytical solutions of diffusion equation with boundary conditions of the first and third kind. These solutions were coupled with an optimizer to determine the process parameters, using experimental data. Three models were proposed to describe the osmotic dehydration of radish slices in brine at low temperature. The two-dimensional model with boundary condition of the third kind well described the kinetics of mass transfers, and it enabled prediction of moisture and solid distributions at any given time.

Organogenesis involves extensive and dynamic changes of tissue shape during development. It is associated with complex morphogenetic events that require enormous tissue plasticity and generate a large variety of transient three-dimensional geometries that are achieved by global tissue responses. Nevertheless, such global responses are driven by tight spatiotemporal regulation of the behaviours of individual cells composing these tissues. Therefore, the development of image analysis tools that allow for extraction of quantitative data concerning individual cell behaviours is central to study tissue morphogenesis. There are many image analysis tools available that permit extraction of cell parameters. Unfortunately, the majority are developed for tissues with relatively simple geometries such as flat epithelia. Problems arise when the tissue of interest assumes a more complex three-dimensional geometry. Here, we use the endothelium of the developing zebrafish dorsal aorta as an example of a tissue with cylindrical geometry and describe the image analysis routines developed to extract quantitative data on individual cells in such tissues, as well as the image acquisition and sample preparation methodology. This article is part of the Theo Murphy meeting issue 'Mechanics of development'. [ABSTRACT FROM AUTHOR]

We study the behaviour of the isentropic flow of a gas in both a straight tube of constant cross section and a cone, open at one end and forced at or near resonance at the other. A continuous transition between these configurations is provided through the introduction of a geometric parameter k associated with the opening angle of the cone where the tube corresponds to $$k=0$$ . The primary objective is to find long-time resonant and near-resonant approximate solutions for the open tube, i.e. $$k\rightarrow 0$$ . Detailed analysis for both the tube and cone in the limit of small forcing $$(O(\varepsilon ^{3}))$$ is carried out, where $$\varepsilon ^{3}$$ is the Mach number of the forcing function and the resulting flow has Mach number $$O(\varepsilon )$$ . The resulting approximate solutions are compared with full numerical simulations. Interesting distinctions between the cone and the tube emerge. Depending on the damping and detuning, the responses for the tube are continuous and of $$O(\varepsilon )$$ . In the case of the cone, the resonant response involves an amplification of the fundamental resonant mode, usually called the dominant first-mode approximation. However, higher modes must be included for the tube to account for the nonlinear generation of higher-order resonances. Bridging these distinct solution behaviours is a transition layer of $$O(\varepsilon ^{2})$$ in k. It is found that an appropriately truncated set of modes provides the requisite modal approximation, again comparing well to numerical simulations. [ABSTRACT FROM AUTHOR]

The efficiency of any numerical scheme measures on the accuracy of the scheme and its computational time. In this work, an efficient augmented local radial basis function finite difference (RBF-FD) scheme has been developed for steady convection-diffusion problems. The accommodative full approximation scheme-full multigrid (FAS-FMG) analogy with local refinement is adopted to achieve this efficiency. The efficiency of the method is illustrated through linear convection-diffusion equation, coupled nonlinear equation and for the incompressible Navier-Stokes (NS) equations. Numerical studies are made by using multiquadric (MQ) RBF. The upwind type nodes are considered to handle convective terms effectively for NS equations. The developed scheme saves 70-80% of the CPU time for the first two model problems and at least 50% of the CPU time for NS equations than the usual RBF-FD method found in the literature. It is found that there are optimum sets of nodes which give accurate and stable solution. It is also found that the value of the shape parameter will be lower when the number of nodes is higher. But higher the value of Re, higher will be the shape parameter. [ABSTRACT FROM AUTHOR]

The cylindrical nodal expansion method (NEM) is promising for improving the efficiency of HTGR whole-core neutronics calculation. In HTGR, there generally exists large void cavity above the pebble-bed core. The traditional cylindrical NEM is formulated by the surface-partial-current (SPC) coupling form, which is unable to solve the problem containing void cavity. Aiming to overcome this problem, this paper proposes three types of improved calculation schemes for cylindrical NEMs to treat the void cavity. The first one is employing a modified SPC coupling form; the second one is employing the surface-net-current (SNC) coupling form combined with advanced matrix solvers; the third one is directly using the surface-flux (SF) coupling form. All of the three types of cylindrical NEM schemes have been developed in PANGU code. Numerical results demonstrate the capability and performance of the improved NEMs in calculating the HTGR containing void cavity. [ABSTRACT FROM AUTHOR]

The gas breakdown in the Inertial Electrostatic Confinement (IEC) device has been studied by DC discharge of Nitrogen gas in the pressure range between 0.03 Torr and 0.7 Torr. Paschen curves and Townsend coefficients are obtained in a cylindrical system with inner anode in shape of rods and the outer grid cathode. The breakdown occurs and the plasma is formed at the center of the electrodes. Townsend coefficients are calculated for anode transparencies of 84%, 92%, and 96% (24, 12, and 6 rods respectively). On the left hand side of Paschen curves, the high transparency has a higher breakdown voltage. More electrons are allowed to pass the anode. So, the collisions increase, leading to a slight increment of the first Townsend coefficient (α) decreasing of the second Townsend coefficient (γ) while the ionization efficiency (η) is not affected. For high reduced electric field E/P, i.e. a high electric field with low pressure, the probability of ionizing collision events decreases. This causes a decrement of α while γ is raised because the ions' energy is not exhausted in collisions at the center. The coefficients α and γ are working in harmony to balance the rates of electrons generated in gas ionization so that the relation between the two coefficients is satisfied. [ABSTRACT FROM AUTHOR]

The nodal methods are significantly more accurate than the traditional methods such as finite difference method (FDM), finite element method (FEM) etc. However, these methods can be used only for the nodes of only a few limited shapes such as rectangular (in 2D) or cuboidal (in 3D). In this paper the approach for solving neutron diffusion equation in 2-d cylindrical polar geometry (r,θ) using nodal method is discussed. The analytic nodal method using transverse integration process is used to solve the 2-d neutron diffusion equation in polar coordinate. The problem involved with the transverse integration in θ-direction has been resolved by approximating the average of products by product of averages . This approximation leads to three different formulations of the scheme. A detailed study of the numerical error for source problems having analytical solutions is carried out. Using this analysis it has been shown that the error is second order for these problems irrespective of the formulation used. In addition to that the methodology is used to solve criticality problems for which analytical or benchmark solutions are available. The solution of the source problem shows that method maintains its accuracy and order even with approximations made to deal with transverse integration in θ-direction. The comparison of eigenvalues obtained with current methodology with those obtained analytically or available as benchmark show that the method is capable of accurately predicting the eigenvalues. [ABSTRACT FROM AUTHOR]