Polar Mesh Research Articles

Re-patterning of cylindrical packing of diblock copolymers under confinement and curvature effects by using approximations of PDE’s involved in the CDS model on polar mesh system

Soft materials, including diblock copolymers, are advancing nanotechnology due to their unique properties, applications materials include energy harvesting, water sanitation, environmental treatment, nanosensors, drug delivery and nanolithography. These materials are light, cheap, efficient, sensitive, durable and more functional, whose new morphologies have been predicted by mathematicians through simulation. This work produces and predicts the pattern of packing of nano-cylinders by using confinement to appreciate the frustration in the packing of nano-cylinders under the influence of curvature. In this contribution, the cell dynamic simulations model is used to examine the impact of circular annular pore confinement on system orientation toward cylindrical morphologies. A 9-point stencil approximates the isotropic Laplacian by finite-difference discretization on a polar grid to meet the requirement of a cell dynamic simulation model. FORTRAN codes are generated for the set of PDEs included in the CDS model. OPEN DX is used to visualise the predicted cylindrical patterns. The consistency of our results with experimental observations makes our research valid and significant.

Morphological Investigation of Lamellae Patterns in Diblock Copolymers under Change of Thickness and Confinement in Polar Geometry

Mathematicians collaborate computationally with practical scientists to predict the novel morphologies soft materials, having potential applications in nanotechnology. This work presents the novel lamellae morphology that the confined diblock copolymer system displayed inside the polar geometry in order to achieve this, cell dynamics simulations (CDS) model is employed on a polar mesh to study the special impacts of confinement on the self-assembly behavior of diblock copolymer systems. Under the numerical formulation of the isotropic Laplacian with polar geometry employed in the CDS model, the set of 9-point stencil is considered to discretize the Laplacian operator on polar mesh system. FTCS finite difference mechanism is taken into consideration for discretization of Laplacian operator due to its computational efficiency and accuracy. Cell dynamics simulation is efficient simulation model for large-scale simulation as compared to the other models available in literature. In the simulation, evolutionary lamella morphology is presented with and without confinement for various sizes of the circular annular pores. A comparative review of lamella morphologies achieved by simulation in comparison of experimental study is presented, coupled with a thorough investigation of symmetric and asymmetric morphologies in captivity.

Comparative Study of Numerical Approximation Schemes for Laplacian Operator in Polar Mesh System On 9-points Stencil Including Mixed Partial Derivative By Finite Difference Method

In this resech paper we have discussed on two different types of discretization schmes for laplacian operator which carries nine points stencil including two pair of lines symmeical and asymmetrical from planned molecule. it is well organized and modicfied five point scheme which has been devloped on finite difference method. FDM is a method which is being used for decretization of of PDES as well as ODES. In this paper we will use two different types of scheme which are devloped by FDM on polar coordinate system and in last we will discuss on error analysis, staibility and graphically behaviour of both scheme.

Cell-centered Lagrangian Lax-Wendroff HLL hybrid scheme in cylindrical geometry

Lagrangian hydrodynamics as described by the Euler equations is treated by an improved version of the basic predictor-corrector Lax-Wendroff method that also has added HLL-type dissipative fluxes, including both artificial viscosity and energy dissipation. The method in Cartesian geometry is enhanced by a different weighting of the conservative variables in the predictor and a new treatment of material interfaces. The basic method is extended to cylindrical r,z geometry, where it satisfies the geometric conservation law and keeps exact spherical symmetry on equiangular polar meshes. The added viscosity does not preserve symmetry, so in order to achieve that we have added a symmetry correction. Numerical hydrodynamic tests, including the Noh, Sedov, spherical Sod, free expansion and Kidder problems show a reasonable performance of the method. In addition, the spherical Noh problem was simulated on an initially rectangular mesh with a very good result regarding its symmetry.

Symmetry-preserving WENO-type reconstruction schemes in Lagrangian hydrodynamics

WENO (Weighted essentially non-oscillatory)-type reconstruction schemes are used to work as a limiter that preserves cylindrical symmetry for radial shock flows on an equal-angle polar mesh in Lagrangian hydrodynamics. These WENO-type limiters are tested with a high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method. In order to guarantee the compactness of the underlying DG methods, for strong shock problems, canonical WENO and Hermite WENO reconstructions are used for second-order DG(P1) and third-order DG(P2) respectively, using the known DG solution for the target cell as the central stencil. A salient feature of Hermite WENO lies in taking advantage of available information, namely the derivatives that are evolved with the DG method. With a local orthonormal basis or a local characteristic decomposition, the WENO-type reconstruction schemes can guarantee the cylindrical symmetry for Lagrangian hydrodynamics. A suite of challenging test problems are calculated to demonstrate the accuracy and robustness of these WENO-type schemes.

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Many important problems in astrophysics, space physics, and geophysics involve flows of (possibly ionized) gases in the vicinity of a spherical object, such as a star or planet. The geometry of such a system naturally favors numerical schemes based on a spherical mesh. Despite its orthogonality property, the polar (latitude-longitude) mesh is ill suited for computation because of the singularity on the polar axis, leading to a highly non-uniform distribution of zone sizes. The consequences are (a) loss of accuracy due to large variations in zone aspect ratios, and (b) poor computational efficiency from a severe limitations on the time stepping. Geodesic meshes, based on a central projection using a Platonic solid as a template, solve the anisotropy problem, but increase the complexity of the resulting computer code. We describe a new finite volume implementation of Euler and MHD systems of equations on a triangular geodesic mesh (TGM) that is accurate up to fourth order in space and time and conserves the divergence of magnetic field to machine precision. The paper discusses in detail the generation of a TGM, the domain decomposition techniques, three-dimensional conservative reconstruction, and time stepping.

A highly-accurate finite element method with exponentially compressed meshes for the solution of the Dirichlet problem of the generalized Helmholtz equation with corner singularities

In this study, a highly-accurate, conforming finite element method is developed and justified for the solution of the Dirichlet problem of the generalized Helmholtz equation on domains with re-entrant corners. The k − t h order Lagrange elements are used for the discretization of the variational form of the problem on exponentially compressed polar meshes employed in the neighbourhood of the corners whose interior angle is α π , α ≠ 1 ∕ 2 , and on the triangular and curved mesh formed in the remainder of the polygon. The exponentially compressed polar meshes are constructed such that they are transformed to square meshes using the Log-Polar transformation, simplifying the realization of the method significantly. For the error bound between the exact and the approximate solution obtained by the proposed method, an accuracy of O ( h k ) , h mesh size and k ≥ 1 an integer, is obtained in the H 1 -norm. Numerical experiments are conducted to support the theoretical analysis made. The proposed method can be applied for dealing with the corner singularities of general nonlinear parabolic partial differential equations with semi-implicit time discretization.

Computational Analysis and Validation of the Cylindrical TLM Approach on IMCP Antennas

Numerical computational methods on a Cartesian grid and analytical approaches face many problems and limitations when modeling antenna structures of circular/cylindrical geometry is concerned. This paper presents possibilities and computational demands of the cylindrical TLM approach with a wire modeling ability, implemented through in-house developed 3DTLMcyl_cw solver, when it is used as an accurate and efficient alternative for the design and analysis of inverted configurations of coaxially fed circular patch antennas. To emphasize the approach advantages, a thorough investigation of a polar mesh extension and resolution influence on simulated results has been carried out including an optimization of a coaxial feed position. The approach has been used to design IMCP and CB-IMCP antennas in order to determine antennas parameters and to study an influence of a substrate permittivity, an air-gap height and a metallic cavity presence on antennas’ performances. Accuracy of the approach has been experimentally verified, while its advantages have been pointed out with regards to the corresponding Cartesian TLM mesh and the approximate cavity model.

Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates

We present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem in 2D Cartesian geometry. This area-weighted approach, in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. A suite of test problems are calculated to demonstrate stable mesh motion, the expected second order accuracy of these methods, and that the new area-weighted DG method preserves spherical symmetry on 1D radial flow problems with equal-angle polar meshes.

Planet–disc interactions with discontinuous Galerkin methods using GPUs

We present a two-dimensional Cartesian code based on high order discontinuous Galerkin methods, implemented to run in parallel over multiple GPUs. A simple planet-disc setup is used to compare the behaviour of our code against the behaviour found using the FARGO3D code with a polar mesh. We make use of the time dependence of the torque exerted by the disc on the planet as a mean to quantify the numerical viscosity of the code. We find that the numerical viscosity of the Keplerian flow can be as low as a few $10^{-8}r^2\Omega$, $r$ and $\Omega$ being respectively the local orbital radius and frequency, for fifth order schemes and resolution of $\sim 10^{-2}r$. Although for a single disc problem a solution of low numerical viscosity can be obtained at lower computational cost with FARGO3D (which is nearly an order of magnitude faster than a fifth order method), discontinuous Galerkin methods appear promising to obtain solutions of low numerical viscosity in more complex situations where the flow cannot be captured on a polar or spherical mesh concentric with the disc.

A note on a conservative finite volume approach to address numerical stiffness in polar meshes

A polar coordinate system introduces a singularity at the pole, r=0, where terms with a factor 1/r can be ill-defined. While there are several approaches to eliminate this pole singularity in finite difference methods, finite volume methods largely bypass this issue by not storing or computing data at the pole. However, all methods face a very restrictive time step when using an explicit time advancement scheme in the azimuthal direction, where cell sizes are of the order O(Δr(rΔθ)). We use a conservative finite volume approach of merging cells on a structured O-mesh to remove this time step limit imposed by the CFL condition. The cell-merging procedure is implemented as a corrector step and incurs no changes to the underlying data structure for a structured grid. This short note describes the procedure and presents the validation and application of the algorithm to various problems. The algorithm is shown to be inexpensive and scalable. In addition, the cell-merging procedure is easily coupled with a line implicit scheme in the radial direction.

Efficient Modeling of a Circular Patch-Ring Antenna Using the Cylindrical TLM Approach

This letter addresses the ability of the TLM method with an incorporated compact wire model, both adjusted to the cylindrical grid, to accurately model a coaxially fed patch-ring antenna together with an investigation of the computational efficiency in a modeling procedure. Design of the antenna has included an optimization of the coaxial feed location to achieve a wider bandwidth. A prototype of the antenna was fabricated and measured. It has been shown that the patch-ring antenna exhibits more than three times wider bandwidth than the classic circular patch antenna. Advantages of the cylindrical-grid-based TLM approach related to a better conformity of the orthogonal polar mesh with the structure being modeled and a better efficiency associated with a number of TLM cells in the modeling procedure have been demonstrated with regards to the rectangular-grid-based TLM method. Also, a limitation of the rectangular solver related to an inadequate modeling of narrow slots that is dependent on the mesh resolution applied has been accentuated and proved.

On physical and numerical instabilities arising in simulations of non-stationary radiatively cooling shocks

We describe our modelling of the radiatively cooling shocks and their thin shells with various numerical tools in different physical and calculational setups. We inspect structure of the dense shell, its formation and evolution, pointing out physical and numerical factors that sustain its shape and also may lead to instabilities. We have found that under certain physical conditions, the circular shaped shells show a strong bending instability and successive fragmentation on Cartesian grids soon after their formation, while remain almost unperturbed when simulated on polar meshes. We explain this by physical Rayleigh--Taylor like instabilities triggered by corrugation of the dense shell surfaces by numerical noise. Conditions for these instabilities follow from both the shell structure itself and from episodes of transient acceleration during re-establishing of dynamical pressure balance after sudden radiative cooling onset. They are also easily excited by physical perturbations of the ambient medium. The widely mentioned Non-linear Thin Shell Instability, in contrast, in tests with physical perturbations is shown to have only limited chances to develop in real radiative shocks, as it seems to require a special spatial arrangement of fluctuations to be excited efficiently. The described phenomena also set new requirements on further simulations of the radiatively cooling shocks in order to be physically correct and free of numerical artefacts.

Solving the guiding-center model on a regular hexagonal mesh

This paper introduces a Semi-Lagrangian solver for the Vlasov-Poisson equations on a uniform hexagonal mesh. The latter is composed of equilateral triangles, thus it doesn't contain any singularities, unlike polar meshes. We focus on the guiding-center model, for which we need to develop a Poisson solver for the hexagonal mesh in addition to the Vlasov solver. For the interpolation step of the Semi-Lagrangian scheme, a comparison is made between the use of Box-splines and of Hermite finite elements. The code will be adapted to more complex models and geometries in the future.

Gyroaverage operator for a polar mesh

In this work, we are concerned with numerical approximation of the gyroaverage operators arising in plasma physics to take into account the effects of the finite Larmor radius corrections. The work initiated in reference [N. Crouseilles, M. Mehrenberger, H. Sellama, CiCP 8, 484 (2010)] is extended here to polar geometries. A direct method is proposed in the space configuration which consists in integrating on the gyrocircles using interpolation operator (Hermite or cubic splines). Numerical comparisons with a standard method based on a Padé approximation are performed: (i) with analytical solutions; (ii) considering the 4D drift-kinetic model with one Larmor radius and (iii) on the classical linear DIII-D benchmark case [A. Dimits et al., Phys. Plasmas 7, 969 (2000)]. In particular, we show that in the context of a drift-kinetic simulation, the proposed method has similar computational cost as the standard method and its precision is independent of the radius.

Compatible, total energy conserving and symmetry preserving arbitrary Lagrangian–Eulerian hydrodynamics in 2D rz – Cylindrical coordinates

We present a new discretization for 2D arbitrary Lagrangian–Eulerian hydrodynamics in rz geometry (cylindrical coordinates) that is compatible, total energy conserving and symmetry preserving.In the first part of the paper, we describe the discretization of the basic Lagrangian hydrodynamics equations in axisymmetric 2D rz geometry on general polygonal meshes. It exactly preserves planar, cylindrical and spherical symmetry of the flow on meshes aligned with the flow. In particular, spherical symmetry is preserved on polar equiangular meshes. The discretization conserves total energy exactly up to machine round-off on any mesh. It has a consistent definition of kinetic energy in the zone that is exact for a velocity field with constant magnitude.The method for discretization of the Lagrangian equations is based on ideas presented in [2,3,7], where the authors use a special procedure to distribute zonal mass to corners of the zone (subzonal masses). The momentum equation is discretized in its “Cartesian” form with a special definition of “planar” masses (area-weighted).The principal contributions of this part of the paper are as follows: a definition of “planar” subzonal mass for nodes on the z axis (r=0) that does not require a special procedure for movement of these nodes; proof of conservation of the total energy; formulated for general polygonal meshes.We present numerical examples that demonstrate the robustness of the new method for Lagrangian equations on a variety of grids and test problems including polygonal meshes. In particular, we demonstrate the importance of conservation of total energy for correctly modeling shock waves.In the second part of the paper we describe the remapping stage of the arbitrary Lagrangian–Eulerian algorithm. The general idea is based on the following papers [25–28], where it was described for Cartesian coordinates. We describe a distribution-based algorithm for the definition of remapped subzonal densities and a local constrained-optimization-based approach for each zone to find the subzonal mass fluxes.In this paper we give a systematic and complete description of the algorithm for the axisymmetric case and provide justification for our approach.1 The ALE algorithm conserves total energy on arbitrary meshes and preserves symmetry when remapping from one equiangular polar mesh to another.The principal contributions of this part of the paper are the extension of this algorithm to general polygonal meshes and 2D rz geometry with requirement of symmetry preservation on special meshes.We present numerical examples that demonstrate the robustness of the new ALE method on a variety of grids and test problems including polygonal meshes and some realistic experiments. We confirm the importance of conservation of total energy for correctly modeling shock waves.

A Multi-Material CCALE-MOF Approach in Cylindrical Geometry

AbstractIn this paper we present recent developments concerning a Cell-Centered Arbitrary Lagrangian Eulerian (CCALE) strategy using the Moment Of Fluid (MOF) interface reconstruction for the numerical simulation of multi-material compressible fluid flows on unstructured grids in cylindrical geometries. Especially, our attention is focused here on the following points. First, we propose a new formulation of the scheme used during the Lagrangian phase in the particular case of axisymmetric geometries. Then, the MOF method is considered for multi-interface reconstruction in cylindrical geometry. Subsequently, a method devoted to the rezoning of polar meshes is detailed. Finally, a generalization of the hybrid remapping to cylindrical geometries is presented. These explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method.

Symmetry- and essentially-bound-preserving flux-corrected remapping of momentum in staggered ALE hydrodynamics

We present a new flux-corrected approach for remapping of velocity in the framework of staggered arbitrary Lagrangian–Eulerian methods. The main focus of the paper is the definition and preservation of coordinate invariant local bounds for velocity vector and development of momentum remapping method such that the radial symmetry of the radially symmetric flows is preserved when remapping from one equiangular polar mesh to another. The properties of this new method are demonstrated on a set of selected numerical cyclic remapping tests and a full hydrodynamic example.

Symmetry-preserving momentum remap for ALE hydrodynamics

In this paper, a symmetry-preserving remapping algorithm for vectors respecting their local bounds by components and in magnitude is presented. First, description of the Vector Image Polygon (VIP) limiter for a piece-wise linear velocity reconstruction is presented. Numerical fluxes obtained from this reconstruction lead to symmetry- and bounds- preserving remap of momentum for staggered Arbitrary Lagrangian-Eulerian (ALE) hydrodynamical methods. A novel bounds definition for vectors and corresponding modification of the VIP limiter is introduced, which fixes undershoots in a radial velocity component for polar meshes. Comparison with standard scalar-based limiters is given. Cyclic remapping is performed to numerically verify properties of the methods.

Discontinuous Galerkin Transport on the Spherical Yin–Yang Overset Mesh

Abstract A discontinuous Galerkin (DG) transport scheme is presented that employs the Yin–Yang grid on the sphere. The Yin–Yang grid is a quasi-uniform overset mesh comprising two notched latitude–longitude meshes placed at right angles to each other. Surface fluxes of conserved scalars are obtained at the overset boundaries by interpolation from the interior of the elements on the complimentary grid, using high-order polynomial interpolation intrinsic to the DG technique. A series of standard tests are applied to evaluate its performance, revealing it to be robust and its accuracy to be competitive with other global advection schemes at equivalent resolutions. Under p-type grid refinement, the DG Yin–Yang method exhibits spectral error convergence for smooth initial conditions and third-order geometric convergence for C1 continuous functions. In comparison with finite-volume implementations of the Yin–Yang mesh, the DG implementation is less complex, as it does not require a wide halo region of elements for accurate boundary value interpolation. With respect to DG cubed-sphere implementations, the Yin–Yang grid exhibits similar accuracy and appears to be a viable alternative suitable for global advective transport. A variant called the Yin–Yang polar (YY-P) mesh is also examined and is shown to have properties similar to the original Yin–Yang mesh while performing better on tests with strictly zonal flow.