Irregular Domains Research Articles

Wavefront phase representation by Zernike and spline models: a comparison.

A comparative analysis of spline and Zernike models is presented for wavefront phase construction. The techniques are analyzed on the basis of representation accuracy, computational costs, and the number of samples used for representation. The strengths and weaknesses of each model over a set of various wavefront phases with different domain shapes are analyzed. The findings show that both models efficiently represent a simple wavefront phase at irregular domain shapes. On the other hand, when complex wavefront phases at irregular domain shapes are represented, the spline model performs much better than the Zernike model. Further, results show that the spline model evaluation speed is significantly faster than the Zernike model.

A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains

A new generalised two-dimensional time and space variable-order fractional Bloch-Torrey equation is developed in this study. The variable-order Riesz fractional derivative and variable diffusion coefficient are introduced to simulate diffusion phenomena in heterogeneous, irregularly shaped biological tissues. The fractional Bloch-Torrey equation is discretised by the weighted and shifted Grünwald-Letnikov formula with respect to time and by finite volume method with respect to space. Additionally, to improve the accuracy of the numerical method for dealing with non-smooth solutions, some appropriate correction terms are introduced in the time approximation. Numerical examples on different irregular domains with various non-smooth solutions are explored to verify the effectiveness of the presented numerical scheme. Furthermore, we also solve the coupled variable-order fractional Bloch-Torrey equation on a human brain-like domain which is composed of white matter and grey matter. The solution behaviour of this model is compared with that of the constant-order fractional model, and the transverse magnetisation in magnetic resonance imaging on different biological micro-environments are graphically analysed. Results suggest that incorporation of the non-local property and spatial heterogeneity in the model by use of fractional operators can lead to a better capability for capturing the complexities of diffusion phenomena in biological tissues. This research may provide a basis for further research on the application of fractional calculus to clinical research and medical imaging.

The direct interpolation boundary element method and the domain superposition technique applied to piecewise Helmholtz's problems with internal heterogeneity

This work presents the combination of the direct interpolation boundary element method (DIBEM) and the domain superposition technique (DST) to address piecewise inhomogeneous two dimensional Helmholtz problems, in which the internal constitutive property in each sector varies smoothly according to a known function. The domain integrals generated by the medium's heterogeneity are transformed into boundary integrals according to the DIBEM strategy where radial basis functions are used. The DST is applied to generate the influence coefficients related to the several sectors and compute them in the final matrix system. Thus, the proposed methodology preserves the main features and advantages of the Boundary Element Method. Concerning the evaluation of the numerical results, three different tests are performed, considering regular and irregular domains. For each example, benchmarks are generated by correlate simulations using the Finite Element Method.

The Development of interfaces in a Parabolic p-Laplacian type diffusion equation with weak convection

This work has the objective to analyse the initial growth of interface and structure of nonnegative weak solution for one-dimensional parabolic p-Laplacian type diffusion-convection with non-positive convection coefficient c. In this situation, the interfaces may expand, shrink or remain stationary relying on the competition between these two factors. In this paper, we concentrate on three regions to classify the behavior of local solutions near the asymptotic interface in the irregular domain. In the first and second regions, the slow diffusion dominates over the convection term with expanding interfaces under some restrictions. In the third region, the slow diffusion dominates over the convection, but the interfaces have a waiting time. In our proof, the rescaling method and blow-up techniques are applied.

Tensorial elastodynamics for anisotropic media

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse time migration and full-waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we have developed a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid (FSG) with a mimetic FD (MFD) (FSG + MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation in which we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG + MFD numerical method has a computational complexity comparable with Cartesian scenarios using a similar FSG + MFD numerical approach. Numerical examples demonstrate that our solution can simulate anisotropic elastodynamic field solutions on irregular geometries; thus, it is a reliable tool for anisotropic elastic modeling, imaging, and inversion applications in generalized computational domains including handling free-surface topography.

Optimal local truncation error method for solution of wave and heat equations for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes

Recently we have developed the optimal local truncation error method (OLTEM) for PDEs with constant coefficients on irregular domains and unfitted Cartesian meshes. However, many important engineering applications include domains with different material properties (e.g., different inclusions, multi-material structural components, etc.) for which this technique cannot be directly applied. In the paper OLTEM is extended to a much more general case of PDEs with discontinuous coefficients and can treat the above-mentioned applications. We show the development of OLTEM for the 1-D and 2-D scalar wave equation as well as the heat equation using compact 3-point (in the 1-D case) and 9-point (in the 2-D case) stencils that are similar to those for linear quadrilateral finite elements. Trivial unfitted Cartesian meshes are used for OLTEM with complex interfaces between different materials. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at a given width of stencil equations. In contrast to the second order of accuracy for linear finite elements, OLTEM provides the fourth order of accuracy in the 1-D case and in the 2-D case for horizontal interfaces as well as the third order of accuracy for the general geometry of smooth interfaces. The numerical results for the domains with complex smooth interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than quadratic finite elements and yields the results close to those for cubic finite elements with much wider stencils. The wave and heat equations are uniformly treated with OLTEM. OLTEM can be directly applied to other partial differential equations.

Medusa

Medusa, a novel library for implementation of non-particle strong form mesh-free methods, such as GFDM or RBF-FD, is described. We identify and present common parts and patterns among many such methods reported in the literature, such as node positioning, stencil selection, and stencil weight computation. Many different algorithms exist for each part and the possible combinations offer a plethora of possibilities for improvements of solution procedures that are far from fully understood. As a consequence there are still many unanswered questions in the mesh-free community resulting in vivid ongoing research in the field. Medusa implements the core mesh-free elements as independent blocks, which offers users great flexibility in experimenting with the method they are developing, as well as easily comparing it with other existing methods. The article describes the chosen abstractions and their usage, illustrates aspects of the philosophy and design, offers some executions time benchmarks and demonstrates the application of the library on cases from linear elasticity and fluid flow in irregular 2D and 3D domains.

From Time–Frequency to Vertex–Frequency and Back

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

Radial point interpolation collocation method based approximation for 2D fractional equation models

In this work, we studied the radial point interpolation collocation method (RIPCM) for the solution of the partial differential models with fractional derivatives. In the present meshless method, polynomial basis function and two novel types local support fields were considered. The (time-)space fractional differential equations with single-term or multi-term time derivatives in irregular domain were well simulated numerically by the proposed RIPCM method. Numerical tests were carried out to verify the accuracy and demonstrate the strength of the RIPCM in solving fractional derivative models in irregular domain.

Graph-signal Reconstruction and Blind Deconvolution for Structured Inputs

Key to successfully deal with complex contemporary datasets is the development of tractable models that account for the irregular structure of the information at hand. This paper provides a comprehensive and unifying view of several sampling, reconstruction, and recovery problems for signals defined on irregular domains that can be accurately represented by a graph. The workhorse assumption is that the (partially) observed signals can be modeled as the output of a graph filter to a structured (parsimonious) input graph signal. When either the input or the filter coefficients are known, this is tantamount to assuming that the signals of interest live on a subspace defined by the supporting graph. When neither is known, the model becomes bilinear. Upon imposing different priors and additional structure on either the input or the filter coefficients, a broad range of relevant problem formulations arise. The goal is then to leverage those priors, the shift operator of the supporting graph, and the samples of the signal of interest to recover: the signal at the non-sampled nodes (graph-signal interpolation), the input (deconvolution), the filter coefficients (system identification), or any combination thereof (blind deconvolution).

Numerical Study of Grain Growth in Laser Powder Bed Fusion Additive Manufacturing of Metals

Laser Powder Bed Fusion (LPBF) is an additive manufacturing method that manufactures high density and quality metal products. We present a coupled grain growth and heat transfer modeling technique to understand the materials microstructure evolution in metals during the cooling process of LPBF. The phase-field model is combined with a transient heat transfer equation to simulate the solidification and crystallization of the melt pool simultaneously. Specifically, the variable domain and driving force of the order parameters in the phase-field calculation are defined using current temperature distribution. Additionally, the latent heat generated by crystallization is introduced as a heat source to affect temperature evolution in the cooling process. The finite element method with a staggering strategy is employed to solve the coupled governing equations on an irregular computational domain. The computational framework is verified in a one-dimensional solidification problem by comparing the velocity of the fluid-solid interface. The two-way coupling solution of solidification and crystallization is studied in an example of LPBF of Aluminum alloys.

ПОРОМАСШТАБНОЕ МОДЕЛИРОВАНИЕ ПОТОКА ЖИДКОСТИ В ПРОНИЦАЕМЫХ СФЕРАХ С ИСПОЛЬЗОВАНИЕМ ПРОЕКЦИОННОГО МЕТОДА ДЛЯ НЕСЖИМАЕМЫХ УРАВНЕНИЙ НАВЬЕ-СТОКСА

The direct numerical simulation (DNS) is an effective and useful tool in the two-phase fluid flow studying. The projection method on the staggered grid was applied in this paper to solve the incompressible Navier-Stokes equations in irregular domains at the pore-scale level (irregular boundary is presented by its level-set function). The permeability of porous medium which was constructed by the random positioning of penetrable spheres of equal radii were numerically calculated and validated by comparing with theoretical estimations of permeability based on the numerical solution of the lattice-Boltzmann equation in irregular domains in previous works. All numerical calculations were performed using PARIS simulator.

Heatline and Massline Visualization of Hydromagnetic Double Diffusive Convective Flow of Nanofluid Within a Porous Trapezoidal Cavity

Double diffusive convective flow of nanofluid within a porous trapezoidal cavity of various aspect ratios consisting of Al2O3 nanoparticle in the presence of applied magnetic field in the direction perpendicular to the parallel top and bottom walls is analysed. The side walls of the cavity are maintained at constant temperature and concentration while its horizontal walls are insulated and impermeable. The irregular physical domain of the problem is transformed to a regular unit square computational domain. The governing equations have been solved by second order of finite difference method (FDM). Based upon numerical predictions, the effects of pertinent parameters such as Rayleigh number, Darcy number, aspect ratio, solid volume fraction and inclination angle on the flow and temperature fields and the heat transfer performance of the enclosure are examined. It is found that the intensity of heat and mass transfer increases with the increase in the Darcy number and aspect ratio. It is also observed that as the solid volume fraction increases there is increase in the average Nusselt number but reverse effect is observed on the average Sherwood number.

Numerical Solution of the Multiterm Time-Fractional Model for Heat Conductivity by Local Meshless Technique

Fractional partial differential equation models are frequently used to several physical phenomena. Despite the ability to express many complex phenomena in different disciplines, researchers have found that multiterm time-fractional PDEs improve the modeling accuracy for describing diffusion processes in contrast to the results of a single term. Nowadays, it attracts the attention of the active researchers. The aim of this work is concerned with the approximate numerical solutions of the three-term time-fractional Sobolev model equation using computationally attractive and reliable technique, known as a local meshless method. Because of the meshless character and the simple application in higher dimensions, there is a growing interest in meshless techniques. To assess the reliability and accuracy of the proposed method, three test problems and two types of irregular domains are taken into account.

Prioritization of the influence of different exchange interactions and uniaxial anisotropy on the generation and morphology of skyrmions

Different magnetic interactions and anisotropy in the multi-layer structure have different effects on the formation and morphology of skyrmions under zero field. In this paper, we show that the Ruderman–Kittel–Kasuya–Yosida (RKKY) coupling plays a dominant role in the generation of skyrmions but has no significant effect on the size of skyrmions. When the RKKY coupling is weak, if the Dzyaloshinskii–Moriya interaction (DMI) is strong enough, some irregular cruciform domain states can be formed. With the increase of the DMI, the shape of skyrmions is altered from Néel skyrmions to skyrmioniums, and when the Heisenberg exchange is relatively low, complex domain states, such as the labyrinth are formed. However, when the Heisenberg coupling is comparatively high, the spin states of the Néel-skyrmions are reversed. Both the Heisenberg exchange interaction and uniaxial anisotropy have a positive influence on reducing the diameters of skyrmions. However, skyrmions are difficult to generate when the Heisenberg exchange and anisotropy are strong enough. We find the prioritization of the effects of different magnetic interactions and uniaxial anisotropy on the formation and morphology of skyrmions by analyzing the energy diagrams, combined with the magnetization configuration in different pairs of the exchange interactions and anisotropy, in which the RKKY coupling has the greatest influence, followed by the DMI, then the Heisenberg exchange and finally the uniaxial anisotropy.

Tensor solutions in irregular domains: Eigenvalue problems

This work develops a numerical method to solve eigenvalue problems considering irregular two-dimension convex domains. It is based on the application of the Jordan spectral method to domains sectionally delimited by C1 class convex Jordan curves. The approach is straightforward in Cartesian coordinates and eliminates the use of polar coordinates or conformal maps, revealing an expressive accuracy gain compared to finite differences approach, obtaining solutions with more than ten digits of precision. Being computer-oriented and presenting spectral accuracy, the method is effective for several geometries, allowing to calculate margins of error in eigenvalues and eigenfunctions, being able to identify spurious results. Besides, the approach can be applied to either Dirichlet or Neumann boundary conditions. Helmholtz equation is used as an example, but the methodology is general and can be applied to any other eigenvalue partial differential equation. The approach does not assume a priori symmetry considerations, which allows the emergence of solutions to non-symmetrical vibration modes.

High precision simulation of thermal-mechanical problems in functionally graded materials by spectral element differential method

Our purpose is to establish a numerical method meeting the requirements of accuracy and easy-using for thermal-mechanical analysis of functionally graded structures. Faced with these demands, a new point collocation pseudo-spectral method named spectral element differential method (SEDM), is proposed in this paper. In this paper the Chebyshev polynomial is used to obtain the derivatives of the variables with respect to intrinsic coordinates. By using the analytical expressions of the differentiation for the shape functions which are used for geometry mapping, the first- and second- derivatives of the variables with respect to global coordinates can be obtained directly. Besides, the local equilibrium equation technique is proposed to connect the elements to form the final system of equations in this paper. Based on the spectral derivative matrix and the element mapping technique, an accurate and efficient strong-form numerical method without any variational principles or energy principles can be obtained lastly for irregular domains. Some examples with complex material properties or geometries are calculated by the proposed method to illustrate the accuracy and convergence. The thermoelastic performances of functionally graded structures with different material distribution and temperature-dependent properties are investigated.

Two-step MPS-MFS ghost point method for solving partial differential equations

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs is problem dependent, several known procedures on how to determine a good shape parameter are introduced for solving various types of problems. The proposed method is not only highly accurate but also fairly stable due to the use of the particular solutions and fundamental solutions. To demonstrate the effectiveness of the proposed method, five numerical examples in highly complicate and irregular domains, which include second and fourth order elliptic partial differential equations in 2D and 3D, are presented.

Microstructure of quasi-one-dimensional superconductor KCr3As3 prepared by K-ion deintercalation

The microstructure of quasi-one-dimensional KCr3As3 (133) superconductors, which were prepared by chemical cation deintercalation from their counterpart K2Cr3As3 (233) compounds, are investigated using scanning transmission electron microscopy. The nominal KCr3As3 crystals generally exhibit irregular nanoscale 133-phase domains accompanied by an amorphous As-deficient phase and cracks as a result of alkali cation deintercalation processes. Analysis of local defective structures reveals the existence of an intermediate state in the transformation from 233 to 133 phase and a possible K-deficient 233-type structure as a nanoscale cluster. Our microscopic investigations offer insight into the microstructure of KCr3As3 and the alkali metal cation deintercalation processes.

An efficient quadrature method for vibration analysis of thin elliptical plates with continuous and discontinuous edge conditions

Mapping an irregular domain into a square one is a common technique in analyzing problems of plates and shells with irregular shapes. For the irregular shape without four corners such as the elliptical shape, the difficulty arises that the Jacobian determinant is zero at the corner points. An efficient quadrature method is presented to analyze the transverse vibration of thin plates with an elliptical shape. To circumvent the above-mentioned difficulty, Gauss quadrature is used in numerical integration. Besides, derivative degrees of freedom are not used, and a boundary point is modeled by two nodes separated by a very small distance. Since the nodes are not coinciding with integration points, a way indirectly using the differential quadrature law is employed to derive the explicit formulas to ease the programming. A convergence study is performed. Free vibration of elliptical plates with continuous and discontinuous edge conditions is analyzed to demonstrate the efficiency of the developed rotation-free weak-form quadrature method.