Curvilinear Domains Research Articles

Early Neoproterozoic Deformation Kinematics in the Chottanagpur Gneiss Complex (Eastern India): Evidence from the Curvilinear Hundru Falls Shear ZoneAnalysis

Abstract Curvilinear steep shear zones originate in different tectonic environments. In the Chottanagpur Gneiss Complex (CGC), the steeply dipping, left-lateral and transpressive Early Neoproterozoic Hundru Falls Shear Zone (HFSZ) with predominantly north-down kinematics comprises two domains, e.g., an arcuate NW-striking (in the west) to W-striking (in the east) domain with gently plunging stretching lineation that curves into a W-striking straight-walled domain with down-dip lineation. The basement-piercing HFSZ truncates a carapace of flat-lying amphibolite facies paraschist and granitoid mylonites, and recumbently folded anatectic gneisses. The carapace—inferred to be a midcrustal regional-scale low-angle detachment zone—structurally overlies an older basement of Early Mesoproterozoic anatectic gneisses intruded by Mid-Mesoproterozoic/Early Neoproterozoic granitoids unaffected by the Early Neoproterozoic extensional tectonics. The mean kinematic vorticity values in the steep HFSZ-hosted granitoids computed using the porphyroclast aspect ratio method are 0.74–0.83 and 0.51–0.65 in domains with shallow and steep lineations, respectively. The granitoid mylonites show a chessboard subgrain microstructure, but lack evidence for suprasolidus deformation. The timing relationship between the two domains is unclear. If the two HFSZ domains were contemporaneous, the domain of steep lineations with greater coaxial strain relative to the curvilinear domain formed due to strain partitioning induced by variations in mineralogy and/or temperature of the cooling granitoid plutons. Alternately, the domain of gently plunging lineations in the HFSZ was a distinct shear zone that curved into a subsequent straight-walled shear zone with steeply plunging lineation due to a northward shift in the convergence direction during deformation contemporaneous with the Early Neoproterozoic accretion of the CGC and the Singhbhum Craton.

Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials

An isogeometric numerical procedure based on non-uniform rational B-splines is applied to solve the free vibration of carbon nanotubes reinforced and functionally graded material sector cylindrical shells. A typical isogeometric analysis cylindrical shell element is introduced to transform curvilinear domain to parametric domain. The effects of the shear deformation, rotary inertia and axis extensibility are taken into consideration. The material properties are estimated by a mixture rule and effective carbon nanotube parameters. The weak form is derived in detail by applying virtual principle. The novelty of this paper is to use the isogeometric finite element method to analyze the dynamic behaviors of sector cylindrical shells with functionally graded material and four types of carbon nanotubes distributed material. By numerical examples, the convergence and accuracy of the current method are validated. Then, a series of natural frequencies and mode shapes are presented to serve as benchmark solutions in future researches. Moreover, the effects of boundary conditions, geometric proprieties as well as material parameters on the frequencies of the carbon nanotubes reinforced and functionally graded material sector cylindrical shells are examined.

Geometric regularity criteria for incompressible Navier–Stokes equations with Navier boundary conditions

We study the regularity criteria for weak solutions to the 3D incompressible Navier–Stokes equations in terms of the direction of vorticity, taking into account the boundary conditions. A boundary regularity theorem is proved on regular curvilinear domains with a family of oblique derivative boundary conditions, provided that the directions of vorticity are coherently aligned up to the boundary. As an application, we establish the boundary regularity for weak solutions to Navier–Stokes equations in round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.

Mathematical modeling of nonlinear displacement processes in oil LEF-layer by methods of complex analysis and summary representations

The approach to the modeling of nonlinear displacement processes (one and two-phase filtration) in heterogeneous oil deformable layers is developed, taking into account the inverse effect of the potential of the velocity field and the flow function on the conductivity of the medium. We constructed a method and computational technology for solving the corresponding boundary value problems for nonlinear-layered triple-connected curvilinear domains, bounded by equipotential lines and flow lines, on the basis of the synthesis of numerical methods of quasiconformal mappings and summary representations for differential equations with discontinuous coefficients in combination with domain decomposition by Schwartz method. Quasi-ideal processes in nonlinearly double-layered horizontal LEF-layers, whose geometry of heterogeneity zones is unknown in advance, is described by the corresponding boundary value problems obtained on the basis of the Darcy law and the continuity equation with the coefficient of layer permeability, which is given by a piecewise-constant function with ruptures along the searched equipotentials and lines of flow. The coefficient of conductivity the medium is given as a piecewise-constant function, that is dependent on the searched quasipotential and function of flow, with unknown line dividing layers (lines gap conductance coefficient) that is along the searched equipotential lines and flow lines and that is finding in the process of solving the problem. The proposed algorithms automatically solve the problem of choice of nodes and building a dynamic grid, finding of the unknown dividing lines of areas constancy coefficient of conductivity the medium, the calculation of the velocity field and calculate other characteristic parameters of the process. The decomposition of the domain on the layers of the constancy of permeability coefficient allows us to solve problems in more «comfortable» subdomain than the original problem the whole domain, and allows make in parallel the computational process, since calculations in the subdomain at each iterative step are independent of each other and can be done in parallel with the use of modern computer technology.

The solution of the heat conduction problem for a rectangular area with mixed boundary conditions and internal source in general by using fast expansions method

The fundamentals of the fast expansions method are outlined, provided that the mixed Dirichlet-Neumann boundary conditions are specified. Then, using the fast expansions method, the solution of the problem in an explicit analytical form with an accuracy of quadrature of Fourier coefficients is obtained. Such a solution allows to explore temperature fields in a rectangle, depending on the given input data of the task, which include the size of the rectangle, the boundary conditions and the internal source. The resulting solution for a rectangle can also be used to construct a solution to a problem in a curvilinear domain, for a two-phase Stefan problem with a curvilinear boundary, and for other applied problems.

The influence of an oscillating wall on the position of oblique shocks in a 2D channel

In the paper we solve the problem of supersonic gas flow in a two-dimensional channel with the moving upper wall making oscillations according to the harmonic law. In order to obtain a numerical solution for gas dynamics equations we have implemented two difference schemes: the scheme with the space and time approximation of the first order and the scheme with the space approximation of the second order. The fluxes were computed using Van Leer’s method. A special form of fluxes in the gas dynamics equations is given, which enables to calculate fluxes on cell faces of difference mesh using Van Leer’s method. Depending on a type of the harmonic law and initial gas inflow conditions, the peculiarities of oblique-shock wave propagation in moving curvilinear domains have been investigated. It has been determined that under a particular oscillation frequency the presence of wall oscillation practically doesn’t have an effect on the flow regime inside the domain. The convergence of the obtained solution is shown by calculations using difference grids with different numbers of cells. While comparing the numerical solution obtained due to our program with the one obtained with Ansys Fluent solver we found that the constructed code operates correctly.

Hierarchical p‐version C1 finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

SummaryThis work focuses on the construction of p‐version finite elements that have curve boundaries for C1 problems. Both triangular and quadrilateral elements are constructed based on the C1‐version blending function interpolation methods that are developed in this work and in the literature. Orthogonal hierarchical bases are constructed and subsequently transformed into interpolative nodal bases to facilitate the imposition of boundary conditions and the implementation of C1 conformity on curvilinear domains. Nodal collocation strategies are also studied for improving the numerical performance, and novel nonuniformly distributed nodes, namely, Gauss‐Jacobi (GJ) points, are proposed. For parallelograms and straight‐sided triangular elements, C1 continuity is exactly satisfied between neighboring elements. The difficulty of C1 conformity for elements that have curved boundaries is circumvented by interpolating the normal derivatives at Gauss‐Lobatto nodes. Moreover, with the help of the blending function interpolation method, the bases on edges and the internal modes can differ in terms of approximation order. Therefore, local p‐refinements can be easily performed by these elements. Numerical results demonstrated that these elements are computationally inexpensive and converge fast for problems with regular and irregular domains.

A differential quadrature hierarchical finite element method using Fekete points for triangles and tetrahedrons and its applications to structural vibration

A differential quadrature hierarchical finite element method (DQHFEM) using Fekete points was formulated for triangles and tetrahedrons and applied to structural vibration analyses. First, orthogonal polynomials on triangles and tetrahedrons that can be used as bases of the hierarchical finite element method (HFEM) were derived and simple formulas of transforming one dimensional non-uniform nodes to simplexes were presented. Then the non-uniform nodes were used as initial guesses to solve the Fekete points on simplexes through Newton–Raphson’s method together with the orthogonal polynomials. New differential quadrature (DQ) rules on simplexes were formulated using the HFEM bases and the Fekete points. The numbers of nodes or bases on different edges and faces and inside the body of the new DQ elements do not relate with each other like the HFEM that can freely assign different numbers of bases on different edges and faces and inside the body. So the new DQ method was named as a differential quadrature hierarchical (DQH) method that uses either interpolation functions or orthogonal polynomials as bases inside the element. Its weak form was named as the DQHFEM. Besides the DQH method and its weak form, a simple method of generating high quality linear and high order triangular and tetrahedral meshes from a single NURBS patch was presented. Numerical tests of the DQHFEM through structural vibration analyses showed that high accuracy results can be obtained using only a few nodes even on curvilinear domains using the DQH bases on both physical and geometric fields. It was concluded that wide applications of the DQH method and the DQHFEM to science and engineering are possible and commercial codes based on them are deserved to be developed.

Optimised High-Order Compact Difference Schemes for Internal Acoustics Problems On Curvilinear Domains

A numerical framework is presented for the solution of 2D and 3D internal acoustics problems using a high-order accurate fully staggered formulation on curvilinear domains. Optimised compact finite difference schemes previously obtained in our previous paper are used for spatial discretisation, while a free parameter linear multistep method is used for temporal discretisation. The resulting scheme does not require any numerical filtering, and several benchmark cases are provided which demonstrate the significantly reduced phase velocity errors, and greater resolving efficiency compared to existing methods. Curvilinear domains are generated with the CRDT algorithm by Driscoll, with an 8th order accurate ODE solver. The governing equations for the curvilinear problem are based on a novel transformation of the decoupled velocity pressure wave equations, with simplifications made to reduce the need to interpolate derivatives at undefined locations which occur on staggered grids. The resulting transformed equations are valid only for orthogonal grids, but are computationally efficient and do not result in loss of accuracy or stability due to grid skewness. Finally, a potential application is shown, demonstrating the solution of a generated acoustic field within a crucible of liquid aluminium by a top loaded electromagnetic induction coil. Generated pressure fields agree with results shown in previous work, and demonstrate the potential use of this contactless electromagnetic excitation method as an alternative to the immersed sonotrode for the ultrasonic treatment of alloys.

Estimation of stochastic behaviour in cardiac myocytes: I. Ca2+ movements inside the cytosol and sarcoplasmic reticulum on curvilinear domains.

BackgroundSince the discovery of Ca2+ sparks and their stochastic behaviour in cardiac myocytes, models have focused on the inclusion of stochasticity in their studies. While most models pay much attention to the stochastic modelling of cytosolic Ca2+ concentration the coupling of Ca2+ sparks and blinks in a stochastic model has not been explored fully. The cell morphology in in silico studies in the past is assumed to be Cartesian, spherical or cylindrical. The application on curvilinear grids can easily address certain restrictions posed by such grid set up and provide more realistic cell morphology. In this paper, we present a stochastic reaction–diffusion model that couples Ca2+ sparks and blinks in realistic shapes of cells in curvilinear domains.Methodology: Transformation of the model was performed to the curvilinear coordinate system. The set of equations is used to produce Ca2+ waves initiated from sparks and blinks. A non-buffered and non-dyed version as well as a buffered and dyed version of these equations were studied in light of observing the dynamics on the two different systems. For comparison, results for both the Cartesian and curvilinear grids are provided.Results and conclusions: A successful demonstration of the application of curvilinear grids serving as basis for future developments.

Smoothness in the Dini Space of a Single Layer Potential for a Parabolic System in the Plane

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

A higher‐order equilibrium finite element method

SummaryIn this paper, a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, ie, surface force components. As a result, the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier, which enforces equilibrium of forces, is the displacement field and the Lagrange multiplier, which enforces equilibrium of moments, is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains, and an example with a point singularity is given.

A High-Order Staggered Meshless Method for Elliptic Problems

We present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points ${\em x}_i$. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and th...

Investigation of Nonreciprocal Dispersion Phenomena in Anisotropic Periodic Structures Based on a Curvilinear FDFD Method

The aim of this paper is the investigation of nonreciprocal phenomena in anisotropically loaded 2-D periodic structures. For this purpose, our well-established 2-D curvilinear finite difference frequency domain method is combined with periodic boundary conditions and extended toward the eigenanalysis of periodic structures loaded with both isotropic and general anisotropic materials. The periodic structures are simulated in a 2-D domain, while uniformity is considered along the third axis. The propagation constant along the third axis can either be zero (in-plane-propagation) or nonzero (out-of-plane propagation). Particular effort was devoted to the identification of the appropriate irreducible Brillouin zone to be scanned during the eigenanalysis. It was herein realized that similar to geometrically artificial crystal anisotropy, the wave propagation directional asymmetries modify the irreducible Brillouin zone in the microwave regime as well. Both gyrotropic and particularly magnetized ferrite as well as full tensor anisotropic (arbitrarily biased ferrite) material loadings are investigated through the eigenanalysis of different periodic structures, including strip grating. Interesting nonreciprocal backward wave and unidirectional phenomena are justified as expected.

On the Quasi-unconditional Stability of BDF-ADI Solvers for the Compressible Navier--Stokes Equations and Related Linear Problems

The companion paper "Higher-order in time quasi-unconditionally stable ADI solvers for the compressible Navier-Stokes equations in 2D and 3D curvilinear domains", which is referred to as Part I in what follows, introduces ADI (Alternating Direction Implicit) solvers of higher orders of temporal accuracy (orders $s = 2$ to $6$) for the compressible Navier-Stokes equations in two- and three-dimensional space. The proposed methodology employs the backward differentiation formulae (BDF) together with a quasilinear-like formulation, high-order extrapolation for nonlinear components, and the Douglas-Gunn splitting. A variety of numerical results presented in Part I demonstrate in practice the theoretical convergence rates enjoyed by these algorithms, as well as their excellent accuracy and stability properties for a wide range of Reynolds numbers. In particular, the proposed schemes enjoy a certain property of "quasi-unconditional stability": for small enough (problem-dependent) fixed values of the time-step $\Delta t$, these algorithms are stable for arbitrarily fine spatial discretizations. The present contribution presents a mathematical basis for the performance of these algorithms. Short of providing stability theorems for the full BDF-ADI Navier-Stokes solvers, this paper puts forth proofs of unconditional stability and quasi-unconditional stability for BDF-ADI schemes as well as some related un-split BDF schemes, for a variety of related linear model problems in one, two and three spatial dimensions, and for schemes of orders $2\leq s\leq 6$ of temporal accuracy. Additionally, a set of numerical tests presented in this paper for the compressible Navier-Stokes equation indicate that quasi-unconditional stability carries over to the fully non-linear context.

Problems of Identification of the Parameters of Quasiideal Filtration Processes in Nonlinear Layered Porous Media

On the basis of synthesis of the numerical methods of quasiconformal mappings and summary representations in combination with the domain decomposition by the Schwartz method, we develop a constructive approach to the numerical solution of nonlinear model problems of the theory of complex quasipotential for nonlinear layered curvilinear domains under the conditions of identification of the parameters (conductivity coefficient and the characteristic values of the potential on the equipotential lines of separation of layers).

A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

SummaryA differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non‐uniform rational B‐splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss–Lobatto–Legendre points should be used as nodes, (2) the recursion formula should be used to compute high‐order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high‐frequency vibration analysis. Copyright © 2016 John Wiley & Sons, Ltd.

In-plane vibration analysis of plates in curvilinear domains by a differential quadrature hierarchical finite element method

Free in-plane vibration analysis of plates is carried out by a differential quadrature hierarchical finite element method (DQHFEM). The NURBS (Non-Uniform Rational B-Splines) patches of geometries were first transformed into differential quadrature hierarchical (DQH) patches, and then the elastic field was discretized by the same DQH basis. The DQHFEM solved the compatibility problem caused by different parametrization of neighbouring patches of isogeometric analysis using NURBS. And mesh refinement in DQHFEM does not propagate from patch to patch. The DQHFEM matrices also have the embedding property as the hierarchical finite element method (HFEM). In-plane vibration analyses of plates of several planforms showed that the DQHFEM is similar as the fixed interface mode synthesis method that can analyse a structure using a few nodes on the boundary of substructure elements and only several clamped modes inside each substructure element, but the DQHFEM does not need modal analysis and is of high accuracy. The accuracy and convergence of the DQHFEM were validated through comparison with exact and approximate results in literatures and computed by the authors.

Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier-Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples $(h,\Delta t)$ of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form $(0,M_h)\times (0,M_t)$. In other words, for each fixed value of $\Delta t$ below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Static and dynamic response of sector-shaped graphene sheets

ABSTRACTNonlocal elasticity theory is presented for the free vibration and bending analysis of nano-scaled graphene sheets having a sector shape. An eight-node curvilinear domain is used for transformation of the governing equation of motion of sector graphene from physical region to computational region in conjunction with the Kirchhoff plate theory. The discrete singular convolution method is employed for numerical solutions of resulting nonlocal governing differential equations and related boundary conditions. Then, the effects of nonlocal parameters, mode numbers, sector angles, and radius ratios on the static and vibration results of nano-scaled sector-shaped graphene sheets are discussed.