Curvilinear Domains Research Articles

The linear hyperbolic initial and boundary value problems in a domain with corners

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.

Generalization of tangential trace spaces of H (curl ,Ω) for curvilinear Lipschitz polyhedral domains Ω

For curvilinear Lipschitz polyhedral domains Ω, explicit characterizations of the tangential trace spaces of H1(Ω) are presented. These extend the original characterizations given by Buffa and Ciarlet that hold on Lipschitz polyhedral domains with plane faces. The tangential trace spaces of H1(Ω) are fundamental for the definition, analysis and intuitive understanding of the trace spaces of H(curl,Ω) and therefore, more general characterizations of the latter are obtained at the same time. Copyright © 2013 John Wiley & Sons, Ltd.

On the growth mode of two-lobed curvilinear graphene domains at atmospheric pressure

We demonstrate the chemical vapor deposition (CVD) growth of 2-lobed symmetrical curvilinear graphene domains specifically on Cu{100} surface orientations at atmospheric pressure. We utilize electron backscattered diffraction, scanning electron microscopy and Raman spectroscopy to determine an as-yet unexplored growth mode producing such a shape and demonstrate how its growth and morphology are dependent on the underlying Cu crystal structure especially in the high CH4:H2 regime. We show that both monolayer and bilayer curvilinear domains are grown on Cu{100} surfaces; furthermore, we show that characteristic atmospheric pressure CVD hexagonal domains are grown on all other Cu facets with an isotropic growth rate which is more rapid than that on Cu{100}. These findings indicate that the Cu-graphene complex is predominant mechanistically at atmospheric pressure, which is an important step towards tailoring graphene properties via substrate engineering.

A Generalized MAC Scheme on Curvilinear Domains

We propose a simple finite difference scheme for Navier--Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contravariant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, it can be shown that this spatially compatible discretization leads to uniform inf-sup condition, which plays a crucial role in the pressure approximation of both dynamic and steady state calculations. Numerical experiments demonstrate the robustness and efficiency of our scheme.

A Low-Storage Curvilinear Discontinuous Galerkin Method for Wave Problems

The low-storage curvilinear discontinuous Galerkin (LSC-DG) method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous Galerkin method. We perform a semidiscrete, a priori convergence analysis of LSC-DG and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind DG method on curvilinear domains. Computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions. Additional results show that the sufficient conditions for optimal order convergence of LSC-DG may also be necessary conditions.

Summation-by-parts operators and high-order quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. The SBP operator definition includes a weight matrix that is used formally for discrete integration; however, the accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. For diagonal weight matrices, the accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; in particular, the diagonal norm accurately approximates the L2 norm for functions, and multi-dimensional SBP discretizations accurately approximate the divergence theorem.

A horizontally curvilinear non-hydrostatic model for simulating nonlinear wave motion in curved boundaries

SUMMARY A horizontally curvilinear non-hydrostatic free surface model that embeds the second-order projection method, the so-called θ scheme, in fractional time stepping is developed to simulate nonlinear wave motion in curved boundaries. The model solves the unsteady, Navier–Stokes equations in a three-dimensional curvilinear domain by incorporating the kinematic free surface boundary condition with a top-layer boundary condition, which has been developed to improve the numerical accuracy and efficiency of the non-hydrostatic model in the standard staggered grid layout. The second-order Adams–Bashforth scheme with the third-order spatial upwind method is implemented in discretizing advection terms. Numerical accuracy in terms of nonlinear phase speed and amplitude is verified against the nonlinear Stokes wave theory over varying wave steepness in a two-dimensional numerical wave tank. The model is then applied to investigate the nonlinear wave characteristics in the presence of dispersion caused by reflection and diffraction in a semicircular channel. The model results agree quantitatively with superimposed analytical solutions. Finally, the model is applied to simulate nonlinear wave run-ups caused by wave-body interaction around a bottom-mounted cylinder. The numerical results exhibit good agreement with experimental data and the second-order diffraction theory. Overall, it is shown that the developed model, with only three vertical layers, is capable of accurately simulating nonlinear waves interacting within curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.

Improving accuracy of high-order discontinuous Galerkin method for time-domain electromagnetics on curvilinear domains

The paper discusses high-order geometrical mapping for handling curvilinear geometries in high-accuracy discontinuous Galerkin simulations for time-domain Maxwell problems. The proposed geometrical mapping is based on a quadratic representation of the curved boundary and on the adaptation of the nodal points inside each curved element. With high-order mapping, numerical fluxes along curved boundaries are computed much more accurately due to the accurate representation of the computational domain. Numerical experiments for two-dimensional and three-dimensional propagation problems demonstrate the applicability and benefits of the proposed high-order geometrical mapping for simulations involving curved domains.

Inverse singularly perturbed problems of the convection-diffusion type in quadrangular curvilinear domains

We have constructed an algorithm for the asymptotic approximation of the solutions of inverse singularly perturbed boundary-value problems of the convection-diffusion type with unknown diffusion coefficient, depending on the coordinates of a quadrangular curvilinear domain of filtration. The case of sufficient smoothness and consistency of the overdetermination, initial, and boundary conditions is considered. Unlike the construction of an algorithm for the solution of similar problems in doubly connected domains, here, in the corresponding relations, there appear corrections taking into account the influence of “lateral sources of pollution.” With the help of this algorithm, we have carried out a computer experiment, the results of which confirm the well-known fact of “strong sensitivity” of the model to assignment of the overdetermination condition. In particular, we have revealed the specific character of influence of this condition on the required diffusion coefficient depending on the filtration velocity.

Systems Analysis of Filtration Processes in Multiconnected Curvilinear Domains

Systems Analysis of Filtration Processes in Multiconnected Curvilinear Domains

Boundary layers in smooth curvilinear domains: Parabolic problems

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in $\mathbb{R}^{d}$, $d \geq 2$, when the diffusivity parameter e is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to e, of the heat solution is obtained at all orders. It appears that unlike the case of a straight boundary, because of the curvature of the boundary, two correctors in powers of e and e1/2 must be introduced at each order. The convergence results, between the exact and approximate solutions, seem optimal. Beside the intrinsic interest of the results presented in the article, we believe that some of the methods introduced here should be useful to study boundary layers for other problems involving curved boundaries.

Investigation of piston bowl geometry and speed effects in a motored HSDI diesel engine using a CFD against a quasi-dimensional model

Abstract The present work investigates the effect of varying the combustion chamber geometry and engine rotational speed on the gas flow and temperature field, using a new quasi-dimensional engine simulation model in conjunction with an in-house developed computational fluid dynamics (CFD) code served to validate the predicted in-cylinder flow field and gas temperature distribution calculated by the quasi-dimensional model, for three alternative piston bowl geometries and three rotational speeds. This CFD code can simulate three-dimensional curvilinear domains using the finite volume method in a collocated grid; it solves the generalized transport equation for the conservation of mass, momentum and energy, and incorporates the standard k–e turbulence model with some slight modifications to introduce the compressibility of a fluid in generalized coordinates. On the other hand, the quasi-dimensional model solves the general transport equation for the conservation of mass and energy by a finite volume method throughout the entire in-cylinder volume, while for the estimation of the flow field a new simplified three dimensional air motion model is used. To compare these two models the in-cylinder spatial and temporal temperature distribution, the mean cylinder pressure diagram, as well as the mean in-cylinder radial and axial velocity are examined, for the three piston bowl geometries and the three speeds, for a high speed direct injection (HSDI) diesel engine operating under motoring conditions. From the comparison of calculated results, it becomes apparent that the two models predict similar in-cylinder temperature distributions and mean air velocity fields at each crank angle, for all cases examined. Thus, it is shown that the quasi-dimensional model with the proposed simplified air motion model is capable of capturing the physical effect of combustion chamber geometry and speed on the in-cylinder velocity and temperature field, while needing significantly lower computing time compared to the more detailed and accurate CFD model. On the other hand, the CFD model is more suitable when detailed simulation of the in-cylinder geometry is required and the way the corresponding transport phenomena are affected.

Use of Eight-node Curvilinear Domains in Discrete Singular Convolution Method for Free Vibration Analysis of Annular Sector Plates with Simply Supported Radial Edges

Free vibration analysis of annular sector plate is studied using an eight-node curvilinear domain. The method of discrete singular convolution (DSC) is used for computations. Radial edges of plates are simply supported; however, various edge conditions are taken into account for circumferential edge. In the proposed approach, an irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation. The DSC procedures are then applied to discretization of the transformed set of governing equations and boundary conditions. The effects of mode numbers, vertex angle, boundary conditions and outer-to-inner radius ratio on the vibration frequencies are investigated. It is seen that the results obtained by DSC method shows very good agreement with that of other available analytical and numerical solutions for the plate problems under investigation.

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

AbstractBased on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem.The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices.By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.

BiGlobal stability computations on curvilinear meshes

A numerical method is developed to perform BiGlobal stability computations on structured curvilinear meshes. The Linearized Euler Equations for an incompressible planar flow are considered. Perturbations are sought for in normal form, leading to a differential eigenvalue problem, which can be discretized on a Cartesian computational domain through a spectral collocation method based on Chebyshev polynomials. The Jacobian of the non-analytical coordinate change from the computational domain to the physical curvilinear domain is also calculated numerically using the same spectral method. This procedure is tested on several test cases with comparison to reference solutions obtained by 1D stability calculations. To cite this article: F. Longueteau, J.-P. Brazier, C. R. Mecanique 336 (2008).

The D-bar method, inversion of certain integrals and integrability in 4 + 2 and 3 + 1 dimensions

We first review a method for deriving linear and nonlinear transform pairs, which is based on the spectral analysis of an eigenvalue equation and on the formulation of a d-bar problem. Then, we present two applications of this method: (a) we derive a certain linear transform pair in one dimension, which appears in the characterization of the Dirichlet-to-Neumann map of the Laplace equation in the interior of a convex two-dimensional curvilinear domain. (b) We derive a nonlinear Fourier transform pair in four dimensions, which can be used for the solution of the Cauchy problem of an integrable generalization of the Kadomtsev–Petviashvilli equation in 4 + 2, i.e. in four spatial and two temporal dimensions. The question of reducing this equation form 4 + 2 to 3 + 1 dimensions is also discussed.

G 1-Interpolation and geometry reconstruction for higher order finite elements

The paper proposes a new G 1-interpolation and geometry reconstruction scheme for regular surfaces approximated with unstructured quadrilateral grids. The parametrization is based on C 1 transfinite interpolation and sixth order polynomials. The G 1-conforming approximation of surfaces guarantees continuity of surface normals and it is critical for higher order finite and boundary element discretizations on curvilinear domains. The proposed interpolation–reconstruction scheme is illustrated with a number of numerical examples including the geometry of human head.

Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries, Journal of Computational Physics 225 (2007) 1782–1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions the FSI algorithm is unconditionally unstable even when strong coupling FSI is employed. For such cases, however, combining the strong coupling iteration with under-relaxation in conjunction with the Aitken’s acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and the upper bound of the under-relaxation coefficient, required for stability, is derived.

On mean exit time from a curvilinear domain

Abstract In this short communication, we consider a mean exit time problem for a non-degenerate, two-dimensional, coupled diffusion process M t = ( x t , y t ) in the interior of a curvilinear domain D ψ = { ( x , y ) ∈ R + 2 : y > ψ ( x ) } with a C 2 -boundary, where x t is any arbitrary diffusion process and y t is a geometric Brownian motion evolving under non-explosive conditions, and ψ ( . ) is a real-valued, positive, increasing, continuous function such that ψ ( 0 ) ≥ 0 . It is proved that, under certain conditions, the mean exit time is a logarithmic function associated with a certain second-order nonlinear ordinary differential equation. At the end of the note, we shall present several examples to illustrate our main result.

The finite differences method for solving systems on irregular shapes

A relatively simple and efficient symbolic-numerical procedure based on the finite differences method for solving partial differential equations on systems of irregular shapes is presented. The new concept is based on the spline parameterization of the irregular domain. The curvilinear domain of the real system is transformed to the rectangular domain by spline functions where the finite differences method is used to solve the transformed system of depended variables. The numerical results are then transported back to the original irregular shape of the system. In order to present the symbolic-numerical technique effectively, the Laplace's equation of heat transfer with the Dirichlet and the Neumann boundary conditions in different 2D curvilinear domains is considered. The proposed technique is applied for the non-steady-state heat transfer by conduction as well. Numerical experiments were performed to justify the proposed method.