Local refinement techniques for elliptic problems on cell-centered grids

Summary Algebraic multilevel analogues of the BEPS preconditioner designed for solving discrete elliptic problems on grids with local refinement are formulated, and bounds on their relative condition numbers, with respect to the composite-grid matrix, are derived. TheV-cycle and, more generally,v-foldV-cycle multilevel BEPS preconditioners are presented and studied. It is proved that for 2-D problems theV-cycle multilevel BEPS is almost optimal, whereas thev-foldV-cycle algebraic multilevel BEPS is optimal under a mild restriction on the composite cell-centered grid. For thev-fold multilevel BEPS, the variational relation between the finite difference matrix and the corresponding matrix on the next-coarser level is not necessarily required. Since they are purely algebraically derived, thev-fold (v>1) multilevel BEPS preconditioners perform without any restrictionson the shape of subregions, unless the refinement is too fast. For theV-cycle BEPS preconditioner (2-D problem), a variational relation between the matrices on two consecutive grids is required, but there is no restriction on the method of refinement on the shape, or on the size of the subdomains.     

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with $h$-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.     

Supersonic flow past a flat lattice of cylindrical rods

Two-dimensional supersonic laminar ideal gas flows past a regular flat lattice of identical circular cylinders lying in a plane perpendicular to the free-stream velocity are numerically simulated. The flows are computed by applying a multiblock numerical technique with local boundary-fitted curvilinear grids that have finite regions overlapping the global rectangular grid covering the entire computational domain. Viscous boundary layers are resolved on the local grids by applying the Navier–Stokes equations, while the aerodynamic interference of shock wave structures occurring between the lattice elements is described by the Euler equations. In the overlapping grid regions, the functions are interpolated to the grid interfaces. The regimes of supersonic lattice flow are classified. The parameter ranges in which the steady flow around the lattice is not unique are detected, and the mechanisms of hysteresis phenomena are examined.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.     

Algorithms of wavelet compression of linear spline spaces

Splines and wavelets have been finding increasing use in the theory of information. Wavelet decompositions are used in designing efficient algorithms for processing (compression) of large information flows. If one succeeds in establishing the embeddability of spaces of splines on a sequence of sparsing/refining grids, in representing the chain of embedded spaces as a direct sum of wavelet spaces, and in realizing the base functions with the minimum length of their support, then this suggests a wavelet decomposition of the information flow, leading, in turn, to substantial savings in the computational cost. This being so, it proves possible to resolve the initial information flow into components to single out the principal and refining information flows, depending on the needs. For uniform grids on the real line, wavelet decompositions are well known. In this case, there applies the powerful technique of harmonic analysis, as well as the lifting scheme or the wavelet scheme. However, many applications require considering bounded intervals and nonuniform grids. For example, for efficient compression of nonuniform flows of information (featuring singularities or rapidly fluctuating characteristics), it is expedient to employ an adaptive nonuniform grid, which takes account of the singularities of the flow being processed. This renders possible to improve approximation of functions without complicating the computations. The previously obtained results pertained to splines on infinite grids. Making both the grid and the corresponding numerical flow infinite renders theoretical studies simpler; however, in practice, one has to deal with finite flows. This paper continues the studies initiated for finite-dimensional spaces. The purpose of this work is to built a wavelet decomposition (compression) on a nonuniform grid and develop the corresponding decomposition and reconstruction algorithms for infinite flows (with a grid on an open interval) and finite flows (with a grid on a segment) for linear spaces of splines of Lagrange type.     

Computing the maximum amplification of the solution norm of differential-algebraic systems

We describe an effective approach for computing the maximum amplification of the solution norm of linear differential-algebraic systems that arise, in particular, when approximating with respect to space variables the linearized viscous incompressible flow equations for disturbances of laminar flows. In this context the square of the maximum amplification is the largest amplification of the kinetic energy of the disturbances whose knowledge is important in stability investigations and laminar-turbulent transition analysis. First, we reduce such a differential-algebraic system to an ordinary differential one. Then, the maximum amplification is computed as the matrix exponential norm for which a special low-rank approximation is used. To obtain an additional decrease in the computational cost, we use two initial differential-algebraic systems corresponding to coarse and fine grid approximations. The first one is used to compute a rough value of the maximum amplification, and the second one is used to refine the computation. We illustrate the efficiency of this approach with two sample flows of grooved-channel and boundary-layer types.     

The Logarithmic finite element method in a multigrid setting

The Logarithmic finite element (“LogFE”) method is a novel finite element approach for solving boundary-value problems proposed in [1]. In contrast to the standard Ritz-Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function, which is obtained by the exponentiation of the linear combination of the shape functions given by the degrees of freedom. Unlike many existing multigrid formulations, the LogFE method allows for a very smooth interpolation between nodal values on the coarse grid. It can thus avoid problems with regard to locking and convergence that appear in multigrid applications using only linear interpolation, especially for larger corsening factors. We illustrate the use of the LogFE method as a coarse grid algorithm, in conjunction with an atomistic finite element method on the fine grid, for calculating the mechanical response of super carbon nanotubes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Block mesh refinement for incompressible flows in curvilinear domains

A method for the solution of the Navier–Stokes equation for the prediction of flows inside domains of arbitrary shaped bounds by the use of Cartesian grids with block-refinement in space is presented. In order to avoid the complexity of the body fitted numerical grid generation procedure, we use a saw tooth method for the curvilinear geometry approximation. By using block-nested refinement, we achieved the desired geometry Cartesian approximation in order to find an accurate solution of the N–S equations. The method is applied to incompressible laminar flows and is based on a cell-centred approximation. We present the numerical simulation of the flow field for two geometries, driven cavity and stenosed tubes. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm. The Cartesian block refinement algorithm can be used in any complex curvilinear geometry simulation, to accomplish a reduction in memory requirements and the computational time effort.     

A hybrid linking approach for solving the conservation equations with an adaptive mesh refinement method

Accurate numerical computation of complex flows on a single grid requires very fine meshes to capture phenomena occurring at both large and small scales. The use of adaptive mesh refinement (AMR) methods, significantly reduces the involved computational time and memory. In the present article, a hybrid linking approach for solving the conservation equations with an AMR method is proposed. This method is essentially a coupling between the h-AMR and the multigrid methods. The efficiency of the present approach has been demonstrated by solving species and Navier–Stokes equations.     

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.     

Wiring edge-disjoint layouts

We consider the wiring or layer assignment problem for edge-disjoint layouts. The wiring problem is well understood for the case that the underlying layout graph is a square grid (Lipski Jr. and Preparata, 1987). In this paper, we introduce a more general approach to this problem. For an edge-disjoint layout in the plane, respectively in an arbitrary planar layout graph, we give equivalent conditions for k-layer wirability. Based on these conditions, we obtain linear-time algorithms to wire every layout in a tri-hexagonal grid or a tri-square-hexagonal grid, respectively, using at most five layers.     

An adaptive grid technique for the vertical structure of shallow water models

The efficient resolution of the boundary layers occurring in geophysical flows has motivated the search for criteria to optimize the vertical nodal placement in three-dimensional (3D) shallow water models. This paper describes the implementation and testing of an adaptive grid technique for the internal mode of shallow water models. The technique uses an r-method in which the nodes are moved vertically based on the velocity gradients between consecutive nodes. One-dimensional (1D) tests show that the method behaves well in tidal- and wind-driven flows, both in well-mixed and stratified conditions. Average accuracy improvements of 50% were obtained relative to uniform grids, with a 15% CPU time increase. The adaptive technique accounts accurately for the space and time variability of the flow, thus being attractive for any type of problem. Furthermore, the technique does not require an a priori knowledge of the flow conditions, thus simplifying greatly the modeling procedure.     

The method of lines for the computation of incompressible viscous flows

A common approach to finding numerical solutions of the time-dependent incompressible Navier-Stokes equations is considered within the method of lines framework [1]. For the determination of viscous incompressible flows the stream-function formulation for the fourth-order equation [2, 3], an artificial compressibility method [4], and a modified velocity correction method [5] are designed. Some improved and extended results of numerical simulations obtained by the author in the previous works are presented. Test calculations have been done for various flows inside square, triangular and semicircular cavities with one moving wall, the backward-facing step, double bent channels and for the flow around an aerofoil at large angle of attack. An alternative and practical methodology for resolving the Navier-Stokes equations in arbitrarily complex geometries using Cartesian meshes is proposed. Some of complex geometrical configurations can be decomposed into a set of subdomains. The simplest approach for specifying boundary conditions near curved or irregular boundaries is to transfer all the variables from the boundaries to the nearest grid knots. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of grid boundary value problems for functions defined on grid cells and faces

Properties of a version of MFD method are studied for a grid problem on a polyhedral grid in which the grid scalars are defined on grid cells and the grid flows are specified by their local normal coordinates on the plane faces of cells. In a domain with curvilinear boundary, a grid inhomogeneous boundary value problem for stationary diffusion-type equations is considered. An operator statement of the grid problem is given, and a local approximation of the equations and boundary conditions is studied.     

On the interface boundary conditions between two interacting incompressible viscous fluid flows

We consider two incompressible viscous fluid flows interacting through thin non-Newtonian boundary layers of higher Reynolds? number. We study the asymptotic behaviour of the problem, with respect to the vanishing thickness of the layers, using Γ-convergence methods. We derive general interfacial boundary conditions between the two fluid flows. These boundary conditions are specified for some particular cases including periodic or fractal structures of layers.     

自适应多重网格法与超松弛法的比较

多重网格法(Multiple Grid Method,简称M-G方法)是近年来出现的快速方法之一,本文在M-G方法中采用自适应控制层间转换的技术,并将自适应M-G方法与G-S迭代方法及SOR迭代方,法进行了比较。其计算结果表明,自适应M-G方法的计算量比G-S迭代及SOR迭代少得多,当M-G方法所用层数为4-6层,这种优越性就更加明显,且自适应M-G方法中选取控制参数有很大的灵活性。     

Model reduction on inertial manifolds for N–S equations approached by multilevel finite element method

Approximate Inertial Manifolds (AIMs) is approached by multilevel finite element method, which can be referred to as a Post-processed nonlinear Galerkin finite element method, and is applied to the model reduction for fluid dynamics, a typical kind of nonlinear continuous dynamic system from viewpoint of nonlinear dynamics. By this method, each unknown variable, namely, velocity and pressure, is divided into two components, that is the large eddy and small eddy components. The interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially by AIMs, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. As an example, the flow field of incompressible flows around airfoil is simulated numerically and discussed, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less essential degrees-of-freedom which can dominate the dynamic behaviors of the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. In a sense, the small eddy component can be captured by AIMs with fewer grids, and an accurate result can also be obtained.     

一种守恒型间断跟踪法中对任意多个间断的移动和相互作用的处理

对一种守恒型间断跟踪法设计了一种技巧来处理任意多个间断的移动和相互作用．由此技巧我们就可以建立一个“一般的强健的”间断跟踪法．由于采用了此技巧就会使得算法在某时刻在某网格上会存在非相容性并且会产生O(1)-强度的误差．但不管怎样,这些误差在后续的计算中会被算法的守恒性所消除．还给出了几个数值例子来显示这一技巧的有效性．     

Lagrange-projection scheme for two-phase flows 1. Applications to realistic test cases

We are concerned with the numerical simulation of two-phase flows representing oil transportation along a 1-D pipeline. More specifically, we wish to show that, for this kind of problems, it is highly recommended to make use of the Lagrange-Projection formalism. Indeed, this approach naturally splits the effects of fast (acoustic) and slow (kinematic) waves, thus enabling us to design numerical schemes with many desirable properties: explicit with respect to slow waves and implicit with respect to fast waves. The design of the overall explicit-implicit Lagrange-Projection method involves many interesting features, among which the positivity of the partial densities is of utmost interest to us. In this paper, the emphasis will be laid on the motivation and the general philosophy of the Lagrange-Projection method we propose. Extensive numerical results for realistic test cases are then presented in order to illustrate the capabilities of the method     

Vortex design problem

In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity. The problem is formulated as a PDE-constrained optimization in which the solutions are found using a gradient-based descent method. Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients. In explicit tracking of interfaces (vortex boundaries) this method offers an alternative to grid-based techniques, such as the level-set methods, and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique. We develop and validate this approach using the design of 2D equilibrium Euler flows with finite-area vortices as a model problem. It is also discussed how the proposed methodology can be applied to Euler flows featuring other vorticity distributions, such as vortex sheets, and to time-dependent phenomena.