Non-periodic Domains Research Articles

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

Dynamics of meandering rivers in finite-length channels: linear theory

Meandering channels are dynamic landforms that arise as a result of fluid mechanic and sedimentary processes. Their evolution has been described by the meander-morphodynamic equations, which dictate that channel curvature and bed topography give rise to local perturbations in streamwise fluid velocity, prompting the preferential erosion and sediment deposition that constitute meander behaviour. Novel mathematical conditions are presented to guarantee unique solutions for the linearized equations in non-periodic domains with finite boundaries. With the boundary condition specification sufficient for the uniqueness proof one finds a well-posed initial-boundary-value problem amenable to standard numerical techniques for partial differential equations. This provides a pathway for improved numerical algorithms for simulations of meandering river dynamics. Previous theoretical analysis for linear stability theory in meandering dynamics has been restricted to spatially periodic systems. The present effort develops new results for linear stability theory in non-periodic systems with temporal driving at system boundaries as well as non-homogeneous initial conditions. Predictions for temporal driving at the inlets for non-periodic finite domains provide clarification for observed behaviour in laboratory flumes where driven conditions at the inlet avoids the long-term decay of all meanders observed in flumes with fixed entry conditions. Linear stability theory for finite domains confirms that a continuous perturbation is required for sustained meandering. Original scaling arguments are presented for the dependence of the meander migration rate on geological parameters, showing that the rate of channel migration increases with increased width, downreach slope and bank erodibility, and decreases with increased volumetric flow rate.

Homogenization of the Stokes System in a Non-Periodically Perforated Domain

In our recent work [8], we have studied the homogenization of the Poisson equation in a class of non periodically perforated domains. In this paper, we examine the case of the Stokes system. We consider a porous medium in which the characteristic distance between two holes, denoted by $\epsilon$, is proportional to the characteristic size of the holes. It is well known (see [1],[17] and [19]) that, when the holes are periodically distributed in space, the velocity converges to a limit given by the Darcy's law when the size of the holes tends to zero. We generalize these results to the setting of [8]. The non-periodic domains are defined as a local perturbation of a periodic distribution of holes. We obtain classical results of the homogenization theory in perforated domains (existence of correctors and regularity estimates uniform in $\epsilon$) and we prove $H^2$--convergence estimates for particular force fields.

Higher Order Multiscale Finite Element Method for Heat Transfer Modeling.

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

Fourier continuation method for incompressible fluids with boundaries

We present a Fourier Continuation-based parallel pseudospectral method for incompressible fluids in cuboid non-periodic domains. The method produces dispersionless and dissipationless derivatives with fast spectral convergence inside the domain, and with very high order convergence at the boundaries. Incompressibility is imposed by solving a Poisson equation for the pressure. Being Fourier-based, the method allows for fast computation of spectral transforms. It is compatible with uniform grids (although refined or nested meshes can also be implemented), which in turn allows for explicit time integration at sufficiently high Reynolds numbers. Using a new parallel code named SPECTER we illustrate the method with two problems: channel flow, and plane Rayleigh–Bénard convection under the Boussinesq approximation. In both cases the method yields results compatible with previous studies using other high-order numerical methods, with mild requirements on the time step for stability.

Multigrid treatment of implicit continuum diffusion

Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms are handled efficiently by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for a large number of processors. We built and present here a multigrid algorithm for these types of problems that outperforms a spectral solution employing the highly optimized FFTW library. This solver is suitable for high performance computing and may be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with harmonic test functions as well as turbulent 2D magnetohydrodynamic simulations. It is also shown that an anisotropic operator without mixed-derivative terms improves model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation.

An Immersed Boundary method with divergence-free velocity interpolation and force spreading

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid–structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least C1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

A finite element approach to self-consistent field theory calculations of multiblock polymers

Self-consistent field theory (SCFT) has proven to be a powerful tool for modeling equilibrium microstructures of soft materials, particularly for multiblock polymers. A very successful approach to numerically solving the SCFT set of equations is based on using a spectral approach. While widely successful, this approach has limitations especially in the context of current technologically relevant applications. These limitations include non-trivial approaches for modeling complex geometries, difficulties in extending to non-periodic domains, as well as non-trivial extensions for spatial adaptivity. As a viable alternative to spectral schemes, we develop a finite element formulation of the SCFT paradigm for calculating equilibrium polymer morphologies. We discuss the formulation and address implementation challenges that ensure accuracy and efficiency. We explore higher order chain contour steppers that are efficiently implemented with Richardson Extrapolation. This approach is highly scalable and suitable for systems with arbitrary shapes. We show spatial and temporal convergence and illustrate scaling on up to 2048 cores. Finally, we illustrate confinement effects for selected complex geometries. This has implications for materials design for nanoscale applications where dimensions are such that equilibrium morphologies dramatically differ from the bulk phases.

A unified pore-network algorithm for dynamic two-phase flow

This paper describes recent work on image-based network modeling of multiphase flow. The algorithm expands the range of flow scenarios and boundary conditions that can be implemented using dynamic network modeling, the most significant advance being the ability to model simultaneous injection of immiscible fluids under either transient or steady-state conditions using non-periodic domains. Pore-scale saturation distributions are solved rigorously from two-phase mass conservation equations simultaneously within each pore. Results show that simulations using a periodic network fail to track saturation history because periodic domains limit how the bulk saturation can evolve over time. In contrast, simulations using a non-periodic network with fractional flow as the boundary condition can account for behavior associated with both hysteresis and saturation history, and can capture phenomena such as the long pressure and saturation tails that are observed during dynamic drainage processes. Results include a sensitivity analysis of relative permeability to different model variables, which may provide insight into mechanisms for a variety of transient, viscous dominated flow processes.

The local response in structures using the Embedded Unit Cell Approach

This paper presents the development of a new concept, the Embedded Unit Cell (EUC) approach, to calculate local responses in elastic media. The EUC approach is based on a multi-scale formulation of non-periodic domains to evaluate the local/micro response where stress concentrations are expected. The formulation is based on alternative boundary conditions which is not restricted to periodic assumptions of the unit cell response that is required in the classical theory. This approach provides a reduced computational cost model of the macroscopic/global problem while preserving the accuracy at the microscale problem. We conclude with a numerical verification study.

Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of geophysical turbulence, where the net transfer of energy is from small to large scales. One approach to modeling backscatter in underresolved simulations is to add a stochastic forcing term. This study, set in the idealized context of the inverse cascade of two-layer quasigeostrophic turbulence, focuses on the importance of spatial and temporal correlation in numerical stochastic backscatter schemes when used with low-order finite-difference spatial discretizations. A minimal stochastic backscatter scheme is developed as a stripped-down version of stochastic superparameterization [Grooms and Majda, J. Comput. Phys., 271, (2014), pp. 78--98]. This simplified scheme allows detailed numerical analysis of the spatial and temporal correlation structure of the modeled backscatter. Its essential properties include a local formulation amenable to implementation in finite difference codes and nonperiodic domains, and tunable spatial and temporal correlations. Experiments with this scheme in the idealized context of homogeneous two-layer quasigeostrophic turbulence demonstrate the need for stochastic backscatter to be smooth at the coarse grid scale when used with low-order finite-difference schemes in an inverse-cascade regime. In contrast, temporal correlation of the backscatter is much less important for achieving realistic energy spectra. It is expected that the spatial and temporal correlation properties of the simplified backscatter schemes examined here will inform the development of stochastic backscatter schemes in more realistic models.

Hybrid simulation of whistler excitation by electron beams in two-dimensional non-periodic domains

We present a two-dimensional hybrid fluid-PIC scheme for the simulation of whistler wave excitation by relativistic electron beams. This scheme includes a number of features which are novel to simulations of this type, including non-periodic boundary conditions and fresh particle injection. Results from our model suggest that non-periodicity of the simulation domain results in the development of fundamentally different wave characteristics than are observed in periodic domains.

A spectral quadrilateral multidomain penalty method model for high Reynolds number incompressible stratified flows

SUMMARYA two‐dimensional quadrilateral spectral multidomain penalty method (SMPM) model has been developed for the simulation of high Reynolds number incompressible stratified flows. The implementation of higher‐order quadrilateral subdomains renders this model a nontrivial extension of a one‐dimensional subdomain SMPM model built for the simulation of the same type of flows in vertically nonperiodic domains (Diamessis et al., J. Comp. Phys, 202:298‐322, 2005). The nontrivial aspects of this extension consist of the implementation of subdomain corners, the penalty formulation of the pressure Poisson equation (PPE), and, most importantly, the treatment of specific challenges that arise in the iterative solution of the SMPM‐discretized PPE. The two primary challenges within the framework of the iterative solution of the PPE are its regularization to ensure the consistency of the associated linear system of equations and the design of an appropriate two‐level preconditioner. A qualitative and quantitative assessment of the accuracy, efficiency, and stability of the quadrilateral SMPM solver is provided through its application to the standard benchmarks of the Taylor vortex, lid‐driven cavity, and double shear layer. The capacity of the flow solver for the study of environmental stratified flow processes is shown through the simulation of long‐distance propagation of an internal solitary wave of depression in a manner that is free of numerical dispersion and dissipation. The methods and results presented in this paper make it a point of reference for future studies oriented toward the reliable application of the quadrilateral SMPM model to more complex environmental stratified flow process studies. Copyright © 2014 John Wiley & Sons, Ltd.

A comparison of Fourier pseudospectral method and finite volume method used to solve the Burgers equation

The Burger’s equation serves as a useful mathematical model to be applied in fluid dynamic problems. Besides, this equation is often used to test and compare numerical techniques. In the present paper, the numerical solutions determined by both the finite volume method (FVM) and the Fourier pseudospectral method (FPM), and the results are compared with the analytical, exact solution. Furthermore, performance in obtaining the results is properly reported. The analysis includes periodic and non-periodic domains, for both methods. The non-linear term in the FVM, is discretion by using upwind and central-differencing schemes, showing a rate of convergence for the second-order accurate. Due requirement of the domains in FPM to be periodic, in the present work, the immersed boundary methodology is used, to solve the equation at non-periodic domains. It will also be shown that this approach damages the accuracy and the rate convergence of the FPM.

Inventory model of deteriorating items on non-periodic discrete-time domains

In this paper, we demonstrate how to model a discrete-time dynamic process on a non-periodic time domain with applications to operations research. We introduce a discrete-time model of inventory with deterioration on domains where time points may be unevenly spaced over a time interval. We formalize the average cost function composed of storage, depreciation and back-ordering costs. The optimal condition is given to locate the optimal point that minimizes the average cost function. Finally, we present simulations to demonstrate how a manager can use this model to make inventory decisions.

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A LONG PERIODIC DOMAIN

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in <TEX>$H^1$</TEX> and the forcing function in <TEX>$L^p$</TEX>([0; T);<TEX>$L^2$</TEX>), T > 0, <TEX>$2{\leq}p{\leq}+\infty$</TEX> satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain <TEX>$\epsilon$</TEX> tends to zero.

Coupled Navier-Stokes/molecular dynamics simulations in nonperiodic domains based on particle forcing

SUMMARY This paper focuses on coupling methods for hybrid Navier–Stokes/molecular dynamics (MD) simulations. The computational domain is split in a continuum flow region, where a finite-volume discretisation of the Navier–Stokes equations is used, and one or more particle domains, where molecular level modelling of the flow is employed. The domains are defined with a partial overlap, in which the flow states are coupled through an exchange of the velocity components. For the steady flows considered, an under-relaxed Newton iteration method is used to drive the coupled system to convergence. The main focus of the present work is on methods to impose nonperiodic boundary conditions on the particle domain(s). A particle forcing is applied in the direction normal to the particle domain boundary to impose the boundary normal velocity component. A novel aspect of the present work is the extension of this method to more general nonplanar particle domain boundaries. The main contribution of the paper is the development of a particle forcing method in the direction tangential to the domain boundary, which is based on the equivalent continuum-flow boundary shear stresses along with an iterative forcing strength adjustment based on the extrapolated particle boundary velocity. Furthermore, an adaptation scheme is presented, which uses the finite-volume flux residuals of the particle bin averaged velocity field as a truncation criterion for the iterative force-update scheme. It is demonstrated that by comparing the residual reduction for the momentum equation in the nonhomogeneous directions during the molecular dynamics simulations with that for a homogeneous direction, the forcing iteration at which the statistical noise in the velocity field dominates the uncertainty in the forcing strength can be determined. At this point the iteration can be truncated. It is shown that with adaptive schemes of this type, the total number of MD evaluations required in a coupled Navier–Stokes/MD simulation can be reduced relative to a hybrid scheme with a fixed number of forcing-strength updates. Copyright © 2011 John Wiley & Sons, Ltd.

Spatiotemporal system identification on nonperiodic domains using Chebyshev spectral operators and system reduction algorithms

A system identification methodology based on Chebyshev spectral operators and an orthogonal system reduction algorithm is proposed, leading to a new approach for data-driven modeling of nonlinear spatiotemporal systems on nonperiodic domains. A continuous model structure is devised allowing for terms of arbitrary derivative order and nonlinearity degree. Chebyshev spectral operators are introduced to realm of inverse problems to discretize that continuous structure and arrive with spectral accuracy at a discrete form. Finally, least squares combined with an orthogonal system reduction algorithm are employed to solve for the parameters and eliminate the redundancies to achieve a parsimonious model. A numerical case study of identifying the Allen-Cahn metastable equation demonstrates the superior accuracy of the proposed Chebyshev spectral identification over its finite difference counterpart.

Construction and practical use of two-scaled extensions for rapidly oscillating functions

This paper suggests several methods for constructing two-scaled extensions for arbitrary rapidly oscillating functions. Special attention is paid to the continuity (smoothness) of extensions. The author discusses the scheme of applications of extensions to homogenization problems in non-periodic perforated domains and describes the difficulties arising in this case.

Eigenvalue stability of radial basis function discretizations for time-dependent problems

Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collocation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.