Spurious Pressure Research Articles

Polygonal finite elements for incompressible fluid flow

SUMMARYWe discuss the use of polygonal finite elements for analysis of incompressible flow problems. It is well‐known that the stability of mixed finite element discretizations is governed by the so‐called inf‐sup condition, which, in this case, depends on the choice of the discrete velocity and pressure spaces. We present a low‐order choice of these spaces defined over convex polygonal partitions of the domain that satisfies the inf‐sup condition and, as such, does not admit spurious pressure modes or exhibit locking. Within each element, the pressure field is constant while the velocity is represented by the usual isoparametric transformation of a linearly‐complete basis. Thus, from a practical point of view, the implementation of the method is classical and does not require any special treatment. We present numerical results for both incompressible Stokes and stationary Navier–Stokes problems to verify the theoretical results regarding stability and convergence of the method. Copyright © 2013 John Wiley & Sons, Ltd.

Sharp-interface immersed boundary lattice Boltzmann method with reduced spurious-pressure oscillations for moving boundaries

A sharp-interface immersed boundary lattice Boltzmann method (IBLBM) is developed to reduce spurious-pressure oscillations in moving boundary problems. We adopt a cut-cell-based method, i.e., the partially saturated computational cell method, because the primary cause of spurious-pressure oscillations is the failure to obey the geometric conservation law near the boundary in the sharp-interface IBLBM. We modify a sharp-interface IBLBM (ghost fluid method) to fit the cut-cell approach. This boundary condition can guarantee the Dirichlet and Neumann boundary conditions for velocity and pressure. Some simulations are shown to test the validity of the method, including a circular cylinder with motions that are at rest, moving, oscillatory, and neutrally buoyant. The results illustrate that the method reduces effectively spurious pressure and can simulate moving boundary problems, especially when pressure field accuracy is a key concern.

An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree‐type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress‐conservative finite difference constraints which are enforced at the “hanging” velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.

A solution-adaptive method for efficient compressible multifluid simulations, with application to the Richtmyer–Meshkov instability

The evolution of high-speed initially laminar multicomponent flows into a turbulent multi-material mixing entity, e.g., in the Richtmyer–Meshkov instability, poses significant challenges for high-fidelity numerical simulations. Although high-order shock- and interface-capturing schemes represent such flows well at early times, the excessive numerical dissipation thereby introduced and the resulting computational cost prevent the resolution of small-scale features. Furthermore, unless special care is taken, shock-capturing schemes generate spurious pressure oscillations at material interfaces where the specific heats ratio varies. To remedy these problems, a solution-adaptive high-order central/shock-capturing finite difference scheme is presented for efficient computations of compressible multi-material flows, including turbulence. A new discontinuity sensor discriminates between smooth and discontinuous regions. The appropriate split form of (energy preserving) central schemes is derived for flows of smoothly varying specific heats ratio, such that spurious pressure oscillations are prevented. High-order accurate weighted essentially non-oscillatory (WENO) schemes are applied only at discontinuities; the standard approach is followed for shocks and contacts, but material discontinuities are treated by interpolating the primitive variables. The hybrid nature of the method allows for efficient and accurate computations of shocks and broadband motions, and is shown to prevent pressure oscillations for varying specific heats ratios. The method is assessed through one-dimensional problems with shocks, sharp interfaces and smooth distributions of specific heats ratio, and the two-dimensional single-mode inviscid and viscous Richtmyer–Meshkov instability with re-shock.

Correcting Fast-Mode Pressure Errors in Storm-Scale Ensemble Kalman Filter Analyses

A typical storm-scale ensemble Kalman filter (EnKF) analysis/forecast system is shown to introduce imbalances into the ensemble posteriors that generate acoustic waves in subsequent integrations. When the EnKF is used to research storm-scale dynamics, the resulting spurious pressure oscillations are large enough to impact investigation of processes driven by nonhydrostatic pressure gradient forces. Fortunately, thermodynamic retrieval techniques traditionally applied to dual-Doppler wind analyses can be adapted to diagnose the balanced portion of an EnKF pressure analysis, thereby eliminating the fast-mode pressure oscillations. The efficacy of this approach is demonstrated using a high-resolution supercell thunderstorm simulation as well as EnKF analyses of a simulated and a real supercell.

Two‐phase dynamic analysis by material point method

SUMMARYThis paper extends the material point method to analyze coupled dynamic, two‐phase boundary‐valued problems via a velocity formulation, in which solid and fluid phase velocities are the variables. Key components of the proposed approach are the adoption of Verruijt's sequence of update steps when integrating over time and the enhancement of volumetric strains. The connection between fractional step method and the time‐stepping algorithm presented in this paper is addressed.Enhancement of volumetric strains allows lower order variations in pressure and mitigates spurious pressure fields and locking that plague low‐order finite‐element implementations.A stress averaging technique to smoothen stress variations is proposed, and the local damping procedure adopted by FLAC is extended to handle two‐phase problems. Special Kelvin‐Voigt boundaries are developed to suppress reflections at artificial boundaries. Idealized examples are presented to demonstrate the capability of the proposed framework to accurately capture the physics of wave propagation, consolidation and wave attack on a sea dike. Copyright © 2012 John Wiley & Sons, Ltd.

A collocated method for the incompressible Navier–Stokes equations inspired by the Box scheme

We present a new finite-difference numerical method to solve the incompressible Navier–Stokes equations using a collocated discretization in space on a logically Cartesian grid. The method shares some common aspects with, and it was inspired by, the Box scheme. It uses centered second-order-accurate finite-difference approximations for the spatial derivatives combined with semi-implicit time integration. The proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity–pressure form of the equations. The continuity equation is enforced exactly (to machine accuracy) at the collocated locations, whereas the momentum equations are evaluated in a staggered manner. This formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure. The method shows uniform order of accuracy, both in space and time, for velocity and pressure. In addition, the skew-symmetric form of the non-linear advection term of the Navier–Stokes equations improves discrete conservation of kinetic energy in the inviscid limit, to within the order of the truncation error of the time integrator. The method has been formulated to accommodate different types of boundary conditions; fully periodic, periodic channel, inflow-outflow and lid-driven cavity; always ensuring global mass conservation. A novel aspect of this finite-difference formulation is the derivation of the discretization near boundaries using the weak form of the equations, as in the finite element method. The method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver. To assess the applicability of the new method presented in this paper, four realistic flow problems have been simulated and results are compared with those in the literature. These cases include a lid-driven cavity, backward-facing step, Kovasznay flow, and fully developed turbulent channel.

Mixed finite elements for numerical weather prediction

We show how mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: (a) energy conservation, (b) mass conservation, (c) no spurious pressure modes, and (d) steady geostrophic modes on the f-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RTk-Q(k-1) element pairs on quadrilaterals and the BDFM1-P1DG element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM). The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

Levelset based fluid topology optimization using the extended finite element method

This study focuses on finding the optimal layout of fluidic devices subjected to incompressible flow at low Reynolds numbers. The proposed approach uses a levelset method to describe the fluid-solid interface geometry. The flow field is modeled by the incompressible Navier–Stokes equations and discretized by the extended finite element method (XFEM). The no-slip condition along the fluid-solid interface is enforced via a stabilized Lagrange multiplier method. Unlike the commonly used porosity approach, the XFEM approach does not rely on a material interpolation scheme, which allows for more flexibility in formulating the design problems. Further, it mitigates shortcomings of the porosity approach, including spurious pressure diffusion through solid material, strong dependency of the accuracy of the boundary enforcement with respect to the model parameters which may affect the optimization results, and poor boundary resolution. Numerical studies verify that the proposed method is able to recover optimization results obtained with the porosity approach. Further, it is demonstrated that the XFEM approach yields physical results for problems that cannot be solved with the porosity approach.

A modified SPH method for simulating motion of rigid bodies in Newtonian fluid flows

A weakly compressible smoothed particle hydrodynamics (WCSPH) method is used along with a new no-slip boundary condition to simulate movement of rigid bodies in incompressible Newtonian fluid flows. It is shown that the new boundary treatment method helps to efficiently calculate the hydrodynamic interaction forces acting on moving bodies. To compensate the effect of truncated compact support near solid boundaries, the method needs specific consistent renormalized schemes for the first and second-order spatial derivatives. In order to resolve the problem of spurious pressure oscillations in the WCSPH method, a modification to the continuity equation is used which improves the stability of the numerical method. The performance of the proposed method is assessed by solving a number of two-dimensional low-Reynolds fluid flow problems containing circular solid bodies. Wherever possible, the results are compared with the available numerical data.

Hydrodynamic simulations with the Godunov smoothed particle hydrodynamics

We present results based on an implementation of the Godunov Smoothed Particle Hydrodynamics (GSPH), originally developed by Inutsuka (2002), in the GADGET-3 hydrodynamic code. We first review the derivation of the GSPH discretization of the equations of moment and energy conservation, starting from the convolution of these equations with the interpolating kernel. The two most important aspects of the numerical implementation of these equations are (a) the appearance of fluid velocity and pressure obtained from the solution of the Riemann problem between each pair of particles, and (b the absence of an artificial viscosity term. We carry out three different controlled hydrodynamical three-dimensional tests, namely the Sod shock tube, the development of Kelvin-Helmholtz instabilities in a shear flow test, and the "blob" test describing the evolution of a cold cloud moving against a hot wind. The results of our tests confirm and extend in a number of aspects those recently obtained by Cha (2010): (i) GSPH provides a much improved description of contact discontinuities, with respect to SPH, thus avoiding the appearance of spurious pressure forces; (ii) GSPH is able to follow the development of gas-dynamical instabilities, such as the Kevin--Helmholtz and the Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl structures in the shear-flow test and the dissolution of the cold cloud in the "blob" test. We also discuss in detail the effect on the performances of GSPH of changing different aspects of its implementation. The results of our tests demonstrate that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled to an N-body solver, for astrophysical and cosmological applications. [abridged]

A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations

A method for reducing the spurious pressure oscillations observed when simulating moving boundary flow problems with sharp-interface immersed boundary methods (IBMs) is proposed. By first identifying the primary cause of these oscillations to be the violation of the geometric conservation law near the immersed boundary, we adopt a cut-cell based approach to strictly enforce geometric conservation. In order to limit the complexity associated with the cut-cell method, the cut-cell based discretization is limited only to the pressure Poisson and velocity correction equations in the fractional-step method and the small-cell problem tackled by introducing a virtual cell-merging technique. The method is shown to retain all the desirable properties of the original finite-difference based IBM while at the same time, reducing pressure oscillations for moving boundaries by roughly an order of magnitude.

A comparison of the GWCE and mixed P − P1 formulations in finite‐element linearized shallow‐water models

SUMMARYThe appearance of spurious pressure modes in early shallow‐water (SW) models has resulted in two common strategies in the finite element (FE) community: using mixed primitive variable and generalized wave continuity equation (GWCE) formulations of the SW equations. One FE scheme in particular, the P − P1 pair, combined with the primitive equations may be advantageously compared with the wave equation formulations and both schemes have similar data structures. Our focus here is on comparing these two approaches for a number of measures including stability, accuracy, efficiency, conservation properties, and consistency. The main part of the analysis centres on stability and accuracy results via Fourier‐based dispersion analyses in the context of the linear SW equations. The numerical solutions of test problems are found to be in good agreement with the analytical results. Copyright © 2011 John Wiley & Sons, Ltd.

Comparison of Pressure-Controlled Provocation Discography Using Automated Versus Manual Syringe Pump Manometry in Patients with Chronic Low Back Pain

The study compares the rate of positive discograms using an automated versus a manual pressure-controlled injection devise and compares the pressure and volume values at various pressures and initial evoked pain and 6/10 or greater evoked pain. A retrospective study prospectively collected patient study data used in a prior prospective study and with prospectively collected data which is routinely collected per our institutional standardized audit protocol. Two custom-built disc manometers (automated injection speed control; manual injection speed control) were sequentially employed during provocation discography in 510 discs of 151 consecutive patients. Two hundred thirty-seven discs of 67 patients with chronic low back pain were evaluated using the automated manometer (automated group) and 273 discs of 84 patients were evaluated with a manual manometer (manual group). No significant differences in positive discogram rates were found between the automated and manual groups (32.1% vs 32.6% per disc, respectively, P>0.05). No significant differences in low-pressure positive discogram rates were found (16.0% vs 15.0% per disc, automated group versus manual group, respectively, P>0.05). However, there were significantly increased volumes and lower pressures at initial and "bad" pain provocation. The study results found equivalent positive discogram rates following a series of pressure-controlled discography using either an automated or manual pressure devise. There were, however significant increases in volume at both initial onset of evoked pain and at 6/10 pain when using the automated injection devise that may have caused the observed lower opening pressure and lower pressure values at initial evoked pain. Assuming increased volumes are innocuous, automated injection is inherently more controlled and may better reduce unintended and often unrecorded spurious high dynamic pressure peaks thereby reducing conscious and unconscious operator bias.

Stabilization procedures in coupled poromechanics problems: A critical assessment

AbstractNumerical solutions for problems in coupled poromechanics suffer from spurious pressure oscillations when small time increments are used. This has prompted many researchers to develop methods to overcome these oscillations. In this paper, we present an overview of the methods that in our view are most promising. In particular we investigate several stabilized procedures, namely the fluid pressure Laplacian stabilization (FPL), a stabilization that uses bubble functions to resolve the fine‐scale solution within elements, and a method derived by using finite increment calculus (FIC). On a simple one‐dimensional test problem, we investigate stability of the three methods and show that the approach using bubble functions does not remove oscillations for all time step sizes. On the other hand, the analysis reveals that FIC stabilizes the pressure for all time step sizes, and it leads to a definition of the stabilization parameter in the case of the FPL‐stabilization. Numerical tests in one and two dimensions on 4‐noded bilinear and linear triangular elements confirm the effectiveness of both the FPL‐ and the FIC‐stabilizations schemes for linear and nonlinear problems. Copyright © 2010 John Wiley & Sons, Ltd.

Stable and unstable cross-grid [formula omitted] mixed finite elements for the Stokes problem

In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so the new finite elements are called cross-grid P k Q l . A stability analysis of these approximations is provided, based on the macroelement technique of Stenberg. The lowest order P 1 Q 1 and P 2 Q 1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases are also presented, which confirm the existence of the spurious pressure mode for the P 1 Q 1 element and the stability of the P 2 Q 1 element.

An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model

We propose a discretization method of a five-equation model with isobaric closure for the simulation of interfaces between compressible fluids. This numerical solver is a Lagrange–Remap scheme that aims at controlling the numerical diffusion of the interface between both fluids. This method does not involve any interface reconstruction procedure. The solver is equipped with built-in stability and consistency properties and is conservative with respect to mass, momentum, total energy and partial masses. This numerical scheme works with a very broad range of equations of state, including tabulated laws. Properties that ensure a good treatment of the Riemann invariants across the interface are proven. As a consequence, the numerical method does not create spurious pressure oscillations at the interface. We show one-dimensional and two-dimensional classic numerical tests. The results are compared with the approximate solutions obtained with the classic upwind Lagrange–Remap approach, and with experimental and previously published results of a reference test case.

Prevention of Pressure Oscillations in Modeling a Cavitating Acoustic Fluid

Cavitation effects play an important role in the UNDEX loading of a structure. For far-field UNDEX, the structural loading is affected by the formation of local and bulk cavitation regions, and the pressure pulses resulting from the closure of the cavitation regions. A common approach to numerically modeling cavitation in far-field underwater explosions is Cavitating Acoustic Finite Elements (CAFE) and more recently Cavitating Acoustic Spectral Elements (CASE). Treatment of cavitation in this manner causes spurious pressure oscillations which must be treated by a numerical damping scheme. The focus of this paper is to investigate the severity of these oscillations on the structural response and a possible improvement to CAFE, based on the original Boris and Book Flux-Corrected Transport algorithm on structured meshes (6), to limit oscillations without the energy loss associated with the current damping schemes.

On Exact Conservation for the Euler Equations with Complex Equations of State

Conservative numerical methods are often used for simulations of fluid flows involving shocks and other jumps with the understanding that conservation guarantees reasonable treatment near discontinuities. This is true in that convergent conservative approximations converge to weak solutions and thus have the correct shock locations. However, correct shock location results from any discretizationwhose violation of conservation approaches zero as the mesh is refined. Here we investigate the case of the Euler equations for a single gas using the Jones-Wilkins-Lee (JWL) equa- tion of state. We show that a quasi-conservative method can lead to physically realistic solutions which are devoid of spurious pressure oscillations. Furthermore, we demon- strate that under certain conditions, a quasi-conservative method can exhibit higher rates of convergence near shocks than a strictly conservative counterpart of the same formal order. AMS subject classifications: 65N08, 35L60, 35L65, 76N15