Spurious Pressure Research Articles

Mesh topology-based spurious pressure stabilization in 3D finite elasticity using Voronoi tessellations

AbstractIn this paper, we present a mesh topology-based stabilization approach to suppress spurious pressure modes in 3D nearly-incompressible finite elasticity. The focus lies on a mixed formulation with lowest-order approximation for the displacement and pressure fields. Motivated by the fact that the popular H1/P0 element does not fulfill the inf-sup condition, all possible local spurious pressure modes are derived on a patch of elements. The nullspace method is used to determine all spurious pressure solutions. From this, the topological requirements of the finite element mesh are established. We conclude that no more than four elements are allowed to intersect in the same vertex to overcome local checkerboarding. To fulfill this requirement, we employ non-degenerate 3D Voronoi diagrams with several different site distributions. These result in random, centroidal, and honeycomb Voronoi meshes. The resulting convex polyhedral elements are discretized by a polyhedral mixed finite element based on the lowest possible interpolation pair. The numerical examples illustrate that spurious pressure modes do not occur for any degree of mesh refinement as long as the topological mesh requirements are met. Furthermore, it is shown that the numerical inf-sup test is passed. By violating the topological requirements, it is shown that a stable pressure field cannot be guaranteed and the checkerboard phenomenon is provoked.

Adaptation of the Low Dissipation Low Dispersion Scheme for Reactive Multicomponent Flows on Unstructured Grids Using Density-Based Solvers

Abstract The low dissipation low dispersion (LD2) second-order accurate scheme is effective for scale-resolving simulations in finite volume computational fluid dynamics (CFD) solvers, thanks to its combination of a skew-symmetric split form for convective terms and matrix-valued artificial dissipation fluxes. However, reactive flow simulations face challenges due to steep gradients and varying gas properties. This study replaces the skew-symmetric scheme with the kinetic energy and entropy preserving (KEEP) scheme, utilizing quadratic and cubic split forms for convective terms, enhancing stability. The nonsmooth fluid interfaces in reactive flow simulations necessitate upwind fluxes for reactive scalars to limit total variation (TV), also requiring upwind fluxes for the mixture-dependent internal energy fluxes. Other convective terms use central discretizations from the KEEP scheme, leveraging LD2’s spatial reconstruction to minimize dispersive errors. Numerical assessments show this approach reduces spurious pressure oscillations in single and multicomponent flows. The absolute flux Jacobian for dissipation flux calculation is efficiently computed using an expanded Turkel’s approach for thermally perfect gas mixtures. Partial pressure derivatives are approximated when using the flamelet generated manifolds (FGM) combustion model. The proposed scheme is evaluated through scale-resolving simulations of the Cambridge burner flame SWB1 on an unstructured grid using the density-based solver TRACE, employing both finite rate chemistry (FRC) and FGM combustion models. Comparative analysis with the all-speed SLAU2 scheme shows the superior performance of the proposed scheme in handling turbulent reactive multicomponent flows.

المحاكاة العددية للتدفقات المضغوطة المكو نة من عنصرين مع المعادلة العامة للحالة

Numerical simulations of compressible two-component flows with a general equation of state (EOS), written in the form of the Mie-Gru¨neisen EOS, have been conducted using the diffuse interface method on a structured mesh. The unsteady and inviscid one-dimensional sixequation model of Kapila is employed to describe compressible two-component flows. The model is hyperbolic and non-conservative. The solution of the hyperbolic equations, including the non-conservative equation for the volume fraction evolution, is obtained using an extended Harten-Lax-Leer (HLL) approximate Riemann solver. A general formulation for various EOSs is proposed to enable simulations of a wide range of applications. In which the two constituents are either governed by the same EOS or different types of EOS. In this model, both fluids have the same velocity, but each fluid has its own pressure. Therefore, the pressure relaxation process is performed instantaneously to drive both constituents' pressures towards equilibrium. Several computational test problems were conducted in one and two dimensions to show the ability of the numerical method to simulate such problems. The computed results are presented without introducing spurious pressure oscillations in the method.

A stabilised Total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics

AbstractThis paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks.

A super-convergent quasi-discontinuous Galerkin method for approximating inertia-gravity and Rossby waves in geophysical flows

The numerical approximation of the shallow-water equations, which support geophysical wave propagation, namely the fast external Poincaré and Kelvin waves and the slow large-scale planetary Rossby waves, is a delicate and difficult problem. Indeed, the coupling between the momentum and the continuity equations may lead to the presence of erratic or spurious solutions for these waves, e.g. the spurious pressure and inertial modes. In addition, the presence of spurious branches and the emergence of spectral gaps in the dispersion relation at specific wavenumbers may lead to anomalous dissipation/dispersion in the representation of both fast and slow waves. The aim of the present study is to propose a class of possible discretization schemes, via the discontinuous Galerkin method, that is not affected by the above-mentioned problems. A Fourier/stability analysis of the 2D shallow-water model is performed by analysing the finite volume method and the linear discontinuous P1DG and non conforming P1NC Galerkin discretizations. These are stabilized by means of a family of numerical fluxes via the Polynomial Viscosity Matrix approach. A long time stability result is proven for all schemes and fluxes examined here. Further, a super-convergent result is demonstrated for the P1NC discrete frequencies compared to the P1DG ones, except for the slow mode in the Roe flux case. Indeed, we show that the Roe flux yields spurious frequencies and sub-optimal rates of convergence for the slow mode, for both the P1DG and P1NC methods. Finally, numerical solutions of linear and non-linear test problems confirm the theoretical results and reveal the computational cost efficiency of the P1NC approach.

A balanced outflow boundary condition for swirling flows

In open flow simulations, the dispersion characteristics of disturbances near synthetic boundaries can lead to unphysical boundary scattering interactions that contaminate the resolved flow upstream by propagating numerical artifacts back into the domain interior. This issue is exacerbated in flows influenced by real or apparent body forces, which can significantly disrupt the normal stress balance along outflow boundaries and generate spurious pressure disturbances. To address this problem, this paper develops a zero-parameter, physics-based outflow boundary condition (BC) designed to minimize pressure scattering from body forces and pseudo-forces and enhance transparency of the artificial boundary. This “balanced outflow BC” is then compared against other common BCs from the literature using example axisymmetric and three-dimensional open swirling flow computations. Due to centrifugal and Coriolis forces, swirling flows are known to be particularly challenging to simulate in open geometries, as these apparent forces induce non-trivial hydrostatic stress distributions along artificial boundaries that cause scattering issues. In this context, the balanced outflow BC is shown to correspond to a geostrophic hydrostatic stress correction that balances the induced pressure gradients. Unlike the alternatives, the balanced outflow BC yields accurate results in truncated domains for both linear and nonlinear computations without requiring assumptions about wave characteristics along the boundary.

Overlapping finite elements for the Navier-Stokes equations

The purpose of this paper is to investigate the performance of a certain family of low-order quadrilateral finite elements in the solution of the stationary Navier-Stokes equations governing incompressible fluid flows. These finite elements are derived from the ‘overlapping finite elements’, first developed for the solution of problems in solid mechanics [5,9,48] and incorporate features of both meshfree and traditional finite element methods. One of their most remarkable properties is the insensitivity to mesh distortions. Also, since the Shepard functions are replaced by a suitable interpolation, the resulting basis functions are entirely polynomial, which allows the numerical integration of the weak forms to be performed with few integration points [6,49]. We also discuss the theoretical reasons for the stability of the solutions, that is, the absence of spurious pressure modes, and show that the proposed discretization scheme passes the relevant inf-sup test.

A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions: hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.

Performance Assessment of Pneumatic-Driven Automatic Valves to Improve Pipeline Fault Detection Procedure by Fast Transient Tests.

The use of fast transients for fault detection in long transmission networks makes the generation of controlled transients crucial. In order to maximise the information that can be extracted from the measured pressure time history (pressure signal), the transients must meet certain requirements. In particular, the manoeuvre that generates the transient must be fast and repeatable, and must produce a pressure wave that is as sharp as possible, without spurious pressure oscillations. This implies the use of small-diameter valves and often pneumatically operated automatic valves. In the present work, experimental transient tests are carried out at the Water Engineering Laboratory (WEL) of the University of Perugia using a butterfly valve and a ball pneumatic-driven valve to generate pressure waves in a pressurised copper pipe. A camera is used to monitor the valve displacement, while the pressure is measured by a pressure transducer close to the downstream end of the pipe where the pneumatic valve is installed. The experimental data are analysed to characterise the valve performance and to compare the two geometries in terms of valve closing dynamics, the sharpness of the generated pressure wave and the stability of the pressure time history. The present work demonstrates how the proposed approach can be very effective in easily characterising the transient dynamics.

Stabilized mixed material point method for incompressible fluid flow analysis

This paper proposes novel and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated-Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.

On the stability of mixed polygonal finite element formulations in nonlinear analysis

AbstractThis article discusses the accuracy and stability of the pressure field in nonlinear mixed displacement‐pressure finite element formulations in solid mechanics. We focus on two‐dimensional mixed polygonal finite element formulations with linear displacement and constant pressure approximations in particular. The inf‐sup stability of these formulations is assessed and compared with classical mixed finite element formulations. An analytical proof is presented, which concludes that the occurrence of spurious pressure modes depends on the chosen meshing strategy. It is shown that these spurious modes are successfully suppressed on any Voronoi mesh in both linear elasticity and nonlinear hyperelasticity without the need for any kind of stabilization. Several linear and nonlinear nearly‐incompressible examples with different discretization strategies and boundary conditions are considered to validate the analytical proof. A mixed polygonal finite element formulation based on the scaled boundary parameterization is used to approximate the field variables, however, the derivations presented herein hold for any lowest‐order mixed polygonal finite element formulation. The nonexistence of checkerboard modes on linear elastic Voronoi discretizations is shown graphically. By evaluating the incremental pressure in each Newton–Raphson iteration, the stabilization effect of the Voronoi discretization is demonstrated for nonlinear problems. In addition, the analytical proof is validated by the numerical (generalized) inf‐sup test.

Kinetic-energy- and pressure-equilibrium-preserving schemes for real-gas turbulence in the transcritical regime

Numerical simulations of compressible turbulent flows governed by real-gas equations of state, such as high-pressure transcritical flows, are strongly susceptible to instabilities. In addition to the inherent multi-scale nature of the flow, the presence of a pseudo-interface can generate spurious pressure oscillations when conventional schemes are utilized. This study proposes a general framework to derive and analyze discretization methods that are able to preserve kinetic energy by convection, and simultaneously maintain pressure equilibrium in discontinuity-free compressible real-gas flows. The formal analysis reveals that the discrete pressure-equilibrium condition can be fulfilled at most to second-order accuracy, as it requires the spatial differential operator to satisfy a discrete chain rule when total or internal energy are directly discretized. A novel class of schemes based on the solution of a pressure equation is thus proposed, which preserves mass, momentum, kinetic energy and pressure equilibrium, but not total energy. Extensive numerical tests of increasing complexity confirm the theoretical predictions, and show that the proposed scheme is capable of providing non-dissipative, stable and oscillation-free simulations, unlike existing methods tailored for the transcritical regime.

Gradient Robust Mixed Methods for Nearly Incompressible Elasticity

Within the last years pressure robust methods for the discretization of incompressible fluids have been developed. These methods allow the use of standard finite elements for the solution of the problem while simultaneously removing a spurious pressure influence in the approximation error of the velocity of the fluid, or the displacement of an incompressible solid. To this end, reconstruction operators are utilized mapping discretely divergence free functions to divergence free functions. This work shows that the modifications proposed for Stokes equation by Linke (Comput Methods Appl Mech Eng 268:782–800, 2014) also yield gradient robust methods for nearly incompressible elastic materials without the need to resort to discontinuous finite elements methods as proposed in Fu et al. (J Sci Comput 86(3):39–30, 2021).

Fully conservative and pressure-equilibrium preserving scheme for compressible multi-component flows

In compressible flows with a variable ratio of specific heats, such as multi-component flows, the conventional conservative schemes generate spurious pressure oscillations at multi-component interfaces even when the interfaces are smooth. In this study, we propose a novel spatial discretization scheme that maintains both the primary conservation and pressure equilibrium in discontinuity-free compressible multi-component flows. The key to this study is the compatibility condition, which is the discrete condition for implicitly satisfying the pressure equilibrium at the discrete level. The compatibility condition is derived based on the analytical relation between the mass of species and the ratio of specific heats in the governing equations. The proposed scheme is constructed by the spatial discretization that satisfies the compatibility condition and can be applied to the arbitrary number of species components. The inviscid smooth interfaces advection problems verify that the proposed scheme satisfies both the pressure equilibrium and primary conservation only by solving the conservation equations, unlike the existing schemes that solve non-conservative or overspecified equations.

On the inf-sup stability of Crouzeix-Raviart Stokes elements in 3D

We consider discretizations of the stationary Stokes equation in three spatial dimensions by non-conforming Crouzeix-Raviart elements. The original definition in the seminal paper by M. Crouzeix and P.-A. Raviart in 1973 [Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), pp. 33–75] is implicit and also contains substantial freedom for a concrete choice. In this paper, we introduce basic Crouzeix-Raviart spaces in 3D in analogy to the 2D case in a fully explicit way. We prove that this basic Crouzeix-Raviart element for the Stokes equation is inf-sup stable for polynomial degree k = 2 k=2 (quadratic velocity approximation). We identify spurious pressure modes for the conforming ( k , k − 1 ) \left ( k,k-1\right ) 3D Stokes element and show that these are eliminated by using the basic Crouzeix-Raviart space.

A finite particle method based on a Riemann solver for modeling incompressible flows

The weakly compressible SPH (WCSPH) method has been widely used for solving hydrodynamic problems, in which the fluid is assumed to be weakly compressible through limiting the density deviation to a small value. Despite its simplicity and robustness in implementation, the WCSPH method is generally hindered by its poor accuracy and spurious oscillations in the pressure field which may lead to numerical instability. In order to dampen the spurious oscillations without introducing excessive dissipation, a Roe-type approximate Riemann solver with a low-dissipation limiter was proposed and applied to the WCSPH method by Zhang et al. (2017). The present paper aims to extend this low-dissipation Riemann solver to the finite particle method (FPM), which can be referred to an enhanced or modified version of the conventional SPH method, for reducing spurious pressure oscillations and meanwhile achieving higher order of accuracy. Numerical results demonstrate that the Riemann solver based FPM is effective in modeling incompressible flows and it produces a more accurate flow field than the conventional WCSPH and Riemann solver based WCSPH methods.

The stability criterion based on the spurious pressure oscillation analysis of MPS method

The Moving Particle Semi-implicit (MPS) method has been widely used in nuclear engineering due to its unique advantages in capturing moving interfaces. However, the unphysical numerical oscillation in MPS method is still a hotspot and difficulty in current study, and also a bottleneck hindering its high-accuracy numerical simulation of nuclear engineering. To obtain more accurate and stable numerical simulation, the spurious pressure oscillation behavior is studied theoretically and numerically. Through theoretical analysis, it is found that the incompressible compensation term is the main cause of spurious pressure oscillation. Besides, three mainstream forms of PPE are analyzed in detail. The simulations are carried out to study the effects of corrective matrix, time step, external force, numerical resolution and velocity on pressure oscillation. Finally, the stability criterion of spurious oscillation phenomenon is newly proposed to determine the time step, and reliable simulation results are obtained as well.

Performance of different inflow turbulence methods for wind engineering applications

Defining the correct inlet boundary conditions for large eddy simulations is a critical issue in computational wind engineering. Since synthetic inflow turbulence does not require costly prior flow simulations like recycling or precursor methods, it is a preferable approach. In this study, different synthetic turbulence generator methods are considered to investigate their performance in wind engineering applications. The considered methods are a) Digital Filter Methods (DFM), b) Synthetic eddy methods (SEM) with different shape functions, c) Divergence Free Synthetic Eddy Method (DFSEM), and d) two types of Anisotropy Turbulent Spot Method (ATSM). These methods are provided in Turbulence Inflow Tool (TInF) from the SimCenter (https://simcenter.designsafe-ci.org/backend-components/tinf/). Additionally, velocity spectrum at the inlet and building location is compared to the Von Karman spectrum for different inflow methods to determine how well the energy is carried from the inlet to the building location. Furthermore, different methods are evaluated to see whether they produce spurious pressure in the domain. It is concluded that spurious pressure exists in all the considered methods except SEM method with the Gaussian shape function (SEM-G). In addition, SEM-G is found to be a suitable method for peak pressure prediction on buildings with upmost 30% error.

A novel coupled NS‐PFEM with stable nodal integration and polynomial pressure projection for geotechnical problems

AbstractThe node‐based smoothed particle finite element method (NS‐PFEM) offers high computational efficiency but is numerically unstable due to possible spurious low‐energy mode in direct nodal integration (NI). Moreover, the NS‐PFEM has not been applied to hydromechanical coupled analysis. This study proposes an implicit stabilised T3 element‐based NS‐PFEM (stabilised node‐based smoothed particle finite element method [SNS‐PFEM]) for solving fully hydromechanical coupled geotechnical problems that (1) adopts the stable NI based on multiple stress points over the smooth domain to resolve the NI instability of NS‐PFEM, (2) implements the polynomial pressure projection (PPP) technique in the NI framework to cure possible spurious pore pressure oscillation in the undrained or incompressible limit and (3) expresses the NI for assembling coefficient matrices and calculating internal force in SNS‐PFEM with PPP as closed analytical expressions, guaranteeing computational accuracy and efficiency. Four classical benchmark tests (1D Terzaghi's consolidation, Mandel's problem, 2D strip footing consolidation and foundation on a vertical cut) are simulated and compared with analytical solutions or results from other numerical methods to validate the correctness and efficiency of the proposed approach. Finally, penetration of strip footing into soft soil is investigated, showing the outstanding performance the proposed approach can offer for large deformation problems. All results demonstrate that the proposed SNS‐PFEM with PPP is capable of tracking hydromechanical coupled geotechnical problems under small and large deformation with different drainage capacities.

Maximum grid spacing effect on peak pressure computation using inflow turbulence generators

The peak pressures are computed using computational fluid dynamics (CFD) with the synthetic inflow turbulence generator and compared with 1:6 scale Texas Tech University (TTU) wind tunnel measurements. The inflow turbulence is calculated using the Consistent Discrete Random Flow Generation Method (CDRFG) method. The maximum and minimum frequencies from the field or experimental measurements as input to the inflow turbulence generator without considering the largest grid spacing used in the CFD model leads to high pressure error. For one case, more than 100% error in peak pressure results is observed. In addition, spurious pressures are observed at the building location without building. By varying maximum frequencies systematically for each computational mesh size and comparing the velocities and pressures at the inflow and the building location without building, possible causes of the error are explained. From the investigation, it is suggested not to use the maximum frequency in the inflow turbulence generator beyond the frequency that can be transported by the largest grid spacing.