Discontinuous Pressure Research Articles

A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete unknowns, we reconstruct inside each element a Stokes velocity one degree higher than face unknowns, and a Darcy velocity in the Raviart–Thomas–Nédélec space. These reconstructed velocities are respectively used to formulate the discrete versions of the Stokes and Darcy terms in the momentum equation, along with suitably designed penalty contributions. The proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local (Stokes- or Darcy-dominated) regimes, as identified by a dimensionless number which can be interpreted as a friction coefficient. The singular limit corresponding to the Darcy equation is also fully supported by the method. Numerical examples corroborate the theoretical results. This paper also contains two contributions whose interest goes beyond the specific method and application treated in this work: an investigation of the dependence of the constant in the second Korn inequality on star-shaped domains and its application to the study of the approximation properties of the strain projector in general Sobolev seminorms.

Generalized finite element systems for smooth differential forms and Stokes’ problem

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of finite element systems, and the examples include conforming mixed finite elements for Stokes’ equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.

A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction

Abstract The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart–Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

INFLUENCE OF MODELING METHODS FOR CARTILAGE LAYER ON SIMULATION OF PERIACETABULAR OSTEOTOMY USING FINITE ELEMENT CONTACT ANALYSIS

Finite element (FE) analysis has been used in the simulation of periacetabular osteotomy (PAO) to predict the improvement of contact pressure concentration in dysplastic hip joint. Since the cartilage layer is difficult to be segmented from CT or MRI images, most hip joint models were assumed to be a simple perfect ball and socket joint. However, the influence of different cartilage modeling methods on the reliability of the simulation has not been assessed. The objective of this study is to elucidate the influence of different cartilage modeling methods on predictions of cartilage layers’ contact pressure by FE contact analysis. In this study, the cartilage layer was generated by applying three typical kinds of modeling methods (spherical, uniform thickness, and midline-based). After comparisons with these cartilage modeling methods, the computational results demonstrate that the cartilage modeling methods have a dramatic influence on predictions of contact pressure in the PAO. The relatively continuous contact pressure distribution and lower peak contact pressure are observed in spherical cartilage modeling method. The discontinuous contact pressure distribution and higher peak contact pressure are obtained in uniform thickness and midline-based cartilage modeling methods. And the degree of discontinuous pressure distribution is even worse in the midline-based cartilage modeling method.

Efficient stress–velocity least-squares finite element formulations for the incompressible Navier–Stokes equations

In this contribution three different mixed least-squares finite element methods (LSFEMs) are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies. The deliberations are made for the incompressible Navier–Stokes equations. The generally known first-order div–grad system in terms of the (total) stress, velocity and pressure (SVP) formulation is the basis for two further div–grad least-squares formulations in terms of (total) stress and velocity (SV), whereby both formulations are derived from different fountainheads. The extended SV formulation is an enrichment of the known stress–velocity formulation proposed by Cai et al. (2004). The second SV formulation is based on the substitution of the pressure in the SVP formulation, thus it is based on a discontinuous pressure interpolation. Advantage of the SV formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom. Nevertheless, the approximation quality has to last the demand of the established least-squares formulations. The drawback of a poor mass conservation is well-investigated for the LSFEM and known to overcome by using high-order interpolations for all solution variables. Therefore, the main attention of this contribution is focused on accuracy, while aiming for high efficiency, which is intrinsic for the fundamental idea in here. In the light of efficient LSFEM solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h-type adaptive mesh refinement.

Formation of Regions with High Energy and Pressure Gradients at the Free Surface of Liquid Dielectric in a Tangential Electric Field

Nonlinear dynamics of the free surface of an ideal incompressible non-conducting fluid with high dielectric constant subjected by strong horizontal electric field is simulated on the base of the method of conformal transformations. It is demonstrated that interaction of counter-propagating waves leads to formation of regions with steep wave front at the fluid surface; angles of the boundary inclination tend to {\pi}/2, and the curvature of surface extremely increases. A significant concentration of the energy of the system occurs at these points. From the physical point of view, the appearance of these singularities corresponds to formation of regions at the fluid surface where pressure exerted by electric field undergoes a discontinuity and dynamical pressure increases almost an order of magnitude.

Implicit-Modal Discontinuous Galerkin Scheme for Two-Phase Flow with Discontinuous Capillary Pressure

In this paper, we propose an implicit-modal discontinuous Galerkin scheme for solving an incompressible and immiscible oil-water flow through a heterogeneous reservoir with different rock types. The scheme uses the modal discontinuous Galerkin method for spatial discretization of both pressure and saturation equations and the backward Euler method for temporal discretization. In the modal discontinuous Galerkin method, we apply the well-localized bases that are constructed by both the nodal and modal basis functions. To find the approximate solution for the saturation equation, we propose the upwind technique. In the stability analysis of the scheme, we obtain the constant of the trace inequality for the approximate function and its first derivatives using the modal basis functions. In this way, we give suitable ranges for penalty terms. To guarantee the convergence of the approximate solution, we find suitable ranges of the time step. To demonstrate the applicability of the proposed scheme, we apply it to the dense nonaqueous phase liquids and the Versuchseinrichtung zur Grundwasser und Altlastensanierung infiltration problems. Numerical results are presented.

Mixed Aggregated Finite Element Methods for the Unfitted Discretization of the Stokes Problem

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that this kind of methods lead to arbitrarily ill-conditioned systems. In order to solve this issue, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes, and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with non-trivial geometries.

Recent developments of miniature ion trap mass spectrometers

With outstanding analytical performance and portability, miniature mass spectrometer is one of the most powerful tools for in-situ analysis. The miniaturization of mass spectrometers has lasted for more than ten years, during which a number of miniature mass spectrometers employing different techniques have been developed. Small-in-size, working at relatively high pressure region and capable of performing tandem mass spectrometry, ion trap is the most widely used mass analyzer in miniature mass spectrometer systems. The recent development of miniature ion trap mass spectrometer systems in the last ten years was reviewed in this paper. These instruments adopt different atmospheric pressure interfaces (APIs), which are membrane inlets (MIs), discontinuous atmospheric pressure interface (DAPI) and continuous atmospheric pressure interface (CAPI). This review emphasizes on the mini mass spectrometry (MS) system that can be handheld by one person, but not the field-able ones that are too large to be hand-portable.

A new continuous/discontinuous formulation for Stokes problem with analysis of InfSup stability

In this paper we present a new continuous/discontinuous finite element method with continuous trace space for Stokes problem. This new method has two new bi-linear forms which are responsible for the properties of the method. One of these two new bi-linear forms allows to eliminate completely the discontinuous pressure at element level preserving the convergence rates. The other bi-linear form allows interpolation of any order for pressure to be possible since the order of interpolation for the velocity and for the trace variable or hybrid variable satisfies a limit well determined. The InfSup stability condition is well established for any order of interpolation of the pressure and for arbitrary meshes. Numerical experiments indicate optimal rates of convergence, and that, for the Stokes problem, the method presented in this paper has computational cost lower than that of the Galerkin method with continuous or discontinuous pressure and of the HDG method with discontinuous trace space, for solving the global system using direct solver. Finally, it must be observed that the formulation and the analysis presented here are naturally extended if we used discontinuous trace space.

Hybrid-dimensional modelling of two-phase flow through fractured porous media with enhanced matrix fracture transmission conditions

In this work, we extend, to two-phase flow, the single-phase Darcy flow model proposed in [26], [12] in which the (d−1)-dimensional flow in the fractures is coupled with the d-dimensional flow in the matrix. Three types of so called hybrid-dimensional two-phase Darcy flow models are proposed. They all account for fractures acting either as drains or as barriers, since they allow pressure jumps at the matrix–fracture interfaces. The models also permit to treat gravity dominated flow as well as discontinuous capillary pressure at the material interfaces. The three models differ by their transmission conditions at matrix fracture interfaces: while the first model accounts for the nonlinear two-phase Darcy flux conservations, the second and third ones are based on the linear single phase Darcy flux conservations combined with different approximations of the mobilities. We adapt the Vertex Approximate Gradient (VAG) scheme to this problem, in order to account for anisotropy and heterogeneity aspects as well as for applicability on general meshes. Several test cases are presented to compare our hybrid-dimensional models to the generic equi-dimensional model, in which fractures have the same dimension as the matrix, leading to deep insight about the quality of the proposed reduced models.

Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media

This paper presents a study of immiscible incompressible two-phase flow through fractured porous media. The results obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikelic (1996) and L. M. Yeh (2006) are revisited. The main goal is to incorporate some of the most recent improvements in the convergence of the solutions in the homogenization of such models. The microscopic model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat law. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. The fractured medium consists of periodically repeating homogeneous blocks and fractures, the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by \begin{document}${\varepsilon }^θ$\end{document} , where \begin{document}$\varepsilon$ \end{document} is the size of a typical porous block and \begin{document}$θ>0$\end{document} is a parameter. The model involves highly oscillatory characteristics and internal nonlinear interface conditions. Under some realistic assumptions on the data, the convergence of the solutions, and the macroscopic models corresponding to various range of contrast are constructed using the two-scale convergence method combined with the dilation technique. The results improve upon previously derived effective models to highly heterogeneous porous media with discontinuous capillary pressures.

Developing a Vacuum Electrospray Source To Implement Efficient Atmospheric Sampling for Miniature Ion Trap Mass Spectrometer.

The performance of a miniature mass spectrometer in atmospheric analysis is closely related to the design of its sampling system. In this study, a simplified vacuum electrospray ionization (VESI) source was developed based on a combination of several techniques, including the discontinuous atmospheric pressure interface, direct capillary sampling, and pneumatic-assisted electrospray. Pulsed air was used as a vital factor to facilitate the operation of electrospray ionization in the vacuum chamber. This VESI device can be used as an efficient atmospheric sampling interface when coupled with a miniature rectilinear ion trap (RIT) mass spectrometer. The developed VESI-RIT instrument enables regular ESI analysis of liquid, and its qualitative and quantitative capabilities have been characterized by using various solution samples. A limit of detection of 8 ppb could be attained for arginine in a methanol solution. In addition, extractive electrospray ionization of organic compounds can be implemented by using the same VESI device, as long as the gas analytes are injected with the pulsed auxiliary air. This methodology can extend the use of the proposed VESI technique to rapid and online analysis of gaseous and volatile samples.

Implicit Hybrid Upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity

We consider the numerical solution of the partial differential equations governing multiphase flow in porous media. For highly nonlinear problems, the temporal discretization of choice is often the unconditionally stable fully implicit method. However, the nonlinear systems, often solved with Newton’s method, are difficult to solve. Thus, the computational cost is strongly dependent on the nonlinear convergence rate, and enhancing this convergence property is key to speed up subsurface flow simulation.We focus on the case of spatially discontinuous capillary pressure between rock regions. To efficiently and accurately simulate the flow dynamics in heterogeneous porous media, the flux computation combines Implicit Hybrid Upwinding with transmission conditions between different rock regions. This leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence.We extend our previous results (Hamon et al., 2016 [18]) by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions. The generalized scheme is supported by an analysis of its mathematical properties. Our multidimensional numerical examples, which range from buoyancy-driven flow with capillary barriers to viscous-dominated flow, demonstrate that the Implicit Hybrid Upwinding scheme improves the accuracy compared to the standard phase-based upwinding scheme, while leading to significant reductions in the number of nonlinear iterations in multiple dimensions.

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

SummaryAn arbitrary Lagrangian‐Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier‐Stokes equations are discretized using the side‐centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision. The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy. The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Fully implicit time-space discretizations applied to the two-phase Darcy flow problem lead to the systems of nonlinear equations, which are traditionally solved by some variant of Newton's method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton's method is not invariant with respect to a nonlinear change of variable. In this regard the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two phase water-gas Darcy flow in fractured porous media for which the fractures are modeled as interfaces of co-dimension one. The problem is discretized using an extension of Vertex Approximate Gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.

Development of Electrospray/Photoionization Miniature Ion Trap Mass Spectrometer

Abstract An electrospray/ultraviolet lamp dual-source ion trap mass spectrometer was developed for the rapid detection of gas and liquid samples. A discontinuous atmospheric pressure sampling system was used in the instrument, in which both the electrospray ions and gaseous analytes were sampled and transferred by a pinch valve device. The two ionization sources used here were generally suitable for different kinds of analytes and provide complementary applications. Electrospray was used for the ionization of polar compounds in solution, while the UV ionization source was mainly applied to the analysis of gaseous organic compounds. A variety of samples such as anisole, toluene, 2,4-dimethylaniline, arginine, reserpine and aspartame were employed to test the performance in different working mode of the instrument. The result showed that the two sources was suitable for ionization of different samples, and different types of molecular ions were generated when analyzing 2,4-dimethylaniline. Two ionization sources could be used alternately or simultaneously without interference with the other and the working mode could also be switched to fit the application requirements. Dual-source configuration was an effective way to extend the applications for miniature mass spectrometers. It did not significantly increase the size of the instrument, but provided more versatile analysis to meet the need for the measurement of different types of samples.

Effect of rainfall on a colluvial landslide in a debris flow valley

A colluvial landslide in a debris flow valley is a typical phenomena and is easily influenced by rainfall. The direct destructiveness of this kind of landslide is small, however, if failure occurs the resulting blocking of the channel may lead to a series of magnified secondary hazards. For this reason it is important to investigate the potential response of this type of landslide to rainfall. In the present paper, the Goulingping landslide, one of the colluvial landslides in the Goulingping valley in the middle of the Bailong River catchment in Gansu Province, China, was chosen for the study. Electrical Resistivity Tomography (ERT), Terrestrial Laser Scanning (TLS), together with traditional monitoring methods, were used to monitor changes in water content and the deformation of the landslide caused by rainfall. ERT was used to detect changes in soil water content induced by rainfall. The most significant findings were as follows:(1) the water content in the centralupper part (0~41 m) of the landslide was greater than in the central-front part (41~84 m) and (2) there was a relatively high resistivity zone at depth within the sliding zone. The deformation characteristics at the surface of the landslide were monitored by TLS and the results revealed that rainstorms caused three types of deformation and failure: (1) gully erosion at the slope surface; (2) shallow sliding failure; (3) and slope foot erosion. Subsequent monitoring of continuous changes in pore-water pressure, soil pressure and displacement (using traditional methods) indicated that long duration light rainfall (average 2.22 mm/d) caused the entire landslide to enter a state of creeping deformation at the beginning of the rainy season. Shear-induced dilation occurred for the fast sliding (30.09 mm/d) during the critical failure sub-phase (EF). Pore-water pressure in the sliding zone was affected by rainfall. In addition, the sliding L1 parts of the landslide exerted a discontinuous pressure on the L2 part. Through the monitoring and analysis, we conclude that this kind of landslide may have large deformation at the beginning and the late of the rainy season.

Modelling hydraulic fractures in porous media using flow cohesive interface elements

This paper revisits the problem of computational modelling of a fluid-driven fracture propagating in a permeable porous medium using zero-thickness flow cohesive interface elements. Both cases of continuous and discontinuous pressure field across the fractures are implemented in a unified formulation. The paper provides computational aspects of hydraulic fracture modelling such as mesh generation, execution time, convergence and numerical integration issues. We show that Newton-Cotes quadrature must be used for quadratic flow cohesive interface elements at least for the presented problems. Our simulations exhibit the so-called intermittent crack tip advancement as recently confirmed in the literature. This paper is addressed to researchers who would like to have a quick working implementation of the zero-thickness flow cohesive interface elements for simulating hydraulic fracturing processes with finite elements.

A stabilized finite element method for finite-strain three-field poroelasticity

We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.