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Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ⁢ (Ω) . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered. [ABSTRACT FROM AUTHOR]

In this study, a non-local partial Peridynamic mesh-free incompressible method is proposed for simulating dry dense granular flows such as granular column collapse. In the method, a predictor-and-corrector time splitting scheme is used, and Peridynamic theory is incorporated only in the predictor to solve an integrated momentum equation, in which part of the pressure gradient force, viscous force and external force are included. An incompressible explicit solver is employed to obtain the pressure field, and the corrector allows for the remaining part of the pressure gradient force to be calculated and updates the flow field. The proposed method is then validated by simulating granular flows in several configurations, including a steady granular flow down an inclined slope, granular column collapses at both one side and two sides, and collision of two adjacent granular columns. The simulated velocity profiles are in good agreement with the analytical solution in the steady granular down an inclined slope in which sensitivity of the particle distance, a coefficient by incorporating Peridynamics, and Peridynamic horizon are examined. The simulation of the granular column collapses shows that the method can reproduce the final deposit, free surface, and velocity in the flows. The method can capture interface variations between two granular columns during their collision in good agreement with experimental observations. [ABSTRACT FROM AUTHOR]

For the simulation of fluid–structure interaction (FSI) problems, the monolithic method provides more robust convergence than the partitioned method but requires solving a larger matrix for a saddle-point problem that is difficult to precondition. We extend the existing semi-implicit method to propose a new semi-monolithic method for FSI simulation that uses a velocity boundary equation of a pressure Poisson equation so that only the pressure variables of the fluid domain are coupled with the displacement variables of the solid domain in a monolithic manner. The fluid domain is solved by employing a fractional four-step method for the incompressible Navier–Stokes equations based on an arbitrary Lagrangian–Eulerian (ALE) formulation, and the solid domain is solved by an updated Lagrangian method for simulating large nonlinear deformations. We applied the proposed method to 2D/3D benchmark problems with various time steps and density ratios, and the results confirmed that FSI problems with not only a strong added-mass effect but also a large deformation are simulated well. The proposed method is faster than the existing monolithic method because it solves a smaller matrix whose diagonal blocks are diagonally dominant matrices, which are much easier to precondition. • A new semi-monolithic formulation for FSI problems is proposed. • Fluid domain is solved by ALE based on the incompressible Navier–Stokes equations. • Solid domain is solved by updated Lagrangian method for nonlinear large deformation. • The method can simulate FSI with a strong added-mass effect and a large deformation. • The method is faster than an existing monolithic method requiring less memory. [ABSTRACT FROM AUTHOR]

Performance of simple, fluidic harvesters consisting of a piezoelectric cantilever strongly relies on a time-dependent fluid forcing they experience. To quantify this forcing, an analytical solution of a pressure Poisson equation (PPE) is presented that uses particle image velocimetry (PIV) data to calculate pressure around the harvester. This analytical solution is based on a modified Green's function approach and provides a favorable method of calculating the pressure field from PIV data. It eliminates a need to compute higher-order derivatives of velocity that are present in the viscous terms and it eliminates the need to integrate Navier–Stokes equations to find the pressure along the boundaries of interest. An experiment was carried out to validate this solution. Pressure distribution along the piezoelectric cantilever was calculated by solving PPE analytically as a single vortex passed over it. This distribution was then integrated to calculate the net force acting on the beam. Euler–Bernoulli theory was then used to predict the beam's dynamic response based on the calculated pressure. Using dielectric properties of the piezoelectric harvester, the voltage and total power output were computed from the motion of the beam. These values were compared to the voltage and power directly measured during the experiment. [ABSTRACT FROM AUTHOR]

As a Lagrangian mesh-free method, the moving particle semi-implicit (MPS) method can easily handle complex incompressible flow with a free surface. However, some deficiencies of the MPS method, such as inaccurate results, unphysical pressure oscillation, and particle thrust near the free surface, still need to be further resolved. Here, we propose a modified MPS method that uses the following techniques: (1) a modified MPS scheme with a split-pressure Poisson equation is proposed to reproduce hydrostatic pressure stably; (2) a new virtual particle technique is developed to ensure the symmetrical distribution of particles on the free surface; (3) a Laplacian operator that is consistent with the original gradient operator is introduced to replace the original Laplacian operator. In addition, a two-judgment technique for distinguishing free surface particles is introduced in the proposed MPS method. Four free surface flows were adopted to verify the proposed MPS method, including two hydrostatic problems, a dam-breaking problem, and a violent sloshing problem. The enhancement of accuracy and stability by these improvements was demonstrated. Moreover, the numerical results of the proposed MPS method showed good agreement with analytical solutions and experimental results. [ABSTRACT FROM AUTHOR]

Summary: The mechanism of many cardiovascular diseases can be understood by studying the pressure distribution in blood vessels. Direct pressure measurements, however, require invasive probing and provide only single‐point data. Alternatively, relative pressure fields can be reconstructed from imaging‐based velocity measurements by considering viscous and inertial forces. Both contributions can be potential troublemakers in pressure reconstruction: the former due to its higher‐order derivatives, and the latter because of the quadratic nonlinearity in the convective acceleration. Viscous and convective terms can be treated in various forms, which, although equivalent for ideal measurements, can perform differently in practice. In fact, multiple versions are often used in literature, with no apparent consensus on the more suitable variants. In this context, the present work investigates the performance of different versions of relative pressure estimators. For viscous effects, in particular, two new modified estimators are presented to circumvent second‐order differentiation without requiring surface integrals. In‐silico and in‐vitro data in the typical regime of cerebrovascular flows are considered, allowing a systematic noise sensitivity study. Convective terms are shown to be the main source of error, even for flows with pronounced viscous component. Moreover, the conservation (often integrated) form of convection exhibits higher noise sensitivity than the standard convective description, in all three families of estimators considered here. For the classical pressure Poisson estimator, the present modified version of the viscous term tends to yield better accuracy than the (recently introduced) integrated form, although this effect is in most cases negligible when compared to convection‐related errors. [ABSTRACT FROM AUTHOR]

As a Lagrangian mesh-free method, the moving particle semi-implicit (MPS) method can easily handle complex incompressible flow with a free surface. However, some deficiencies of the MPS method, such as inaccurate results, unphysical pressure oscillation, and particle thrust near the free surface, still need to be further resolved. Here, we propose a modified MPS method that uses the following techniques: (1) a modified MPS scheme with a split-pressure Poisson equation is proposed to reproduce hydrostatic pressure stably; (2) a new virtual particle technique is developed to ensure the symmetrical distribution of particles on the free surface; (3) a Laplacian operator that is consistent with the original gradient operator is introduced to replace the original Laplacian operator. In addition, a two-judgment technique for distinguishing free surface particles is introduced in the proposed MPS method. Four free surface flows were adopted to verify the proposed MPS method, including two hydrostatic problems, a dam-breaking problem, and a violent sloshing problem. The enhancement of accuracy and stability by these improvements was demonstrated. Moreover, the numerical results of the proposed MPS method showed good agreement with analytical solutions and experimental results.

Poisson equation lies in the heart of the numerical solutions of incompressible Navier–Stokes equations (NSEs) based on the popular projection methods, which decouple the pressure and velocities fields. This paper presents enhanced numerical solutions of the 2D incompressible NSEs inspired by a newly-developed Generalized Harmonic Polynomial Cell (GHPC) method for the Poisson equation, which has a fourth-order spatial accuracy. To achieve that, the original GHPC method is adapted to accommodate immersed boundaries, necessary when a staggered grid for velocities and pressure is applied. The finite difference method (FDM) is used in the spatial discretization of the diffusion and advection terms. Numerical analyses of lid-driven cavity flow, a Taylor–Green vortex, and flow around a smooth circular cylinder all show encouraging results, confirming the accuracy and efficiency of the new solver. Through comparison with an NSEs solver which applies second-order FDM for the Poisson equation, we demonstrate that the accuracy of the pressure solution can be greatly improved by applying the more accurate GHPC method for the Poisson equation, even though the accuracy of the velocities solutions are limited by the numerical schemes for advection–diffusion equations. The accuracy in the pressure solution can also be translated into CPU time saving to achieve a predefined accuracy. The present immersed-boundary GHPC Poisson equation solver can easily be 'plugged' into other existing NSEs solvers utilizing staggered grids. • A new 2D Navier–Stokes equations solver based on generalized harmonic polynomial cell (GHPC) method. • Immersed-boundary strategy introduced in the original GHPC method. • Demonstration of the enhanced pressure field solution by using the new solver. [ABSTRACT FROM AUTHOR]

Computing pressure fields from given flow velocities is a task frequently arising in engineering, biomedical, and scientific computing applications. The so‐called pressure Poisson equation (PPE) derived from the balance of linear momentum provides an attractive framework for such a task. However, the PPE increases the regularity requirements on the pressure and velocity spaces, thereby imposing theoretical and practical challenges for its application. In order to stay within a Lagrangian finite element framework, it is common practice to completely neglect the influence of viscosity and compressibility when computing the pressure, which limits the practical applicability of the pressure Poisson method. In this context, we present a mixed finite element framework which enables the use of this popular technique with generalized Newtonian fluids and compressible flows, while allowing standard finite element spaces to be employed for the unknowns and the given data. This is attained through the use of appropriate vector calculus identities and simple projections of certain flow quantities. In the compressible case, the mixed formulation also includes an additional equation for retrieving the density field from the given velocities so that the pressure can be accurately determined. The potential of this new approach is showcased through numerical examples. [ABSTRACT FROM AUTHOR]

Due to simplicity in implementation and data structure, elements with equal‐order interpolation of velocity and pressure are very popular in finite‐element‐based flow simulations. Although such pairs are inf‐sup unstable, various stabilization techniques exist to circumvent that and yield accurate approximations. The most popular one is the pressure‐stabilized Petrov–Galerkin (PSPG) method, which consists of relaxing the incompressibility constraint with a weighted residual of the momentum equation. Yet, PSPG can perform poorly for low‐order elements in diffusion‐dominated flows, since first‐order polynomial spaces are unable to approximate the second‐order derivatives required for evaluating the viscous part of the stabilization term. Alternative techniques normally require additional projections or unconventional data structures. In this context, we present a novel technique that rewrites the second‐order viscous term as a first‐order boundary term, thereby allowing the complete computation of the residual even for lowest‐order elements. Our method has a similar structure to standard residual‐based formulations, but the stabilization term is computed globally instead of only in element interiors. This results in a scheme that does not relax incompressibility, thereby leading to improved approximations. The new method is simple to implement and accurate for a wide range of stabilization parameters, which is confirmed by various numerical examples. [ABSTRACT FROM AUTHOR]

The discrete element method (DEM) coupled with computational fluid dynamics (CFD) is regarded to be a standard approach in a numerical calculation of gas-solid flow. When the DEM-CFD method is applied to the simulation of industrial systems, we cannot shy away from the modeling of arbitrary shape wall boundaries. Very recently, we have developed an innovative wall boundary model in the DEM-CFD simulations, where the signed distance function (SDF) and the immersed boundary method (IBM) are used for the wall boundary in the DEM and CFD. The SDF makes it possible to create an arbitrary shape wall boundary based on the scalar field. The IBM can be used for the wall boundary model by reflecting the number of the SDF points on the local volume fraction in CFD grid. Thus, the arbitrary shape wall boundary can be easily created in the DEM-CFD method by combining the SDF with the IBM. On the other hand, the authors' group has pointed out the necessity of the density scale technique in CFD. In the density scale technique, the density scale coefficient is considered in the calculation of the pressure Poisson equation. When the SDF/IBM is applied to CFD without density scaling technique, an unnatural fluid flow might occur through the wall boundary object. The effect of the density scaling coefficient on a gas-solid flow has not been examined so far. From this background, the influence of the density scale coefficient on a gas-solid flow is investigated in the present study. Through this study, introduction of the density scaling technique is designated to be crucial when the IBM is used for the wall boundary in the DEM-CFD simulation. The density scaling coefficient is shown to be decided not by theory but by experience, and the value should become 100 to 1000 in a typical gas-solid flow system. Besides, it is illustrated that the calculation cost is not influenced by the value of the density scale coefficient in the DEM-CFD simulations. Unlabelled Image • The density scaling technique is indispensable in the Advanced DEM-CFD method. • In the Advanced DEM-CFD method, wall boundary can be created by SDF/IBM. • Influence of the density scale coefficient on a gas solid flow is investigated. • Unrealistic gas-solid flow is observed without the density scale technique. • The calculation cost is hardly influenced even if the density scale technique was employed. [ABSTRACT FROM AUTHOR]

Severity of aortic coarctation (CoA) is currently assessed by estimating trans-coarctation pressure drops through cardiac catheterization or echocardiography. In principle, more detailed information could be obtained non-invasively based on space- and time-resolved magnetic resonance imaging (4D flow) data. Yet the limitations of this imaging technique require testing the accuracy of 4D flow-derived hemodynamic quantities against other methodologies. With the objective of assessing the feasibility and accuracy of this non-invasive method to support the clinical diagnosis of CoA, we developed an algorithm (4DF-FEPPE) to obtain relative pressure distributions from 4D flow data by solving the Poisson pressure equation. 4DF-FEPPE was tested against results from a patient-specific fluid-structure interaction (FSI) simulation, whose patient-specific boundary conditions were prescribed based on 4D flow data. Since numerical simulations provide noise-free pressure fields on fine spatial and temporal scales, our analysis allowed to assess the uncertainties related to 4D flow noise and limited resolution. 4DF-FEPPE and FSI results were compared on a series of cross-sections along the aorta. Bland-Altman analysis revealed very good agreement between the two methodologies in terms of instantaneous data at peak systole, end-diastole and time-averaged values: biases (means of differences) were +0.4 mmHg, −1.1 mmHg and +0.6 mmHg, respectively. Limits of agreement (2 SD) were ±0.978 mmHg, ±1.06 mmHg and ±1.97 mmHg, respectively. Peak-to-peak and maximum trans-coarctation pressure drops obtained with 4DF-FEPPE differed from FSI results by 0.75 mmHg and −1.34 mmHg respectively. The present study considers important validation aspects of non-invasive pressure difference estimation based on 4D flow MRI, showing the potential of this technology to be more broadly applied to the clinical practice. [ABSTRACT FROM AUTHOR]

Zheng, X.; Ma, Q.; Shao, S.; Hu, Z., and Gui, Q., 2019. An improved 2D + t incompressible smoothed particle hydrodynamics approach for high-speed vessel waves. Journal of Coastal Research, 35(5), 1106–1119. Coconut Creek (Florida), ISSN 0749-0208. The wave-generated resistance force is substantial for high-speed vessels. A thorough understanding of the wave pattern as well as its drag effect is important for improving vessel performance and optimal design. Owing to the existence of highly nonlinear waves and large deformations of the free surface, the mesh-free smoothed particle hydrodynamics (SPH) approach would provide a useful solution technique for such a practical problem. Based on the pioneering study on the combination of weakly compressible SPH with the two-dimensional (2D) + t (time) theory, the present paper carries out a further investigation on the complex wave patterns generated by a high-speed vessel, through coupling the 2D + t theory with an improved incompressible SPH (ISPH) solver. For the solid vessel boundary, a new treatment method of calculating the velocity and acceleration of the regenerated particles near the curved hull is proposed. In addition, a high accuracy numerical scheme based on the simplified finite difference interpolation (SFDI) approach is used to solve the pressure Poisson equation in the ISPH framework. The robustness of the proposed 2D + t ISPH model is demonstrated through the benchmark tests of water entry and practical applications to the self-designed laboratory experiment. [ABSTRACT FROM AUTHOR]

The moving particle semi-implicit method is a meshless particle method for incompressible fluid and has proven useful in a wide variety of engineering applications of free-surface flows. Despite its wide applicability, the moving particle semi-implicit method has the defects of spurious unphysical pressure oscillation. Three various divergence approximation formulas, including basic divergence approximation formula, difference divergence approximation formula, and symmetric divergence approximation formula are proposed in this paper. The proposed three divergence approximation formulas are then applied for discretization of source term in pressure Poisson equation. Two numerical tests, including hydrostatic pressure problem and dam-breaking problem, are carried out to assess the performance of different formulas in enhancing and stabilizing the pressure calculation. The results demonstrate that the pressure calculated by basic divergence approximation formula and difference divergence approximation formula fluctuates severely. However, application of symmetric divergence approximation formula can result in a more accurate and stabilized pressure. [ABSTRACT FROM AUTHOR]

Solving the pressure Poisson equation dominates a large portion of computational time for incompressible fluid flow simulations. To solve the pressure Poisson equation efficiently, geometric multigrid methods are used directly or used as the preconditioner for the conjugate gradient (CG) method. Conventionally, the V-cycle multigrid method is widely employed, and little attention has been paid to other cycles. In this paper, we introduce the sawtooth cycle multigrid method and its simple extension called N-cycle, which provides better convergence in equal time comparison with the V-cycle as a direct solver of the pressure Poisson equation. We also apply the N-cycle to the preconditioner of the CG method and show that the N-cycle multigrid CG method can provide better convergence than the V-cycle multigrid CG method. [ABSTRACT FROM AUTHOR]