Pressure Poisson Solver Research Articles

Wavelet Potentials: An Efficient Potential Recovery Technique for Pointwise Incompressible Fluids

AbstractWe introduce an efficient technique for recovering the vector potential in wavelet space to simulate pointwise incompressible fluids. This technique ensures that fluid velocities remain divergence‐free at any point within the fluid domain and preserves local volume during the simulation. Divergence‐free wavelets are utilized to calculate the wavelet coefficients of the vector potential, resulting in a smooth vector potential with enhanced accuracy, even when the input velocities exhibit some degree of divergence. This enhanced accuracy eliminates the need for additional computational time to achieve a specific accuracy threshold, as fewer iterations are required for the pressure Poisson solver. Additionally, in 3D, since the wavelet transform is taken in‐place, only the memory for storing the vector potential is required. These two features make the method remarkably efficient for recovering vector potential for fluid simulation. Furthermore, the method can handle various boundary conditions during the wavelet transform, making it adaptable for simulating fluids with Neumann and Dirichlet boundary conditions. Our approach is highly parallelizable and features a time complexity of O(n), allowing for seamless deployment on GPUs and yielding remarkable computational efficiency. Experiments demonstrate that, taking into account the time consumed by the pressure Poisson solver, the method achieves an approximate 2x speedup on GPUs compared to state‐of‐the‐art vector potential recovery techniques while maintaining a precision level of 10−6 when single float precision is employed. The source code of ‘Wavelet Potentials’ can be found in https://github.com/yours321dog/WaveletPotentials.

A parallel unstructured multi-color SOR solver for 3D Navier–Stokes equations on graphics processing units

In this paper, we develop a GPU parallelization technique to accelerate the pressure Poisson solver. We apply it in three-dimensional Navier–Stokes simulations with arbitrary geometries. The numerical method is based on a projection method, and the discretization is obtained by combining a second-order unstructured finite-volume method and a σ transformation. The pressure solution is responsible for this formulation’s most time-consuming computation tasks. This is because the Poisson equation must be approximated. This work proposes a GPU-based Multi-Color Successive Over-Relaxation (MC-SOR) method to solve the resulting sparse linear system. The proposed parallel finite-volume code is verified and validated by solving the Poisson problem. We employ analytical solutions and well-known benchmark problems with complex flows. These include the Taylor–Green vortex problem and the problem of turbulent flow around a circular cylinder.

An implicit conformation tensor decoupling approach for viscoelastic flow simulation within the monolithic projection framework

The highly nonlinear nature of the system governing equations makes it difficult to simulate viscoelastic flows efficiently. In this paper, an implicit decoupling approach is proposed for the viscoelastic flow simulation with a monolithic projection method. The decoupling approach can be derived from the approximate block factorization at the matrix level, which has been successfully applied into the Newtonian flow simulations. In our work, we extend this approach to decouple the pressure, conformation tensor and velocity from the viscoelastic flow system sequentially. Firstly, the pressure-conformation tensor-velocity decoupling is realized by a two-step approximate factorization. By combining the conformation tensor and velocity together at the first-step factorization, the original efficient pressure Poisson solver for the Newtonian flow simulation can be directly adopted in the present framework for non-Newtonian flow simulation. Secondly, the conformation tensor components are further decoupled from each other by the present component-decoupling approach. Then all quantities, including pressure, conformation tensor and velocity, can be resolved without iteration. Additionally, the accuracy corrector is proposed to access the prior scheme limiter-estimation problem introduced by the implicit high-resolution scheme. We numerically demonstrate that all quantities preserve the second-order accuracy in time and space as expected by the theoretical splitting errors. Finally, several different types of canonical benchmark flow examples are conducted to validate the present solver, including laminar flow, turbulent drag-reducing flow and the rotation of a spherical particle immersed in a sheared viscoelastic fluid.

A formulation for high-fidelity simulations of pool boiling in low gravity

Direct numerical simulation of pool boiling under varying gravity environments was performed and compared with experimental data. The three dimensional solver is fully explicit in time and uses a fractional step method for Navier-Stokes, energy and mass transfer equations. A level-set technique is used to represent and advect the interface. Sharp jumps are applied to the diffusion terms using a ghost-fluid method and the advection terms are discretized using a third order weighted essentially non-oscillatory scheme. A multi-level, block-structured adaptive mesh refinement is used to selectively refine the grid around the interface and thermal boundary layer. The latter contributes towards accurate calculation of evaporative heat flux. The pressure Poisson solver uses Hypre to solve the linear system on the coarsest level and geometric multi-grid method to prolong-restrict the solution between successive refinements. Parallelization is achieved using MPI on distributed memory systems and a dynamic contact line model is implemented to account for inertial effects on vapor bubbles due to wall adhesion force. Validation cases are presented to test the accuracy of the solver in comparison to analytical and experimental results. Replicability is also demonstrated by comparing results with already published data. A simple mathematical model is used to account for nucleation site density and results of pool boiling simulations are compared with experimental results to illustrate the effects of subcooling and gravity.

Error propagation dynamics of PIV-based pressure field calculations: How well does the pressure Poisson solver perform inherently?

Obtaining pressure field data from particle image velocimetry (PIV) is an attractive technique in fluid dynamics due to its noninvasive nature. The application of this technique generally involves integrating the pressure gradient or solving the pressure Poisson equation using a velocity field measured with PIV. However, very little research has been done to investigate the dynamics of error propagation from PIV-based velocity measurements to the pressure field calculation. Rather than measure the error through experiment, we investigate the dynamics of the error propagation by examining the Poisson equation directly. We analytically quantify the error bound in the pressure field, and are able to illustrate the mathematical roots of why and how the Poisson equation based pressure calculation propagates error from the PIV data. The results show that the error depends on the shape and type of boundary conditions, the dimensions of the flow domain, and the flow type.

Pressure Decimation and Interpolation (PDI) method for a baroclinic non-hydrostatic model

Non-hydrostatic models are computationally expensive in simulating density flows and mass transport problems due to the requirement of sufficient grid resolution to resolve density and flow structures. Numerical tests based on the Non-Hydrostatic Wave Model, NHWAVE (Ma et al., 2012), indicated that up to 70% of the total computational cost may be born by the pressure Poisson solver in cases with high grid resolution in both vertical and horizontal directions. However, recent studies using Poisson solver-based non-hydrostatic models have shown that an accurate prediction of wave dispersion does not require a large number of vertical layers if the dynamic pressure is properly discretized. In this study, we explore the possibility that the solution for the dynamic pressure field may, in general, be decimated to a resolution far coarser than that used in representing velocities and other transported quantities, without sacrificing accuracy of solutions. Following van Reeuwijk (2002), we determine the dynamic pressure field by solving the Poisson equation on a coarser grid and then interpolate the pressure field onto a finer grid used for solving for the remaining dynamic variables. With the Pressure Decimation and Interpolation (PDI) method, computational efficiency is greatly improved. We use three test cases to demonstrate the model’s accuracy and efficiency in modeling density flows.

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.

Scalability studies and large grid computations for surface combatant using CFDShip-Iowa

Scalability studies and computations using the largest grids to date for free-surface flows are performed using message-passing interface (MPI)-based CFDShip-Iowa toolbox curvilinear (V4) and Cartesian (V6) grid solvers on Navy high-performance computing systems. Both solvers show good strong scalability up to 2048 processors, with V6 showing somewhat better performance than V4. V6 also outperforms V4 in terms of the memory requirements and central processing unit (CPU) time per time-step per grid point. The explicit solvers show better scalability than the implicit solvers, but the latter allows larger time-step sizes, resulting in a lower total CPU time. The multi-grid HYPRE solver shows better scalability than the portable, extensible toolkit for scientific computation solver. The main scalability bottleneck is identified to be the pressure Poisson solver. The memory bandwidth test suggests that further scalability improvements could be obtained by using hybrid MPI/open multi-processing (OpenMP) parallelization. V4-detached eddy simulation (DES) on a 300 M grid for the surface combatant model DTMB 5415 in the straight-ahead condition provides a plausible description of the vortical structures and mean flow patterns observed in the experiments. However, the vortex strengths are over predicted and the turbulence is not resolved. V4-DESs on up to 250 M grids for DTMB 5415 at 20° static drift angle significantly improve the forces and moment predictions compared to the coarse grid unsteady Reynolds averaged Navier–Stokes, due to the improved resolved turbulence predictions. The simulations provide detailed resolution of the free-surface and breaking pattern and vortical and turbulent structures, which will guide planned experiments. V6 simulations on up to 276 M grids for DTMB 5415 in the straight-ahead condition predict diffused vortical structures due to poor wall-layer predictions. This could be due to the limitations of the wall-function implementation for the immersed boundary method.

A parallel multilevel preconditioned iterative pressure Poisson solver for the large-eddy simulation of turbulent flow inside a duct

Turbulent Poiseuille flows inside the square duct are simulated by the large-eddy simulation based on the multilevel Schwarz preconditioned conjugate gradient pressure Poisson solver, which was developed on top of the Portable, Extensible Toolkit for Scientific Computation (PESTc). The impact of the five different matrix reordering techniques for an incomplete LU (ILU) decomposition as a subdomain solver on the overall performance of Schwarz-type preconditioners for the solution of the pressure Poisson equation are studied. The numerical results indicate that ILU of two-level fill-ins with the reverse Cuthill–McKee matrix ordering technique produces the best performance. Further investigation on the parallel performance of different multilevel methods was also conducted for two different problem sizes. It was observed that the computational cost saturates at around six-level for both the problem sizes explored. Also, though the one-level method is better for small problem size, for the larger problem size, the six-level method performs best in terms of scalability and compute time; hence, the benefit of a multilevel method is more obviously.

Fast numerical solutions of patient‐specific blood flows in 3D arterial systems

The study of hemodynamics in arterial models constructed from patient-specific medical images requires the solution of the incompressible flow equations in geometries characterized by complex branching tubular structures. The main challenge with this kind of geometries is that the convergence rate of the pressure Poisson solver is dominated by the graph depth of the computational grid. This paper presents a deflated preconditioned conjugate gradients (DPCG) algorithm for accelerating the pressure Poisson solver. A subspace deflation technique is used to approximate the lowest eigenvalues along tubular domains. This methodology was tested with an idealized cylindrical model and three patient-specific models of cerebral arteries and aneurysms constructed from medical images. For these cases, the number of iterations decreased by up to a factor of 16, while the total CPU time was reduced by up to 4 times when compared with the standard PCG solver.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.