Accurate Projection Method Research Articles

Accurate deterministic projection methods for stiff detonation waves

Accurate deterministic projection methods for stiff detonation waves

Inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500

This letter reports inlet flow disturbance effects on direct numerical simulation of incompressible round jet at Reynolds number 2500. The simulation employs an accurate projection method in which a sixth order biased upwind difference scheme is used for spatial discretization of nonlinear convective terms, with a fourth order central difference scheme used in the discretization of the divergence of intermediate velocity. Carefully identifying reveals that the inlet flow disturbance has some influences on the distribution pattern of mean factor of swirling strength intermittency. With the increase of inlet disturbance magnitude jet core cone slightly shortens, observable differences occur in the centerline velocity and its fluctuations, despite the negligible impacts on the least square fitted centerline velocity decay constant (Bu) and distribution parameter (Ku) for velocity profile in self-similar region.

Three novel accurate pixel-driven projection methods for 2D CT and 3D EPR imaging.

This work aims to explore more accurate pixel-driven projection methods for iterative image reconstructions in order to reduce high-frequency artifacts in the generated projection image. Three new pixel-driven projection methods namely, small-pixel-large-detector (SPLD), linear interpolation based (LIB) and distance anterpolation based (DAB), were proposed and applied to reconstruct images. The performance of these methods was evaluated in both two-dimensional (2D) computed tomography (CT) images via the modified FORBILD phantom and three-dimensional (3D) electron paramagnetic resonance (EPR) images via the 6-spheres phantom. Specifically, two evaluations based on projection generation and image reconstruction were performed. For projection generation, evaluation was using a 2D disc phantom, the modified FORBILD phantom and the 6-spheres phantom. For image reconstruction, evaluations were performed using the FORBILD and 6-spheres phantom. During evaluation, 2 quantitative indices of root-mean-square-error (RMSE) and contrast-to-noise-ratio (CNR) were used. Comparing to the use of ordinary pixel-driven projection method, RMSE of the SPLD based least-square algorithm was reduced from 0.0701 to 0.0384 and CNR was increased from 5.6 to 19.47 for 2D FORBILD phantom reconstruction. For 3D EPRI, RMSE of SPLD was also reduced from 0.0594 to 0.0498 and CNR was increased from 3.88 to 11.58. In addition, visual evaluation showed that images reconstructed in both 2D and 3D images suffered from high-frequency line-shape artifacts when using the ordinary pixel-driven projection method. However, using 3 new methods all suppressed the artifacts significantly and yielded more accurate reconstructions. Three proposed pixel-driven projection methods achieved more accurate iterative image reconstruction results. These new and more accurate methods can also be easily extended to other imaging modalities. Among them, SPLD method should be recommended to 3D and four dimensional (4D) EPR imaging.

Moving immersed boundary method

SummaryA novel implicit immersed boundary method of high accuracy and efficiency is presented for the simulation of incompressible viscous flow over complex stationary or moving solid boundaries. A boundary force is often introduced in many immersed boundary methods to mimic the presence of solid boundary, such that the overall simulation can be performed on a simple Cartesian grid. The current method inherits this idea and considers the boundary force as a Lagrange multiplier to enforce the no‐slip constraint at the solid boundary, instead of applying constitutional relations for rigid bodies. Hence excessive constraint on the time step is circumvented, and the time step only depends on the discretization of fluid Navier‐Stokes equations, like the CFL condition in present work. To determine the boundary force, an additional moving force equation is derived. The dimension of this derived system is proportional to the number of Lagrangian points describing the solid boundaries, which makes the method very suitable for moving boundary problems since the time for matrix update and system solving is not significant. The force coefficient matrix is made symmetric and positive definite so that the conjugate gradient method can solve the system quickly. The proposed immersed boundary method is incorporated into the fluid solver with a second‐order accurate projection method as a plug‐in. The overall scheme is handled under an efficient fractional step framework, namely, prediction, forcing, and projection. Various simulations are performed to validate current method, and the results compare well with previous experimental and numerical studies.

A HEAVISIDE-FUNCTION APPROACH FOR THE INTERACTION OF TWO-PHASE FLUID AND NON-DEFORMABLE SOLID

We introduce a Heaviside-function formulation of the interaction between incompressible two-phase fluid and a non-deformable solid. Fluid and solid interact in two ways : fluid satises the Dirichlet boundary condition imposed by the velocity field of solid, and solid is accelerated by the surface traction exerted by fluid. The two-way couplings are formulated by the Heaviside function to the interface between solid and fluid. The cumbersome treatment of interface is taken care of by the Heaviside function, and the interaction is discretized in a simple manner. The discretization results in a stable and accurate projection method.

Study on Mechanical Properties of the Composite Materials Film by Speckle Projection Method

An improved accurate speckle projection method is used for study the mechanical properties of the composite material film in the paper. A system for deformation measurement is developed with the telecentric lenses, in which such conventional lens’ disadvantages such as lens distortion and perspective error will be diminished. Experiments are performed to validate the availability and reliability of the calibration method. The system can also be used to measure the dynamic deformation and then results are also given.

Renormalization of tensor-network states

We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network states/models in two dimensions. A second renormalization scheme is introduced to take into account the environment contribution in the calculation of the partition function of classical tensor network models or the expectation values of quantum tensor network states. It improves significantly the accuracy of the coarse grained tensor renormalization group method. In the study of the quantum tensor-network states, we point out that the renormalization effect of the environment can be efficiently and accurately described by the bond vector. This, combined with the imaginary time evolution of the wavefunction, provides an accurate projection method to determine the tensor-network wavfunction. It reduces significantly the truncation error and enable a tensor-network state with a large bond dimension, which is difficult to be accessed by other methods, to be accurately determined.

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two- and three-dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two-level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two-level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two- and three-dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two- to three-dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non-symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

Stable and accurate pressure approximation for unsteady incompressible viscous flow

How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier-Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C^0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure.

A Novel Semi-Explicit Spatially Fourth Order Accurate Projection Method for Unsteady Incompressible Viscous Flows

This article describes a simple and elegant compact higher order finite-difference based numerical solution technique to the primitive variable formulation of unsteady incompressible Navier Stokes equations (UINSE) on staggered grids. The method exploits the advantages of the D'yakanov ADI-like scheme and a non-iterative pressure correction based fractional step method. Spatial derivatives are discretized to fourth order accuracy and the time integration is realized through the Euler explicit method. The fast and efficient iterative solution to the discretized momentum and pressure Poisson equations is achieved using a variant of conjugate gradient method. Spatial accuracy and robustness of the solver are tested through its application to relevant benchmark problems.

Accurate Projection Method of Source Quantities in Coupled Finite-Element Problems

Projecting discrete quantities from one mesh to another one is a critical operation which is usually required in multiphysical problems. The efficiency of a Galerkin projection method is studied for magnetothermal and magnetoelasticity couplings. As examples of application, two electromagnetic (EM) devices are studied: a three-phase busbar system as a magnetothermal problem and an electromagnet as a magnetomechanical problem.

A level set‐based immersed interface method for solving incompressible viscous flows with the prescribed velocity at the boundary

AbstractA second‐order accurate immersed interface method (IIM) is presented for solving the incompressible Navier–Stokes equations with the prescribed velocity at the boundary, which is an extension of the IIM of Le et al. (J. Comput. Phys. 2006; 220:109–138) to a level set representation of the boundary in place of the Lagrangian representation of the boundary using control points on a uniform Cartesian grid. In order to enforce the prescribed velocity boundary condition, the singular forces at the immersed boundary are applied on the fluid. These forces are related to the jump in pressure and the jumps in the derivatives of both the pressure and velocity, and are approximated via using the local Hermite cubic spline interpolation. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier–Stokes equations are discretized via using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by a second‐order accurate projection method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate the viscous flows in irregular domains. Copyright © 2009 John Wiley & Sons, Ltd.

An Immersed Interface Method for the Incompressible Navier–Stokes Equations with Discontinuous Viscosity Across the Interface

We present an immersed interface method for solving the incompressible Navier–Stokes equations with discontinuous viscosity across the interface and singular forces. The method is based on the augmented strategy proposed by Li, Ito, and Lai [Comput. Fluids, 36 (2007), pp. 622–635] to decouple the jump conditions of the fluid variables through the introduction of two augmented variables. In the proposed method, the immersed interface is represented by a number of Lagrangian control points, and the augmented interface variables applicable along the interface are solved by the LU method or the GMRES iterative method. In addition, we also derive a new second order accurate bilinear interpolating scheme for the discontinuous velocity and a new jump condition for the normal derivative of the pressure. The jumps in both pressure and velocity and the jumps in their derivatives are related to the augmented interface variables and/or the forces which are interpolated using cubic splines. For a flexible interface, the forces that the interface exerts on the fluid are computed from the constitutive relation of the flexible interface and are applied to the fluid through the jump conditions. The position of the flexible interface is updated implicitly within each time step. The Navier–Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method. The numerical results show that the overall scheme is second order accurate.

Closed-form error estimates for the numerical solution of Fredholm integral equations of the second kind

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel orthogonalpolynomial approximation of the exact kernel. It is proved that, under suitable conditions, the discrepancy between our theoretically predicted error and the actual numerical error tends to zero at the rate ‖K − KM‖, where M is the rank of the degenerate-kernel approximation. A leading-order error analysis is validated on three increasingly accurate projection methods applied to both smooth and sharply peaked kernels, our error predictions being demonstrated to be exponentially convergent to experimentally obtained global numerical errors. The new method is demonstrated to be cost-effective relative to standard extrapolation.

An immersed interface method for solving incompressible viscous flows with piecewise constant viscosity across a moving elastic membrane

This paper presents an implementation of the second-order accurate immersed interface method to simulate the motion of the flexible elastic membrane immersed in two viscous incompressible fluids with different viscosities, which further develops the work reported in Tan et al. [Z.-J. Tan, D.V. Le, K.M. Lim, B.C. Khoo, An Immersed Interface Method for the Incompressible Navier–Stokes Equations with Discontinuous Viscosity Across the Interface, submitted for publication] focussing mainly on the fixed interface problems. In this work, we introduce the velocity components at the membrane as two augmented unknown interface variables to decouple the originally coupled jump conditions for the velocity and pressure. Three forms of augmented equation are derived to determine the augmented variables to satisfy the continuous condition of the velocity. The velocity at the membrane, which determine the motion of the membrane, is then solved by the GMRES iterative method. The forces calculated from the configuration of the flexible elastic membrane and the augmented variables are interpolated using cubic splines and applied to the fluid through the jump conditions. The position of the flexible elastic membrane is updated implicitly using a quasi-Newton method (BFGS) within each time step. The Navier–Stokes equations are solved on a staggered Cartesian grid using a second order accurate projection method with the incorporation of spatial and temporal jump conditions. In addition, we also show that the inclusion of the temporal jump contributions has non-negligible effect on the simulation results when the grids are crossed by the membrane. Using the above method, we assess the effect of different viscosities on the flow solution and membrane motion.

Coupling of Evaporation and Thermocapillary Convection in a Liquid Layer with Mass and Heat Exchanging Interface

We propose and analyse a new model of thermocapillary convection with evaporation in a cavity subjected to horizontal temperature gradient, rather than the previously studied model without evaporation. The pure liquid layer with a top free surface in contact with its own vapour is considered in microgravity condition. The computing programme developed for simulating this model integrates the two-dimensional, time-dependent Navier–Stokes equations and energy equation by a second-order accurate projection method. We focus on the coupling of evaporation and thermocapillary convection by investigating the influence of evaporation Biot number and Marangoni number on the interfacial mass and heat transfer. Three different regimes of the coupling mechanisms are found and explained from our numerical results.

A bridge between projection methods and SIMPLE type methods for incompressible Navier–Stokes equations

AbstractA bridge is built between projection methods and SIMPLE type methods (Semi‐Implicit Method for Pressure‐Linked Equation). A general second‐order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non‐linear update of pressure term as Θn∇pn+1+(I−Θn)∇pn, where Θn is a coefficient matrix, which may depend on the grid size, time step and even velocity. It includes three‐ and four‐step projection methods. The standard SIMPLE method is written in a concise formula for steady and unsteady flows. It is proven that SIMPLE type methods have second‐order temporal accuracy for unsteady flows. The classical second‐order projection method and SIMPLE type methods are united within the framework of the general second‐order projection formula. Two iteration algorithms of SIMPLE type methods for unsteady flows are described and discussed. In addition, detailed formulae are provided for general projection methods by using the Runge–Kutta technique to update the convective term and Crank–Nicholson scheme for the diffusion term. Copyright © 2007 John Wiley & Sons, Ltd.

A non‐diffusive, divergence‐free, finite volume‐based double projection method on non‐staggered grids

AbstractSecond‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient.This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52(4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.

A second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive grids

We present an unconditionally stable second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive Cartesian grids. We employ quadtree and octree data structures as an efficient means to represent the grid. We use the supra-convergent Poisson solver of [C.-H. Min, F. Gibou, H. Ceniceros, A supra-convergent finite difference scheme for the variable coefficient Poisson equation on fully adaptive grids, CAM report 05-29, J. Comput. Phys. (in press)], a second order accurate semi-Lagrangian method to update the momentum equation, an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non-graded grids. We sample all the variables at the grid nodes, producing a scheme that is straightforward to implement. We propose two and three-dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the L1 and L∞ norms.

An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries

We present an immersed interface method for the incompressible Navier–Stokes equations capable of handling both rigid and flexible boundaries. The immersed boundaries are represented by a number of Lagrangian control points. In order to ensure that the no-slip condition on the rigid boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep. For flexible boundaries, the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions. The position of the flexible boundary is updated implicitly using a quasi-Newton method (BFGS) within each timestep. The Navier–Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate.