Full Approximation Storage Research Articles

Multilevel Techniques for the Solution of HJB Minimum-Time Control Problems

The solution of minimum-time feedback optimal control problems is generally achieved using the dynamic programming approach, in which the value function must be computed on numerical grids with a very large number of points. Classical numerical strategies, such as value iteration (VI) or policy iteration (PI) methods, become very inefficient if the number of grid points is large. This is a strong limitation to their use in real-world applications. To address this problem, the authors present a novel multilevel framework, where classical VI and PI are embedded in a full-approximation storage (FAS) scheme. In fact, the authors will show that VI and PI have excellent smoothing properties, a fact that makes them very suitable for use in multilevel frameworks. Moreover, a new smoother is developed by accelerating VI using Anderson’s extrapolation technique. The effectiveness of our new scheme is demonstrated by several numerical experiments.

Numerical simulations of capsule deformation using a dual time-stepping lattice Boltzmann method.

In this work a quasisteady, dual time-stepping lattice Boltzmann method is proposed for simulation of capsule deformation. At each time step the steady-state lattice Boltzmann equation is solved using the full approximation storage multigrid scheme for nonlinear equations. The capsule membrane is modeled as an infinitely thin shell suspended in an ambient fluid domain with the fluid structure interaction computed using the immersed boundary method. A finite element method is used to compute the elastic forces exerted by the capsule membrane. Results for a wide range of parameters and initial configurations are presented. The proposed method is found to reduce the computational time by a factor of ten.

Convergence analysis of the Fast Subspace Descent method for convex optimization problems

Author(s): Chen, L; Hu, X; Wise, SM | Abstract: The full approximation storage (FAS) scheme is a widely used multigrid method for nonlinear problems. In this paper, a new framework to design and analyze FAS-like schemes for convex optimization problems is developed. The new method, the fast subspace descent (FASD) scheme, which generalizes classical FAS, can be recast as an inexact version of nonlinear multigrid methods based on space decomposition and subspace correction. The local problem in each subspace can be simplified to be linear and one gradient descent iteration (with an appropriate step size) is enough to ensure a global linear (geometric) convergence of FASD for convex optimization problems.

An efficient nonlinear multigrid scheme for 2D boundary value problems

In this article, a two-dimensional nonlinear boundary value problem which is strongly related to the well-known Gelfand–Bratu model is solved numerically. The numerical results are obtained by employing three different numerical strategies namely: finite difference based method, a Newton multigrid method and a nonlinear multigrid full approximation storage (FAS). We are able to handle the difficulty of unstable convergence behaviour by using MINRES method as a relaxation smoother in multigrid approach with an appropriate sinusoidal approximation as an initial guess. A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate an improvement for the values of λ ∈ (0, λc]. Numerical results illustrate the performance of the proposed numerical methods wherein FAS-MG method is shown to be the most efficient. Further, we present the numerical bifurcation behaviour for two-dimensional Gelfand-Bratu models and find new multiplicity of solutions in the case of a quadratic and cubic approximation of the nonlinear exponential term. Numerical experiments confirm the convergence of the solutions for different values of λ and prove the effectiveness of the nonlinear FAS-MG scheme.

GPU parallelization of multigrid RANS solver for three-dimensional aerodynamic simulations on multiblock grids

In this paper, graphical processing units (GPUs) are leveraged to accelerate Bombardier’s full aircraft Navier–Stokes solver, a finite-volume, cell-centered RANS flow solver for multiblock structured grids. The efficiency of different parallel smoothers on GPUs is studied, in the context of solving the RANS equations with a nonlinear full approximation storage multigrid scheme. Many variants of parallel red–black Gauss–Seidel and Jacobi solvers are investigated and their efficiency compared against sequential algorithms such as the lower–upper symmetric Gauss–Seidel solver on both CPUs and GPUs. Parametric studies on three-dimensional aircraft configurations are performed to identify the optimal smoothers and determine the optimal number of smoothing iterations on each multigrid level. Furthermore, the efficiency of different approaches to overlapping the communication and computation for the MPI-CUDA implementation in the multi-GPU code is investigated. The results show that the best runtime with the GPU code is obtained using a weaker smoother with more sweeps per multigrid level, whereas the best runtime with the CPU code is obtained using a stronger smoother with fewer sweeps per multigrid level. Despite using a weaker smoother and therefore more iterations to converge to the final solutions, the GPU-accelerated code is significantly faster than the CPU code.

Time integration of unsteady nonhydrostatic equations with dual time stepping and multigrid methods

A time integration method based on dual time stepping coupled with a multigrid and implicit iterative solver is discussed for overcoming the temporal stiffness associated with modeling unsteady atmospheric flows. Adopting the dual time stepping allows solving an unsteady flow problem as a steady-state one by introducing a pseudo-time derivative into the governing equations. Consequently, the model uses a large time step and leads to a reduction in computational time. In the framework of the dual time stepping, a full approximation storage multigrid and lower–upper symmetric Gauss–Seidel implicit solver are employed to accelerate convergence in a structured, quadrilateral grid. The discretization of a physical-time derivative is based on a second-order backward differentiation formula or a family of singly diagonally implicit Runge–Kutta methods. The spatial terms of nonhydrostatic equations are discretized with a second-order Roe scheme in the finite volume method. To assess the performance of the model in terms of accuracy and efficiency, which are indicated by the normalized error norms and wall-clock time, respectively, three idealized tests are performed with the unsteady flow problems of isentropic vortex, rising thermal bubble and inertia-gravity waves. The results of numerical experiments consistently demonstrate that the dual time stepping coupled with explicit-first, singly diagonally implicit Runge–Kutta and multigrid methods is more efficient than the three-stage, third-order explicit Runge–Kutta method with comparable accuracy. The numerical experiment also indicates that the dual time stepping is more efficient when a stretched grid is employed. The efficiency in the framework of the dual time stepping is dependent on various numerical factors, including a multigrid level and cycle, Butcher coefficients, an implicit solver and a physical-time step.

A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids

We present a mass-conservative full approximation storage (FAS) multigrid solver for cell-centered finite difference methods on block-structured, locally cartesian grids. The algorithm is essentially a standard adaptive FAS (AFAS) scheme, but with a simple modification that comes in the form of a mass-conservative correction to the coarse-level force. This correction is facilitated by the creation of a zombie variable, analogous to a ghost variable, but defined on the coarse grid and lying under the fine grid refinement patch. We show that a number of different types of fine-level ghost cell interpolation strategies could be used in our framework, including low-order linear interpolation. In our approach, the smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions at the coarse-fine interface. To maintain global mass conservation, we need only modify the usual FAS algorithm by correcting the coarse-level force function at points adjacent to the coarse-fine interface. We demonstrate through simulations that the solver converges geometrically, at a rate that is h-independent, and we show the generality of the solver, applying it to several nonlinear, time-dependent, and multi-dimensional problems. In several tests, we show that second-order asymptotic (h→0) convergence is observed for the discretizations, provided that (1) at least linear interpolation of the ghost variables is employed, and (2) the mass conservation corrections are applied to the coarse-level force term.

Robust Navier-Stokes method for predicting unsteady flowfield and aerodynamic characteristics of helicopter rotor

A robust unsteady rotor flowfield solver CLORNS code is established to predict the complex unsteady aerodynamic characteristics of rotor flowfield. In order to handle the difficult problem about grid generation around rotor with complex aerodynamic shape in this CFD code, a parameterized grid generated method is established, and the moving-embedded grids are constructed by several proposed universal methods. In this work, the unsteady Reynolds-Averaged Navier-Stokes (RANS) equations with Spalart-Allmaras are selected as the governing equations to predict the unsteady flowfield of helicopter rotor. The discretization of convective fluxes is accomplished by employing the second-order central difference scheme, third-order MUSCL-Roe scheme, and fifth-order WENO-Roe scheme. Aimed at simulating the unsteady aerodynamic characteristics of helicopter rotor, the dual-time scheme with implicit LU-SGS scheme is employed to accomplish the temporal discretization. In order to improve the computational efficiency of hole-cells and donor elements searching of the moving-embedded grid technology, the “disturbance diffraction method” and “minimum distance scheme of donor elements method” are established in this work. To improve the computational efficiency, Message Passing Interface (MPI) parallel method based on subdivision of grid, local preconditioning method and Full Approximation Storage (FAS) multi-grid method are combined in this code. By comparison of the numerical results simulated by CLORNS code with test data, it is illustrated that the present code could simulate the aerodynamic loads and aerodynamic noise characteristics of helicopter rotor accurately.

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

We introduce a family of implementations of low-order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers, and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based on the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free. Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realize full approximation storage (FAS) within the additive environment where, amortized, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realize a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as an enabler for full multigrid (FMG) cycling—the grid unfolds throughout the computation—allows us to reduce the cost per unknown. The present work primary contributes toward software realization and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR, and vectorization-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

Free Surface Thin Film Flow of a Sisko’s Fluid over a Surface Topography

The flow of a thin film down an inclined surface over topography is considered for the case of liquids with Sisko’s model viscosity. For the first time lubrication theory is used to reduce the governing equations to a non-linear evolution equation for a current of a Sisko’s model non-Newtonian fluid on an inclined plane under the action of gravity and the viscous stresses. This model is solved numerically using an efficient Full Approximation Storage (FAS) multigrid algorithm. Free surface results are plotted and carefully examined near the topography for different values of power-law index np, viscosity parameter m, the aspect ratio A and for different inclination angle of the plane with the horizontal. Number of complications and additional physical effects are discussed that enrich real situations. It is observed that the flows into narrow trenches develop a capillary ridge just in front of the upstream edge of a trench followed by a small trough. For relatively small width trenches, the free surface is almost everywhere flat as the dimensional width of the trench is much smaller than the capillary length scale. In this region, surface tension dominates the solution and acts so as to stretch a membrane across the trench leading to smaller height deviations. The ridge originates from the topographic forcing which works to force fluid upstream immediately prior to the trench before helping to accelerate it over. The upstream forcing slows down the fluid locally and increases the layer thickness.

A multigrid scheme for 3D Monge–Ampère equations

ABSTRACTThe elliptic Monge–Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three-dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation storage multigrid method. Starting from a second-order accurate centred finite difference approximation, we present a nonlinear Gauss–Seidel iterative method which has a mechanism for selecting the convex solution of the equation. The iterative method is used as an effective smoother, combined with the full-approximation storage multigrid method. Numerical experiments are provided to validate the accuracy of the finite difference scheme and illustrate the computational efficiency of the proposed multigrid solver.

Multi-component Cahn–Hilliard system with different boundary conditions in complex domains

We propose an efficient phase-field model for multi-component Cahn–Hilliard (CH) systems in complex domains. The original multi-component Cahn–Hilliard system with a fixed phase is modified in order to make it suitable for complex domains in the Cartesian grid, along with contact angle or no mass flow boundary conditions on the complex boundaries. The proposed method uses a practically unconditionally gradient stable nonlinear splitting numerical scheme. Further, a nonlinear full approximation storage multigrid algorithm is used for solving semi-implicit formulations of the multi-component CH system, incorporated with an adaptive mesh refinement technique. The robustness of the proposed method is validated through various numerical simulations including multi-phase separations via spinodal decomposition, equilibrium contact angle problems, and multi-phase flows with a background velocity field in complex domains.

A positivity-preserving, implicit defect-correction multigrid method for turbulent combustion

A novel, robust multigrid method for the simulation of turbulent and chemically reacting flows is developed. A survey of previous attempts at implementing multigrid for the problems at hand indicated extensive use of artificial stabilization to overcome numerical instability arising from non-linearity of turbulence and chemistry model source-terms, small-scale physics of combustion, and loss of positivity. These issues are addressed in the current work. The highly stiff Reynolds-averaged Navier-Stokes (RANS) equations, coupled with turbulence and finite-rate chemical kinetics models, are integrated in time using the unconditionally positive-convergent (UPC) implicit method. The scheme is successfully extended in this work for use with chemical kinetics models, in a fully-coupled multigrid (FC-MG) framework.To tackle the degraded performance of multigrid methods for chemically reacting flows, two major modifications are introduced with respect to the basic, Full Approximation Storage (FAS) approach. First, a novel prolongation operator that is based on logarithmic variables is proposed to prevent loss of positivity due to coarse-grid corrections. Together with the extended UPC implicit scheme, the positivity-preserving prolongation operator guarantees unconditional positivity of turbulence quantities and species mass fractions throughout the multigrid cycle. Second, to improve the coarse-grid-correction obtained in localized regions of high chemical activity, a modified defect correction procedure is devised, and successfully applied for the first time to simulate turbulent, combusting flows.The proposed modifications to the standard multigrid algorithm create a well-rounded and robust numerical method that provides accelerated convergence, while unconditionally preserving the positivity of model equation variables. Numerical simulations of various flows involving premixed combustion demonstrate that the proposed MG method increases the efficiency by a factor of up to eight times with respect to an equivalent single-grid method, and by two times with respect to an artificially-stabilized MG method.

A Unified Fractional-Step, Artificial Compressibility and Pressure-Projection Formulation for Solving the Incompressible Navier-Stokes Equations

AbstractThis paper introduces a unified concept and algorithm for the fractional-step (FS), artificial compressibility (AC) and pressure-projection (PP) methods for solving the incompressible Navier-Stokes equations. The proposed FSAC-PP approach falls into the group of pseudo-time splitting high-resolution methods incorporating the characteristics-based (CB) Godunov-type treatment of convective terms with PP methods. Due to the fact that the CB Godunov-type methods are applicable directly to the hyperbolic AC formulation and not to the elliptical FS-PP (split) methods, thus the straightforward coupling of CB Godunov-type schemes with PP methods is not possible. Therefore, the proposed FSAC-PP approach unifies the fully-explicit AC and semi-implicit FS-PP methods of Chorin including a PP step in the dual-time stepping procedure to a) overcome the numerical stiffness of the classical AC approach at (very) low and moderate Reynolds numbers, b) incorporate the accuracy and convergence properties of CB Godunov-type schemes with PP methods, and c) further improve the stability and efficiency of the AC method for steady and unsteady flow problems. The FSAC-PPmethod has also been coupled with a non-linear, full-multigrid and fullapproximation storage (FMG-FAS) technique to further increase the efficiency of the solution. For validating the proposed FSAC-PP method, computational examples are presented for benchmark problems. The overall results show that the unified FSAC-PP approach is an efficient algorithm for solving incompressible flow problems.

Preconditioned Conjugate Gradient and Multigrid Methods for Numerical Solution of Multicomponent Mass Transfer Equations II. Convection-Diffusion-Reaction Equations

This work continues our previous analysis concerning the performances of the nonlinear multigrid (the Full Approximation Storage algorithm) method and modified Picard preconditioned conjugate gradient methods for the numerical solution of the two-dimensional, steady-state, multicomponent mass transfer equations. The present test problems are steady-state, linear and nonlinear, convection-diffusion-reaction equations. The upwind finite difference method was used to discretize the mathematical model equations. The numerical results obtained show good numerical performances.

Preconditioned Conjugate Gradient and Multigrid Methods for Numerical Solution of Multicomponent Mass Transfer Equations I. Diffusion-Reaction Equations

In a series of two papers, we analyze the numerical performances of the preconditioned conjugate gradient algorithms (BICGSTAB and GMRES) and multigrid method in solving two-dimensional multicomponent mass transfer equations. The present test problems are steady-state, linear and nonlinear diffusion-reaction equations. The nonlinear algorithm used in connection with the preconditioned conjugate gradient methods is the modified Picard iteration. Theoretical results about the preconditioning are presented. The multigrid algorithm used is the Full Approximation Storage algorithm. The finite difference method was used to discretize the mathematical model equations. The numerical results obtained show good numerical performances.

Limitation of parallel flow in double diffusive convection: Two- and three-dimensional transitions in a horizontal porous domain

Two-dimensional (2D) and three-dimensional (3D) numerical simulations of double diffusion natural convection in an elongated enclosure filled with a binary fluid saturating a porous medium are carried out in the present work. The Boussinesq approximation is made in the formulation of the problem, and Neumann boundary conditions for temperature and concentration are adopted, respectively, on vertical and horizontal walls of the cavity. The used numerical method is based on the control volume approach, with the third order quadratic upstream interpolation scheme in approximating the advection terms. A semi implicit method algorithm is used to handle the velocity-pressure coupling. To avoid the excessively high computer time inherent to the solution of 3D natural convection problems, full approximation storage with full multigrid method is used to solve the problem. A wide range of the controlling parameters (Rayleigh-Darcy number Ra, lateral aspect ratio Ay, Lewis number Le, and the buoyancy ration N) is investigated. We clearly show that increasing the depth of the cavity (i.e., the lateral aspect ratio) has an important effect on the flow patterns. The 2D perfect parallel flows obtained for small lateral aspect ratio are drastically destabilized by increasing the cavity lateral dimension. This yields a 3D fluid motion with a much more complex flow pattern and the usually considered 2D parallel flow model cannot be applied.

3D Simulation of Flow in an Aeration Tank with Two Pipelines by a Two-Fluid Model

In this paper, a numerical two-fluid flow model combining with the Realizable k–ε turbulent model for compressible viscous fluid is presented for the computation of flow characteristics in an aeration tank; and the equations are solved with the finite volume method. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence. The numerical results for velocity and turbulent kinetic energy distribution in the aeration tank are obtained. It is shown that the Computational Fluid Dynamics (CFD) is a valuable tool to analyze the interaction of flow field and aeration.

A Two-Fluid Model for Water Drop Falling

In this paper, a numerical two-phase flow model combining with the Realizable k–ε turbulent model for compressible viscous fluid is presented for the computation of water drop falling characteristics; and the equations are solved with the finite volume method. The free fluid surface is simulated by the VOF method. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence. The numerical results for water drop suggest that this basic model can be used to study violent aerated flows, especially by providing fast qualitative estimates.

3D Numerical Simulation of Hydraulics in an Oxidation Ditch

A numerical simulation study of the hydrodynamics of an oxidation ditch is presented. The numerical method is based on a pressure-correction algorithm of the SIMPLE-type. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence, while as a turbulence model the RNG κ-ε model with wall functions is used. The numerical results for velocity in the oxidation ditch are obtained．