Development of a Novel Mixing Plane Interface Using a Fully Implicit Averaging for Stage Analysis

This paper describes the development and validation steps of a characteristics-based explicit along with a novel fully implicit mixing plane implementation for turbomachinery applications. The framework is an unstructured 3D RANS in-house modified solver, based on open-source libraries. Particular attention was paid to mass-conservation, accurate variables interpolation, and algorithm stability in order to improve robustness and convergence. By introducing a specific interface, allowing the use of algebraic multigrid solvers together with multiprocessor computation, a speed up of the numerical solution procedure was achieved. The validation of both mixing plane algorithms is carried out on an industrial radial compressor and a cold air 1.5 stages axial turbine.

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Development of a Novel Mixing Plane Interface Using a Fully Implicit Averaging for Stage Analysis

Volume 6B: Turbomachinery ◽

10.1115/gt2013-94390 ◽

2013 ◽

Author(s):

Lucian Hanimann ◽

Luca Mangani ◽

Ernesto Casartelli ◽

Thomas Mokulys ◽

Sebastiano Mauri

Keyword(s):

Mass Conservation ◽

Solution Procedure ◽

Algebraic Multigrid ◽

Radial Compressor ◽

Fully Implicit ◽

Multigrid Solvers ◽

Algorithm Stability ◽

Speed Up ◽

Stage Analysis ◽

Development And Validation

This paper describes the development and validation steps of a characteristics-based explicit as well as a novel fully implicit mixing plane implementation for turbomachinery applications. The framework is an unstructured 3D RANS in-house modified solver, based on open-source libraries. Particular attention was paid to mass-conservation, accurate variables interpolation and algorithm stability in order to improve robustness and convergence. By introducing a specific interface, allowing the use of algebraic multigrid solvers together with multiprocessor computation, a speed up of the numerical solution procedure was achieved. The validation of both mixing plane algorithms is carried out on an industrial radial compressor and a cold air 1.5 stages axial turbine.

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A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽

10.3390/w12061639 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1639

Author(s):

Abdelkrim Aharmouch ◽

Brahim Amaziane ◽

Mustapha El Ossmani ◽

Khadija Talali

Keyword(s):

Finite Volume ◽

Coastal Aquifers ◽

Nonlinear Partial Differential Equations ◽

Time Integration ◽

Mass Conservation ◽

Coupled System ◽

Finite Volume Scheme ◽

Time Step ◽

Volume Scheme ◽

Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

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Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems

The International Journal of High Performance Computing Applications ◽

10.1177/10943420211001940 ◽

2021 ◽

Vol 35 (3) ◽

pp. 211-225

Author(s):

Jonas Dünnebacke ◽

Stefan Turek ◽

Christoph Lohmann ◽

Andriy Sokolov ◽

Peter Zajac

Keyword(s):

Krylov Subspace ◽

Solution Procedure ◽

Grid Size ◽

Waveform Relaxation ◽

Test Problems ◽

Subspace Method ◽

Multigrid Algorithm ◽

Multigrid Solvers ◽

Large Systems ◽

Linear Pdes

We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.

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A Numerical Algorithm for Solving Stiff ODEs and DAEs in Multibody Mechanics

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8239 ◽

1999 ◽

Author(s):

Hajrudin Pasic

Keyword(s):

Algebraic System ◽

Numerical Solutions ◽

Solution Procedure ◽

Differential Algebraic Equations ◽

Good Precision ◽

Algebraic Equations ◽

Constant Matrix ◽

Fully Implicit ◽

Collocation Points ◽

Collocation Technique

Abstract Presented is an algorithm suitable for numerical solutions of multibody mechanics problems. When s-stage fully implicit Runge-Kutta (RK) method is used to solve these problems described by a system of n ordinary differential equations (ODE), solution of the resulting algebraic system requires 2s3 n3 / 3 operations. In this paper we present an efficient algorithm, whose formulation differs from the traditional RK method. The procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. In terms of number of multiplications, the algorithm is about s2 / 2 times faster than the original, nondiagonalized system, as well as s2 times in terms of number of additions/multiplications. With s = 3 the method has the same precision and stability property as the well-known RADAU5 algorithm. However, our method is applicable with any s and not only to the explicit ODEs My′ = f(x, y), where M = constant matrix, but also to the general implicit ODEs of the form f (x, y, y′) = 0. In the solution procedure y is assumed to have a form of the algebraic polynomial whose coefficients are found by using the collocation technique. A proper choice of locations of collocation points guarantees good precision/stability properties. If constructed such as to be L-stable, the method may be used for solving differential-algebraic equations (DAEs). The application is illustrated by a constrained planar manipulator problem.

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A fully distributed unstructured Navier-Stokes solver for large-scale aeroelasticity computations

The Aeronautical Journal ◽

10.1017/s0001924000012392 ◽

2001 ◽

Vol 105 (1050) ◽

pp. 419-426 ◽

Cited By ~ 10

Author(s):

G. Barakos ◽

M. Vahdati ◽

A.I. Sayma ◽

C. Bréard ◽

M. Imregun

Keyword(s):

Large Scale ◽

Numerical Models ◽

Navier Stokes ◽

Multiple Data ◽

Computational Mesh ◽

Blade Row ◽

Scale Modelling ◽

Modelling Methodology ◽

Speed Up ◽

Development And Validation

Abstract This paper presents the development and validation of a parallel unsteady flow and aeroelasticity code for large-scale numerical models used in turbo machinery applications. The work is based on an existing unstructured Navier-Stokes solver developed over the past ten years by the Aeroelasticity Research Group at Imperial College Vibration University Technology Centre. The single-process multiple-data paradigm was adopted for the parallelisation of the solver and several validation cases were considered. The computational mesh was divided into several sub-sections using a domain decomposition technique. The performance and numerical accuracy of the parallel solver was validated across several computer platforms for various problem sizes. In cases where the solution could be obtained on a single CPU, the serial and parallel versions of the code were found to produce identical results. Studies on up to 32 CPUs showed varying levels of parallelisation efficiency, an almost linear speed-up being obtained in some cases. Finally, an industrial configuration, a 17 blade row turbine with a 47 million point mesh, was discussed to illustrate the potential of the proposed large-scale modelling methodology.

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Mathematical Modelling and Field Measurements of Nitrogen Oxides in Metropolitan Areas

Energy & Environment ◽

10.1177/0958305x9200300203 ◽

1992 ◽

Vol 3 (2) ◽

pp. 133-147

Author(s):

M.M. Elkotb ◽

O.M.F. Elbahar ◽

T.A. Abdou Ahmed ◽

T.W. Abou-Arab

Keyword(s):

Mathematical Model ◽

Wind Speed ◽

Three Dimensional ◽

Numerical Procedure ◽

Field Measurements ◽

Mass Conservation ◽

Eddy Diffusivity ◽

Solution Procedure ◽

Pollutant Emissions ◽

Motor Vehicles

A mathematical model for the prediction of pollutant emissions from motor vehicles is presented. The model is based on the numerical solution of the three-dimensional equation representing the mass conservation of dilute diffusing species. The variation of wind speed and eddy diffusivity with height is taken into consideration. The three-dimensional diffusion equation is solved numerically. The numerical procedure involves the discretization of the partial differential equation using the finite volume approach. The resulting set of discretization equation is solved iteratively using a fully implicit solution procedure. Furthermore, field measurements of the concentrations of nitrogen oxide in the downtown area of Cairo were conducted. For this purpose, a mobile air pollution laboratory fitted with gas analyzers, particulate matter sampler and equipment for the measurement of wind speed and direction has been used. This laboratory is also fitted with data recording and monitoring facility. The mathematical model is tested by comparing the computed pollutant concentrations with the experimental data obtained from the field measurements in the Cairo Metropolitan Area.

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Mixed-order interpolation for the Galerkin coarse-grid approximations in algebraic multigrid solvers

International Journal for Numerical Methods in Fluids ◽

10.1002/fld.2341 ◽

2010 ◽

Vol 67 (2) ◽

pp. 175-188 ◽

Cited By ~ 2

Author(s):

R. Webster

Keyword(s):

Coarse Grid ◽

Algebraic Multigrid ◽

Multigrid Solvers ◽

Mixed Order

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Validation of Depth-Averaged Flow Model Using Flat-Bottomed Benchmark Problems

The Scientific World JOURNAL ◽

10.1155/2014/197539 ◽

2014 ◽

Vol 2014 ◽

pp. 1-18 ◽

Cited By ~ 4

Author(s):

Il Won Seo ◽

Young Do Kim ◽

Chang Geun Song

Keyword(s):

Shallow Water ◽

Mass Conservation ◽

Analytic Solutions ◽

Benchmark Problems ◽

Roughness Coefficient ◽

Spanwise Direction ◽

Clear Trend ◽

Recirculating Flow ◽

Fully Implicit ◽

Algorithmic Approaches

In this study, a shallow water flow code was developed and tested against four benchmark problems of practical relevance. The results demonstrated that as the eddy viscosity increased, the velocity slope along the spanwise direction decreased, and the larger roughness coefficient induced a higher flow depth over the channel width. The mass conservation rate was determined to be 99.2%. This value was measured by the variation of the total volume of the fluid after a cylinder break. As the Re increased to 10,000 in the internal recirculating flow problem, the intensity of the primary vortex had a clear trend toward the theoretically infinite Re value of −1.886. The computed values of the supercritical flow evolved by the oblique hydraulic jump agreed well with the analytic solutions within an error bound of 0.2%. The present model adopts the nonconservative form of shallow water equations. These equations are weighted by the SU/PG scheme and integrated by a fully implicit method, which can reproduce physical problems with various properties. The model provides excellent results under various flow conditions, and the solutions of benchmark tests can present criteria for the evaluation of various algorithmic approaches.

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