A pressure-stabilized Lagrange-Galerkin method is implemented in a parallel domain decomposition system in this work, and the new stabilization strategy is proved to be effective for large Reynolds number and Rayleigh number simulations. The symmetry of the stiffness matrix enables the interface problems of the linear system to be solved by the preconditioned conjugate method, and an incomplete balanced domain preconditioner is applied to the flow-thermal coupled problems. The methodology shows good parallel efficiency and high numerical scalability, and the new solver is validated by comparing with exact solutions and available benchmark results. It occupies less memory than classical product-type solvers; furthermore, it is capable of solving problems of over 30 million degrees of freedom within one day on a PC cluster of 80 cores.
The Lagrange-Galerkin method raises wide concern about the finite-element simulation of fluid dynamics. Based on the approximation of the material derivative along the trajectory of fluid particle, the method is natural in the simulation to physical phenomena, and it is demonstrated to be unconditionally stable for a wide class of problems [
The present study is concentrated on improving the solvability of the Lagrange-Galerkin method on large scale and complex problems by domain decompositions. Piecewise linear interpolations are thus employed for velocity, pressure, and temperature; therefore, the so-called inf-sup condition [
The element searching algorithm in a domain decomposition system using unstructured grids is the second difficulty to implement the Lagrange-Galerkin method in a domain decomposition system (cf. [
The remainder of this paper is organized as follows: in Section
Let
The fluid is assumed to be incompressible according to Boussinesq approximation, and the density is assumed to be constant except in the gravity force term where it depends on temperature according to the indicated linear law; see (
Some preliminaries are arranged for the derivation of a finite element scheme of (
Let
Let
For the purpose of large scale computation, a piecewise equal-order interpolation for velocity and pressure is used, as can be seen from (
By adding (
Compute the particle’s coordinates by
Find
Find
Compute the relative error by a
As can be seen from Steps
To begin with the parallel domain decomposition method, the domain decomposition is introduced briefly as follows. The whole domain is decomposed into a number of
From (
The Lagrange-Galerkin method keeps the symmetry of the stiffness matrix, and the GLS pressure-stabilization term in (
The element searching algorithm requires a global-wise element information to determine the position of one particle in the previous time step. However, in the parallel domain decomposition system, the whole domain is split into several
In order to know the position of a particle at initialize: iterate If else if else break; return
A searching algorithm.
The request of the old solutions, which is the by scanning all the particles in the current All PEs check if there is any request to itself. If it exists, PEs will prepare an array of the needed data and send it. The current PE receives the data sent by other PEs.
Data transferred by MPI communication should be packaged properly to avoid the overflow of MPI buffer in case of large-scale computation. Nonblocking communication is employed, and as the 3 steps are performed subsequently, thus the computation time and communication time will be overlapped.
The parallel efficiency of new solver is firstly evaluated in this section, and to validate the scheme, exact solutions and available benchmark results classical computational models are compared. The CG convergence is judged by Euclidian norm with a tolerance of
The BDD serious preconditioners were employed in this work; they are very efficient, and their iteration numbers are about 1
The penalty methods are not consistent since the substitution of an exact solution into the discrete equations (
Convergence of (a) different constant
The parallel efficiency is assessed firstly by freezing the mesh size of test problem and refining the domain decomposition by decreasing the subdomain size and therefore increasing the number of subdomains; the comparison of the numerical scalability of the current scheme with and without the preconditioner is assessed by Figure
Numerical scalability.
Based on the paralyzed Lagrange-Galerkin method, the new solver makes a symmetric stiffness matrix, therefore only the lower/upper triangular matrix needs to be saved. Moreover, nonblocking MPI communication is used instead of constructing global arrays to keep the old solutions, and the current solver is expected to reduce the memory consumption without sacrificing the computation speed. The usage of time and memory of solving the thermal driven cavity problem by different solvers is compared, and the results are given by Figure
Time and memory usages.
The test problem was solved by the new solver and the ADV_sFlow 0.5 [
The parallel scalability of the searching algorithm is also a concern for us, as it characterizes the ability of an algorithm to deliver larger speed-up using a larger number of PEs. To know this, the number of
Numerical scalability.
In this section, a variety of test problems have been presented in order to prove the capability of the parallel Lagrange-Galerkin algorithm. Benchmarks test of Navier-Stokes problems are in Sections
The solver for Navier-Stokes equations in (
A plane Couette flow model.
An unstructured 3D mesh was generated by ADVENTURE_TetMesh [
Numerical results versus exact solutions.
It can be seen form Figure
The Navier-Stokes problems solver was then verified by a lid-driven cavity flow. The ideal gas flows over the upper face of the cube, and no-slip conditions are applied to all other faces, as in Figure
A lid-driven cavity model.
All the faces of the cube were set with Dirichlet boundary conditions, and a zero reference pressure was at the centre of the cube to keep the simulation stable. The pressure profiles of the scheme using localized stabilization parameter in (
Pressure counters (
Figure
The model was run at different Reynolds numbers with a
Velocity and pressure profiles for different Reynolds number:
The solver for Navier-Stokes equations was then tested with backward facing step, the fluid considered was air. The problem definition is shown in Figure
Backward facing steps.
A laminar flow is considered to enter the domain at inlet section, the inlet velocity profile is parabolic, and the Reynolds number is based on the average velocity at the inlet. The total length of the domain is 30 times the step height, so that the zero pressure is set at the outlet. A full 3D simulation of the step geometry for
To determine the reattachment length, the position of the zero-mean-velocity line was measured. The points of detachment and reattachment were taken as the extrapolated zero-velocity line down the wall. The pressure contour in Figure
Pressure counters (a) and velocity vectors (b) (
The comparison of primary reattachment length between current results and other available benchmark results are show in Figure
Primary reattachment lengths.
In order to test the coupled solver of Navier-Stokes equations and the convection-diffusion equation, the third application model was the natural convection between two infinite flat plates. The geometry is given in 3-dimensional by Figure
The model of infinite plates.
The model was run at the size of
Numerical results versus exact solutions.
With the parameter setting of
The new solver is also applied to a 3-dimensional nonlinear thermal driven cavity flow problem, which is cavity full of ideal gas; see Figure
A thermal-driven cavity.
No-slip boundary conditions are assumed to prevail on all the walls of the cavity. Both the horizontal walls are assumed to be thermally insulated, and the left and right sides are kept at different temperatures. The cube is divided into
Steady state of the thermal-driven cavity (
Figures
In order to further validate the new solver, a comparison of temperature and velocity profiles of the current solver and other benchmark results was made. The centreline velocity results
Centerline temperature velocity profiles of the symmetry plane (
Thermal convection problems are believed to be dominated by two dimensionless numbers by many researchers, the Prandtl number and the Rayleigh number. To acquaint ourselves with the solvability of the new solver and to challenge applications of higher difficulty, a wide range of Rayleigh numbers from
Local Nusselt number along the hot wall (a) and the cold wall (b).
The local Nusselt number (
The new solver enables the simulation of large scale problems, thus models of Rayleigh number up to
A pressure-stabilized Lagrange-Galerkin method is implemented in a domain decomposition system in this research. By using localized stabilization parameter, the new scheme shows better control in the pressure field than constant stabilization parameter; therefore it has good solvability at high Reynolds number and high Rayleigh number. The reliability and accuracy of the present numerical results are validated by comparing with the exact solutions and recognized numerical results. Based on a domain decomposition method, the element searching algorithm shows good numerical scalability and parallel efficiency. The new solver reduces the memory consumption and is faster than classical product-type solvers. It is able to solve large scale problems of over 30 million degrees of freedom within one day by a small PC cluster.
This work was supported by the National Science Foundation of China (NSFC), Grants 11202248, 91230114, and 11072272; the China Postdoctoral Science Foundation, Grant 2012M521646, and the Guangdong National Science Foundation, Grant S2012040007687.