104 research outputs found
Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids
Implicit methods based on the Newton’s rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynolds-averaged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of three-dimensional test cases
Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and Fluid-Structure Interaction Problems
Efficient methods for the approximation of solutions to incompressible fluid flow and fluid-structure interaction problems are presented.
In particular, partial differential equations (PDEs) are derived from basic conservation principles.
First, the incompressible Navier-Stokes equations for Newtonian fluids are introduced.
This is followed by a consideration of solid mechanical problems.
Both, the fluid equations and the equation for solid problems are then coupled and a fluid-structure interaction problem is constructed.
Furthermore, a discretization by the finite element method for weak formulations of these problems is described.
This spatial discretization of variables is followed by a discretization of the remaining time-dependent parts.
An implementation of the discretizations and problems in a parallel C++ software environment is described.
This implementation is based on the software package Trilinos.
The parallel execution of a program is the essence of High Performance Computing (HPC).
HPC clusters are, in general, machines with several tens of thousands of cores. The fastest current machine, as of the TOP500 list from November 2019, has over 2.4 million cores, while the largest machine possesses over 10 million cores.
To achieve sufficient accuracy of the approximate solutions, a fine spatial discretization must be used.
In particular, fine spatial discretizations lead to systems with large sparse matrices that have to be solved.
Iterative preconditioned Krylov methods are among the most widely used and efficient solution strategies for these systems.
Robust and efficient preconditioners which possess good scaling behavior for a parallel execution on several thousand cores are the main component.
In this thesis, the focus is on parallel algebraic preconditioners for fluid and fluid-structure interaction problems.
Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes problems are presented.
Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on block factorizations.
In order to obtain numerically scalable algorithms, coarse spaces obtained from the Generalized Dryja-Smith-Widlund (GDSW) and the Reduced dimension GDSW (RGDSW) approach are used.
These coarse spaces can be constructed in an essentially algebraic way.
Numerical results of the parallel implementation are presented for various incompressible fluid flow problems.
Good scalability for up to 11 979 MPI ranks, which
corresponds to the largest problem configuration fitting on the employed supercomputer, were achieved.
A comparison of these monolithic approaches and commonly used block preconditioners with respect to time-to-solution is made.
Similarly, the most efficient construction of two-level overlapping Schwarz preconditioners with GDSW and RGDSW coarse spaces for solid problems is reported.
These techniques are then combined to efficiently solve fully coupled monolithic fluid-strucuture interaction problems
Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks
Computational fluid dynamics (CFD) simulations of viscous fluids described by
the Navier-Stokes equations are considered. Depending on the Reynolds number of
the flow, the Navier-Stokes equations may exhibit a highly nonlinear behavior.
The system of nonlinear equations resulting from the discretization of the
Navier-Stokes equations can be solved using nonlinear iteration methods, such
as Newton's method. However, fast quadratic convergence is typically only
obtained in a local neighborhood of the solution, and for many configurations,
the classical Newton iteration does not converge at all. In such cases,
so-called globalization techniques may help to improve convergence.
In this paper, pseudo-transient continuation is employed in order to improve
nonlinear convergence. The classical algorithm is enhanced by a neural network
model that is trained to predict a local pseudo-time step. Generalization of
the novel approach is facilitated by predicting the local pseudo-time step
separately on each element using only local information on a patch of adjacent
elements as input. Numerical results for standard benchmark problems, including
flow through a backward facing step geometry and Couette flow, show the
performance of the machine learning-enhanced globalization approach; as the
software for the simulations, the CFD module of COMSOL Multiphysics is
employed
A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD
The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations. This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations.
We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical non-dimensional parameters and with respect to mesh refinement
Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Saddle Point Problems
Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes, Navier-Stokes, and mixed linear elasticity ty e are presented. For the first time, coarse spaces obtained from the GDSW (Generalized Dryja-Smith-Widlund) approach are used in such a setting. Numerical results of our parallel implementation are presented for several model problems. In particular, cases are considered where the problem cannot or should not b e reduced using local static condensation, e.g., Stokes, Navier-Stokes or mixed elasticity problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the FROSch (Fast and Robust Overlapping Schwarz) library, which is part of the Trilinos package ShyLU. The implementation is algebraic in the sense that the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Parallel scalability results for several thousand cores for Stokes, Navier-Stokes, and mixed linear elasticity model problems are reported. Each of the local problems is solved using a direct solver in serial mo de, whereas the coarse problem is solved using a direct solver in serial or MPI-parallel mode or using an MPI-parallel iterative Krylov solve
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
Parallel Solution Methods for Aerostructural Analysis and Design Optimization
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/83550/1/AIAA-2010-9308-579.pd
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