448 research outputs found
A Two-Stage Preconditioner for Multiphase Poromechanics in Reservoir Simulation
Many applications involving porous media--notably reservoir engineering and
geologic applications--involve tight coupling between multiphase fluid flow,
transport, and poromechanical deformation. While numerical models for these
processes have become commonplace in research and industry, the poor
scalability of existing solution algorithms has limited the size and resolution
of models that may be practically solved. In this work, we propose a two-stage
Newton-Krylov solution algorithm to address this shortfall. The proposed solver
exhibits rapid convergence, good parallel scalability, and is robust in the
presence of highly heterogeneous material properties. The key to success of the
solver is a block-preconditioning strategy that breaks the fully-coupled system
of mass and momentum balance equations into simpler sub-problems that may be
readily addressed using targeted algebraic methods. Numerical results are
presented to illustrate the performance of the solver on challenging benchmark
problems
A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media
Simulation of multiphase poromechanics involves solving a multi-physics
problem in which multiphase flow and transport are tightly coupled with the
porous medium deformation. To capture this dynamic interplay, fully implicit
methods, also known as monolithic approaches, are usually preferred. The main
bottleneck of a monolithic approach is that it requires solution of large
linear systems that result from the discretization and linearization of the
governing balance equations. Because such systems are non-symmetric,
indefinite, and highly ill-conditioned, preconditioning is critical for fast
convergence. Recently, most efforts in designing efficient preconditioners for
multiphase poromechanics have been dominated by physics-based strategies.
Current state-of-the-art "black-box" solvers such as algebraic multigrid (AMG)
are ineffective because they cannot effectively capture the strong coupling
between the mechanics and the flow sub-problems, as well as the coupling
inherent in the multiphase flow and transport process. In this work, we develop
an algebraic framework based on multigrid reduction (MGR) that is suited for
tightly coupled systems of PDEs. Using this framework, the decoupling between
the equations is done algebraically through defining appropriate interpolation
and restriction operators. One can then employ existing solvers for each of the
decoupled blocks or design a new solver based on knowledge of the physics. We
demonstrate the applicability of our framework when used as a "black-box"
solver for multiphase poromechanics. We show that the framework is flexible to
accommodate a wide range of scenarios, as well as efficient and scalable for
large problems
Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity
We present a weighted BFBT approximation (w-BFBT) to the inverse Schur
complement of a Stokes system with highly heterogeneous viscosity. When used as
part of a Schur complement-based Stokes preconditioner, we observe robust fast
convergence for Stokes problems with smooth but highly varying (up to 10 orders
of magnitude) viscosities, optimal algorithmic scalability with respect to mesh
refinement, and only a mild dependence on the polynomial order of high-order
finite element discretizations (, order ).
For certain difficult problems, we demonstrate numerically that w-BFBT
significantly improves Stokes solver convergence over the widely used inverse
viscosity-weighted pressure mass matrix approximation of the Schur complement.
In addition, we derive theoretical eigenvalue bounds to prove spectral
equivalence of w-BFBT. Using detailed numerical experiments, we discuss
modifications to w-BFBT at Dirichlet boundaries that decrease the number of
iterations. The overall algorithmic performance of the Stokes solver is
governed by the efficacy of w-BFBT as a Schur complement approximation and, in
addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG)
method, which we use to approximate the inverses of the viscous block and
variable-coefficient pressure Poisson operators within w-BFBT. Building on the
scalability of HMG, our Stokes solver achieves a parallel efficiency of 90%
while weak scaling over a more than 600-fold increase from 48 to all 30,000
cores of TACC's Lonestar 5 supercomputer.Comment: To appear in SIAM Journal on Scientific Computin
Multi-Stage Preconditioners for Thermal-Compositional-Reactive Flow in Porous Media
We present a family of multi-stage preconditioners for coupled
thermal-compositional-reactive reservoir simulation problems. The most common
preconditioner used in industrial practice, the Constrained Pressure Residual
(CPR) method, was designed for isothermal models and does not offer a specific
strategy for the energy equation. For thermal simulations, inadequate treatment
of the temperature unknown can cause severe convergence degradation. When
strong thermal diffusion is present, the energy equation exhibits significant
elliptic behavior that cannot be accurately corrected by CPR's second stage. In
this work, we use Schur-complement decompositions to extract a temperature
subsystem and apply an Algebraic MultiGrid (AMG) approximation as an additional
preconditioning stage to improve the treatment of the energy equation. We
present results for several two-dimensional hot air injection problems using an
extra heavy oil, including challenging reactive In-Situ Combustion (ISC) cases.
We show improved performance and robustness across different thermal regimes,
from advection dominated (high Peclet number) to diffusion dominated (low
Peclet number). The number of linear iterations is reduced by 40-85% compared
to standard CPR for both homogeneous and heterogeneous media, and the new
methods exhibit almost no sensitivity to the thermal regime
A quantitative performance analysis for Stokes solvers at the extreme scale
This article presents a systematic quantitative performance analysis for
large finite element computations on extreme scale computing systems. Three
parallel iterative solvers for the Stokes system, discretized by low order
tetrahedral elements, are compared with respect to their numerical efficiency
and their scalability running on up to parallel threads. A genuine
multigrid method for the saddle point system using an Uzawa-type smoother
provides the best overall performance with respect to memory consumption and
time-to-solution. The largest system solved on a Blue Gene/Q system has more
than ten trillion () unknowns and requires about 13 minutes
compute time. Despite the matrix free and highly optimized implementation, the
memory requirement for the solution vector and the auxiliary vectors is about
200 TByte. Brandt's notion of "textbook multigrid efficiency" is employed to
study the algorithmic performance of iterative solvers. A recent extension of
this paradigm to "parallel textbook multigrid efficiency" makes it possible to
assess also the efficiency of parallel iterative solvers for a given hardware
architecture in absolute terms. The efficiency of the method is demonstrated
for simulating incompressible fluid flow in a pipe filled with spherical
obstacles
A note on robust preconditioners for monolithic fluid-structure interaction systems of finite element equations
In this note, we consider preconditioned Krylov subspace methods for discrete
fluid-structure interaction problems with a nonlinear hyperelastic material
model and covering a large range of flows, e.g, water, blood, and air with
highly varying density. Based on the complete factorization of the
coupled system matrix, the preconditioner is constructed in form of
, where , and are proper
approximations to , and , respectively. The inverse of the
corresponding Schur complement is approximated by applying one cycle of a
special class of algebraic multigrid methods to the perturbed fluid
sub-problem, that is obtained by modifying corresponding entries in the
original fluid matrix with an explicitly constructed approximation of the exact
perturbation coming from the sparse matrix-matrix multiplications
Unified computational framework for the efficient solution of n-field coupled problems with monolithic schemes
In this paper, we propose and evaluate the performance of a unified
computational framework for preconditioning systems of linear equations
resulting from the solution of coupled problems with monolithic schemes. The
framework is composed by promising application-specific preconditioners
presented previously in the literature with the common feature that they are
able to be implemented for a generic coupled problem, involving an arbitrary
number of fields, and to be used to solve a variety of applications. The first
selected preconditioner is based on a generic block Gauss-Seidel iteration for
uncoupling the fields, and standard algebraic multigrid (AMG) methods for
solving the resulting uncoupled problems. The second preconditioner is based on
the semi-implicit method for pressure-linked equations (SIMPLE) which is
extended here to deal with an arbitrary number of fields, and also results in
uncoupled problems that can be solved with standard AMG. Finally, a more
sophisticated preconditioner is considered which enforces the coupling at all
AMG levels, in contrast to the other two techniques which resolve the coupling
only at the finest level. Our purpose is to show that these methods perform
satisfactory in quite different scenarios apart from their original
applications. To this end, we consider three very different coupled problems:
thermo-structure interaction, fluid-structure interaction and a complex model
of the human lung. Numerical results show that these general purpose methods
are efficient and scalable in this range of applications
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics
Motivated by the need for efficient and accurate simulation of the dynamics
of the polar ice sheets, we design high-order finite element discretizations
and scalable solvers for the solution of nonlinear incompressible Stokes
equations. We focus on power-law, shear thinning rheologies used in modeling
ice dynamics and other geophysical flows. We use nonconforming hexahedral
meshes and the conforming inf-sup stable finite element velocity-pressure
pairings or . To solve the nonlinear equations, we
propose a Newton-Krylov method with a block upper triangular preconditioner for
the linearized Stokes systems. The diagonal blocks of this preconditioner are
sparse approximations of the (1,1)-block and of its Schur complement. The
(1,1)-block is approximated using linear finite elements based on the nodes of
the high-order discretization, and the application of its inverse is
approximated using algebraic multigrid with an incomplete factorization
smoother. This preconditioner is designed to be efficient on anisotropic
meshes, which are necessary to match the high aspect ratio domains typical for
ice sheets. We develop and make available extensions to two libraries---a
hybrid meshing scheme for the p4est parallel AMR library, and a modified
smoothed aggregation scheme for PETSc---to improve their support for solving
PDEs in high aspect ratio domains. In a numerical study, we find that our
solver yields fast convergence that is independent of the element aspect ratio,
the occurrence of nonconforming interfaces, and of mesh refinement, and that
depends only weakly on the polynomial finite element order. We simulate the ice
flow in a realistic description of the Antarctic ice sheet derived from field
data, and study the parallel scalability of our solver for problems with up to
383M unknowns.Comment: 31 page
A practical factorization of a Schur complement for PDE-constrained Distributed Optimal Control
A distributed optimal control problem with the constraint of a linear
elliptic partial differential equation is considered. A necessary optimality
condition for this problem forms a saddle point system, the efficient and
accurate solution of which is crucial. A new factorization of the Schur
complement for such a system is proposed and its characteristics discussed. The
factorization introduces two complex factors that are complex conjugate to each
other. The proposed solution methodology involves the application of a parallel
linear domain decomposition solver---FETI-DPH---for the solution of the
subproblems with the complex factors. Numerical properties of FETI-DPH in this
context are demonstrated, including numerical and parallel scalability and
regularization dependence. The new factorization can be used to solve Schur
complement systems arising in both range-space and full-space formulations. In
both cases, numerical results indicate that the complex factorization is
promising
A robust and efficient iterative method for hyper-elastodynamics with nested block preconditioning
We develop a robust and efficient iterative method for hyper-elastodynamics
based on a novel continuum formulation recently developed. The numerical scheme
is constructed based on the variational multiscale formulation and the
generalized- method. Within the nonlinear solution procedure, a block
factorization is performed for the consistent tangent matrix to decouple the
kinematics from the balance laws. Within the linear solution procedure, another
block factorization is performed to decouple the mass balance equation from the
linear momentum balance equations. A nested block preconditioning technique is
proposed to combine the Schur complement reduction approach with the fully
coupled approach. This preconditioning technique, together with the Krylov
subspace method, constitutes a novel iterative method for solving
hyper-elastodynamics. We demonstrate the efficacy of the proposed
preconditioning technique by comparing with the SIMPLE preconditioner and the
one-level domain decomposition preconditioner. Two representative examples are
studied: the compression of an isotropic hyperelastic cube and the tensile test
of a fully-incompressible anisotropic hyperelastic arterial wall model. The
robustness with respect to material properties and the parallel performance of
the preconditioner are examined
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