3,205 research outputs found
Flexible Complementarity Solvers for Large-Scale Applications
Discretizations of infinite-dimensional variational inequalities lead to
linear and nonlinear complementarity problems with many degrees of freedom. To
solve these problems in a parallel computing environment, we propose two
active-set methods that solve only one linear system of equations per
iteration. The linear solver, preconditioner, and matrix structures can be
chosen by the user for a particular application to achieve high parallel
performance. The parallel scalability of these methods is demonstrated for some
discretizations of infinite-dimensional variational inequalities.Comment: 17 pages; 2 figure
Semi-smooth Newton methods for nonlinear complementarity formulation of compositional two-phase flow in porous media
Simulating compositional multiphase flow in porous media is a challenging
task, especially when phase transition is taken into account. The main problem
with phase transition stems from the inconsistency of the primary variables
such as phase pressure and phase saturation, i.e. they become ill-defined when
a phase appears or disappears. Recently, a new approach for handling phase
transition has been developed, whereby the system is formulated as a nonlinear
complementarity problem (NCP). Unlike the widely used primary variable
switching (PVS) method which requires a drastic reduction of the time step size
when a phase appears or disappears, this approach is more robust and allows for
larger time steps. One way to solve an NCP system is to reformulate the
inequality constraints as a non-smooth equation using a complementary function
(C-function). Because of the non-smoothness of the constraint equations, a
semi-smooth Newton method needs to be developed. In this work, we consider two
methods for solving NCP systems used to model multiphase flow: (1) a
semi-smooth Newton method for two C-functions: the minimum and the
Fischer-Burmeister functions, and (2) a new inexact Newton method based on the
Jacobian smoothing method for a smooth version of the Fischer-Burmeister
function. We show that the new method is robust and efficient for standard
benchmark problems as well as for realistic examples with highly heterogeneous
media such as the SPE10 benchmark
Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions
Multiphase flow is a critical process in a wide range of applications,
including oil and gas recovery, carbon sequestration, and contaminant
remediation. Numerical simulation of multiphase flow requires solving of a
large, sparse linear system resulting from the discretization of the partial
differential equations modeling the flow. In the case of multiphase
multicomponent flow with miscible effect, this is a very challenging task. The
problem becomes even more difficult if phase transitions are taken into
account. A new approach to handle phase transitions is to formulate the system
as a nonlinear complementarity problem (NCP). Unlike in the primary variable
switching technique, the set of primary variables in this approach is fixed
even when there is phase transition. Not only does this improve the robustness
of the nonlinear solver, it opens up the possibility to use multigrid methods
to solve the resulting linear system. The disadvantage of the complementarity
approach, however, is that when a phase disappears, the linear system has the
structure of a saddle point problem and becomes indefinite, and current
algebraic multigrid (AMG) algorithms cannot be applied directly. In this study,
we explore the effectiveness of a new multilevel strategy, based on the
multigrid reduction technique, to deal with problems of this type. We
demonstrate the effectiveness of the method through numerical results for the
case of two-phase, two-component flow with phase appearance/disappearance. We
also show that the strategy is efficient and scales optimally with problem
size
A semi-smooth Newton method for solving convex quadratic programming problem under simplicial cone constraint
In this paper the simplicial cone constrained convex quadratic programming
problem is studied. The optimality conditions of this problem consist in a
linear complementarity problem. This fact, under a suitable condition, leads to
an equivalence between the simplicial cone constrained convex quadratic
programming problem and the one of finding the unique solution of a nonsmooth
system of equations. It is shown that a semi-smooth Newton method applied to
this nonsmooth system of equations is always well defined and under a mild
assumption on the simplicial cone the method generates a sequence that
converges linearly to its solution. Besides, we also show that the generated
sequence is bounded for any starting point and a formula for any accumulation
point of this sequence is presented. The presented numerical results suggest
that this approach achieves accurate solutions to large problems in few
iterations.Comment: 17 page
A semi-smooth Newton method for a special piecewise linear system with application to positively constrained convex quadratic programming
In this paper a special piecewise linear system is studied. It is shown that,
under a mild assumption, the semi-smooth Newton method applied to this system
is well defined and the method generates a sequence that converges linearly to
a solution. Besides, we also show that the generated sequence is bounded, for
any starting point, and a formula for any accumulation point of this sequence
is presented. As an application, we study the convex quadratic programming
problem under positive constraints. The numerical results suggest that the
semi-smooth Newton method achieves accurate solutions to large scale problems
in few iterations.Comment: arXiv admin note: substantial text overlap with arXiv:1503.0275
A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity
Mathematical models for flow through porous media typically enjoy the
so-called maximum principles, which place bounds on the pressure field. It is
highly desirable to preserve these bounds on the pressure field in predictive
numerical simulations, that is, one needs to satisfy discrete maximum
principles (DMP). Unfortunately, many of the existing formulations for flow
through porous media models do not satisfy DMP. This paper presents a robust,
scalable numerical formulation based on variational inequalities (VI), to model
non-linear flows through heterogeneous, anisotropic porous media without
violating DMP. VI is an optimization technique that places bounds on the
numerical solutions of partial differential equations. To crystallize the
ideas, a modification to Darcy equations by taking into account
pressure-dependent viscosity will be discretized using the lowest-order
Raviart-Thomas (RT0) and Variational Multi-scale (VMS) finite element
formulations. It will be shown that these formulations violate DMP, and, in
fact, these violations increase with an increase in anisotropy. It will be
shown that the proposed VI-based formulation provides a viable route to enforce
DMP. Moreover, it will be shown that the proposed formulation is scalable, and
can work with any numerical discretization and weak form. Parallel scalability
on modern computational platforms will be illustrated through strong-scaling
studies, which will prove the efficiency of the proposed formulation in a
parallel setting. Algorithmic scalability as the problem size is scaled up will
be demonstrated through novel static-scaling studies. The performed
static-scaling studies can serve as a guide for users to be able to select an
appropriate discretization for a given problem size
Parallel contact-aware simulations of deformable particles in 3D Stokes flow
We present a parallel-scalable method for simulating non-dilute suspensions
of deformable particles immersed in Stokesian fluid in three dimensions. A
critical component in these simulations is robust and accurate collision
handling. This work complements our previous work [L. Lu, A. Rahimian, and D.
Zorin. Contact-aware simulations of particulate Stokesian suspensions. Journal
of Computational Physics 347C: 160-182] by extending it to 3D and by
introducing new parallel algorithms for collision detection and handling. We
use a well-established boundary integral formulation with spectral Galerkin
method to solve the fluid flow. The key idea is to ensure an interference-free
particle configuration by introducing explicit contact constraints into the
system. While such constraints are typically unnecessary in the formulation
they make it possible to eliminate catastrophic loss of accuracy in the
discretized problem by preventing contact explicitly. The incorporation of
contact constraints results in a significant increase in stable time-step size
for locally-implicit time-stepping and a reduction in the necessary number of
discretization points for stability. Our method maintains the accuracy of
previous methods at a significantly lower cost for dense suspensions and the
time step size is independent from the volume fraction. Our method permits
simulations with high volume fractions; we report results with up to 60% volume
fraction. We demonstrated the parallel scaling of the algorithms on up to 16K
CPU cores
Projection onto simplicial cones by Picard's method
By using Moreau's decomposition theorem for projecting onto cones, the
problem of projecting onto a simplicial cone is reduced to finding the unique
solution of a nonsmooth system of equations. It is shown that Picard's method
applied to the system of equations associated to the problem of projecting onto
a simplicial cone generates a sequence that converges linearly to the solution
of the system. Numerical experiments are presented making the comparison
between Picard's and semi-smooth Newton's methods to solve the nonsmooth system
associated with the problem of projecting a point onto a simplicial cone.Comment: 22 pages. arXiv admin note: text overlap with arXiv:1404.242
Rigid Body Motion Prediction with Planar Non-convex Contact Patch
We present a principled method for motion prediction via dynamic simulation
for rigid bodies in intermittent contact with each other where the contact is
assumed to be a planar non-convex contact patch. The planar non-convex contact
patch can either be a topologically connected set or disconnected set. Such
algorithms are useful in planning and control for robotic manipulation. Most
works in rigid body dynamic simulation assume that the contact between objects
is a point contact, which may not be valid in many applications. In this paper,
by using the convex hull of the contact patch, we build on our recent work on
simulating rigid bodies with convex contact patches, for simulating the motion
of objects with planar non-convex contact patches. We formulate a discrete-time
mixed complementarity problem where we solve the contact detection and
integration of the equations of motion simultaneously. Thus, our method is a
geometrically-implicit method and we prove that in our formulation, there is no
artificial penetration between the contacting rigid bodies. We solve for the
equivalent contact point (ECP) and contact impulse of each contact patch
simultaneously along with the state, i.e., configuration and velocity of the
objects. We provide empirical evidence to show that our method can seamlessly
capture the transition between different contact modes like patch contact to
multiple or single point contact during the simulation.Comment: arXiv admin note: text overlap with arXiv:1809.0555
Blended Cured Quasi-Newton for Geometric Optimization
Optimizing deformation energies over a mesh, in two or three dimensions, is a
common and critical problem in physical simulation and geometry processing. We
present three new improvements to the state of the art: a barrier-aware
line-search filter that cures blocked descent steps due to element barrier
terms and so enables rapid progress; an energy proxy model that adaptively
blends the Sobolev (inverse-Laplacian-processed) gradient and L-BFGS descent to
gain the advantages of both, while avoiding L-BFGS's current limitations in
geometry optimization tasks; and a characteristic gradient norm providing a
robust and largely mesh- and energy-independent convergence criterion that
avoids wrongful termination when algorithms temporarily slow their progress.
Together these improvements form the basis for Blended Cured Quasi-Newton
(BCQN), a new geometry optimization algorithm. Over a wide range of problems
over all scales we show that BCQN is generally the fastest and most robust
method available, making some previously intractable problems practical while
offering up to an order of magnitude improvement in others
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