3,205 research outputs found

    Flexible Complementarity Solvers for Large-Scale Applications

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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

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    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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