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A block preconditioner for an exact penalty formulation for stationary MHD

SIAM Journal on Scientific Computing

Phillips, Edward G.; Elman, Howard C.; Cyr, Eric C.; Shadid, John N.; Pawlowski, Roger P.

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 nonself-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 posed in two-dimensional domains. 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 nondimensional parameters and with respect to mesh refinement.

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TYPE Journal Article YEAR 2014
ScopusOSTI

Enhancing least-squares finite element methods through a quantity-of-interest

SIAM Journal on Numerical Analysis

Chaudhry, Jehanzeb H.; Cyr, Eric C.; Liu, Kuo; Manteuffel, Thomas A.; Olson, Luke N.; Tang, Lei

In this paper we introduce an approach that augments least-squares finite element formulations with user-specified quantities-of-interest. The method incorporates the quantity-ofinterest into the least-squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity-of-interest. We establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions. Central to the approach is that it offers an element-level estimate of the error in the quantity-ofinterest. As a result, we introduce an adaptive approach that yields efficient, adaptively refined approximations. Several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology. Notably, the results show that the new approach is effective at improving the accuracy per total computational cost.

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TYPE Journal Article YEAR 2014
ScopusOSTIDOI

Thermal hydraulic simulations, error estimation and parameter sensitivity studies in Drekar::CFD

Shadid, John N.; Pawlowski, Roger P.; Cyr, Eric C.; Wildey, Timothy M.

This report describes work directed towards completion of the Thermal Hydraulics Methods (THM) CFD Level 3 Milestone THM.CFD.P7.05 for the Consortium for Advanced Simulation of Light Water Reactors (CASL) Nuclear Hub effort. The focus of this milestone was to demonstrate the thermal hydraulics and adjoint based error estimation and parameter sensitivity capabilities in the CFD code called Drekar::CFD. This milestone builds upon the capabilities demonstrated in three earlier milestones; THM.CFD.P4.02 [12], completed March, 31, 2012, THM.CFD.P5.01 [15] completed June 30, 2012 and THM.CFD.P5.01 [11] completed on October 31, 2012.

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TYPE SAND Report YEAR 2014
OSTIDOI

Spatially varying embedded stochastic galerkin methods for steady-state PDEs

Cyr, Eric C.

Existing discretizations for stochastic PDEs, based on a tensor product between the deterministic basis and the stochastic basis, treat the required resolution of uncertainty as uniform across the physical domain. However, solutions to many PDEs of interest exhibit spatially localized features that may result in uncertainty being severely over or under-resolved by existing discretizations. In this report, we explore the mechanics and accuracy of using a spatially varying stochastic expansion. This is achieved through an adaptive refinement algorithm where simple error estimates are used to independently drive refinement of the stochastic basis at each point in the physical domain. Results are presented comparing the accuracy of the adaptive techinque to the accuracy achieved using uniform refinement.

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TYPE SAND Report YEAR 2013
OSTIDOI

A comparison of adjoint and data-centric verification techniques

Cyr, Eric C.; Shadid, John N.; Pawlowski, Roger P.

This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. We compare the adjoint-based a posteriori error estimation approach with a recent variant of a data-centric verification technique. We provide a brief overview of each technique and then we discuss their relative advantages and disadvantages. We use Drekar::CFD to produce numerical results for steady-state Navier Stokes and SARANS approximations. 3

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TYPE SAND Report YEAR 2013
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Results 151–175 of 187
Results 151–175 of 187