A Lagrangian method for the numerical solution of the Euler equations for transonic flows
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Klopfer, G. H.↗
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Application of augmented-Lagrangian methods in meteorology: Comparison of different conjugate-gradient codes for large-scale minimization
A Lagrange multiplier method using techniques developed by Bertsekas (1982) was applied to solving the problem of enforcing simultaneous conservation of the nonlinear integral invariants of the shallow water equations on a limited area domain. This application of nonlinear constrained optimization is of the large dimensional type and the conjugate gradient method was found to be the only computationally viable method for the unconstrained minimization. Several conjugate-gradient codes were tested and compared for increasing accuracy requirements. Robustness and computational efficiency were the principal criteria.
Navon, I. M.↗
A combined Eulerian-volume of fraction-Lagrangian method for atomization simulation
The tracking of free surfaces between liquid and gas phases and analysis of the interfacial phenomena between the two during the atomization and breakup process of a liquid fuel jet is modeled. Numerical modeling of liquid-jet atomization requires the resolution of different conservation equations. Detailed formulation and validation are presented for the confined dam broken problem, the water surface problem, the single droplet problem, a jet breakup problem, and the liquid column instability problem.
Seung, S. P.↗
A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations
A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.
Ho, Lee-Wing↗
A high-order Shifted Interface Method for Lagrangian shock hydrodynamics
Here, we present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.
97 MATHEMATICS AND COMPUTING↗
Model-free simulations of turbulent reactive flows
The current computational methods for solving transport equations of turbulent reacting single-phase flows are critically reviewed, with primary attention given to those methods that lead to model-free simulations. In particular, consideration is given to direct numerical simulations using spectral (Galerkin) and pseudospectral (collocation) methods, spectral element methods, and Lagrangian methods. The discussion also covers large eddy simulations and turbulence modeling.
Givi, Peyman↗
Simulating Marine Stratocumulus Clouds using Lagrangian Superdroplet Method in ERF
This study investigates the simulation of marine stratocumulus (stratoCu) clouds – common in many offshore and coastal wind energy regions - using Lagrangian Superdroplet Method (SDM) implemented in Energy Research and Forecasting (ERF) model. Developed by the Wind Energy Technologies Office (WETO), ERF is a high-resolution atmospheric modeling tool that relies on traditional bulk microphysics schemes. Under an LLNL-led Laboratory Directed Research and Development (LDRD) project, SDM was integrated into ERF to improve cloud process fidelity by explicitly tracking individual “superdroplets”. This enables a realistic representation of the droplet size spectrum, cloud-atmosphere interactions, and potential feedbacks from wind plants on atmospheric flow and cloud structure.
54 ENVIRONMENTAL SCIENCES↗
The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates
The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the true stress, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Rayleigh-Ritz method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz, but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems the Lagrangian multiplier method of finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.
Budiansky, Bernard↗
The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates
The theory of Lagrangian multipliers is applied to the problem of finding both upper and lower limits to the true compressive buckling stress of a clamped rectangular plate. The upper and lower limits thus bracket the truss, which cannot be exactly found by the differential-equation approach. The procedure for obtaining the upper limit, which is believed to be new, presents certain advantages over the classical Raleigh-Rite method of finding upper limits. The theory of the lower-limit procedure has been given by Trefftz but, in the present application, the method differs from that of Trefftz in a way that makes it inherently more quickly convergent. It is expected that in other buckling problems and in some vibration problems problems the Lagrangian multiplier method finding upper and lower limits may be advantageously applied to the calculation of buckling stresses and natural frequencies.
Budiansky, Bernard↗
Unsteady Cascade Aerodynamic Response Using a Multiphysics Simulation Code
The multiphysics code Spectrum(TM) is applied to calculate the unsteady aerodynamic pressures of oscillating cascade of airfoils representing a blade row of a turbomachinery component. Multiphysics simulation is based on a single computational framework for the modeling of multiple interacting physical phenomena, in the present case being between fluids and structures. Interaction constraints are enforced in a fully coupled manner using the augmented-Lagrangian method. The arbitrary Lagrangian-Eulerian method is utilized to account for deformable fluid domains resulting from blade motions. Unsteady pressures are calculated for a cascade designated as the tenth standard, and undergoing plunging and pitching oscillations. The predicted unsteady pressures are compared with those obtained from an unsteady Euler co-de refer-red in the literature. The Spectrum(TM) code predictions showed good correlation for the cases considered.
Lawrence, C.↗
The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows
We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems.
Lufkin, Eric A.↗
Lagrangian computation of inviscid compressible flows
A Lagrangian method is developed to solve the Euler equations of gas dynamics. The solution of the equations is obtained by a numerical computation with the well-known Flux-Corrected-Transport (FCT) numerical method. This procedure is modified so that the boundary treatment is accurate and relatively simple. Shock waves and other flow discontinuities are captured monotonically without any type of fitting procedures. The Lagrangian method is employed so that the problem of mesh generation is completely avoided. The method is applicable to all Mach numbers except the low subsonic range where compressibility effects are small. The method is applied to a one-dimensional Riemann problem (shock tube) and to a two-dimensional supersonic channel flow with reflecting shock waves.
Klopfer, G. H.↗
The Lagrangian multiplier method of finding upper and lower limits to critical stresses of clamped plates
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Budiansky, Bernard↗
Notes on the Lagrangian multiplier method in elastic-stability analysis
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EQUATIONS - MOTION - LAGRANGE↗
Nonrecursive formulations of multibody dynamics and concurrent multiprocessing
Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. Historically, much of this research can be traced to applications of low dimensionality in mechanism and vehicle dynamics. Indeed, there is little doubt that recursive order N methods are the method of choice for this class of systems. This approach has the advantage that a minimal number of coordinates are utilized, parallelism can be induced for certain system topologies, and the method is of order N computational cost for systems of N rigid bodies. Despite the fact that many authors have dismissed redundant coordinate formulations as being of order N(exp 3), and hence less attractive than recursive formulations, we present recent research that demonstrates that at least three distinct classes of redundant, nonrecursive multibody formulations consistently achieve order N computational cost for systems of rigid and/or flexible bodies. These formulations are as follows: (1) the preconditioned range space formulation; (2) penalty methods; and (3) augmented Lagrangian methods for nonlinear multibody dynamics. The first method can be traced to its foundation in equality constrained quadratic optimization, while the last two methods have been studied extensively in the context of coercive variational boundary value problems in computational mechanics. Until recently, however, they have not been investigated in the context of multibody simulation, and present theoretical questions unique to nonlinear dynamics. All of these nonrecursive methods have additional advantages with respect to recursive order N methods: (1) the formalisms retain the highly desirable order N computational cost; (2) the techniques are amenable to concurrent simulation strategies; (3) the approaches do not depend upon system topology to induce concurrency; and (4) the methods can be derived to balance the computational load automatically on concurrent multiprocessors. In addition to the presentation of the fundamental formulations, this paper presents new theoretical results regarding the rate of convergence of order N constraint stabilization schemes associated with the newly introduced class of methods.
Kurdila, Andrew J.↗
The Role of Multiphysics Simulation in Multidisciplinary Analysis
This article describes the applications of the Spectrum(Tm) Solver in Multidisciplinary Analysis (MDA). Spectrum, a multiphysics simulation software based on the finite element method, addresses compressible and incompressible fluid flow, structural, and thermal modeling as well as the interaction between these disciplines. Multiphysics simulation is based on a single computational framework for the modeling of multiple interacting physical phenomena. Interaction constraints are enforced in a fully-coupled manner using the augmented-Lagrangian method. Within the multiphysics framework, the finite element treatment of fluids is based on Galerkin-Least-Squares (GLS) method with discontinuity capturing operators. The arbitrary-Lagrangian-Eulerian method is utilized to account for deformable fluid domains. The finite element treatment of solids and structures is based on the Hu-Washizu variational principle. The multiphysics architecture lends itself naturally to high-performance parallel computing. Aeroelastic, propulsion, thermal management and manufacturing applications are presented.
Rifai, Steven M.↗
Using Filter Methods to Guide Convergence for ADMM, with Applications to Nonnegative Matrix Factorization Problems
Nonconvex, nonlinear optimization problems arise naturally in parameter fitting and machine learning. While augmented Lagrangian methods have demonstrated robust convergence for classes of these problems, their convergence for block updates has been relatively unexplored outside of the context of the alternating direction method of multipliers (ADMM). ADMM has seen extensive use in these applications, but may exhibit uncertain convergence behavior in many practical nonconvex settings, and struggles with general nonlinear constraints. In contrast, filter methods have proved effective in enforcing convergence for sequential quadratic programming methods and interior point methods with feasibility criteria. We develop an ADMM-filter method for highly nonlinear and nonconvex problems. Here, we show convergence under mild assumptions for several types of coordinate descent schemes, and demonstrate our algorithm on nonnegative matrix factorization and completion problems in imaging and chemical spectrum analysis.
Nonconvex optimization↗
An Empirical Quantile Estimation Approach for Chance-Constrained Nonlinear Optimization Problems
We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function and hence the quantile-function gradient approximation is easier to implement and there are less accuracy-control parameters to tune. Furthermore, we demonstrate the effectiveness of this approach and compare it with a smoothing method for the quantile-gradient estimation. Numerical investigation shows that the two approaches are competitive for certain problem instances.
Applied Probability↗