Search NASA⌕ Search

SEARCH · Search NASA

Results for “Nonlinear preconditioning”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1983

Stratospheric sudden coolings and the role of nonlinear wave interactions in preconditioning the circumpolar flow

The mechanisms responsible for the transition of the circumpolar flow from its normal midwinter state to the preconditioned state that should evolve before a wavenumber-2 major warming are investigated, through a combination of observational, numerical and theoretical studies. Observations of Eliassen-Palm flux cross sections indicate that while wave zonal mean flow interaction theory could account for the qualitative evolution of the circumpolar flow during the warming, substantial nonlinear wave interactions were active during the cooling period, and these interactions significantly influenced the evolution of the circumpolar flow. Numerical experiments employing a truncated, semispectral model indicate that this cooling phenomenon is realistically reproducible in an idealized integration in which wave-wave interactions are present. Two different mechanisms are proposed to account for these nonlinearities.

Palmer, T. N.↗
NASA NTRS1998

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

Koppenhoefer, Kyle C.↗
NASA NTRS1989

Parallel iterative methods for sparse linear and nonlinear equations

As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.

Saad, Youcef↗
NASA NTRS1991

Implicit solvers for unstructured meshes

Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.

Venkatakrishnan, V.↗
NASA NTRS1991

Implicit solvers for unstructured meshes

Implicit methods for unstructured mesh computations are developed and tested. The approximate system which arises from the Newton-linearization of the nonlinear evolution operator is solved by using the preconditioned generalized minimum residual technique. These different preconditioners are investigated: the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over-relaxation (SSOR). The preconditioners have been optimized to have good vectorization properties. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also investigated. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.

Venkatakrishnan, V.↗
NASA NTRS2014

Optic Nerve Sheath Mechanics and Permeability in VIIP Syndrome

Long-duration space flight carries the risk of developing Visual Impairment and Intracranial Pressure (VIIP) syndrome, a spectrum of ophthalmic changes including posterior globe flattening, choroidal folds, distension of the optic nerve sheath (ONS), optic nerve kinking and potentially permanent degradation of visual function. The slow onset of VIIP, its chronic nature, and certain clinical features strongly suggest that biomechanical factors acting on the ONS play a role in VIIP. Here we measure several relevant ONS properties needed to model VIIP biomechanics. The ONS (meninges) of fresh porcine eyes (n7) was reflected, the nerve proper was truncated near the sclera, and the meninges were repositioned to create a hollow cylinder of meningeal connective tissue attached to the posterior sclera. The distal end was cannulated, sealed, and pressure clamped (mimicking cerebrospinal fluid [CSF] pressure), while the eye was also cannulated for independent control of intraocular pressure (IOP). The meninges were inflated (CSF pressure cycling 7-50 mmHg) while ONS outer diameter was imaged. In another set of experiments (n4), fluid permeation rate across the meninges was recorded by observing the drainage of an elevated fluid reservoir (30 mmHg) connected to the meninges. The ONS showed behavior typical of soft tissues: viscoelasticity, with hysteresis in early preconditioning cycles and repeatable behavior after 4 cycles, and nonlinear stiffening, particularly at CSF pressures 15 mmHg (Figure). Tangent moduli measured from the loading curve were 372 101, 1199 358, and 2050 379 kPa (mean SEM) at CSF pressures of 7, 15 and 30 mmHg, respectively. Flow rate measurements through the intact meninges at 30mmHg gave a permeability of 1.34 0.46 lmincm2mmHg (mean SEM). The ONS is a tough, strain-stiffening connective tissue that is surprisingly permeable. The latter observation suggests that there could be significant CSF drainage through the ONS into the orbit, likely important for CSF transport in the optic nerve. These experimental measurements, extended to human eyes, are informing computational models of the pathophysiology and biomechanics of the ONS in VIIP syndrome.

biomechanics↗
NASA NTRS2014

Biomechanics of the Optic Nerve Sheath in VIIP Syndrome

Long-duration space flight carries the risk of developing Visual Impairment and Intracranial Pressure (VIIP) syndrome, a spectrum of ophthalmic changes including posterior globe flattening, choroidal folds, distension of the optic nerve sheath (ONS), optic nerve kinking and potentially permanent degradation of visual function. The slow onset of VIIP, its chronic nature, and certain clinical features strongly suggest that biomechanical factors acting on the ONS play a role in VIIP. Here we measure several relevant ONS properties needed to model VIIP biomechanics. The ONS (meninges) of fresh porcine eyes (n7) was reflected, the nerve proper was truncated near the sclera, and the meninges were repositioned to create a hollow cylinder of meningeal connective tissue attached to the posterior sclera. The distal end was cannulated, sealed, and pressure clamped (mimicking cerebrospinal fluid [CSF] pressure), while the eye was also cannulated for independent control of intraocular pressure (IOP). The meninges were inflated (CSF pressure cycling 7-50 mmHg) while ONS outer diameter was imaged. In another set of experiments (n4), fluid permeation rate across the meninges was recorded by observing the drainage of an elevated fluid reservoir (30 mmHg) connected to the meninges. The ONS showed behavior typical of soft tissues: viscoelasticity, with hysteresis in early preconditioning cycles and repeatable behavior after 4 cycles, and nonlinear stiffening, particularly at CSF pressures 15 mmHg (Figure). Tangent moduli measured from the loading curve were 372 101, 1199 358, and 2050 379 kPa (mean SEM) at CSF pressures of 7, 15 and 30 mmHg, respectively. Flow rate measurements through the intact meninges at 30mmHg gave a permeability of 1.34 0.46 lmincm2mmHg (mean SEM). The ONS is a tough, strain-stiffening connective tissue that is surprisingly permeable. The latter observation suggests that there could be significant CSF drainage through the ONS into the orbit, likely important for CSF transport in the optic nerve. These experimental measurements, extended to human eyes, are informing computational models of the pathophysiology and biomechanics of the ONS in VIIP syndrome.

fluid shifts↗
NASA NTRS1984

Simultaneous analysis and design

Optimization techniques are increasingly being used for performing nonlinear structural analysis. The development of element by element (EBE) preconditioned conjugate gradient (CG) techniques is expected to extend this trend to linear analysis. Under these circumstances the structural design problem can be viewed as a nested optimization problem. There are computational benefits to treating this nested problem as a large single optimization problem. The response variables (such as displacements) and the structural parameters are all treated as design variables in a unified formulation which performs simultaneously the design and analysis. Two examples are used for demonstration. A seventy-two bar truss is optimized subject to linear stress constraints and a wing box structure is optimized subject to nonlinear collapse constraints. Both examples show substantial computational savings with the unified approach as compared to the traditional nested approach.

Haftka, R. T.↗
NASA NTRS1985

Simultaneous analysis and design

Optimization techniques are increasingly being used for performing nonlinear structural analysis. The development of element by element (EBE) preconditioned conjugate gradient (CG) techniques is expected to extend this trend to linear analysis. Under these circumstances the structural design problem can be viewed as a nested optimization problem. There are computational benefits to treating this nested problem as a large single optimization problem. The response variables (such as displacements) and the structural parameters are all treated as design variables in a unified formulation which performs simultaneously the design and analysis. Two examples are used for demonstration. A seventy-two bar truss is optimized subject to linear stress constraints and a wing box structure is optimized subject to nonlinear collapse constraints. Both examples show substantial computational savings with the unified approach as compared to the traditional nested approach.

Haftka, R. T.↗
NASA NTRS1998

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized demoralization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin demoralization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS, and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobean linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Discrete maximum principle theory will be presented for general finite volume approximations on unstructured meshes. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc, will. be addressed as needed.

Barth, Timothy J.↗
NASA NTRS1997

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

Several stabilized discretization procedures for conservation law equations on triangulated domains will be considered. Specifically, numerical schemes based on upwind finite volume, fluctuation splitting, Galerkin least-squares, and space discontinuous Galerkin discretization will be considered in detail. A standard energy analysis for several of these methods will be given via entropy symmetrization. Next, we will present some relatively new theoretical results concerning congruence relationships for left or right symmetrized equations. These results suggest new variants of existing FV, DG, GLS and FS methods which are computationally more efficient while retaining the pleasant theoretical properties achieved by entropy symmetrization. In addition, the task of Jacobian linearization of these schemes for use in Newton's method is greatly simplified owing to exploitation of exact symmetries which exist in the system. These variants have been implemented in the "ELF" library for which example calculations will be shown. The FV, FS and DG schemes also permit discrete maximum principle analysis and enforcement which greatly adds to the robustness of the methods. Some prevalent limiting strategies will be reviewed. Next, we consider embedding these nonlinear space discretizations into exact and inexact Newton solvers which are preconditioned using a nonoverlapping (Schur complement) domain decomposition technique. Elements of nonoverlapping domain decomposition for elliptic problems will be reviewed followed by the present extension to hyperbolic and elliptic-hyperbolic problems. Other issues of practical relevance such the meshing of geometries, code implementation, turbulence modeling, global convergence, etc. will be addressed as needed.

Barth, Timothy↗
NASA NTRS2020

Multigrid Preconditioning for a Space-Time Spectral-Element Discontinuous-Galerkin Solver

In this work we examine a multigrid preconditioning approach in the context of a high- order tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scale- resolving solutions of the Navier-Stokes equations: separated low-Mach unsteady ow over an airfoil from laminar to turbulent ow. A reduction in the number of ne space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined.

Franciolini, Matteo↗
NASA NTRS1990

Application of the TRANAIR rectangular grid approach to the aerodynamic analysis of complex configurations

A numerical method is described which uses a rectangular grid to solve the nonlinear full potential equation about complex configurations. The grid is locally refined to resolve high velocity gradients arising from leading edge expansions or shock waves. The grid penetrates the boundary (described by networks of quadrilateral panels) and is generated automatically. Discrete operators are constructed using the finite element method. The system of nonlinear discrete equations is solved iteratively using a Krylov subspace method preconditioned by an exterior Poisson solver and a direct sparse solver. The primary emphasis is to provide design engineers with an aerodynamic analysis tool (the TRANAIR code) which is accurate, reliable, economical, and flexible to use. Computational results for many interesting configurations are presented.

Johnson, Forrester T.↗
NASA NTRS1993

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.↗
NASA NTRS1985

GMRES acceleration of computational fluid dynamics codes

The generalized minimal residual algorithm (GMRES) is a conjugate-gradient like method that applies directly to nonsymmetric linear systems of equations. In this paper, GMRES is modified to handle nonlinear equations characteristic of computational fluid dynamics. Attention is devoted to the concept of preconditioning and the role it plays in assuring rapid convergence. A formulation is developed that allows GMRES to be preconditioned by the solution procedures already built into existing computer codes. Examples are provided that demonstrate the ability of GMRES to greatly improve the robustness and rate of convergence of current state-of-the-art fluid dynamics codes. Theoretical aspects of GMRES are presented that explain why it works. Finally, the advantage GMRES enjoys over related methods such as conjugate gradients are discussed.

Wigton, L. B.↗
NASA NTRS1985

Iterative methods for mixed finite element equations

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Nakazawa, S.↗
NASA NTRS2002

P1 Nonconforming Finite Element Method for the Solution of Radiation Transport Problems

The simulation of radiation transport in the optically thick flux-limited diffusion regime has been identified as one of the most time-consuming tasks within large simulation codes. Due to multimaterial complex geometry, the radiation transport system must often be solved on unstructured grids. In this paper, we investigate the behavior and the benefits of the unstructured P(sub 1) nonconforming finite element method, which has proven to be flexible and effective on related transport problems, in solving unsteady implicit nonlinear radiation diffusion problems using Newton and Picard linearization methods. Key words. nonconforrning finite elements, radiation transport, inexact Newton linearization, multigrid preconditioning

Kang, Kab S.↗
NASA NTRS1992

A study of equation solvers for linear and non-linear finite element analysis on parallel processing computers

Concurrent computing environments provide the means to achieve very high performance for finite element analysis of systems, provided the algorithms take advantage of multiple processors. The authors have examined several algorithms for both linear and nonlinear finite element analysis. The performance of these algorithms on an Alliant FX/80 parallel supercomputer has been studied. For single load case linear analysis, the optimal solution algorithm is strongly problem dependent. For multiple load cases or nonlinear analysis through a modified Newton-Raphson method, decomposition algorithms are shown to have a decided advantage over element-by-element preconditioned conjugate gradient algorithms.

Watson, Brian C.↗