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NASA NTRS1989

Least-squares finite element method for fluid dynamics

An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples.

Jiang, Bo-Nan↗

NASA NTRS1989

Least-squares finite element method for fluid dynamics

Jiang, Bo-Nan↗

DOE OSTI2019

A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

In this paper we propose a space-time adaptive finite element method for the phase field model of pitting corrosion, which is a parabolic partial differential equation system consisting of a phase variable and a concentration variable. A major challenge in solving this phase field model is that the problem is very stiff, which makes the time step size extremely small for standard temporal discretizations. Another difficulty is that a high spatial resolution is required to capture the steep gradients within the diffused interface, which results in very large number of degrees of freedom for uniform meshes. To overcome the stiffness of this model, we combine the Rosenbrock–Euler exponential integrator with Crank–Nicolson scheme for the temporal discretization. Moreover, by exploiting the fact that the speed of the corroding interface decreases with time, we derive an adaptive time stepping formula. For the spatial approximation, we propose a simple and efficient strategy to generate adaptive meshes that reduces the computational cost significantly. Thus, the proposed method utilizes local adaptivity and mesh refinement for efficient simulation of the corrosive dissolution over long times in heterogeneous media with complex microstructures. We also present an extensive set of numerical experiments in both two and three dimensional spaces to demonstrate efficiency and robustness of the proposed method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

A fast matrix-free approach to the high-order control volume finite element method with application to low-Mach flow

Here, a fast matrix-free formulation of the control volume finite element method is presented, requiring much less memory and computational work than previous efforts. The method is implemented and evaluated as a solver for low-Mach flow, including the evaluation of a preconditioning strategy for the pressure Poisson equation. The efficiency and scaling with polynomial order is evaluated on simple turbulent flows of interest, with appropriate solution quality metrics, and compared with a reference node-centered finite volume discretization. For a turbulent channel flow test, we show improvement in computational work for a given accuracy with the high-order scheme. The performance on a GPU accelerated platform is also investigated, with benefit shown for the matrix-free discretization.

42 ENGINEERING↗

NASA NTRS1987

Multiple-mode nonlinear free and forced vibrations of beams using finite element method

Multiple-mode nonlinear free and forced vibration of a beam is analyzed by the finite element method. The geometric nonlinearity is investigated. Inplane displacement and inertia (IDI) are also considered in the formulation. Harmonic force matrix is derived and explained. Nonlinear free vibration can be simply treated as a special case of the general forced vibration by setting the harmonic force matrix equal to zero. The effect of the higher modes is more pronouced for the clamped supported beam than the simply supported one. Beams without IDI yield more effect of the higher modes than the one with IDI. The effects of IDI are to reduce nonlinearity. For beams with end supports restrained from axial movement (immovable cases), only the hardening type nonlinearity is observed. However, beams of small slenderness ratio (L/R = 20) with movable end supports, the softening type nonlinearity is found. The concentrated force case yields a more severe response than the uniformly distributed force case. Finite element results are in good agreement with the solution of simple elliptic response, harmonic balance method, and Runge-Kutte method and experiment.

Mei, Chuh↗

NASA NTRS1985

Application of the finite element method in the calculation of transmission loss of flat and curved panels

This investigation represents an extension of a study of Roussos (1985) who considered the noise transmission loss of a rectangular plate in an infinite baffle. Roussos, who employed an analytical formulation, considered an unstiffened plate. While it is difficult to consider stiffeners by means of analytical methods, the difficulties can be avoided by employing a finite element procedure. For this reason, the present study is concerned with the implementation of a finite element method. The representation of the panel transmission loss is discussed, and the determination of the panel motion by means of the finite element technique is described, taking into account an isotropic flat panel, the exciting force, an eigenvalue problem, the radiation pressure, a plate element, and a cylindrical shell element. Numerical results are considered for a flat panel, a curved panel, and a stiffened flat panel.

Koval, L. R.↗

NASA NTRS1993

Helicopter trim analysis by shooting and finite element methods with optimally damped Newton iterations

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

Achar, N. S.↗

NASA NTRS1985

Finite element methods for high speed flows

An explicit finite element based solution procedure for solving the equations of compressible viscous high speed flow is presented. The method uses domain splitting to advance the solution with different timesteps on different portions of the mesh. For steady inviscid flows, adaptive mesh refinement procedures are successfully employed to enhance the definition of discontinuities. Preliminary ideas on the application of adaptive mesh refinement to the solution of problems involving steady viscous flow are presented. Sample timings are given for the performance of the finite element code on modern supercomputers.

Loehner, R.↗

NASA NTRS1989

Finite element analysis in fluids; Proceedings of the Seventh International Conference on Finite Element Methods in Flow Problems, University of Alabama, Huntsville, Apr. 3-7, 1989

Recent advances in computational fluid dynamics are examined in reviews and reports, with an emphasis on finite-element methods. Sections are devoted to adaptive meshes, atmospheric dynamics, combustion, compressible flows, control-volume finite elements, crystal growth, domain decomposition, EM-field problems, FDM/FEM, and fluid-structure interactions. Consideration is given to free-boundary problems with heat transfer, free surface flow, geophysical flow problems, heat and mass transfer, high-speed flow, incompressible flow, inverse design methods, MHD problems, the mathematics of finite elements, and mesh generation. Also discussed are mixed finite elements, multigrid methods, non-Newtonian fluids, numerical dissipation, parallel vector processing, reservoir simulation, seepage, shallow-water problems, spectral methods, supercomputer architectures, three-dimensional problems, and turbulent flows.

Chung, T. J.↗

NASA NTRS2018

An Efficient Inverse Finite Element Method for Shape and Stress Sensing of Laminated Composite and Sandwich Plates and Shells

The inverse Finite Element Method (iFEM) is a revolutionary methodology for real-time reconstruction of full-field structural displacements and stresses in structures that are instrumented with strain sensors. This inverse problem is commonly referred to as shape and stress sensing, which is well-recognized as an enabling technology for structural health monitoring systems. In this study, an improved iFEM formulation is proposed for shape and stress sensing of laminated composite and sandwich plates and shells. The formulation includes the kinematics of a shear deformation plate theory known as Refined Zigzag Theory (RZT) as its baseline. The present iFEM formulation is based upon the minimization of a weighted-least-squares functional that uses the complete set of section strains of RZT. The improved iFEM methodology is applicable for shape and stress sensing of thin and moderately thick plate and shell structures involving a relatively small number of strain gauges. The main advantage of the current formulation is that highly accurate through-the-thickness distributions of displacements, strains, and stresses are attainable using an element based on simple C0-continuous displacement interpolation functions. A three-node inverseshell element, named i3-RZT, is developed. Two example problems are examined in detail: (1) a simply supported rectangular laminated composite plate and (2) a wedge structure with a hole near one of the clamped ends. For both problems, the experimental strain data are generated numerically by the direct finite element analysis using high-fidelity discretizations. These strains are then regarded as the experimental strains obtained from surface mounted strain gauges or embedded fiber Bragg grating (FBG) sensors. The numerical results demonstrate the superior capability and potential applicability of the i3- RZT/iFEM methodology for performing accurate shape and stress sensing of complex composite structures.

Kefal, Adnan↗

NASA NTRS1971

Acoustic Analysis of Solid Rocket Motor Cavities by a Finite Element Method

Adaptation of NASTRAN program to solve acoustic mode problems of solid rocket motor cavities by finite element method

Herting, D. N.↗

NASA NTRS1983

A posteriori error indicators for the p-version of the finite element method

The existence of local a posteriori error indicators for the p-version of the finite element method is demonstrated through numerical examples. It is shown that it is possible to construct reliable error indicators from the residuums and tractions which have the same rate of convergence as the strain energy. The error indicators contribution of an element can be estimated by considering only the element and its immediate neighbors. The optimal sequence of p-distributions can be closely followed by the indicators even when there are only a very few elements, the elements vary greatly in size, the polynomial orders vary, the mesh grading is poor, and the smoothness indices and relative errors vary between wide limits.

Dunavant, D. A.↗

DOE OSTI2019

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Mathematics↗

DOE OSTI2020

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1968

Duality in finite element methods.

Duality between basic equations and dependent variables in problems of plate stretching and bending applied to finite element method

Elias, Z. M.↗

NASA NTRS1972

Leading-edge force features of the aerodynamic finite element method.

Description of a practical procedure for computing the wing leading-edge thrust distribution by the finite element method. When incorporated into a wing-body aerodynamic computer program, the technique is capable of predicting (at subsonic and supersonic speeds) the leading-edge thrust distribution (and therefore, the lateral-directional stability derivatives due to roll) and the nonlinear aerodynamic characteristics of low aspect-ratio wings with leading edge separation due to the application of suction technology.

Lan, C.-T.↗

DOE OSTI2018

A high-order multiscale finite-element method for time-domain acoustic-wave modeling

Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS2005

Structural Health Monitoring Using High-Density Fiber Optic Strain Sensor and Inverse Finite Element Methods

In an effort to mitigate accidents due to system and component failure, NASA s Aviation Safety has partnered with industry, academia, and other governmental organizations to develop real-time, on-board monitoring capabilities and system performance models for early detection of airframe structure degradation. NASA Langley is investigating a structural health monitoring capability that uses a distributed fiber optic strain system and an inverse finite element method for measuring and modeling structural deformations. This report describes the constituent systems that enable this structural monitoring function and discusses results from laboratory tests using the fiber strain sensor system and the inverse finite element method to demonstrate structural deformation estimation on an instrumented test article

Vazquez, Sixto L.↗