Results for “mixed finite element method” — Page 3

NASA NTRS1981

Recent advances in reduction methods for nonlinear problems

Status and some recent developments in the application of reduction methods to nonlinear structural mechanics problems are summarized. The aspects of reduction methods discussed herein include: (1) selection of basis vectors in nonlinear static and dynamic problems, (2) application of reduction methods in nonlinear static analysis of structures subjected to prescribed edge displacements, and (3) use of reduction methods in conjunction with mixed finite element models. Numerical examples are presented to demonstrate the effectiveness of reduction methods in nonlinear problems. Also, a number of research areas which have high potential for application of reduction methods are identified.

Noor, A. K.↗

NASA NTRS1989

A weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1991

Weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1990

A weak Hamiltonian finite element method for optimal control problems

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Hodges, Dewey H.↗

NASA NTRS1989

A weak Hamiltonian finite element method for optimal guidance of an advanced launch vehicle

A temporal finite-element method based on a mixed form of the Hamiltonian weak principle is presented for optimal control problems. The mixed form of this principle contains both states and costates as primary variables, which are expanded in terms of nodal values and simple shape functions. Time derivatives of the states and costates do not appear in the governing variational equation; the only quantities whose time derivatives appear therein are virtual states and virtual costates. Numerical results are presented for an elementary trajectory optimization problem; they show very good agreement with the exact solution along with excellent computational efficiency and self-starting capability. The feasibility of this approach for real-time guidance applications is evaluated. A simplified model for an advanced launch vehicle application that is suitable for finite-element solution is presented.

Hodges, Dewey H.↗

NASA NTRS1992

Thermomechanical buckling and postbuckling of multilayered composite panels

A study is made of the thermomechanical buckling and postbuckling responses of flat unstiffened composite panels. The panels are subjected to combined temperature change and applied edge displacement. The analysis is based on a first-order shear deformation, von Karman type nonlinear plate theory. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. An efficient multiple-parameter reduction method is used in conjunction with mixed finite element models, for determining the stability boundary and postbuckling response. The reduction method is also used for evaluating the sensitivity coefficients which measure the sensitivity of the buckling and postbuckling responses to variations in the different lamination and material parameters of the panel. Numerical results are presented showing the effects of variations in the laminate stacking sequence, fiber orientation, number of layers and aspect ratio of the panels on their thermomechanical buckling and postbuckling responses and their sensitivity coefficients.

Noor, Ahmed K.↗

NASA NTRS1994

The least-squares finite element method for low-mach-number compressible viscous flows

The present paper reports the development of the Least-Squares Finite Element Method (LSFEM) for simulating compressible viscous flows at low Mach numbers in which the incompressible flows pose as an extreme. Conventional approach requires special treatments for low-speed flows calculations: finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique; and, finite element methods rely on the mixed method and the operator-splitting method. In this paper, however, we show that such difficulty does not exist for the LSFEM and no special treatment is needed. The LSFEM always leads to a symmetric, positive-definite matrix through which the compressible flow equations can be effectively solved. Two numerical examples are included to demonstrate the method: first, driven cavity flows at various Reynolds numbers; and, buoyancy-driven flows with significant density variation. Both examples are calculated by using full compressible flow equations.

Yu, Sheng-Tao↗

NASA NTRS1992

Computational aspects of helicopter trim analysis and damping levels from Floquet theory

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.

Gaonkar, Gopal H.↗

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 NTRS1991

Thermal postbuckling of thin-walled composite stiffeners

A study is made of the thermal postbuckling response of composite stiffeners subjected to prescribed edge displacement and a temperature rise. The flanges and web of the stiffeners are modeled by using two-dimensional plate finite elements. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. A reduction method is used in conjunction with mixed finite element models for determining the postbuckling response of the stiffeners. Sensitivity derivatives are evaluated and used to study the effects of variations in the different lamination and material parameters of the stiffeners on their postbuckling response characteristics. Numerical studies are presented for anisotropic stiffeners with Zee and channel sections.

Noor, Ahmed K.↗

NASA NTRS1991

Optimal least-squares finite element method for elliptic problems

An optimal least squares finite element method is proposed for two dimensional and three dimensional elliptic problems and its advantages are discussed over the mixed Galerkin method and the usual least squares finite element method. In the usual least squares finite element method, the second order equation (-Delta x (Delta u) + u = f) is recast as a first order system (-Delta x p + u = f, Delta u - p = 0). The error analysis and numerical experiment show that, in this usual least squares finite element method, the rate of convergence for flux p is one order lower than optimal. In order to get an optimal least squares method, the irrotationality Delta x p = 0 should be included in the first order system.

Jiang, Bo-Nan↗

NASA NTRS1988

An approach to probabilistic finite element analysis using a mixed-iterative formulation

An efficient algorithm for computing the response sensitivity of finite element problems based on a mixed-iterative formulation is proposed. This method does not involve explicit differentiation of the tangent stiffness array and can be used with formulations for which a consistent tangent stiffness is not readily available. The method has been successfully applied to probabilistic finite element analysis of problems using the proposed mixed formulation, and this exercise has provided valuable insights regarding the extension of the method to a more general class of problems to include material and geometric nonlinearities.

Dias, J. B.↗

NASA NTRS1988

Finite elements: Theory and application

Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.

Dwoyer, D. L.↗

NASA NTRS1997

Application of the Finite-Element Z-Matrix Method to e-H2 Collisions

The present study adapts the Z-matrix formulation using a mixed basis of finite elements and Gaussians. This is a energy-independent basis which allows flexible boundary conditions and is amenable to efficient algorithms for evaluating the necessary matrix elements with molecular targets.

Huo, Winifred M.↗

DOE OSTI2021

Reconstruction of concrete microstructure using complementarity of X-ray and neutron tomography

The concrete microstructure was successfully reconstructed using the complementarity of X-ray and neutron computed tomography (CT). Neither tomogram alone was found to be suitable to properly describe the microstructure of concrete under this study. However, by merging the information revealed by the two modalities, and using image segmentation, noise reduction, and image registration techniques we reconstruct the concrete microstructure. Void, aggregate, and cement paste phases are successfully captured down to the images' spatial resolution, even though the aggregate consists of multiple minerals. The coarse-aggregate volume fraction of the reconstructed microstructure was similar to that of the mixing proportions. Furthermore, image-based finite element analysis is performed to demonstrate the effects of microstructure on stress concentration and strain localization.

36 MATERIALS SCIENCE↗

NASA NTRS1991

Transient modeling/analysis of hyperbolic heat conduction problems employing mixed implicit-explicit alpha method

This paper describes the evaluation of mixed implicit-explicit finite element formulations for hyperbolic heat conduction problems involving non-Fourier effects. In particular, mixed implicit-explicit formulations employing the alpha method proposed by Hughes et al. (1987, 1990) are described for the numerical simulation of hyperbolic heat conduction models, which involves time-dependent relaxation effects. Existing analytical approaches for modeling/analysis of such models involve complex mathematical formulations for obtaining closed-form solutions, while in certain numerical formulations the difficulties include severe oscillatory solution behavior (which often disguises the true response) in the vicinity of the thermal disturbances, which propagate with finite velocities. In view of these factors, the alpha method is evaluated to assess the control of the amount of numerical dissipation for predicting the transient propagating thermal disturbances. Numerical test models are presented, and pertinent conclusions are drawn for the mixed-time integration simulation of hyperbolic heat conduction models involving non-Fourier effects.

Tamma, Kumar K.↗

NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗

NASA NTRS1992

Postbuckling of multilayered composite plates subjected to combined axial and thermal loads

A study is made of the postbuckling response of composite plates subjected to combined axial and thermal loadings. The analysis is based on a first-order shear deformation, von Karman type of nonlinear plate theory. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. An efficient reduction method is used in conjunction with mixed finite element models for determining the stability boundary and the postbuckling response of the plate. Sensitivity derivatives are evaluated and used to study the sensitivity of the postbuckling response to variations in the different lamination and material parameters of the plate. For quasi-isotropic plates, numerical results are presented showing the effects of variations in the material characteristics and fiber orientation of individual layers on the postbuckling response of the plate.

Noor, Ahmed K.↗