Results for “nodal formulation” — Page 5

NASA NTRS1992

A three-dimensional nonlinear Timoshenko beam based on the core-congruential formulation

A three-dimensional, geometrically nonlinear two-node Timoshenkoo beam element based on the total Larangrian description is derived. The element behavior is assumed to be linear elastic, but no restrictions are placed on magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for the bending-stretching and bending-torsional coupling effects without special provisions. The core-congruential formulation (CCF) is used to derived the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples is presented to test and validate the present element.

Crivelli, Luis A.↗

NASA NTRS1984

Formulation of blade-flutter spectral analyses in stationary reference frame

Analytic representations are developed for the discrete blade deflection and the continuous tip static pressure fields in a stationary reference frame. Considered are the sampling rates equal to the rotational frequency, equal to blade passing frequency, and for the pressure, equal to a multiple of the blade passing frequency. For the last two rates the expressions for determining the nodal diameters from the spectra are included. A procedure is presented for transforming the complete unsteady pressure field into a rotating frame of reference. The determination of the true flutter frequency by using two sensors is described. To illustrate their use, the developed procedures are used to interpret selected experimental results.

Kurkov, A. P.↗

NASA NTRS1981

On the shape and orientation control of an orbiting shallow spherical shell structure

A proposed method for controlling the shape and orientation of very large shallow dish type receiver/reflectors to be used in communication, radiometry and in electronic orbital based mail systems involves connecting a rigid light weight dumbell with heavy tip masses to the shell at its apex by a spring loaded double gimballed joint with dampling. To completely damp the system transient motion in all of the important lower frequency modes, an active control system is required. A mathematical model is extended to include the effects of point actuators located at preselected positions on the shell surface. The formulation of the uncontrolled dynamics assumes an a priori knowledge of the frequencies of all the elastic modes to be incorporated within the system model. As an example, three rigid body modes and six elastic modes are included in the model and six actuators are assumed, none of which lies on a nodal line or circle.

Bainum, P. M.↗

NASA NTRS1974

Quantized vortices around wavefunction nodes. II

The expectation of quantized vortices around the nodes of wavefunctions is formulated in terms of Schroedinger quantum mechanics, and the derivation is shown to depend only on the requirement that the wavefunction be single-valued and continuous. The implications of circulation for the topology of the nodes are discussed, and consideration is given to the nodes and circulation of a smooth wavefunction in D dimensions and to the significance of exceptional nodal points. The theoretical results are illustrated by an example of an infinite field of regularly spaced vortices.

Hirschfelder, J. O.↗

NASA NTRS1974

Quantum mechanical streamlines. I - Square potential barrier

Exact numerical calculations are made for scattering of quantum mechanical particles hitting a square two-dimensional potential barrier (an exact analog of the Goos-Haenchen optical experiments). Quantum mechanical streamlines are plotted and found to be smooth and continuous, to have continuous first derivatives even through the classical forbidden region, and to form quantized vortices around each of the nodal points. A comparison is made between the present numerical calculations and the stationary wave approximation, and good agreement is found between both the Goos-Haenchen shifts and the reflection coefficients. The time-independent Schroedinger equation for real wavefunctions is reduced to solving a nonlinear first-order partial differential equation, leading to a generalization of the Prager-Hirschfelder perturbation scheme. Implications of the hydrodynamical formulation of quantum mechanics are discussed, and cases are cited where quantum and classical mechanical motions are identical.

Hirschfelder, J. O.↗

DOE OSTI2023

Recent Hydrodynamic Modeling Enhancements in OpenFAST

To improve the hydrodynamic modeling capabilities of the offshore wind design and modeling tool OpenFAST, a restructuring of the HydroDyn hydrodynamics module was undertaken with several new features implemented. The generation of the wave field is now separated from HydroDyn into a new module called SeaState. Unlike previous versions of HydroDyn, which precomputed the wave kinematics at the undisplaced positions of the hydrodynamic nodes, the new SeaState module computes the complete time history of the wave field at the vertices of a user-defined wave grid. During the simulation, the wave kinematics at any point within the grid can be efficiently interpolated, allowing the strip-theory wave loads to be evaluated based on the wave kinematics at the displaced structure position. A phase correction was also implemented for the potential-flow wave-exciting loads to account for large structure displacements. Several wave-stretching methods were implemented for the strip-theory solution along with a wave-load redistribution method that ensures smooth variation of the nodal loads as the nodes of a discrete hydrodynamic mesh enter and exit the water. The load smoothing is included to avoid the excitation of unphysical high-frequency structural vibration during hydroelastic simulations. The MacCamy-Fuchs diffraction correction for the strip-theory solution and the capability to insert a constrained NewWave into extreme stochastic sea states with directional spreading were also added. The present article documents the formulations of the new improvements to HydroDyn with example applications and numerical results.

constrained wave↗

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.↗

DOE OSTI2022

Recent Hydrodynamic Modeling Enhancements in OpenFAST: Preprint

To improve the hydrodynamic modeling capabilities of the offshore wind design and modeling tool OpenFAST, a restructuring of the HydroDyn hydrodynamics module was undertaken with several new features implemented. The generation of the wave field is now separated from HydroDyn into a new module called SeaState. Unlike previous versions of HydroDyn, which precomputed the wave kinematics at the undisplaced positions of the hydrodynamic nodes, the new SeaState module computes the complete time history of the wave field at the vertices of a user-defined wave grid. During the simulation, the wave kinematics at any point within the grid can be efficiently interpolated, allowing the strip-theory wave loads to be evaluated based on the wave kinematics at the displaced structure position. A phase correction was also implemented for the potential-flow wave-exciting loads to account for large structure displacements. Several wave-stretching methods were implemented for the strip-theory solution along with a wave-load redistribution method that ensures smooth variation of the nodal loads as the nodes of a discrete hydrodynamic mesh enter and exit the water. The load smoothing is included to avoid the excitation of unphysical high-frequency structural vibration during hydroelastic simulations. The MacCamy-Fuchs diffraction correction for the strip-theory solution and the capability to insert a constrained NewWave into extreme stochastic sea states with directional spreading were also added. The present article documents the formulations of the new improvements to HydroDyn with example applications and numerical results.

constrained wave↗

DOE OSTI2026

PDE-constrained high-order mesh optimization

Here, we present a novel framework for PDE-constrained r-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regularization. We adopt the Target-Matrix Optimization Paradigm to control geometric properties across the mesh, independent of the PDE of interest. To incorporate the accuracy of the PDE solution, we introduce error measures that control the finite element discretization error. The implicit dependence of these error measures on the mesh nodal positions is accurately captured by adjoint sensitivity analysis. Additionally, a convolution-based gradient regularization strategy is used to ensure stable and effective adaptation of high-order meshes. We demonstrate that the proposed framework can improve mesh quality and reduce the error by up to 10 times for the solution of Poisson and linear elasto-static problems. The approach is general with respect to the dimensionality, the order of the mesh, the types of mesh elements, and can be applied to any PDE that admits well-defined adjoint operators.

Computer science↗

DOE OSTI2024

Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. In conclusion, we demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2011

Extension of Ko Straight-Beam Displacement Theory to Deformed Shape Predictions of Slender Curved Structures

The Ko displacement theory originally developed for shape predictions of straight beams is extended to shape predictions of curved beams. The surface strains needed for shape predictions were analytically generated from finite-element nodal stress outputs. With the aid of finite-element displacement outputs, mathematical functional forms for curvature-effect correction terms are established and incorporated into straight-beam deflection equations for shape predictions of both cantilever and two-point supported curved beams. The newly established deflection equations for cantilever curved beams could provide quite accurate shape predictions for different cantilever curved beams, including the quarter-circle cantilever beam. Furthermore, the newly formulated deflection equations for two-point supported curved beams could provide accurate shape predictions for a range of two-point supported curved beams, including the full-circular ring. Accuracy of the newly developed curved-beam deflection equations is validated through shape prediction analysis of curved beams embedded in the windward shallow spherical shell of a generic crew exploration vehicle. A single-point collocation method for optimization of shape predictions is discussed in detail

Ko, William L.↗

DOE OSTI2024

Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagrangian–Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. Here, the efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1987

A simple and efficient shear-flexible plate bending element

A shear-flexible triangular element formulation, which utilizes an assumed quadratic displacement potential energy approach and is numerically integrated using Gauss quadrature, is presented. The Reissner/Mindlin hypothesis of constant cross-sectional warping is directly applied to the three-dimensional elasticity theory to obtain a moderately thick-plate theory or constant shear-angle theory (CST), wherein the middle surface is no longer considered to be the reference surface and the two rotations are replaced by the two in-plane displacements as nodal variables. The resulting finite-element possesses 18 degrees of freedom (DOF). Numerical results are obtained for two different numerical integration schemes and a wide range of meshes and span-to-thickness ratios. These, when compared with available exact, series or finite-element solutions, demonstrate accuracy and rapid convergence characteristics of the present element. This is especially true in the case of thin to very thin plates, when the present element, used in conjunction with the reduced integration scheme, outperforms its counterpart, based on discrete Kirchhoff constraint theory (DKT).

Chaudhuri, Reaz A.↗

NASA NTRS2002

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

Atluri, Satya N.↗

DOE OSTI2022

An Immersed Finite Element Lagrangian-Eulerian Code-Coupling Framework

This report presents an assessment of immersed Eulerian-Lagrangian code-coupling techniques suitable for use in a broad range of mechanics applications. The coupling algorithm is based on an immersed finite element method that considers the Lagrangian and Eulerian overlap regions in the overall variational formulation. In this report the basic formulation details are presented followed by various aspects of the code-coupling algorithm using OpenIFEM as the Lagrangian/coupling framework. A series of representative test cases that illustrate the code-coupling algorithm are discussed. The current work provides an in-depth investigation into the immersed finite element method for the purposes of providing a rigorous coupling technique that is minimally invasive in the respective Eulerian and Lagrangian codes. A number of extensions to the base immersed finite element method have been examined. These extension include nodal and quadrature-based indicator functions, a Lagrangian volume-fraction calculation in regions of overlap, and the use of penalty constraints between the Lagrangian and Eulerian domains. A unique MPI-based coupling strategy that retains the independent MPI structure of each code has been demonstrated.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Resilient Operation of Power Distribution Systems Using MPC-Based Critical Service Restoration

Power distribution systems are more prone to disruptions and cause most power system outages. We propose a service restoration technique to recover the system service (electricity delivery) following an extreme event-triggered substation outage. The proposed technique considers the problem of controlling distributed energy resources (DERs) of a distribution system with the objective of achieving maximum load pick up while satisfying network flow and voltage constraints. The problem is formulated as a model predictive control (MPC), where a linearized optimal power flow (OPF) model is employed to describe the network. The formulation is augmented with a ramping (up) reserve product for the DERs to ensure an upward monotonic load restoration as time evolves. We perform simulations considering the IEEE 13-bus test feeder integrated with wind, solar, microturbine, and energy storage devices. We demonstrate the efficacy of the devised technique in restoring the system loads monotonically, without shedding previously restored loads. We also show the benefit of co-optimization of power and reserve products for DERs on service restoration. In addition, the capability of the technique in regulating nodal voltages and reducing renewable power curtailment is demonstrated.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗