Search NASA⌕ Search

SEARCH · Search NASA

Results for “Elasticity equations”

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
DOE OSTI2025

10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. 5 x 5 x 5 = 125-point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of 0.1% stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by 360 - 8000 times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.

elasticity equations↗
DOE OSTI2018

Convergence of the multigrid reduction in time algorithm for the linear elasticity equations

This paper presents some recent advances for parallel-in-time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel-in-time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel-in-time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF-relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel-in-time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. Finally, the paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2011

An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials

An in-depth tutorial on the constitutive equations for elastic, anisotropic materials is presented. Basic concepts are introduced that are used to characterize materials, and notions about how anisotropic material deform are presented. Hooke s law and the Duhamel-Neuman law for isotropic materials are presented and discussed. Then, the most general form of Hooke s law for elastic anisotropic materials is presented and symmetry requirements are given. A similar presentation is also given for the generalized Duhamel-Neuman law for elastic, anisotropic materials that includes thermal effects. Transformation equations for stress and strains are presented and the most general form of the transformation equations for the constitutive matrices are given. Then, specialized transformation equations are presented for dextral rotations about the coordinate axes. Next, concepts of material symmetry are introduced and criteria for material symmetries are presented. Additionally, engineering constants of fully anisotropic, elastic materials are derived from first principles and the specialized to several cases of practical importance.

Nemeth, Michael P.↗
DOE OSTI2014

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

Abstract We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

58 GEOSCIENCES↗
NASA NTRS1983

Thermal elastohydrodynamic lubrication of line contacts

A numerical solution to the problem of thermal elastohydrodynamic lubrication of line contacts was obtained by using a finite difference formulation. The solution procedure consists of simultaneous solution of the thermal Reynolds equation, the elasticity equation, and the energy equation subject to appropriate boundary conditions. Pressure distribution, film shape, and temperature distribution were obtained for fully flooded conjunctions, a paraffinic lubricant, and various dimensionless speed parameters while the dimensionless load and materials parameters were held constant. Reduction in the minimum film thickness due to thermal effects (as a ratio of thermal to isothermal minimum film thickness) is given by a simple formula as a function of the thermal loading parameter Q: H(min)/H(min,I) = 10/10+ Q(0.4). Plots of pressure distribution, film shape, temperature distribution, and flow are shown for some representative cases.

Ghosh, M. K.↗
NASA NTRS1985

Thermal elastohydrodynamic lubrication of line contacts

A numerical solution to the problem of thermal elastohydrodynamic lubrication of line contacts was obtained by using a finite difference formulation. The solution procedure consists of simultaneous solution of the thermal Reynolds equation, the elasticity equation, and the energy equation subject to appropriate boundary conditions. Pressure distribution, film shape, and temperature distribution were obtained for fully flooded conjunctions, a paraffinic lubricant, and various dimensionless speed parameters while the dimensionless load and materials parameters were held constant. Reduction in the minimum film thickness due to thermal effects (as a ratio of thermal to isothermal minimum film thickness) is given by a simple formula as a function of the thermal loading parameter Q: H(min)/H(min, I) = 10/10 + Q(0.4). Plots of pressure distribution, film shape, temperature distribution, and flow are shown for some representative cases.

Ghosh, M. K.↗
NASA NTRS1992

Acoustic intensity calculations for axisymmetrically modeled fluid regions

An algorithm for calculating acoustic intensities from a time harmonic pressure field in an axisymmetric fluid region is presented. Acoustic pressures are computed in a mesh of NASTRAN triangular finite elements of revolution (TRIAAX) using an analogy between the scalar wave equation and elasticity equations. Acoustic intensities are then calculated from pressures and pressure derivatives taken over the mesh of TRIAAX elements. Intensities are displayed as vectors indicating the directions and magnitudes of energy flow at all mesh points in the acoustic field. A prolate spheroidal shell is modeled with axisymmetric shell elements (CONEAX) and submerged in a fluid region of TRIAAX elements. The model is analyzed to illustrate the acoustic intensity method and the usefulness of energy flow paths in the understanding of the response of fluid-structure interaction problems. The structural-acoustic analogy used is summarized for completeness. This study uncovered a NASTRAN limitation involving numerical precision issues in the CONEAX stiffness calculation causing large errors in the system matrices for nearly cylindrical cones.

Hambric, Stephen A.↗
NASA NTRS1986

Multigrid method for the equilibrium equations of elasticity using a compact scheme

A compact difference scheme is derived for treating the equilibrium equations of elasticity. The scheme is inconsistent and unstable. A multigrid method which takes into account these properties is described. The solution of the discrete equations, up to the level of discretization errors, is obtained by this method in just two multigrid cycles.

Taasan, S.↗
NASA NTRS1987

Multigrid method for the equilibrium equations of elasticity using a compact scheme

A compact difference scheme is derived for treating the equilibrium equations of elasticity. The scheme is inconsistent and unstable. A multigrid method which takes into account these properties is described. The solution of the discrete equations, up to the level of discretization errors, is obtained by this method in just two multigrid cycles.

Taasan, Shlomo↗
DOE OSTI2018

Comparison of pseudo-static limit equilibrium and elastic wave equation analyses of dynamic earth pressures on retaining structures

The seismic earth pressure increment is typically computed using either pseudo-static limit equilibrium methods or elastic wave equation analyses of the interaction between a retaining structure and backfill material, yet current interpretations of the two methods provide conflicting recommendations. The focus of this study is to compare the seismic earth pressure increment computed using the two methods. This approach is demonstrated by subjecting an initially uniform prototype site selected from standard site classifications to harmonic excitation in one-dimensional equivalent linear analyses. Then, the seismic earth pressure resultant for a rigid wall is computed using the two methods. The limit equilibrium approach utilizes the acceleration records from the equivalent linear analysis to compute a seismic coefficient, whereas the elastic solution incorporates the reduced modulus and damping from the final iteration of the analysis, as well as the relative displacement records. The results presented herein corroborate the findings of recent centrifuge experiments and associated analyses.

58 GEOSCIENCES↗
DOE OSTI2025

High-pressure elasticity and equation of state of the fluoroelastomer Viton® A-500

Viton® A is a semi-crystalline copolymer of polyvinylidene fluoride and hexafluoropropylene used in various engineering applications due to its mechanical properties and chemical inertness. In situ ultrasonic spectroscopy and x-ray radiography measurements were performed in a Paris–Edinburgh press to measure the pressure dependence of the transverse and longitudinal acoustic velocity of the fluoroelastomer A-500 from 2.7 to 5.7 GPa at 296 K. In addition, we performed high-pressure Brillouin scattering measurements to obtain acoustic velocities from ambient pressure to 5.7 GPa to supplement the ultrasonic measurements, especially at low pressures. The acoustic velocities were then used to calculate a pressure–volume (P–V) equation of state, the bulk and shear moduli, and the Poisson's ratio. These quantities are compared with the reported pressure-dependent properties of related polymers over this range of pressures.

Acoustical properties↗
DOE OSTI2023

Acoustic properties, elasticity, and equation of state of glycerol under pressure

Herein we employed high-pressure Brillouin scattering to study the pressure dependencies of acoustic modes of glycerol up to 14 GPa at 300 K. We observed longitudinal acoustic velocities and transverse acoustic velocities for the first time from 5 to 14 GPa. The results allow the determination of a complete set of elastic properties and an accurate determination of the pressure–volume (P–V) equation of state (EOS). EOS parameters, K 0 = 14.9 ± 1.8 GPa and K' 0 = 5.6 ± 0.5, were determined from fits to the data from ambient pressure to 14 GPa. Direct volume measurements of the P–V EOS are consistent with those determined by Brillouin scattering. A deviation from a Cauchy-like relationship for elastic properties was observed, and the pressure dependencies of the photoelastic constants and relaxation times were documented from 5 to 14 GPa. These results have broad implications for glass-forming liquids, viscoelastic theory, and mode coupling theory.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1973

Effect of surface asperity on elastohydrodynamic lubrication

The important aspects of elastohydrodynamic lubrication, with a single, one-dimensional asperity, have been found by solving numerically the coupled transient Reynolds equation and the elasticity equation. Even though the assumption of a single asperity is highly ideal, this study sheds some light on the effect of surface roughness on elastohydrodynamic lubrication. The results show that the film pressure tends to increase more than the steady state pressure, and in particular, the increase in pressure reaches a maximum as the asperity approaches the inlet of the contact region. The asperity height and the pressure increase above the steady state pressure are closely related to each other; the higher the asperity height, the larger the pressure increase. In the pure rolling case, it has been found that a local pressure peak is not developed. However, in the cases of sliding and rolling, a small, local pressure peak is developed on the pressure profile when the asperity moves into the contact region. In general, the overall film thickness profile increases with increasing asperity height, but is not significantly affected by the asperity width. Moreover, the slope of the overall film thickness profile for the transient cases is much greater than the steady state profile, which is approximately constant across the contact width. The increase in the center film thickness also depends upon the width and height of the asperity.

Lee, K.↗
NASA NTRS1985

Elastohydrodynamic lubrication calculations used as a tool to study scuffing

A new Reynolds equation was developed that takes into account the nonlinear viscous behavior of the fluid. The new Reynolds equation considers the nonlinear viscous fluid model of Eyring, the equilibrium equation, the constant mass flow, and the kinematic boundary condition. The new Reynolds equation and the elasticity equation are solved simultaneously by using a system approach and a Newton-Raphson technique. Comparisons are made with results obtained from the classical Reynolds equation. The effects of sliding speed and introducing a bump or a groove within the conjunction are studied. Results are shown for both moderate and heavy loads.

Houpert, L. G.↗