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NASA NTRS2010

Refined Zigzag Theory for Homogeneous, Laminated Composite, and Sandwich Plates: A Homogeneous Limit Methodology for Zigzag Function Selection

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is presented from a multi-scale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse kinematic field is that of first-order shear-deformation theory, whereas the fine kinematic field has a piecewise-linear zigzag distribution through the thickness. The condition of limiting homogeneity of transverse-shear properties is proposed and yields four distinct sets of zigzag functions. By examining elastostatic solutions for highly heterogeneous sandwich plates, the best-performing zigzag functions are identified. The RZT predictive capabilities to model homogeneous and highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy ; and a wide range of applicability. The present theory, which is derived from the virtual work principle, is well-suited for developing computationally efficient CO-continuous finite elements, and is thus appropriate for the analysis and design of high-performance load-bearing aerospace structures.

Tessler, Alexander↗

DOE OSTI2023

Automated Symbolic Upscaling: 2. Model Generation for Extended Applicability Regimes

Abstract In this second part of the two paper series, we detail an algorithmic procedure for systematically implementing the generalized closure form strategy presented in Part 1. This strategy extends the applicability of homogenized models with respect to classical homogenization theory, as demonstrated in Part 1 where upscaled models are rigorously derived in moderately reactive physical regimes. After encoding the algorithm into Symbolica, an automated upscaling framework, we upscale two reactive mass transport problems and numerically validate the resulting nonlinear homogenized models by showing the absolute error estimates predicted by homogenization theory are satisfied. In both problems, nontrivial closure forms and closure problems are automatically formulated using the encoded strategy with no human interaction, nor prior knowledge regarding the closure required for the systems. We hope these demonstrations spark further interest in automated analytical frameworks for multiscale modeling, as such capabilities are invaluable for generating rigorous multiscale models of complex phenomena in porous media.

Pietrzyk, Kyle↗

DOE OSTI2023

Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

Abstract In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy‐scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore‐scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Pietrzyk, Kyle↗

NASA NTRS2012

Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates

This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C0-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite elements provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.

Tessler, Alexander↗

NASA NTRS2012

Analytic and Computational Perspectives of Multi-Scale Theory for Homogeneous, Laminated Composite, and Sandwich Beams and Plates

This paper reviews the theoretical foundation and computational mechanics aspects of the recently developed shear-deformation theory, called the Refined Zigzag Theory (RZT). The theory is based on a multi-scale formalism in which an equivalent single-layer plate theory is refined with a robust set of zigzag local layer displacements that are free of the usual deficiencies found in common plate theories with zigzag kinematics. In the RZT, first-order shear-deformation plate theory is used as the equivalent single-layer plate theory, which represents the overall response characteristics. Local piecewise-linear zigzag displacements are used to provide corrections to these overall response characteristics that are associated with the plate heterogeneity and the relative stiffnesses of the layers. The theory does not rely on shear correction factors and is equally accurate for homogeneous, laminated composite, and sandwich beams and plates. Regardless of the number of material layers, the theory maintains only seven kinematic unknowns that describe the membrane, bending, and transverse shear plate-deformation modes. Derived from the virtual work principle, RZT is well-suited for developing computationally efficient, C(sup 0)-continuous finite elements; formulations of several RZT-based elements are highlighted. The theory and its finite element approximations thus provide a unified and reliable computational platform for the analysis and design of high-performance load-bearing aerospace structures.

Tessler, Alexander↗

DOE OSTI2024

Theories of homogeneous and electrochemical electron transfer in complex media and interfaces (Final Technical Report)

This project makes the next step in establishing practical theories of charge transfer in complex media. The development of formal models is supported by extensive atomistic simulations, quantum calculations of force-field parameters, and direct measurements of charge-transfer spectra. All theory development is supported by experiment, extensive numerical simulations, and through external collaborations.

14 SOLAR ENERGY↗

NASA NTRS1973

Repeated cascade theory of homogeneous turbulence.

The problem of turbulent spectrum engenders two coupled hierarchies: one originates from the development of stress, leading to a transfer function, and the other from the development of an eddy viscosity. In order to incorporate physical roles among scales, the turbulent velocity fluctuation is decomposed into a series of ranks in the increasing order of randomness, contributing successively to energy or stress, eddy viscosity, relaxation frequency, and higher-rank frequencies in the memory chain. As a result, the first hierarchy mentioned above becomes closed at the quadrupole correlation. The second hierarchy governs the eddy viscosities of different ranks, related to relaxation frequencies of such ranks, in the form of a memory chain. It is cut off by an implicit viscous mechanism. For zero wind gradient, the spectrum in the inertial subrange recovers the Kolmogoroff k to the minus 5/3 law with a numerical constant 1.58, in good agreement with experiments. For a strong wind gradient, the spectrum in the production subrange has a k to the minus 1 law.

Tchen, C. M.↗

NASA NTRS1986

Theory of homogeneous nucleation - A chemical kinetic view

A simple function with two undetermined parameters has been used in place of the Thomson-Gibbs relation to relate the activation energy of the vaporization reaction to cluster size. The parameters are iterated to assume optimum values in numerical computation so experimental data may be correlated. Calculations show this approach closely predicts and correlates available data for water, benzene, and ethanol. The nucleation formulism is redeveloped with an emphasis on the chemical kinetic view. Surface tension of the liquid and free energy of droplet formation are not used in its derivation.

Yang, C. H.↗

NASA NTRS1990

Mathematical theory of a relaxed design problem in structural optimization

Various attempts have been made to construct a rigorous mathematical theory of optimization for size, shape, and topology (i.e. layout) of an elastic structure. If these are represented by a finite number of parametric functions, as Armand described, it is possible to construct an existence theory of the optimum design using compactness argument in a finite dimensional design space or a closed admissible set of a finite dimensional design space. However, if the admissible design set is a subset of non-reflexive Banach space such as L(sup infinity)(Omega), construction of the existence theory of the optimum design becomes suddenly difficult and requires to extend (i.e. generalize) the design problem to much more wider class of design that is compatible to mechanics of structures in the sense of variational principle. Starting from the study by Cheng and Olhoff, Lurie, Cherkaev, and Fedorov introduced a new concept of convergence of design variables in a generalized sense and construct the 'G-Closure' theory of an extended (relaxed) optimum design problem. A similar attempt, but independent in large extent, can also be found in Kohn and Strang in which the shape and topology optimization problem is relaxed to allow to use of perforated composites rather than restricting it to usual solid structures. An identical idea is also stated in Murat and Tartar using the notion of the homogenization theory. That is, introducing possibility of micro-scale perforation together with the theory of homogenization, the optimum design problem is relaxed to construct its mathematical theory. It is also noted that this type of relaxed design problem is perfectly matched to the variational principle in structural mechanics.

Kikuchi, Noboru↗

NASA NTRS1974

On the marginally stable saturation spectrum of unstable type I equatorial electrojet irregularities

Formulation of a self-consistent convective nonlinear theory of type I irregularities in the equatorial electrojet. It is found that a combination of three mechanisms - convective amplification, quasi-linear polarization electric field reduction, and nonlinear particle orbit diffusion damping - accounts for radar backscatter observations of a ubiquitous marginally stable (or 'constant ion-acoustic Doppler shift') saturation spectrum better than any of the three mechanisms treated separately. In particular, no spatially homogeneous theory without wave refraction can account for the observations. Wave refraction alone or with quasi-linear polarization electric field reduction is also inadequate. Wave refraction, quasi-linear polarization reduction, and particle orbit diffusion theory appear to account for type I observations at radar elevation angles less than 60 deg. Vertical type I backscatter cannot be explained without modifying the present laminar electrojet model.

Lee, K.↗

DOE OSTI2024

Enforcing global constraints for the dispersion closure problem: τ 2 -SIMPLE algorithm

Permeability and effective dispersion tensors are critical parameters to characterize flow and transport in porous media at the continuum scale. Homogenization theory defines a framework in which such effective properties are first computed from solving a closure problem in a repeating unit cell of the periodic microstructure and then used in a macroscopic formulation for efficient computation. The closure problem is formulated as a local boundary value problem subjected to global constraints, which guarantee the uniqueness of the solution and can be difficult to satisfy for complex geometries and at high flow conditions. These constraints also ensure that pore-scale pressure, velocity, and concentration fields can be accurately reconstructed from the closure variable. Building on a previous work, here we present a framework that allows to satisfy global constraints associated to both the permeability and the dispersion closure problems by introducing two artificial time scales. The algorithm, called τ 2 -SIMPLE, computes both permeability and effective dispersion given an arbitrarily complex geometry and flow condition. Furthermore, this algorithm is demonstrated to be accurate for both 2D and 3D geometries across varying flow conditions, and thus it can be used to quickly characterize effective properties from porous media images in many applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

The Thermo-Elastic Properties and Damping of U-6wt%Nb

While thermal expansion data exists for quenched (as well as aged) U-6wt%Nb, there is wide variation in the reported room temperature elastic moduli. To better understand the room temperature elastic behavior and to address the complete absence of data on the temperature dependence of the elastic moduli, room temperature and in-situ elevated temperature resonant ultrasonic spectroscopy (RUS) was performed along with x-ray diffraction (XRD) and dilatometry using a thermomechanical analyzer (TMA). An in-situ small- and wide-angle x-ray scattering (SAXS/WAXS) experiment was performed on a companion sample to help interpret the results. The room temperature polycrystalline dynamic moduli were measured to be E = 95.4 ± 4.5 GPa and G = 35.2 ± 0.2 GPa, respectively. As the temperature is raised, the stiffness slowly decreases, consistent with the empirical rule proposed by Varshni, up to the point of precipitation of the equilibrium α-U phase, which is stiffer than the martensitic phases. Homogenization theory of composites can be used to rationalize the observed response at high temperatures and after high-temperature exposures. Further, aging of the material at low temperatures (≤200°C) does not affect the linear elastic stiffness, but does impact the damping behavior which can be measured through RUS. This change in damping provides another perspective on the microstructure changes induced by low-temperature aging which also result in significant strengthening.

36 MATERIALS SCIENCE↗

NASA NTRS1985

Some Asymptotic Problems in the Optimal Control of Distributed Systems

The optimal control of structures which consist of composite materials or of perforated materials is discussed. Asymptotic formula, derived from the so-called homogenization theory, are presented which allow the replacement of very complicated problems by much simpler ones.

Lions, J. L.↗

NASA NTRS1993

Topology and layout optimization of discrete and continuum structures

The basic features of the ground structure method for truss structure an continuum problems are described. Problems with a large number of potential structural elements are considered using the compliance of the structure as the objective function. The design problem is the minimization of compliance for a given structural weight, and the design variables for truss problems are the cross-sectional areas of the individual truss members, while for continuum problems they are the variable densities of material in each of the elements of the FEM discretization. It is shown how homogenization theory can be applied to provide a relation between material density and the effective material properties of a periodic medium with a known microstructure of material and voids.

Bendsoe, Martin P.↗

NASA NTRS1993

Multigrid methods for differential equations with highly oscillatory coefficients

New coarse grid multigrid operators for problems with highly oscillatory coefficients are developed. These types of operators are necessary when the characters of the differential equations on coarser grids or longer wavelengths are different from that on the fine grid. Elliptic problems for composite materials and different classes of hyperbolic problems are practical examples. The new coarse grid operators can be constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Convergence analysis based on the homogenization theory is given for elliptic problems with periodic coefficients and some hyperbolic problems. These are classes of equations for which there exists a fairly complete theory for the interaction between shorter and longer wavelengths in the problems. Numerical examples are presented.

Engquist, Bjorn↗

NASA NTRS1974

Nonstationary homogeneous nucleation

The theory of homogeneous condensation is reviewed and equations describing this process are presented. Numerical computer solutions to transient problems in nucleation (relaxation to steady state) are presented and compared to a prior computation.

Harstad, K. G.↗

NASA NTRS1994

Rapid Distortion Theory for Compressible Homogeneous Turbulence Under Isotropic Mean Strain

The case of isotropic compressible turbulence subjected to rapid isotropic compression is studied using inviscid rapid distortion theory and direct numerical simulation. An exact solution to the rapid distortion problem is given, and results are compared to those of direct numerical simulation. Implications for modelling turbulent flows are discussed.

Blaisdell, G. A.↗

NASA NTRS1991

Homogeneous freezing nucleation of stratospheric solution droplets

The classical theory of homogeneous nucleation was used to calculate the freezing rate of sulfuric acid solution aerosols under stratospheric conditions. The freezing of stratospheric aerosols would be important for the nucleation of nitric acid trihydrate particles in the Arctic and Antarctic stratospheres. In addition, the rate of heterogeneous chemical reactions on stratospheric aerosols may be very sensitive to their state. The calculations indicate that homogeneous freezing nucleation of pure water ice in the stratospheric solution droplets would occur at temperatures below about 192 K. However, the physical properties of H2SO4 solution at such low temperatures are not well known, and it is possible that sulfuric acid aerosols will freeze out at temperatures ranging from about 180 to 195 K. It is also shown that the temperature at which the aerosols freeze is nearly independent of their size.

Jensen, Eric J.↗