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NASA NTRS1995

Sensitivity analysis of the non-linear dynamic viscoplastic response of 2-d structures with respect to material parameters

A computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain-Cauchy stress formulation is used, and the geometric non-linearities arising from large strains are incorporated. The Jaumann rate is used as a frame indifferent stress rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi-discretization are integrated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central cutout subjected to an in-plane compressive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.

Kulkarni, Makarand↗

NASA NTRS1999

Recent Development in the Analytical Investigation of Shipboard Rotorcraft Engage and Disengage Operations

This paper presents an overview of recent developments in an effort to predict transient aeroelastic rotor response during shipboard engage and disengage sequences. The blade is modeled as an elastic beam undergoing in flap, lag, extension and torsion. The blade equations of motion are formulated using Hamilton's principle and they are spatially discretized using the finite element method. The discretized blade equations of motion are integrated for a specified rotor speed run-up or run-down profile. Blade element theory is used to calculate quasi-steady or unsteady aerodynamic loads in linear and nonlinear regimes. The analysis is capable of simulating both articulated, hingeless, and gimballed rotor systems. Validation of the rotor code is discussed, including correlation with droop stop impact tests and wind tunnel experiments. Predictions of safe engagement and disengagement envelopes, limited by excessive blade tip deflections or hub moments, are presented. Future directions of study are also discussed.

Smith, Edward C.↗

NASA NTRS1994

Coupled electromechanical response of composite beams with embedded piezoelectric sensors and actuators

Unified mechanics are developed with the capability to model both sensory and active composite laminates with embedded piezoelectric layers. A discrete-layer formulation enables analysis of both global and local electromechanical response. The mechanics include the contributions from elastic, piezoelectric, and dielectric components. The incorporation of electric potential into the state variables permits representation of general electromechanical boundary conditions. Approximate finite element solutions for the static and free-vibration analysis of beams are presented. Applications on composite beams demonstrate the capability to represent either sensory or active structures and to model the complicated stress-strain fields, the interactions between passive/active layers, interfacial phenomena between sensors and composite plies, and critical damage modes in the material. The capability to predict the dynamic characteristics under various electrical boundary conditions is also demonstrated.

Saravanos, D. A.↗

NASA NTRS2006

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

Watson, Willie R.↗

DOE OSTI2013

Final Report. Analysis and Reduction of Complex Networks Under Uncertainty

The project was a collaborative effort among MIT, Sandia National Laboratories (local PI Dr. Habib Najm), the University of Southern California (local PI Prof. Roger Ghanem), and The Johns Hopkins University (local PI Prof. Omar Knio, now at Duke University). Our focus was the analysis and reduction of large-scale dynamical systems emerging from networks of interacting components. Such networks underlie myriad natural and engineered systems. Examples important to DOE include chemical models of energy conversion processes, and elements of national infrastructure—e.g., electric power grids. Time scales in chemical systems span orders of magnitude, while infrastructure networks feature both local and long-distance connectivity, with associated clusters of time scales. These systems also blend continuous and discrete behavior; examples include saturation phenomena in surface chemistry and catalysis, and switching in electrical networks. Reducing size and stiffness is essential to tractable and predictive simulation of these systems. Computational singular perturbation (CSP) has been effectively used to identify and decouple dynamics at disparate time scales in chemical systems, allowing reduction of model complexity and stiffness. In realistic settings, however, model reduction must contend with uncertainties, which are often greatest in large-scale systems most in need of reduction. Uncertainty is not limited to parameters; one must also address structural uncertainties—e.g., whether a link is present in a network—and the impact of random perturbations, e.g., fluctuating loads or sources. Research under this project developed new methods for the analysis and reduction of complex multiscale networks under uncertainty, by combining computational singular perturbation (CSP) with probabilistic uncertainty quantification. CSP yields asymptotic approximations of reduceddimensionality “slow manifolds” on which a multiscale dynamical system evolves. Introducing uncertainty in this context raised fundamentally new issues, e.g., how is the topology of slow manifolds transformed by parametric uncertainty? How to construct dynamical models on these uncertain manifolds? To address these questions, we used stochastic spectral polynomial chaos (PC) methods to reformulate uncertain network models and analyzed them using CSP in probabilistic terms. Finding uncertain manifolds involved the solution of stochastic eigenvalue problems, facilitated by projection onto PC bases. These problems motivated us to explore the spectral properties stochastic Galerkin systems. We also introduced novel methods for rank-reduction in stochastic eigensystems—transformations of a uncertain dynamical system that lead to lower storage and solution complexity. These technical accomplishments are detailed below. This report focuses on the MIT portion of the joint project.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2016

Stochastic-Strength-Based Damage Simulation of Ceramic Matrix Composite Laminates

The Finite Element Analysis-Micromechanics Analysis Code/Ceramics Analysis and Reliability Evaluation of Structures (FEAMAC/CARES) program was used to characterize and predict the progressive damage response of silicon-carbide-fiber-reinforced reaction-bonded silicon nitride matrix (SiC/RBSN) composite laminate tensile specimens. Studied were unidirectional laminates [0] (sub 8), [10] (sub 8), [45] (sub 8), and [90] (sub 8); cross-ply laminates [0 (sub 2) divided by 90 (sub 2),]s; angled-ply laminates [plus 45 (sub 2) divided by -45 (sub 2), ]s; doubled-edge-notched [0] (sub 8), laminates; and central-hole laminates. Results correlated well with the experimental data. This work was performed as a validation and benchmarking exercise of the FEAMAC/CARES program. FEAMAC/CARES simulates stochastic-based discrete-event progressive damage of ceramic matrix composite and polymer matrix composite material structures. It couples three software programs: (1) the Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC), (2) the Ceramics Analysis and Reliability Evaluation of Structures Life Prediction Program (CARES/Life), and (3) the Abaqus finite element analysis program. MAC/GMC contributes multiscale modeling capabilities and micromechanics relations to determine stresses and deformations at the microscale of the composite material repeating-unit-cell (RUC). CARES/Life contributes statistical multiaxial failure criteria that can be applied to the individual brittle-material constituents of the RUC, and Abaqus is used to model the overall composite structure. For each FEAMAC/CARES simulation trial, the stochastic nature of brittle material strength results in random, discrete damage events that incrementally progress until ultimate structural failure.

composite structures↗

NASA NTRS2016

Analysis and Characterization of Damage and Failure Utilizing a Generalized Composite Material Model Suitable for Use in Impact Problems

A material model which incorporates several key capabilities which have been identified by the aerospace community as lacking in state-of-the art composite impact models is under development. In particular, a next generation composite impact material model, jointly developed by the FAA and NASA, is being implemented into the commercial transient dynamic finite element code LS-DYNA. The material model, which incorporates plasticity, damage, and failure, utilizes experimentally based tabulated input to define the evolution of plasticity and damage and the initiation of failure as opposed to specifying discrete input parameters (such as modulus and strength). The plasticity portion of the orthotropic, three-dimensional, macroscopic composite constitutive model is based on an extension of the Tsai-Wu composite failure model into a generalized yield function with a non-associative flow rule. For the damage model, a strain equivalent formulation is utilized to allow for the uncoupling of the deformation and damage analyses. In the damage model, a semi-coupled approach is employed where the overall damage in a particular coordinate direction is assumed to be a multiplicative combination of the damage in that direction resulting from the applied loads in the various coordinate directions. Due to the fact that the plasticity and damage models are uncoupled, test procedures and methods to both characterize the damage model and to covert the material stress-strain curves from the true (damaged) stress space to the effective (undamaged) stress space have been developed. A methodology has been developed to input the experimentally determined composite failure surface in a tabulated manner. An analytical approach is then utilized to track how close the current stress state is to the failure surface.

Finite Element Method↗

NASA NTRS1988

A proposed method for enhanced eigen-pair extraction using finite element methods: Theory and application

The paper covers two distinct parts: theory and application. The goal of this work was the reduction of model size with an increase in eigenvalue/vector accuracy. This method is ideal for the condensation of large truss- or beam-type structures. The theoretical approach involves the conversion of a continuum transfer matrix beam element into an 'Exact' dynamic stiffness element. This formulation is implemented in a finite element environment. This results in the need to solve a transcendental eigenvalue problem. Once the eigenvalue is determined the eigenvectors can be reconstructed with any desired spatial precision. No discretization limitations are imposed on the reconstruction. The results of such a combined finite element and transfer matrix formulation is a much smaller FEM eigenvalue problem. This formulation has the ability to extract higher eigenvalues as easily and as accurately as lower eigenvalues. Moreover, one can extract many more eigenvalues/vectors from the model than the number of degrees of freedom in the FEM formulation. Typically, the number of eigenvalues accurately extractable via the 'Exact' element method are at least 8 times the number of degrees of freedom. In contrast, the FEM usually extracts one accurate (within 5 percent) eigenvalue for each 3-4 degrees of freedom. The 'Exact' element results in a 20-30 improvement in the number of accurately extractable eigenvalues and eigenvectors.

Jara-Almonte, J.↗

DOE OSTI2015

Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method

Here, we have investigated the use of the adaptive high-order finite-element method (FEM) for geoelectromagnetic modeling. Because high-order FEM is challenging from the numerical and computational points of view, most published finite-element studies in geoelectromagnetics use the lowest order formulation. Solution of the resulting large system of linear equations poses the main practical challenge. We have developed a fully parallel and distributed robust and scalable linear solver based on the optimal block-diagonal and auxiliary space preconditioners. The solver was found to be efficient for high finite element orders, unstructured and nonconforming locally refined meshes, a wide range of frequencies, large conductivity contrasts, and number of degrees of freedom (DoFs). Furthermore, the presented linear solver is in essence algebraic; i.e., it acts on the matrix-vector level and thus requires no information about the discretization, boundary conditions, or physical source used, making it readily efficient for a wide range of electromagnetic modeling problems. To get accurate solutions at reduced computational cost, we have also implemented goal-oriented adaptive mesh refinement. The numerical tests indicated that if highly accurate modeling results were required, the high-order FEM in combination with the goal-oriented local mesh refinement required less computational time and DoFs than the lowest order adaptive FEM.

58 GEOSCIENCES↗

NASA NTRS2017

Incorporation of Failure Into an Orthotropic Three-Dimensional Model with Tabulated Input Suitable for Use in Composite Impact Problems

The need for accurate material models to simulate the deformation, damage and failure of polymer matrix composites under impact conditions is becoming critical as these materials are gaining increased use in the aerospace and automotive communities. The aerospace community has identified several key capabilities which are currently lacking in the available material models in commercial transient dynamic finite element codes. To attempt to improve the predictive capability of composite impact simulations, a next generation material model is being developed for incorporation within the commercial transient dynamic finite element code LS-DYNA. The material model, which incorporates plasticity, damage and failure, utilizes experimentally based tabulated input to define the evolution of plasticity and damage and the initiation of failure as opposed to specifying discrete input parameters such as modulus and strength. The plasticity portion of the orthotropic, three-dimensional, macroscopic composite constitutive model is based on an extension of the Tsai-Wu composite failure model into a generalized yield function with a non-associative flow rule. For the damage model, a strain equivalent formulation is used to allow for the uncoupling of the deformation and damage analyses. In the damage model, a semi-coupled approach is employed where the overall damage in a particular coordinate direction is assumed to be a multiplicative combination of the damage in that direction resulting from the applied loads in various coordinate directions. For the failure model, a tabulated approach is utilized in which a stress or strain based invariant is defined as a function of the location of the current stress state in stress space to define the initiation of failure. Failure surfaces can be defined with any arbitrary shape, unlike traditional failure models where the mathematical functions used to define the failure surface impose a specific shape on the failure surface. In the current paper, the complete development of the failure model is described and the generation of a tabulated failure surface for a representative composite material is discussed.

Impact↗

NASA NTRS1984

On accuracy conditions for the numerical computation of waves

The Helmholtz equation (Delta + K(2)n(2))u = f with a variable index of refraction n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. Such problems can be solved numerically by first truncating the given unbounded domain and imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization (for example, using a finite element method). In practical applications, the mesh size h and the wave number K, are not independent but are constrained by the accuracy of the desired computation. It will be shown that the number of points per wavelength, measured by (Kh)(-1), is not sufficient to determine the accuracy of a given discretization. For example, the quantity K(3)h(2) is shown to determine the accuracy in the L(2) norm for a second-order discretization method applied to several propagation models.

Bayliss, A.↗

NASA NTRS1985

On accuracy conditions for the numerical computation of waves

Bayliss, A.↗

DOE OSTI2019

Axial Tensile Failure Analysis of SiC{sub f}/Ti Composite Based on Continuum Cohesive Zone Model

In this paper, the failure behavior of SiC fiber-reinforced Ti-6Al-4V matrix composite (SiC{sub f}/Ti) under longitudinal loads was studied experimentally and theoretically. Experimental results show that the interface of SiC{sub f}/Ti composite is continuous and integral. Through SEM observations, it was found that transverse matrix cracks initiated from the debonding region of the interface. A three-dimensional representative volume element was developed to simulate the failure process of SiC{sub f}/Ti-6Al-4V composite. A continuum cohesive zone model is employed to describe the debonding behavior at the interface, compared with a discrete one. More accurate simulated results were obtained by a continuum cohesive zone model, corresponding to experimental results. The transverse cracking behavior of the matrix was simulated. The stress–strain curves predicted by the continuum cohesive zone model show a good agreement with the experimental results. The fiber volume fraction, interface strength and interface toughness are investigated to exhibit their influences upon the longitudinal mechanical characteristics of SiC{sub f}/Ti-6Al-4V composite.

36 MATERIALS SCIENCE↗

DOE OSTI2015

A stochastic approach to uncertainty in the equations of MHD kinematics

The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to a simpler model than the fully coupled MHD system, this may introduce some epistemic uncertainty into the model. If the velocity of a physical system is not known with certainty, the magnetic field obtained from the model may not be reflective of the magnetic field seen in experiments. Additionally, uncertainty in physical parameters such as the magnetic resistivity may affect the reliability of predictions obtained from this model. By modeling the velocity and the resistivity as random variables in the MHD kinematics model, we seek to quantify the effects of uncertainty in these fields on the induced magnetic field. We develop stochastic expressions for these quantities and investigate their impact within a finite element discretization of the kinematics equations. We obtain mean and variance data through Monte Carlo simulation for several test problems. Toward this end, we develop and test an efficient block preconditioner for the linear systems arising from the discretized equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Fast computation of synthetic seismograms within a medium containing remote localized perturbations: a numerical solution to the scattering problem

We derive a fast discrete solution to the scattering problem. This solution allows us to compute accurate synthetic seismograms or waveforms for arbitrary locations of sources and receivers within a medium containing localized perturbations. The key to efficiency is that wave propagation modelling does not need to be carried out in the entire volume that encompasses the sources and the receivers but only within the subvolume containing the perturbations or scatterers. The proposed solution has important applications, for example, it permits the imaging of remote targets located in regions where no sources or receivers are present. Our solution relies on domain decomposition: within a small volume that contains the scatterers, wave propagation is modelled numerically, while in the surrounding volume, where the medium isn't perturbed, the response is obtained through wavefield extrapolation. The originality of this work is the derivation of discrete formulas for representation theorems and Kirchhof–Helmholtz integrals that naturally adapt to the numerical scheme employed for modelling wave propagation. Our solution applies, for example, to finite difference methods or finite/spectral elements methods. The synthetic seismograms obtained with our solution can be considered ‘exact’ as the total numerical error is comparable to that of the method employed for modelling wave propagation. We detail a basic implementation of our solution in the acoustic case using the finite difference method and present numerical examples that demonstrate the accuracy of the method. We show that ignoring some terms accounting for higher order scattering effects in our solution has a limited effect on the computed seismograms and significantly reduces the computational effort. Finally, we show that our solution can be used to compute localized sensitivity kernels and we discuss applications to target oriented imaging. Extension to the elastic case is straightforward and summarized in a dedicated section.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2018

Simulation of Stochastic Mud-Crack Damage Formation in an Environmental Barrier Coating

The integrated Finite Element Analysis–Micromechanics Analysis Code/Ceramics Analysis and Reliability Evaluation of Structures (FEAMAC/CARES) program was used to simulate the formation of mudflat-cracks from thermomechanical loading on a multi-layered Environmental Barrier Coating (EBC) system deposited on a ceramic substrate. FEAMAC/CARES combines MAC/GMC multiscale composite micromechanics code with CARES/Life probabilistic multiaxial failure criteria code and Abaqusfinite element analysis. In this work, step function elastic modulus reduction of randomly damaged finite elements was used to represent discrete cracking events. The use of many small-sized low-aspect-ratio finite elements enabled the depiction of crack boundaries and formation of mudflat patterned damage. Demonstrated examples include finite element models of button–sized disk–shaped 3-D specimen, and a 2-D model of through-the-thickness cross-section. All models were subjected to a progressive cool down from 1300oC to room temperature. Mudflat crack damage in the coating system resulted from the buildup of residual tensile stresses between the individual material constituents from thermal expansion mismatch. A 2-parameter Weibull distribution characterized the coating layer stochastic strength response and the effect of the Weibull modulus on the formation of damage was studied here.

residual tensile stress↗

NASA NTRS2018

Weibull-Based Stochastic Simulation of Mud-Crack Damage Formation in an Environmental Barrier Coating

The integrated Finite Element Analysis Micromechanics Analysis Code/Ceramics Analysis and Reliability Evaluation of Structures (FEAMAC/CARES) program was used to simulate the formation of mudflat-cracks from thermomechanical loading on a multi-layered Environmental Barrier Coating (EBC) system deposited on a ceramic substrate. FEAMAC/CARES combines MAC/GMC (Micromechanics Analysis Code/Generalized Method of Cells) multiscale composite micromechanics code with CARES/Life probabilistic multiaxial failure criteria code and Abaqusfinite element analysis. In this work, step function elastic modulus reduction of randomly damaged finite elements was used to represent discrete cracking events. The use of many small-sized low-aspect-ratio finite elements enabled the depiction of crack boundaries and formation of mudflat patterned damage. Demonstrated examples include finite element models of button-sized disk-shaped 3-D specimen, and a 2-D model of through-the-thickness cross-section. All models were subjected to a progressive cool down from 1300 degrees Centigrade to room temperature. Mudflat crack damage in the coating system resulted from the buildup of residual tensile stresses between the individual material constituents from thermal expansion mismatch. A 2-parameter Weibull distribution characterized the coating layer stochastic strength response and the effect of the Weibull modulus on the formation of damage was studied here.

Residual Tensile Stress↗

NASA NTRS2016

Mesh Convergence Requirements for Composite Damage Models

The ability of the finite element method to accurately represent the response of objects with intricate geometry and loading renders the finite element method as an extremely versatile analysis technique for structural analysis. Finite element analysis is routinely used in industry to calculate deflections, stress concentrations, natural frequencies, buckling loads, and much more. The method works by discretizing complex problems into smaller, simpler approximations that are valid over small uniform domains. For common analyses, the maximum size of the elements that can be used is often be determined by experience. However, to verify the quality of a solution, analyses with several levels of mesh refinement should be performed to ensure that the solution has converged. In recent years, the finite element method has been used to calculate the resistance of structures, and in particular that of composite structures. A number of techniques such as cohesive zone modeling, the virtual crack closure technique, and continuum damage modeling have emerged that can be used to predict cracking, delaminations, fiber failure, and other composite damage modes that lead to structural collapse. However, damage models present mesh refinement requirements that are not well understood. In this presentation, we examine different mesh refinement issues related to the representation of damage in composite materials. Damage process zone sizes and their corresponding mesh requirements will be discussed. The difficulties of modeling discontinuities and the associated need for regularization techniques will be illustrated, and some unexpected element size constraints will be presented. Finally, some of the difficulties in constructing models of composite structures capable of predicting transverse matrix cracking will be discussed. It will be shown that to predict the initiation and propagation of transverse matrix cracks, their density, and their saturation may require models that are significantly more refined than those that have been contemplated in the past.

Davila, Carlos G.↗