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SAM Theory Manual

The System Analysis Module (SAM) is an advanced and modern system analysis tool under development at Argonne National Laboratory for advanced non-LWR reactor safety analysis. It aims to provide fast-running, modest-fidelity, whole-plant transient analyses capabilities, which are essential for fast turnaround design scoping and engineering analyses of advanced reactor concepts. While SAM is being developed as a system-level modeling and simulation tool, advanced modeling techniques being implemented include a reduced-order three-dimensional module, pseudo 3-D conjugate heat transfer modeling in reactor core, flexible and multi-scale modeling of heat transfer between fluid and structures, in addition to the advances in software environments and design, and numerical methods. SAM aims to be a generic system-level safety analysis tool for advanced non-LWRs, including Liquid-Metal-cooled fast Reactors (LMR), Molten Salt Reactors (MSR), Fluoride-salt-cooled High- temperature Reactors (FHR), and High-Temperature Gas-cooled Reactors (HTGR). SAM takes ad- vantage of advances in physical modeling, numerical methods, and software engineering to enhance its user experience and usability. It utilizes an object-oriented computational framework (MOOSE), and its underlying meshing and finite-element library and linear and non-linear solvers, to leverage the modern advanced software environments and numerical methods. This document provides the theoretical and technical basis of the code to help users understand the underlying physical models (such as governing equations, closure models, and component models), system modeling approaches, numerical discretization and solution methods, and the overall capabilities in SAM. As new code capabilities and features are added, the SAM Theory Manual will be updated periodically to keep it consistent with the state of the development.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

SAM Theory Manual

The System Analysis Module (SAM) is an advanced and modern system analysis tool under development at Argonne National Laboratory for advanced non-LWR reactor safety analysis. It aims to provide fast-running, modest-fidelity, whole-plant transient analyses capabilities, which are essential for fast turnaround design scoping and engineering analyses of advanced reactor concepts. While SAM is being developed as a system-level modeling and simulation tool, advanced modeling techniques being implemented include a reduced-order three-dimensional module, pseudo 3-D conjugate heat transfer modeling in reactor core, flexible and multi-scale modeling of heat transfer between fluid and structures, in addition to the advances in software environments and design, and numerical methods. SAM aims to be a generic system-level safety analysis tool for advanced non-LWRs, including Liquid-Metal-cooled fast Reactors (LMR), Molten Salt Reactors (MSR), Fluoride-salt-cooled High-temperature Reactors (FHR), and High-Temperature Gas-cooled Reactors (HTGR). SAM takes advantage of advances in physical modeling, numerical methods, and software engineering to enhance its user experience and usability. It utilizes an object-oriented computational framework (MOOSE), and its underlying meshing and finite-element library and linear and non-linear solvers, to leverage the modern advanced software environments and numerical methods. This document provides the theoretical and technical basis of the code to help users understand the underlying physical models (such as governing equations, closure models, and component models), system modeling approaches, numerical discretization and solution methods, and the overall capabilities in SAM. As new code capabilities and features are added, the SAM Theory Manual will be updated periodically to keep it consistent with the state of the development.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Sensitivity analysis for the dynamic response of viscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric response of viscoplastic shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. Spatial discretization is performed using finite elements, with discontinuous stress resultants across element interfaces. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) with an implicit constitutive update. The sensitivity coefficients are evaluated using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading, and a circular plate subjected to impulsive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to thickness, mass density, Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients are presented, along with spatial distributions of these quantities at selected times.

Kulkarni, Makarand↗

NASA Tech Briefs, May 2006

T-Shaped Emitter Metal Structures for HBTs Rigorous Estimation of SNR of a PSK Communication Link Advanced Ka-Band Transceiver With Monopulse Tracking EMI Filters for Low-Temperature Applications Lightweight Electronic Camera for Research on Clouds Pilot Weather Advisor System Waveguide Power-Amplifier Module for 80 to 150 GHz Better Back Contacts for Solar Cells on Flexible Substrates Topics covered include:Tunable, Highly Stable Lasers for Coherent Lidar; Improved Photon-Emission-Microscope System; Program Synthesizes UML Sequence Diagrams; Aspect-Oriented Subprogram Synthesizes UML Sequence Diagrams; Updated Computational Model of Cosmic Rays Near Earth; Software for Alignment of Segments of a Telescope Mirror; Simulation of Dropping of Cargo With Parachutes; DAVE-ML Utility Program; Robust Control for the Mercury Laser Altimeter; Thermally Stable Piezoelectric and Pyroelectric Polymers; Combustion Synthesis of Ca3(PO4)2 Net-Shape Surgical Implants; Stochastic Representation of Chaos Using Terminal Attractors; Two High-Temperature Foil Journal Bearings; Using Plates To Represent Fillets in Finite-Element Modeling; Repairing Chipped Silicide Coatings on Refractory Metal Substrates; Simplified Fabrication of Helical Copper Antennas; Graded-Index "Whispering-Gallery" Optical Microresonators; Optical Profilometers Using Adaptive Signal Processing; Manufacture of Sparse-Spectrum Optical Microresonators; Exact Tuning of High-Q Optical Microresonators by Use of UV; Automation for "Direct-to" Clearances in Air-Traffic Control; Improved Traps for Removing Gases From Coolant Liquids; and Lunar Constellation of Frozen Elliptical Inclined Orbits.

Source record↗

Numerical Analysis of Regular Material Point Method and its Application to Multiphase Flows

The material point method (MPM) is gaining wide popularity in engineering research to model and simulate complex multiphase flow dynamics. The method relies on solving the governing equations of motion and transport in a Lagrangian framework using particles also known as material points. The fluid and kinematic properties are stored on the material points while the spatial gradient calculation and temporal integration are performed on a background grid. This Lagrangian framework allows for large deformations, easy integration of constitutive models, and direct import of complex geometries as particles. However, despite their increasing popularity, very few studies have addressed the issues of numerical resolution and stability of MPM techniques. The presence of additional factors such as the number of material points-per-cell, the location of the material points, the CFL-like condition used in time update, and the grid shape functions also increase the complexity of the error analysis when compared to other finite element methods. In this presentation, we analyze the various forms of error incurred in the application of MPM to continuum mechanics and multiphase flows. The effect of the previously mentioned factors on the error dynamics is studied. The application of these principles to canonical and industrial problems is also presented.

high pressure reverse osmosis↗

An unstructured body-of-revolution electromagnetic particle-in-cell algorithm with radial perfectly matched layers and dual polarizations

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell’s equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TE Φ and TM Φ polarizations on the finite element (primal) mesh. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TE Φ polarized fields, (E z , E ρ , B Φ ) and (2) D and H of TM Φ polarized fields, (D Φ , H z , H ρ ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ = 0 when defining the discrete Hodge matrices (generalized finite element mass matrices). Here, a cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TMΦ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

TPSAS-NF1676L-33512-DND

INTRODUCTION: In an effort to quantify the predictive capability of the Livermore Software Technology Corporation (LSTC)-provided Hybrid III (small female, mid-size male, and large male) anthropomorphic test device (ATD) finite element models (FEMs) within aerospace loading environments, the National Aeronautics and Space Administration (NASA) performed a series of sled tests with the isolated head-neck complex of each ATD. These tests and subsequent modeling efforts revealed significant deficiencies in the large male head neck FEM. Differences in both the shape and material properties between the FEM and the physical ATD were identified. In order to improve the predictive response of the FEM, a model development effort was undertaken to update the head-neck complex to better represent the geometric and material properties of the physical ATD.METHODS: To better match the physical geometry, a new neck component model was meshed and implemented into the Hybrid III large male head-neck complex. The updated geometry and weight were verified against measurements of the physical ATD to insure accuracy. The primary source of material discrepancy between the physical ATD and FEM was found to be in the material definition of the rubber neck pucks. Using LS-Opt the material properties of this part were calibrated to match physical response of the head-neck complex in the isolated head-neck testing performed. RESULTS AND DISCUSSION: The updated geometry and calibrated material properties were shown to significantly improve the FEM predictive accuracy under lateral, rear, and frontal impacts with combined horizontal and vertical loading. The results of this study demonstrate an effective means for improving ATD FEM response through isolated component level testing and material parameter optimization. The improved and quantified accuracy of the Hybrid III large male FEM head-neck complex stemming from with work lends to confidence in its future use for occupant protection evaluation of large male occupants

J Putnam↗

CFD Simulation of Floating Body Motion with Mooring Dynamics: Coupling MoorDyn with OpenFOAM

It is increasingly popular to use Computational Fluid Dynamics (CFD) models to study floating structures subjected to ocean waves, especially when it comes to applications of floating offshore wind turbines and Wave Energy Converters (WECs). Mooring dynamics are currently lacking in most of these applications. This paper presents a coupled simulation study of moored floating body motion by coupling two open-source libraries: a finite volume CFD toolbox, OpenFOAM, and a lumped-mass mooring model, MoorDyn. The instantaneous floating body position and velocity are passed from the body motion solver in the CFD model to the mooring model to calculate the fairlead kinematics. The mooring reaction forces, which are calculated by MoorDyn after updating the mooring system states, are then returned to the body motion solver to update the floating body motion. Both mesh deformation and overset mesh methods are used as the mesh motion solver in the CFD model to account for the floating body motion. The coupled model was validated against experimental measurements for a floating box moored with four catenary lines under the action of regular waves, which came from a preliminary test campaign for WECs. Apart from the lumped-mass mooring model, the present work also coupled a quasi-static mooring model and a finite element model with the floating body motion solver in OpenFOAM. The mooring line tensions predicted by these models were compared. The coupled model equipped with three mooring line codes may be further used to carry out survivability studies of FOWTs and WECs subject to severe sea states.

floating body↗

An Integrated Simulation of Multiple-Pass U-10Mo Alloy Hot Rolling and Static Recrystallization

To achieve a desired microstructure and minimize the thickness variation in rolled foils, researchers must understand the effects of foil fabrication process variables on microstructure evolution. We developed an integrated simulation of deformation and recrystallization that employs the finite element method (FEM) and the kinetic Monte Carlo (KMC) Potts model, respectively, to investigate microstructure evolution during multiple-pass hot rolling and heat treatment in polycrystalline U-10Mo fuel. Scanning electron microscopy and electron backscatter diffraction images of microstructures were directly used as input in FEM calculation of deformation, and the calculated strains were used to determine the driving force of nucleation and growth of recrystallized grains in the Potts model. Grain structures predicted by the Potts model were used to update the grain structure and material properties for FEM. Simulation alternated between FEM and the Potts model to simulate grain structure evolution during multiple rolling and heat treatments. The initial model parameters were determined by benchmarking the recrystallization kinetics against experimental data. Then, the model was applied to predict the grain structure evolution. Results showed that our model can capture the coupling between deformation and recrystallization and can quantitatively reproduce the observed U-10Mo recrystallization and grain growth kinetics. The simulation results demonstrated that the developed model can predict U-10Mo grain structures as a function of initial microstructure and foil fabrication parameters.

36 MATERIALS SCIENCE↗

Improvements in Nonconservative Force Modelling for TOPEX/POSEIDON

It was recognized prior to the launch of TOPEX/POSEIDON, that the most important source of orbit error other than the gravity field, was due to nonconservative force modelling. Accordingly, an intensive effort was undertaken to study the nonconservative forces acting on the spacecraft using detailed finite element modelling (Antreasian, 1992; Antreasian and Rosborough, 1992). However, this detailed modelling was not suitable for orbit determination, and a simplified eight plate "box-wing" model was developed that took into account the aggregate effect of the various materials and associated thermal properties of each spacecraft surface. The a priori model was later tuned post launch with actual tracking data [Nerem et al., 1994; Marshall and Luthcke, 1994; Marshall et al., 1995]. More recently, Kubitschek (1997] developed a newer box-wing model for TOPEX/POSEIDON, which included updated material properties, accounted for a solar array deflection, and modelled solar array warping due to thermal effects. We have used this updated model as a basis to retune the macromodel for TOPEX/POSEIDON, and report on preliminary results using at least 36 cycles (one year) of SLR and DORIS data in 1993.

Lemoine, Frank G.↗

Probabilistic Prognosis of Non-Planar Fatigue Crack Growth

Quantifying the uncertainty in model parameters for the purpose of damage prognosis can be accomplished utilizing Bayesian inference and damage diagnosis data from sources such as non-destructive evaluation or structural health monitoring. The number of samples required to solve the Bayesian inverse problem through common sampling techniques (e.g., Markov chain Monte Carlo) renders high-fidelity finite element-based damage growth models unusable due to prohibitive computation times. However, these types of models are often the only option when attempting to model complex damage growth in real-world structures. Here, a recently developed high-fidelity crack growth model is used which, when compared to finite element-based modeling, has demonstrated reductions in computation times of three orders of magnitude through the use of surrogate models and machine learning. The model is flexible in that only the expensive computation of the crack driving forces is replaced by the surrogate models, leaving the remaining parameters accessible for uncertainty quantification. A probabilistic prognosis framework incorporating this model is developed and demonstrated for non-planar crack growth in a modified, edge-notched, aluminum tensile specimen. Predictions of remaining useful life are made over time for five updates of the damage diagnosis data, and prognostic metrics are utilized to evaluate the performance of the prognostic framework. Challenges specific to the probabilistic prognosis of non-planar fatigue crack growth are highlighted and discussed in the context of the experimental results.

Leser, Patrick E.↗

A mechanistic model for creep and thermal aging in Alloy 709

This report describes a physics-based model for creep and thermal aging in Alloy 709. Alloy 709 is an advanced austenitic alloy, targeted for use in future Sodium Fast Reactors (SFRs) and other advanced reactors. The material has superior high temperature properties compared to currently qualified 316 and 304 stainless steels. However, the available creep and thermal aging test database for Alloy 709 is significantly more limited compared to the historical materials. The physics-based model developed here is one way to accelerate the qualification of the material by providing more accurate long-term predictions for creep properties and thermal aging, compared to current empirical time-extrapolate techniques. The crystal plasticity finite element model is used to predict the deformation and failure of alloy 709. The same setup for the CPFE model is used in both the baseline model calibration process and the simulation campaigns for parameter inference. Specific constitutive choices are made for Alloy 709 to capture the primary deformation mechanisms. The dislocation creep formulation developed by Hu and Cocks is extended to account for coupled precipitation formation and the grain boundary cavitation model developed by Sham, Needleman, et al. is used to model grain boundary cavitation-induced failure. A novel update algorithm is proposed to render the semi-discrete constitutive update for the Sham-Needleman model unconditionally stable. A progressive calibration approach is adopted based on the observations that several types of material responses can be effectively decoupled. A surrogate model is trained based on full-fledged CPFE simulations to accelerate the forward model evaluations, and stochastic variational inference (SVI) is used to calibrate the unknown microstructural model parameters. The calibrated mechanistic model is used to predict the long-term creep life of Alloy 709, and the predictions are compared against classical empirical approaches.

36 MATERIALS SCIENCE↗

Incorporation of an Energy Equation into a Pulsed Inductive Thruster Performance Model

A model for pulsed inductive plasma acceleration containing an energy equation to account for the various sources and sinks in such devices is presented. The model consists of a set of circuit equations coupled to an equation of motion and energy equation for the plasma. The latter two equations are obtained for the plasma current sheet by treating it as a one-element finite volume, integrating the equations over that volume, and then matching known terms or quantities already calculated in the model to the resulting current sheet-averaged terms in the equations. Calculations showing the time-evolution of the various sources and sinks in the system are presented to demonstrate the efficacy of the model, with two separate resistivity models employed to show an example of how the plasma transport properties can affect the calculation. While neither resistivity model is fully accurate, the demonstration shows that it is possible within this modeling framework to time-accurately update various plasma parameters.

Polzin, Kurt A.↗

Non-Linear Analysis of the NASA Super Pressure Balloons: Whole Flight Simulations

Tensys have a long-established background in the shape generation and load analysis of architectural stressed membrane structures. Founded upon their inTENS finite element analysis suite, these activities have broadened to encompass 'lighter than air' structures such as aerostats, airships, hybrid air vehicles and stratospheric balloons. Tensys have acted as consultants to the NASA Super Pressure Balloon (SPB) Program since 2004. Previous papers have focussed upon the application of inTENS to the overall structural and stability analysis of pumpkin type balloons as used in the SPB Program. Particular emphasis has been placed upon the ability to study both stress and stability at all stages of a flight. As the program has developed, increasing modelling fidelity has been introduced as design refinement has moved emphasis from the overall shape to the performance of individual details. Examples include the introduction of a contact capability into inTENS to better represent the separate cap layer of film. Localised investigations have considered the consequence of local geometric anomalies introduced during fabrication and the effects of debonding of the PBO tendon within its sleeve. Analysis to date has used a material model for the polyethylene shell film developed by Dr Rand of Winzen Engineering, supported by a program of fine resolution material tests at the Balloon Research and Development Laboratory (BRDL) at the GSFC Wallops Island Facility. Based upon work first reported by Schapery, this model has been incorporated into so-called 'snapshot' analyses within inTENS. For a given elapsed time point and temperature distribution, non-linear material properties are iteratively updated for the current stress state in each individual element until changes in those material properties are insignificant. This process has to be iterative as the film stresses and material properties are interdependent. Attempts to derive a time-stepping incremental viscoelastic capability using the Schapery Rand model encountered problems when dealing with the varying temperatures associated with balloon deployment, pressurisation and diurnal behaviour. An alternative approach has been developed by Pellegrino et al at Caltech, again with support from the NASA Balloon Program Office. This is a large strain non-linear viscoelastic model that includes film out-of-plane mechanical and thermal effects. Working from a unified base, this model can be utilised for time-stepping analyses in both a modulus or compliance mode, or a combination of both. This new capability is intended to enhance the current NASA balloon design process. This paper presents the implementation of the Caltech model into a specialist finite element suite, inTENS and its application to whole flight simulations.

Wakefield, David↗

Parallel projection—An improved return mapping algorithm for finite element modeling of shape memory alloys

Here, we present a novel finite element analysis of inelastic structures containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for SMAs lead to material nonlinearities, that require substantial computational effort to resolve. Finite element analysis methods, which rely on Gauss quadrature integration schemes, must solve two sets of coupled differential equations: one at the global level and the other at the local, i.e. Gauss point level. In contrast to the conventional return mapping algorithm, which solves these two sets of coupled differential equations separately using a nested Newton procedure, we propose a scheme to solve the local and global differential equations simultaneously. In the process we also derive closed-form expressions used to update the internal/constitutive state variables, and unify the popular closest-point and cutting plane methods with our formulas. Numerical testing indicates that our method allows for larger thermomechanical loading steps and provides increased computational efficiency, over the standard return mapping algorithm.

42 ENGINEERING↗

Iterative methods for mixed finite element equations

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Nakazawa, S.↗

A General Interface Method for Aeroelastic Analysis of Aircraft

The aeroelastic analysis of an aircraft requires an accurate and efficient procedure to couple aerodynamics and structures. The procedure needs an interface method to bridge the gap between the aerodynamic and structural models in order to transform loads and displacements. Such an interface method is described in this report. This interface method transforms loads computed by any aerodynamic code to a structural finite element (FE) model and converts the displacements from the FE model to the aerodynamic model. The approach is based on FE technology in which virtual work is employed to transform the aerodynamic pressures into FE nodal forces. The displacements at the FE nodes are then converted back to aerodynamic grid points on the aircraft surface through the reciprocal theorem in structural engineering. The method allows both high and crude fidelities of both models and does not require an intermediate modeling. In addition, the method performs the conversion of loads and displacements directly between individual aerodynamic grid point and its corresponding structural finite element and, hence, is very efficient for large aircraft models. This report also describes the application of this aero-structure interface method to a simple wing and an MD-90 wing. The results show that the aeroelastic effect is very important. For the simple wing, both linear and nonlinear approaches are used. In the linear approach, the deformation of the structural model is considered small, and the loads from the deformed aerodynamic model are applied to the original geometry of the structure. In the nonlinear approach, the geometry of the structure and its stiffness matrix are updated in every iteration and the increments of loads from the previous iteration are applied to the new structural geometry in order to compute the displacement increments. Additional studies to apply the aero-structure interaction procedure to more complicated geometry will be conducted in the second phase of the present contract.

Tzong, T.↗

Dynamic Characterization of NASA Armstrong Test Facility Mechanical Vibration Facility

In preparation for Sierra Nevada Corporation’s (SNC) Dream Chaser spacecraft vibration test campaign at NASA Armstrong Test Facility (ATF) in Sandusky, Ohio, a dynamic characterization of the Mechanical Vibration Facility (MVF) is currently ongoing. The Mechanical Vibration Facility is comprised of an 18 ft diameter annulus table that is driven with sixteen hydraulic vertical actuator assemblies and four hydraulic horizontal actuator assemblies. During a test campaign this allows for single axis vibration testing in the vertical axis and in each of the two orthogonal horizontal axes without the need for reconfiguring the test article. The dynamic characterization of the facility was defined to follow a building-block approach requiring three vibration test configurations: (1) MVF bare table (2) MVF table and head expander (3) MVF table, head expander, and a dummy test article. Following the completion of the facility dynamic characterization, a facility finite element model (FEM) will be test-verified allowing for more accurate pretest analysis used for determining response limits and abort levels. To date two of the three test configurations have been completed. Test results have been used to verify the facility FEM that includes both the 18 ft diameter annulus table and the aluminum head expander that fills in the center opening of the annulus to provide a continuous flat mounting surface. Both the MVF table and the head expander models were previously correlated using test results from free-free boundary condition modal tests so model updates to the facility FEM focused on the uncorrelated vertical actuator assemblies and horizontal actuator assemblies. The accuracy of the test-verified facility FEM will be further determined based on comparison of analytical and test results from the vibration test of the dummy test article. Efforts will also be made to determine fixed base modes of the test article attached to the flexible vibration table using fixed base correction techniques. This paper will discuss test and analytical results from the MVF dynamic characterization test sequence and corresponding model updating effort.

Emma L Pierson↗