Publications

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The SNL/AWE joint mechanics workshop, held in Dartington Hall, Totnes, Devon, UK 26-29 April 2009 was a follow up to another international joints workshop held in Arlington, Virginia, in October 2006. The preceding workshop focused on identifying what length scales and interactions would be necessary to provide a scientific basis for analyzing and understanding joint mechanics from the atomistic scale on upward. In contrast, the workshop discussed in this report, focused much more on identification and development of methods at longer length scales that can have a nearer term impact on engineering analysis, design, and prediction of the dynamics of jointed structures. Also, the 2009 meeting employed less technical presentation and more break out sessions for developing focused strategies than was the case with the early workshop. Several 'challenges' were identified and assignments were made to teams to develop approaches to address those challenges.

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The nonlinear behavior of mechanical joints is a confounding element in modeling the dynamic response of structures. Though there has been some progress in recent years in modeling individual joints, modeling the full structure with myriad frictional interfaces has remained an obstinate challenge. A strategy is suggested for structural dynamics modeling that can account for the combined effect of interface friction distributed spatially about the structure. This approach accommodates the following observations: (1) At small to modest amplitudes, the nonlinearity of jointed structures is manifest primarily in the energy dissipation - visible as vibration damping; (2) Correspondingly, measured vibration modes do not change significantly with amplitude; and (3) Significant coupling among the modes does not appear to result at modest amplitudes. The mathematical approach presented here postulates the preservation of linear modes and invests all the nonlinearity in the evolution of the modal coordinates. The constitutive form selected is one that works well in modeling spatially discrete joints. When compared against a mathematical truth model, the distributed dissipation approximation performs well.

This report describes work performed from October 2007 through September 2009 under the Sandia Laboratory Directed Research and Development project titled 'Reduced Order Modeling of Fluid/Structure Interaction.' This project addresses fundamental aspects of techniques for construction of predictive Reduced Order Models (ROMs). A ROM is defined as a model, derived from a sequence of high-fidelity simulations, that preserves the essential physics and predictive capability of the original simulations but at a much lower computational cost. Techniques are developed for construction of provably stable linear Galerkin projection ROMs for compressible fluid flow, including a method for enforcing boundary conditions that preserves numerical stability. A convergence proof and error estimates are given for this class of ROM, and the method is demonstrated on a series of model problems. A reduced order method, based on the method of quadratic components, for solving the von Karman nonlinear plate equations is developed and tested. This method is applied to the problem of nonlinear limit cycle oscillations encountered when the plate interacts with an adjacent supersonic flow. A stability-preserving method for coupling the linear fluid ROM with the structural dynamics model for the elastic plate is constructed and tested. Methods for constructing efficient ROMs for nonlinear fluid equations are developed and tested on a one-dimensional convection-diffusion-reaction equation. These methods are combined with a symmetrization approach to construct a ROM technique for application to the compressible Navier-Stokes equations.

Advanced computing hardware and software written to exploit massively parallel architectures greatly facilitate the computation of extremely large problems. On the other hand, these tools, though enabling higher fidelity models, have often resulted in much longer run-times and turn-around-times in providing answers to engineering problems. The impediments include smaller elements and consequently smaller time steps, much larger systems of equations to solve, and the inclusion of nonlinearities that had been ignored in days when lower fidelity models were the norm. The research effort reported focuses on the accelerating the analysis process for structural dynamics though combinations of model reduction and mitigation of some factors that lead to over-meshing.

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The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation. © 2008 Elsevier Inc.

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Energy dissipation in mechanical joints occurs as a result of micro-slip motion between contacting rough surfaces. An account of this phenomenon is especially challenging due to the vast differences in the length and time scale differences between the macro-mechanical structure and the micron-scale events at the joint interface. This paper considers the contact between two nominally flat surfaces containing micron-scale roughness. The rough surface interaction is viewed as a multi-sphere elastic interaction subject to a periodic tangential force. It combines the Mindlin's formulation [1,2] for the elastic interaction of two spheres with the Greenwood and Williamson's [3] statistical approach for the contact of two nominally flat rough surfaces so as to develop a model for multi-sphere problem in which sphere radii, contact load and the number of spheres in contact can only be known in a statistical sense and not deterministically. Copyright © 2007 by ASME.

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The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transform leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Coupling with a linearized structural dynamics model is made possible through the solid wall boundary condition. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using piecewise-smooth finite element bases. Stability of the coupled fluid/structure system is examined for the case of uniform flow past a thin plate. Stability of the reduced order model for the fluid is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.

The NSF/SNL joint mechanics workshop, held in Arlington, Virginia, 16-18 October, 2006, attempted to assess the current state of the art for modeling joint mechanics for the purpose of structural dynamics calculation, to identify the underlying physics issues that must be addressed to advance the field, and to propose a path forward. Distinguished participants from several countries representing research communities that focus on very different length and time scales identified multiple challenges in bridging those scales. Additionally, two complementary points of view were developed for addressing those challenges. The first approach - the 'bottom-up' perspective - attempts to bridge scales by starting from the smallest length scale and working up. The other approach starts at the length scale of application and attempts to deduce mechanics at smaller length scales through reconciliation with laboratory observation. Because interface physics is a limiting element of predictive simulation in defense and transportation, this issue will be of continuing importance for the foreseeable future.

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Using fine material meshes in structural dynamics analysis is often impractical due to time step considerations. Unfortunately, fine meshes are typically required to capture the inherent physics in jointed connections. This is especially true in threaded connections which feature numerous contact interfaces and stress singularities. A systematic method is presented here for representing the threaded volume by a continuous, homogeneous, linear elastic, anisotropic equivalent material. The parameters of that equivalent material depend on thread geometry and the assumed contact condition between adjacent threads and are derived from detailed finite element simulations of a characteristic thread-pair unit cell. Numerical simulations using the equivalent material closely match the local stiffness through the load path calculated from the finely meshed thread models and also reproduce classical theoretical and experimental results from the literature. Copyright © 2007 by ASME.

A relatively new concept in the field of mechanical shock analysis has been introduced whereby an analysis is made on the work done on structures by the excitation force. The energy imparted to a structure by the excitation can then be divided into various storage and loss mechanisms within the structure. These energies can be used to both evaluate shock response severity and characterize the underlying excitation. Previous work has illustrated the many advantages of the energy methods over traditional shock response spectrum techniques. This work will show that the energy delivered to a MDOF system is uncoupled between modes. Therefore, the total deformational energy delivered to a MDOF system is a weighted sum of the uncoupled modal contributions. This leads to the ability to compute input energy on a modal basis using uncoupled, SDOF calculations. Further, the internal storage and loss energies are also uncoupled. When the input excitation is broadband, the energy input into a MDOF structure by ground motion is dominated by that mode with the largest fraction of participating mass, often the fundamental mode of the system. This leads to the justification for treating complex structures as SDOF oscillators when using energy methods to evaluate both the underlying excitation and the structural response.

It is shown that for any material or structural model expressible as a Masing model, there exists a unique parallel-series (displacement-based) Iwan system that characterizes that model as a function of displacement history. This poses advantages both in terms of more convenient force evaluation in arbitrary deformation histories as well as in terms of model inversion. Characterization as an Iwan system is demonstrated through the inversion of the Ramberg-Osgood model, a force(stress)-based material model that is not explicitly invertible. An implication of the inversion process is that direct, rigorous comparisons of different Masing models, regardless of the ability to invert their constitutive relationship, can be achieved through the comparison of their associated Iwan distribution densities.

A technique published in SAND Report 2006-1789 ''Model Reduction of Systems with Localized Nonlinearities'' is illustrated in two problems of finite element structural dynamics. That technique, called here the Method of Locally Discontinuous Basis Vectors (LDBV), was devised to address the peculiar difficulties of model reduction of systems having spatially localized nonlinearities. It's illustration here is on two problems of different geometric and dynamic complexity, but each containing localized interface nonlinearities represented by constitutive models for bolted joint behavior. As illustrated on simple problems in the earlier SAND report, the LDBV Method not only affords reduction in size of the nonlinear systems of equations that must be solved, but it also facilitates the use of much larger time steps on problems of joint macro-slip than would be possible otherwise. These benefits are more dramatic for the larger problems illustrated here. The work of both the original SAND report and this one were funded by the LDRD program at Sandia National Laboratories.

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