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Academic Commons Search Resultsen-usAdvances in Multiscale Methods with Applications in Optimization, Uncertainty Quantification and Biomechanics
http://academiccommons.columbia.edu/catalog/ac:202066
Hu, Nanhttp://dx.doi.org/10.7916/D8FX79N8Fri, 19 Aug 2016 19:22:30 +0000Advances in multiscale methods are presented from two perspectives which address the issue of computational complexity of optimizing and inverse analyzing nonlinear composite materials and structures at multiple scales. The optimization algorithm provides several solutions to meet the enormous computational challenge of optimizing nonlinear structures at multiple scales including: (i) enhanced sampling procedure that provides superior performance of the well-known ant colony optimization algorithm, (ii) a mapping-based meshing of a representative volume element that unlike unstructured meshing permits sensitivity analysis on coarse meshes, and (iii) a multilevel optimization procedure that takes advantage of possible weak coupling of certain scales. We demonstrate the proposed optimization procedure on elastic and inelastic laminated plates involving three scales. We also present an adaptive variant of the measure-theoretic approach (MTA) for stochastic characterization of micromechanical properties based on the observations of quantities of interest at the coarse (macro) scale. The salient features of the proposed nonintrusive stochastic inverse solver are: identification of a nearly optimal sampling domain using enhanced ant colony optimization algorithm for multiscale problems, incremental Latin-hypercube sampling method, adaptive discretization of the parameter and observation spaces, and adaptive selection of number of samples. A complete test data of the TORAY T700GC-12K-31E and epoxy #2510 material system from the NIAR report is employed to characterize and validate the proposed adaptive nonintrusive stochastic inverse algorithm for various unnotched and open-hole laminates. Advances in Multiscale methods also provides us a unique tool to study and analyze human bones, which can be seen as a composite material, too. We used two multiscale approaches for fracture analysis of full scale femur. The two approaches are the reduced order homogenization (ROH) and the novel accelerated reduced order homogenization (AROH). The AROH is based on utilizing ROH calibrated to limited data as a training tool to calibrate a simpler, single-scale anisotropic damage model. For bone tissue orientation, we take advantage of so-called Wolff’s law. The meso-phase properties are identified from the least square minimization of error between the overall cortical and trabecular bone properties and those predicted from the homogenization. The overall elastic and inelastic properties of the cortical and trabecular bone microstructure are derived from bone density that can be estimated from the Hounsfield units (HU). For model validation, we conduct ROH and AROH simulations of full scale finite element model of femur created from the QCT and compare the simulation results with available experimental data.Civil engineering, Mechanicsnh2350Civil Engineering and Engineering MechanicsDissertationsInferring the 3D gravitational field of the Milky Way with stellar streams
http://academiccommons.columbia.edu/catalog/ac:199667
Price-Whelan, Adrian Michaelhttp://dx.doi.org/10.7916/D82N52BDTue, 24 May 2016 15:22:12 +0000We develop two new methods to measure the structure of matter around the Milky Way using stellar tidal streams from disrupting dwarf galaxies and globular clusters. The dark matter halo of the Milky Way is expected to be triaxial and filled with substructure, but measurements of the shape and profile of dark matter around the Galaxy are highly uncertain and often contradictory. We demonstrate that kinematic data from near-future surveys for stellar streams or shells produced by tidal disruption of stellar systems around the Milky Way will provide precise measures of the gravitational potential to test these predictions. We develop a probabilistic method for inferring the Galactic potential with tidal streams based on the idea that the stream stars were once close in phase space and test this method on synthetic datasets generated from N-body simulations of satellite disruption with observational uncertainties chosen to mimic current and near-future surveys of various stars. We find that with just four well-measured stream stars, we can infer properties of a triaxial potential with precisions of order 5--7 percent. We then demonstrate that, if the Milky Way's dark matter halo is triaxial and is not fully integrable (as is expected), an appreciable fraction of orbits will be chaotic. We examine the influence of chaos on the phase-space morphology of cold tidal streams and show that streams even in weakly chaotic regions look very different from those in regular regions. We discuss the implications of this fact given that we see several long, thin streams in the Galactic halo; our results suggest that long, cold streams around our Galaxy must exist only on regular (or very nearly regular) orbits and potentially provide a map of the regular regions of the Milky Way potential. We then apply this understanding of stream formation along chaotic orbits to the interpretation of a newly-discovered, puzzling stellar stream near the Galactic bulge. We conclude that the morphology of this stream is consistent with forming along chaotic orbits due to the presence of the time-dependent Galactic bar. These results are encouraging for the eventual goal of using flexible, time-dependent potential models combined with larger data sets to unravel the detailed shape of the dark matter distribution around the Milky Way.Astrophysics, Astronomy, Mechanicsamp2217AstronomyDissertationsThe Stability at the Solid-Solid and Liquid-Solid Interfaces
http://academiccommons.columbia.edu/catalog/ac:198539
Xiao, Junfenghttp://dx.doi.org/10.7916/D8KK9BS1Wed, 04 May 2016 15:33:41 +0000In this thesis, we studied three small subjects in the realm of continuum mechanics: imbibition in fluid mechanics, beam and rod buckling in solid mechanics and shell buckling at the solid-liquid interface.
In chapter 2, we examined the radial imbibition into a homogenous semi-infinite porous media from a point source with infinite liquid supply. We proved that in the absence of gravity (or in the regime while gravity is negligible compared to surface tension), the shape of the wet area is a hemisphere, and the radius of the wet area evolves as a function with respect to time. This new law with respect to time has been verified by Finite Element Method simulation in software COMSOL and a series of experiments using packed glass microsphere as the porous media. We also found that even though the imbibition slows down, the flow rate through the point source remains constant. This new result for three dimensional radial imbibition complements the classic Lucas-Washburn law in one dimension and two dimensional radial imbibition in one plane.
In chapter 3, we studied the elastic beam/rod buckling under lateral constraints in two dimension as well as in three dimension. For the two dimensional case with unique boundary conditions at both ends, the buckled beam can be divided into segments with alternate curved section and straight section. The curved section can be solved by the Euler beam equation. The straight sections, however, are key to the transition between different buckling modes, and the redistributed length of straight sections sets the upper limit and lower limit for the transition. We compared our theoretical model of varying straight sections with Finite Element Method simulation in software ABAQUS, and good agreements are found. We then attempted to employ this model as an explanation with qualitative feasibility for the crawling snake in horizontal plane between parallel walls, which shows unique shape like square wave. For the three dimensional buckling beam/rod confined in cylindrical constraints, three stages are found for the buckling and post buckling processes: initial two dimensional shape, three dimensional spiral/helix shape and final foldup/alpha shape. We characterized the shape at each stage, and then we calculated the transition points between the three stages using geometrical arguments for energy arguments. The theoretical analysis for three dimensional beam/rod are also complemented with Finite Element Method simulations from ABAQUS.
In chapter 4, we investigated the buckling shape of solid shell filled with liquid core in two dimension and three dimension. A material model for liquid is first described that can be readily incorporated in the framework of solid mechanics. We then applied this material model in two dimensional and three dimensional Finite Element Method simulation using software ABAQUS. For the two dimensional liquid core solid shell model, a linear analysis is first performed to identify that ellipse corresponds to lowest order of buckling with smallest elastic energy. Finite Element Method simulation is then performed to determine the nonlinear post-buckling process. We discovered that two dimensional liquid core solid shell structures converge to peanut shape eventually while the evolution process is determined by two dimensionless parameters Kτ/μ and ρR^2/μτ. Amorphous shape exists before final peanut shape for certain models with specific Kτ/μ and ρR^2/μτ. The two dimensional peanut shape is also verified with Lattice Boltzmann simulations. For the three dimensional liquid core solid shell model, the post buckling shape is studied from Finite Element Method simulations in ABAQUS. Depending on the strain loading rate, the deformations show distinctive patterns. Large loading rate induces herringbone pattern on the surface of solid shell which resembles solid core solid shell structure, while small loading rate induces major concave pattern which resemble empty solid shell structure. For both two dimensional and three dimensional liquid core system, small scale ordered deformation pattern can be generated by increasing the shear stress in liquid core.
In the final chapter, we summarized the discoveries in the dissertation with highlights on the role that geometry plays in all of the three subjects. Recommendations for future studies are also discussed.Mechanical engineering, Mechanicsjx2151Mechanical Engineering, Earth and Environmental EngineeringDissertationsThe Multiscale Damage Mechanics in Objected-oriented Fortran Framework
http://academiccommons.columbia.edu/catalog/ac:193936
Yuan, Zifenghttp://dx.doi.org/10.7916/D8KS6RCJWed, 27 Jan 2016 23:20:28 +0000We develop a dual-purpose damage model (DPDM) that can simultaneously model intralayer damage (ply failure) and interlayer damage (delamination) as an alternative to conventional practices that models ply failure by continuum damage mechanics (CDM) and delamination by cohesive elements. From purely computational point of view, if successful, the proposed approach will significantly reduce computational cost by eliminating the need for having double nodes at ply interfaces. At the core, DPDM is based on the regularized continuum damage mechanics approach with vectorial representation of damage and ellipsoidal damage surface. Shear correction factors are introduced to match the mixed mode fracture toughness of an analytical cohesive zone model. A predictor-corrector local-nonlocal regularization scheme, which treats intralayer portion of damage as nonlocal and interlayer damage as local, is developed and verified. Two variants of the DPDM are studied: a single- and two- scale DPDM. For the two-scale DPDM, reduced-order-homogenization (ROH) framework is employed with matrix phase modeled by the DPDM while the inclusion phase modeled by the CDM. The proposed DPDM is verified on several multi-layer laminates with various ply orientations including double-cantilever beam (DCB), end-notch-flexure (ENF), mixed-mode-bending (MMB), and three-point-bending (TPB). The simulation is executed in the platform of FOOF (Finite element solver based on Object-Oriented Fortran).
The objective of FOOF is to develop a new architecture of the nonlinear multiphysics finite element code in object oriented Fortran environment. The salient features of FOOF are reusability, extensibility, and performance. Computational efficiency stems from the intrinsic optimization of numerical computing intrinsic to Fortran, while reusability and extensibility is inherited from the support of object-oriented programming style in Fortran 2003 and its later versions. The shortcomings of the object oriented style in Fortran 2003 (in comparison to C++) are alleviated by introducing the class hierarchy and by utilizing a multilevel programming style.Engineering, Civil engineering, Mechanicszy2134Civil Engineering and Engineering MechanicsDissertationsA nonlocal multiscale discrete-continuum model for predicting mechanical behavior of granular materials
http://academiccommons.columbia.edu/catalog/ac:193162
Liu, Yang; Sun, WaiChing; Yuan, Zifeng; Fish, Jacobhttp://dx.doi.org/10.7916/D8Z89C5PWed, 20 Jan 2016 13:40:46 +0000A three-dimensional nonlocal multiscale discrete-continuum model has been developed for modeling mechanical behavior of granular materials. In the proposed multiscale scheme, we establish an information-passing coupling between the discrete element method, which explicitly replicates granular motion of individual particles, and a finite element continuum model, which captures nonlocal overall responses of the granular assemblies. The resulting multiscale discrete-continuum coupling method retains the simplicity and efficiency of a continuum-based finite element model, while circumventing mesh pathology in the post-bifurcation regime by means of staggered nonlocal operator. We demonstrate that the multiscale coupling scheme is able to capture the plastic dilatancy and pressure-sensitive frictional responses commonly observed inside dilatant shear bands, without employing a phenomenological plasticity model at a macroscopic level. In addition, internal variables, such as plastic dilatancy and plastic flow direction, are now inferred directly from granular physics, without introducing unnecessary empirical relations and phenomenology. The simple shear and the biaxial compression tests are used to analyze the onset and evolution of shear bands in granular materials and sensitivity to mesh density. The robustness and the accuracy of the proposed multiscale model are verified in comparisons with single-scale benchmark discrete element method simulations.Materials science, Civil engineering, Mechanicsyl2683, ws2414, zy2134, jf2695Civil Engineering and Engineering MechanicsArticlesMultiscale analysis of shear failure of thick-walled hollow cylinder in dry sand
http://academiccommons.columbia.edu/catalog/ac:193147
Guo, N.; Zhao, J.; Sun, WaiChinghttp://dx.doi.org/10.7916/D87P8Z4HWed, 20 Jan 2016 12:54:29 +0000A novel hierarchical multiscale model has been applied to simulate the thick-walled hollow cylinder tests in dry sand and to investigate the corresponding shear failures. The combined finite-element method and discrete-element method (FEM/DEM) model employs the FEM as a vehicle to advance the solution for a macroscopic non-linear boundary value problem incrementally. It is, meanwhile, free of conventional macroscopic phenomenological constitutive law, which is replaced by discrete-element simulations conducted with representative volume elements (RVEs) associated with the Gauss quadrature points of the FEM mesh. Numerical simulations proposed by the authors indicate that this multiscale approach is capable of replicating the evolution of cavity pressure during cavity expansion – before and after the onset of strain localisation – in qualitative agreement with laboratory tests. In particular, the curvilinear shear bands observed from experiments have been reproduced numerically. The information provided by the mesoscale DEM and the macroscale FEM reveals a close linkage between significant particle rotations taking place inside the dilative shear bands and the highly anisotropic microstructural attributes of the associated RVEs.Materials science, Mechanics, Civil engineeringws2414Civil Engineering and Engineering MechanicsArticlesMultiscale Modeling of Granular Materials
http://academiccommons.columbia.edu/catalog/ac:189226
Liu, Yanghttp://dx.doi.org/10.7916/D88W3CR1Tue, 06 Oct 2015 15:07:37 +0000Granular materials have a “discrete” nature whose global mechanical behaviors are originated from the grain scale micromechanical mechanisms. The intriguing properties and non-trivial behaviors of a granular material pose formidable challenges to the multiscale modeling of these materials. Some of the key challenges include upscaling of coarse-scale continuum equation form fine-scale governing equations, calibrating material parameters at different scales, alleviating pathological mesh dependency in continuum models, and generating unit cells with versatile morphological details. This dissertation aims to addressing the aforementioned challenges and to investigate the mechanical behavior of granular materials through multiscale modeling.
Firstly, a three-dimensional nonlocal multiscale discrete-continuum model is presented for modeling the mechanical behavior of granular materials. We establish an information-passing coupling scheme between DEM that explicitly replicates granular motion of individual particles and a finite element continuum model, which captures nonlocal overall response of the granular assemblies. Secondly, a new staggered multilevel material identification procedure is developed for phenomenological critical state plasticity models. The emphasis is placed on cases in which available experimental data and constraints are insufficient for calibration. The key idea is to create a secondary virtual experimental database from high-fidelity models, such as discrete element simulations, then merge both the actual experimental data and secondary database as an extended digital database to determine material parameters for the phenomenological macroscopic critical state plasticity model. This expansion of database provides additional constraints necessary for calibration of the phenomenological critical state plasticity models.
Thirdly, a regularized phenomenological multiscale model is investigated, in which elastic properties are computed using direct homogenization and subsequently evolved using a simple three-parameter orthotropic continuum damage model. The salient feature of the model is a unified regularization framework based on the concept of effective softening strain. The unified regularization scheme is employed in the context of constitutive law rescaling and the staggered nonlocal approach to alleviate pathological mesh dependency. Lastly, a robust parametric model is presented for generating unit cells with randomly distributed inclusions. The proposed model is computationally efficient using a hierarchy of algorithms with increasing computational complexity, and is able to generate unit cells with different inclusion shapes.Mechanics, Civil engineering, Geotechnologyyl2683Civil Engineering and Engineering MechanicsDissertationsStructural Identification, Health Monitoring and Uncertainty Quantification under Incomplete Information with Minimal Requirements for Identifiability
http://academiccommons.columbia.edu/catalog/ac:194310
Mukhopadhyay, Suparnohttp://dx.doi.org/10.7916/D8ZG6R1CWed, 21 Jan 2015 12:12:02 +0000Structural identification is the inverse problem of estimating the physical parameters, e.g. element masses and stiffnesses, of a model representing a structural system, using response measurements obtained from the actual structure subjected to operational or well-defined experimental excitations. It is one of the principal focal areas of modal testing and structural health monitoring, with the identified model finding a wide variety of applications, from obtaining reliable response predictions to timely detection of structural damage (location and severity) and consequent planning and validating of maintenance/retrofitting operations. However, incomplete instrumentation of the monitored system and ambient vibration testing generally result in spatially incomplete and arbitrarily normalized measured modal information, often making the inverse problem ill-conditioned and resulting in non-unique identification results. The problem of parameter identifiability addresses the question of whether or not a parameter set of interest can be identified from the available information. The identifiability of any parameter set of interest depends on the number and location of sensors on the monitored system. In this dissertation we study the identifiability of the mass and stiffness parameters of shear-type systems, including 3-dimensional laterally-torsionally coupled rigid floor systems, with incomplete instrumentation, simultaneous to the development of algorithms to identify the complete mass and stiffness matrices of such systems. Both input-output and output-only situations are considered, and mode shape expansion and mass normalization approaches are developed to obtain the complete mass normalized mode shape matrix, starting from the incomplete modal parameters identified using any suitable experimental or operational modal analysis technique. Methods are discussed to decide actuator/sensor locations on the structure which will ensure identifiability of the mass and stiffness parameters. Several possible minimal and near-minimal instrumentation set-ups are also identified. The minimal a priori information necessary in output-only situations is determined, and different scenario of available a priori information are considered. Additionally, tests for identifiability are discussed for both pre- and post-experiment applications. The different theoretical discussions are illustrated using numerical simulations and experimental data. It is shown that the proposed identification algorithms are able to obtain reliably accurate physical parameter estimates even under the constraints of minimal instrumentation, minimal a priori information, and unmeasured input. The different actuator/sensor placement rules and identifiability tests are useful for both experiment design purposes, to determine the necessary number and location of sensors, as well as in identifying possibilities of multiple solutions post-experiment. The parameter identification methods are applied for structural health monitoring using experimental data, and an approach is discussed for probabilistic characterization of structural damage location and severity. A perturbation based uncertainty propagation approach is also discussed for the identification of the distributions of mass and stiffness parameters, reflecting the variability in the test structure, using very limited measured and a priori information.Civil engineering, Mechanics, Mechanical engineeringsm3315Civil Engineering and Engineering MechanicsDissertationsDevelopment of Hierarchical Optimization-based Models for Multiscale Damage Detection
http://academiccommons.columbia.edu/catalog/ac:177209
Sun, Haohttp://dx.doi.org/10.7916/D8GQ6W2JSat, 06 Sep 2014 03:24:46 +0000In recent years, health monitoring of structure and infrastructure systems has become a valuable source of information for evaluating structural integrity, durability and reliability throughout the lifecycle of structures as well as ensuring optimal maintenance planning and operation. Important advances in sensor and computer technologies made possible to process a large amount of data, to extract the characteristic features of the signals, and to link those to the current structural conditions. In general, the process of data feature extraction relates to solving an inverse problem, in either a data-driven or a model-based type setting.
This dissertation explores state-of-the-art hierarchical optimization-based computational algorithms for solving multiscale model-based inverse problems such as system identification and damage detection. The basic idea is to apply optimization tools to quantify an established model or system, characterized by a set of unknown governing parameters, via minimizing the discrepancy between the predicted system response and the measured data. We herein propose hierarchical optimization algorithms such as the improved artificial bee colony algorithms integrated with local search operators to accomplish this task.
In this dissertation, developments in multiscale damage detection are presented in two parts. In the first part, efficient hybrid bee algorithms in both serial and parallel schemes are proposed for time domain input-output and output-only identification of macro-scale linear/nonlinear systems such as buildings and bridges. Solution updating strategies of the artificial bee colony algorithm are improved for faster convergence, meanwhile, the simplex method and gradient-based optimization techniques are employed as local search operators for accurate solution tuning. In the case of output-only measurements, both system parameters and the time history of input excitations can be simultaneously identified using a modified Newmark integration scheme. The synergy between the proposed method and Bayesian inference are proposed to quantify uncertainties of a system. Numerical and experimental applications are investigated and presented for macro-scale system identification, finite element model updating and damage detection.
In the second part, a framework combining the eXtended Finite Element Method (XFEM) and the proposed optimization algorithms is investigated, for nondestructive detection of multiple flaws/defects embedded in meso-scale systems such as critical structural components like plates. The measurements are either static strains or displacements. The number of flaws as well as their locations and sizes can be identified. XFEM with circular and/or elliptical void enrichments is employed to solve the forward problem and alleviates the costly re-meshing along with the update of flaw boundaries in the identification process. Numerical investigations are presented to validate the proposed method in application to detection of multiple flaws and damage regions.
Overall, the proposed multiscale methodologies show a great potential in assessing the structural integrity of building and bridge systems, critical structural components, etc., leading to a smart structure and infrastructure management system.Civil engineering, Mechanicshs2595Civil Engineering and Engineering MechanicsDissertationsRelation between stress heterogeneity and aftershock rate in the rate-and-state model
http://academiccommons.columbia.edu/catalog/ac:162514
Helmstetter, Agnès; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20798Wed, 19 Jun 2013 11:29:23 +0000We estimate the rate of aftershocks triggered by a heterogeneous stress change, using the rate-and-state model of Dieterich. We show that an exponential stress distribution Pτ(τ) ~ exp(−τ/τ0) gives an Omori law decay of aftershocks with time ~ 1/tp, with an exponent p = 1 − Aσn/τ0, where A is a parameter of the rate-and-state friction law and σn is the normal stress. Omori exponent p thus decreases if the stress “heterogeneity” τ0 decreases. We also invert the stress distribution Pτ(τ) from the seismicity rate R(t), assuming that the stress does not change with time. We apply this method to a synthetic stress map, using the (modified) scale invariant “k2” slip model (Herrero and Bernard). We generate synthetic aftershock catalogs from this stress change. The seismicity rate on the rupture area shows a huge increase at short times, even if the stress decreases on average. Aftershocks are clustered in the regions of low slip, but the spatial distribution is more diffuse than for a simple slip dislocation. Because the stress field is very heterogeneous, there are many patches of positive stress changes everywhere on the fault. This stochastic slip model gives a Gaussian stress distribution but nevertheless produces an aftershock rate which is very close to Omori's law, with an effective p ≤ 1, which increases slowly with time. We obtain a good estimation of the stress distribution for realistic catalogs when we constrain the shape of the distribution. However, there are probably other factors which also affect the temporal decay of aftershocks with time. In particular, heterogeneity of Aσn can also modify the parameters p and c of Omori's law. Finally, we show that stress shadows are very difficult to observe in a heterogeneous stress context.Mechanics, Plate tectonics, Geophysicsbes11Lamont-Doherty Earth ObservatoryArticlesMagnitude dependence of radiated energy spectra: Far-field expressions of slip pulses in earthquake models
http://academiccommons.columbia.edu/catalog/ac:162497
Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20793Tue, 18 Jun 2013 17:11:06 +0000We examine the radiated waves emitted by events on a model fault. The model deterministically produces a complex sequence of events, with a wide range of sizes, from a uniform frictional instability. The spontaneous rupture events emit a rich spectrum of radiated waves as they nucleate, propagate, and decelerate within the complex stress field left by previous events. Two model innovations, a new driving boundary condition on the fault and a new radiating boundary condition which allows a spatially varying prestress away from the fault, allow us to directly measure the radiation without problems from boundary reflections in our two-dimensional model. We quantify the radiation by first measuring the energy spectral density and then averaging over events of a similar size to examine the magnitude dependence. Assuming only a physics of the tractions on the fault, we obtain a full spectra of radiated waves for a complex population of events with a wide range of sizes. To quantify the resulting spectra, we consider two different spectral measures. One, the peak amplitude of the spectral energy density, occurs at a period which scales with the rupture length and corresponds with the classical corner frequency measurement. The other, the peak amplitude of the spectral average acceleration or the low-frequency corner in the case of a flat acceleration spectrum, occurs at a period that scales with the duration of slip of points on the fault. The period of the peak spectral acceleration saturates for large events. Looking at the rupture motions on the fault, we find that this spectral behavior corresponds with the behavior of slip pulses in the model. Intense narrow pulses of slip develop for very long rupture events. We quantify this by measuring the mean slip duration as a function of rupture length and show that it is has the same behavior as the peak period of spectral acceleration. Thus the duration of the slip pulses in these ruptures is directly expressed in their radiated spectra. Moreover, we find that these corner periods exhibit a nontrivial dependence on event magnitude for the different frictional instabilities that we have examined, suggesting that any observed dependence of these corner periods on earthquake magnitude might provide insight into the frictional instability of earthquakes.Mechanics, Geophysics, Plate tectonicsbes11Lamont-Doherty Earth ObservatoryArticlesExistence of continuum complexity in the elastodynamics of repeated fault ruptures
http://academiccommons.columbia.edu/catalog/ac:162484
Shaw, Bruce E.; Rice, James R.http://hdl.handle.net/10022/AC:P:20789Tue, 18 Jun 2013 14:38:25 +0000What are the origins of earthquake complexity? The possibility that some aspects of the complexity displayed by earthquakes might be explained by stress heterogeneities developed through the self-organization of repeated ruptures has been suggested by some simple self-organizing models. The question of whether or not even these simple self-organizing models require at least some degree of material heterogeneity to maintain complex sequences of events has been the subject of some controversy. In one class of elastodynamic models, previous work has described complexity as arising on a model fault with completely uniform material properties. Questions were raised, however, regarding the role of discreteness, the relevance of the nucleation mechanism, and special parameter choices, in generating the complexity that has been reported. In this paper, we examine the question of whether or not continuum complexity is achieved under the stringent conditions of continuous loading, and whether the results are similar to previously claimed findings of continuum complexity or its absence. The elastodynamic model that we use consists of a 1-D fault boundary with friction, a steady slowly moving 1-D boundary parallel to the fault, and a 2-D scalar elastic media connecting the two boundaries. The constitutive law used involves a pair of sequential weakening processes, one occurring over a small slip (or velocity) and accomplishing a small fraction of the total strength drop, and the other at larger slip (or velocity) and providing the remaining strength drop. The large-scale process is motivated by a heat weakening instability. Our main results are as follows. (1) We generally find complexity of type I, a broad distribution of large event sizes with nonperiodic recurrence, when the modeled region is very long, along strike, compared to the layer thickness. (2) We find that complexity of type II, with numerous small events showing a power law distribution, is not a generic result but does definitely exist in a restricted range of parameter space. For that, in the slip weakening version of our model, the strength drop and nucleation size in the small slip process must be much smaller than in the large slip process, and the nucleation length associated with the latter must be comparable to layer thickness. This suggests a basis for reconciling different previously reported results. (3) Bulk dispersion appears to be relatively unimportant to the results. In particular, motions on the fault plane are seen to be relatively insensitive to a wide range of changes in the dispersion in the bulk away from the fault, both at long wavelengths and at short wavelengths. In contrast, the fault properties are seen to be very important to the results. (4) Nucleation from slip weakening and time-dependent weakening showed similar large-scale behavior. However, not all constitutive laws are insensitive to all nucleation approximations; those making a model “inherently discrete” and hence grid-dependent, in particular, can affect large scales. (5) While inherent discreteness has been seen to be a source of power law small-event complexity in some fault models, it does not appear to be the cause of the complexity in the attractors examined here, and reported in earlier work, fortuitously in the pecial parameter range, with the same class of continuum fault models and same or very similar constitutive relations. Continuum homogeneous dynamic complexity does indeed exist, although that includes type II small-event complexity only under restricted circumstances.Plate tectonics, Mechanics, Physicsbes11Lamont-Doherty Earth ObservatoryArticlesExperimental Evidence for Different Strain Regimes of Crack Populations in a Clay Model
http://academiccommons.columbia.edu/catalog/ac:162481
Spyropoulos, Chrysanthe; Griffith, William J.; Scholz, Christopher H.; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20788Tue, 18 Jun 2013 14:23:57 +0000We report results from clay extension experiments used as a model for the evolution of fault populations due to stress interactions. At yielding cracks begin to appear and the brittle strain due to them quickly reaches a rate matching the applied stretching rate. The crack density (number of cracks per unit area) initially increases apace, then reaches a maximum at a critical strain, decreasing thereafter. At low strains, where the crack population is dilute, a power law length distribution is observed, which at high strain, gradually transitions to an exponential. This agrees with fault populations data observed in low and high strain settings. These results indicate that fault populations ranging from power law to exponential size-frequency distributions reflect the population evolution with increased strain.Plate tectonics, Mechanicschs2, bes11Applied Physics and Applied Mathematics, Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesPost seismic response of repeating aftershocks
http://academiccommons.columbia.edu/catalog/ac:162478
Schaff, David P.; Beroza, Gregory C.; Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20786Tue, 18 Jun 2013 13:59:02 +0000The recurrence intervals of repeating earthquakes on the San Andreas Fault in the Loma Prieta aftershock zone follow the characteristic 1/t decay of Omori's law. A model in which these earthquakes occur on isolated patches of the fault that fail in stick-slip with creep around them can explain this observation. In this model the recurrence interval is inversely proportional to the loading rate due to creep. Logarithmic velocity strengthening friction predicts 1/t decay in creep rate following the mainshock. The time dependence of recurrence is inconsistent with a simple viscous constitutive relationship, which predicts an exponential decay of loading rate. Thus, our observations imply postseismic slip at seismogenic depth under a power law rheology. The time dependence of postseismic deformation measured geodetically may be diagnostic of whether postseismic deformation is caused by creep or possible viscoelastic deformation at greater depths.Plate tectonics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesUniversality in selection with local perturbations in the Saffman-Taylor problem
http://academiccommons.columbia.edu/catalog/ac:162398
Shaw, Bruce E.http://hdl.handle.net/10022/AC:P:20764Mon, 17 Jun 2013 09:55:40 +0000An analytic theory using WKBJ methods for selection with local perturbations in the SaffmanTaylor [Proc. R. Soc. London, Ser. A 245, 312 (1958)] problem is presented. I obtain qualitative agreement with previously published phenomenology, including symmetric narrowed fingers for local reductions in the surface-tension parameter, narrowed asymmetric fingers for local increases,
and scaling of the tip curvature and asymmetry with the square root of the surface-tension parameter. The source of the universality in the perturbed problem is discussed, giving some explanation of why the experimental perturbations can be modeled by locally varying surface tension. Very good quantitative agreement between theory and a numerical simulation of the same perturbation is shown, with no adjustable parameters to fit. Finally, I outline experiments to test new behavior predicted by the theory; a quantitative prediction observable experimentally is given.Theoretical physics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesFinger narrowing under local perturbations in the Saffman-Taylor problem
http://academiccommons.columbia.edu/catalog/ac:162376
Zocchi, Giovanni; Shaw, Bruce E.; Libchaber, Albert; Kadanoff, Leo P.http://hdl.handle.net/10022/AC:P:20749Fri, 14 Jun 2013 11:45:56 +0000We present an experimental study and a numerical simulation of the effect of time-independent, localized perturbations applied to the interface in the Saffman-Taylor fingering problem. When the perturbation is applied at a specific spot near the tip of the finger, the selection of the steady-state shape is drastically changed. In particular, one can obtain fingers with a width well below λ=1/2. A perturbation applied far away from the tip has no effect. We observe the same behavior in the simulation and in the experiment.Theoretical physics, Mechanicsbes11Lamont-Doherty Earth ObservatoryArticlesModeling and Simulation of Random Processes and Fields in Civil Engineering and Engineering Mechanics
http://academiccommons.columbia.edu/catalog/ac:188460
Benowitz, Brett Alexanderhttp://dx.doi.org/10.7916/D8P26XGNWed, 15 May 2013 10:05:52 +0000This thesis covers several topics within computational modeling and simulation of problems arising in Civil Engineering and Applied Mechanics. There are two distinct parts. Part 1 covers work in modeling and analyzing heterogeneous materials using the eXtended Finite Element Method (XFEM) with arbitrarily shaped inclusions. A novel enrichment function, which can model arbitrarily shaped inclusions within the framework of XFEM, is proposed. The internal boundary of an arbitrarily shaped inclusion is first discretized, and a numerical enrichment function is constructed "on the fly" using spline interpolation. This thesis considers a piecewise cubic spline which is constructed from seven localized discrete boundary points. The enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve. Parametric convergence studies are carried out to show the accuracy of this approach, compared to pointwise and linear segmentation of points, for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity. Moreover, the viability of this approach is illustrated on a Neo-Hookean hyperelastic material with a hole undergoing large deformation. In this case, the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing. Part 2 then moves on to research work in simulation of random processes and fields. Novel algorithms for simulating random processes and fields such as earthquakes, wind fields, and properties of functionally graded materials are discussed. Specifically, a methodology is presented to determine the Evolutionary Spectrum (ES) for non-stationary processes from a prescribed or measured non-stationary Auto-Correlation Function (ACF). Previously, the existence of such an inversion was unknown, let alone possible to compute or estimate. The classic integral expression suggested by Priestley, providing the ACF from the ES, is not invertible in a unique way so that the ES could be determined from a given ACF. However, the benefits of an efficient inversion from ACF to ES are vast. Consider for example various problems involving simulation of non-stationary processes or non-homogeneous fields, including non-stationary seismic ground motions as well as non-homogeneous material properties such as those of functionally graded materials. In such cases, it is sometimes more convenient to estimate the ACF from measured data, rather than the ES. However, efficient simulation depends on knowing the ES. Even more important, simulation of non-Gaussian and non-stationary processes depends on this inversion, when following a spectral representation based approach. This work first examines the existence and uniqueness of such an inversion from the ACF to the ES under a set of special conditions and assumptions (since such an inversion is clearly not unique in the most general form). It then moves on to efficient methodologies of computing the inverse, including some established optimization techniques, as well as proposing a novel methodology. Its application within the framework of translation models for simulation of non-Gaussian, non-stationary processes is developed and discussed. Numerical examples are provided demonstrating the capabilities of the methodology. Additionally in Part 2, a methodology is presented for efficient and accurate simulation of wind velocities along long span structures at a virtually infinite number of points. Currently, the standard approach is to model wind velocities as a multivariate stochastic process, characterized by a Cross-Spectral Density Matrix (CSDM). In other words, the wind velocities are modeled as discrete components of a vector process. To simulate sample functions of the vector process, the Spectral Representation Method (SRM) is used. The SRM involves a Cholesky decomposition of the CSDM. However, it is a well known issue that as the length of the structure, and consequently the size of the vector process, increases, this Cholesky decomposition breaks down (from the numerical point of view). To avoid this issue, current research efforts in the literature center around approximate techniques to simplify the decomposition. Alternatively, this thesis proposes the use of the frequency-wavenumber (F-K) spectrum to model the wind velocities as a stochastic "wave," continuous in both space and time. This allows the wind velocities to be modeled at a virtually infinite number of points along the length of the structure. In this work, the relationship between the CSDM and the F-K spectrum is first examined, as well as simulation techniques for both. The F-K spectrum for wind velocities is then derived. Numerical examples are then carried out demonstrating that the simulated wave samples exhibit the desired spectral and coherence characteristics. The efficiency of this method, specifically through the use of the Fast Fourier Transform, is demonstrated.Mechanics, Civil engineeringbab2140Civil Engineering and Engineering MechanicsDissertationsA novel discrete damage zone model and enhancement of the extended finite element method for fracture mechanics problems
http://academiccommons.columbia.edu/catalog/ac:166527
Liu, Xiahttp://hdl.handle.net/10022/AC:P:14865Tue, 09 Oct 2012 16:38:53 +0000This research develops two novel numerical methods for applications in fracture mechanics: (I) A new crack tip enrichment function in the extended finite element method (XFEM), and (II) a discrete damage zone model for quasi-static and fatigue delamination in composites. The first method improves XFEM when applied to general nonlinear materials when crack tip analytical solutions are not available. For linear elastic materials, Branch functions are commonly used as crack tip enrichments. Typically, these are four functions derived from linear elasticity theory and added as additional degrees of freedom. However, for general inelastic material behavior, where the analytical solution and the order of singularity are unknown, Branch functions are typically not used, and only the Heaviside function is employed. This however may introduce numerical error, such as inconsistency in the position of the crack tip. Hence, a special construction of Ramp function is proposed as tip enrichment, which may alleviate some of the problems associated with the Heaviside function when applied to general nonlinear materials, especially ones with no analytical solutions available. The idea is to linearly ramp down the displacement jump on the opposite sides of the crack to the actual crack tip, which may stop the crack at any point within an element, employing only one enrichment function. Moreover, a material length scale that controls the slope of the ramping is introduced to allow for better flexibility in modeling general nonlinear materials. Numerical examples for ideal and hardening elasto-plastic and elasto-viscoplastic materials are given, and the convergence studies show that a better performance is obtained by the proposed Ramp function in comparison with the Heaviside function. Nevertheless, when analytical functions, such as the Hutchinson-Rice-Rosengren (HRR) fields, do exist (for very limited material models), they indeed perform better than the proposed Ramp function. However, they also employ more degrees of freedom per node and hence are more expensive. The second method developed in this thesis is a discrete damage zone model (DDZM) to simulate delamination in composite laminates. The method is aimed at simulating fracture initiation and propagation within the framework of the finite element method. In this approach, rather than employing specific cohesive laws, we employ damage laws to prescribe both interface spring softening and bulk material stiffness degradation to study crack propagation. For a homogeneous isotropic material the same damage law is assumed to hold in both the continuum and the interface elements. The irreversibility of damage naturally accounts for the reduction in material strength and stiffness if the material was previously loaded beyond the elastic limit. The model parameters for interface element are calculated from the principles of linear elastic fracture mechanics. The model is implemented in Abaqus and numerical results for single-mode as well as mixed-mode delamination are presented. The results are in good agreement with those obtained from the virtual crack closure technique (VCCT) and available analytical solutions, thus, illustrating the validity of this approach. The suitability of the method for studying fracture in fiber-matrix composites involving fiber debonding and matrix cracking is demonstrated. Finally, the DDZM method is extended to account for temperature dependent fatigue delamination in composites. The interface element softening is described by a combination of static and fatigue damage growth laws so as to model delamination under high-cycle fatigue. The dependence of fatigue delamination on the ambient temperature is incorporated by introducing an Arrhenius type relation into the damage evolution law. Numerical results for mode I, mode II and mixed mode delamination growth under cyclic loading are presented and the model parameters are calibrated using previously published experimental data. Then, predictions are made under varying mode mix conditions and are compared with numerical results in the literature.Engineering, Mechanicsxl2201Civil Engineering and Engineering MechanicsDissertationsNanofluidics: Fundamentals and Applications in Energy Conversion
http://academiccommons.columbia.edu/catalog/ac:152478
Liu, Linghttp://hdl.handle.net/10022/AC:P:14671Wed, 12 Sep 2012 11:58:14 +0000As a nonwetting liquid is forced to invade the cavities of nanoporous materials, the liquid-solid interfacial tension and the internal friction over the ultra-large specific surface area (usually billions of times larger than that of bulk materials) can lead to a nanoporous energy absorption system (or, composite) of unprecedented performance. Meanwhile, while functional liquids, e.g. electrolytes, are confined inside the nanopores, impressive mechanical-to-electrical and thermal-to-electrical effects have been demonstrated, thus making the nanoporous composite a promising candidate for harvesting/scavenging energy from various environmental energy sources, including low grade heat, vibrations, and human motion. Moreover, by taking advantage of the inverse process of the energy absorption/harvesting, thermally/electrically controllable actuators can be designed with simultaneous volume memory characteristics and large mechanical energy output. In light of all these attractive functionalities, the nanoporous composite becomes a very promising building block for developing the next-generation multifunctional (self-powered, protective and adaptive) structures and systems, with wide potential consumer, military, and national security applications. In essence, all the functionalities of the proposed nanofluidic energy conversion system are governed by nanofluidics , namely, the behavior of liquid molecules and ions when confined in ultra-small nanopores. Nanofluidics is an emerging research frontier where solid mechanics and fluid mechanics meet at the nanoscale. The complex interactions between liquid molecules/ions and solid atoms at the nanointerface, as well as the unique structural, thermal and electrical characteristics of fluids confined in nanocavities collectively represent an outstanding challenge in physical science. A thorough understanding of the science of nanofluids, in particular the detailed molecular mechanisms as well as the roles of various material and system parameters, does not only underpin the development and optimization of the aforementioned nanofluidic energy conversion system, but it also have broad impact on a number of other areas including environmental engineering, chemical engineering, bioengineering, and energy engineering, etc. This dissertation carries out a systematic computational study to explore the fundamental nanofluidic infiltration and transport mechanisms, as well as the thermal and electrical characteristics of the solid-liquid interface. New physical models describing the unique nanofluidic phenomena will be established, where critical parameters, such as the surface tension, contact angle, and viscosity, will be reinvestigated at the nanoscale. The effects of various material and system parameters, such as the solid phase, liquid phase, pore size and pore geometry, as well as the external thermal, electrical and mechanical loads, etc., will be systematically investigated and bridged with the nanofluidic energy conversion processes. The energy conversion efficiencies under various conditions will be evaluated via a synergy between simulation and experiment. Reverse analysis based on the revealed principles can guide the optimization of the various material and system parameters, which potentially may contribute to the design of highly efficient and sustainable nanofluidic energy conversion devices. Besides the direct impact on the nanofluidic energy conversion, the study is also directly relevant to biological conduction and environmental sustainability, in both of which infiltration and transport play important roles.Mechanics, Alternative energy, Energy, Materials sciencell2405Civil Engineering and Engineering Mechanics, Earth and Environmental EngineeringDissertationsScience of Nanofluidics and Energy Conversion
http://academiccommons.columbia.edu/catalog/ac:148244
Xu, Baoxinghttp://hdl.handle.net/10022/AC:P:13570Thu, 21 Jun 2012 12:12:49 +0000The emerging subject of nanofluidics, where solids and fluids interact closely at the nanoscale, has exhibited radically different from their macroscopic counterparts (and sometimes counterintuitive), and yet relatively less explored. On the other hand, the resulting unique properties may contribute to a number of innovative functions with fascinating applications. Among various exciting potential applications, an important and ever expanding one is to provide alternative solutions to energy conversion with high efficiency, including energy absorption, actuation and harvesting. In this dissertation, we first report a novel protection mechanism of energy capture through which an intensive impact or blast energy can be effectively mitigated based on a nonwetting liquid-nanoporous material system. The captured energy is stored in nanopores in the form of potential energy of intercalated water molecules for a while, and not necessarily converted to other forms of energy (e.g. heat). At unloading stage, the captured energy will be released gradually due to the hydrophobic inner surfaces of nanopores through the diffusion of water molecules out of nanopores, thus making this system reusable. Several key controlling factors including impacting velocity, nanopore size, nanopore structure, and liquid phase have been investigated on the capacity of energy capture. The molecular mechanism is elucidated through the study of water molecular distributions inside nanpores. These molecular dynamic (MD) findings are quantitatively verified by a parallel blast experiment on a zeolite/water system. During the transport of confined liquid molecules, the friction resistance exerted by solid atoms of nanopores to liquid molecules will dissipate part of energy, and is highly dependent of temperature of liquid molecules and wall morphology of nanopores. Using MD simulations, the effects of temperature and wall roughness on the transport resistance of water molecules inside nanopores are investigated in Chapter 3. The effective shear stress and nominal viscosity that dominate the nanofluidic transport resistance are extracted and coupled with the nanopore size, transport rate, and liquid property. The molecular-level mechanisms are revealed through the study of the density profile and hydrogen bonding of confined liquid molecules. A parallel experiment on a nanoporous carbon-liquid system is carried out and qualitatively verifies MD findings. Motived by the well-known thermo- and electro-capillary effect, Chapter 4 and Chapter 5 present a conceptual design of thermal and electric actuation system by adjusting the relative hydrophobicity of a liquid-nanoporous system through a thermal and electric field, respectively. The thermally and electrically dependent infiltration behaviors of liquids into nanopores are analyzed by using MD simulations. The fundamental molecular characteristics, including the density profile, contact angle, and surface tension of the confined liquid molecules, are examined to reveal underlying mechanisms. The energy density, power density, and efficiency of both thermal and electric actuation systems are explored and their variations with pore size, solid phase, and liquid phase are evaluated. Thermally and electrically controlled infiltration experiments on a zeolite-water /electrolyte solution system are performed accordingly to qualitatively validate these findings. These energy actuation systems can also become high density thermal or electric storage devices with proper designs. Energy harvesting by the flow of a hydrochloric acid-water solution through a nanopore is explored using atomistic simulations in the last chapter. Through ion configurations near the pore wall, an averaged ion drifting velocity is determined, and the induced voltage along the axial direction is obtained as a function of key material parameters, including the applied flow rate, environmental temperature, solution concentration and nanopore size. The molecular mechanism of ion hopping and motion is revealed. This study shed light on harvesting wasted thermal and mechanical energy from ambient environmental sources such as wasted heat in power plants. Nanofluidics is a novel and thriving research area, whose couplings with other disciplines such as material, mechanical, physical, chemical, electrical engineering are open.Engineering, Mechanics, Materials sciencebx2107Earth and Environmental EngineeringDissertations