Journal articles
  • R. Abedi, S. Mudaliar. “An Asynchronous Spacetime Discontinuous Galerkin Finite Element Method for Time Domain Electromagnetics”, Journal of Computational Physics, , 351:121–144, 2017.
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    We present an asynchronous spacetime discontinuous Galerkin (aSDG) method for time domain electromagnetics in which space and time are directly discretized. By using differential forms we express Maxwell’s equations and consequently their discontinuous Galerkin discretization for arbitrary domains in spacetime. The elements are discretized with electric and magnetic basis functions that are discontinuous across all inter-element boundaries and can have arbitrary high and per element spacetime orders.  When restricted to unstructured grids that satisfy a specific causality constraint, the method has a local and asynchronous solution procedure with linear solution complexity in terms of the number of elements. We numerically investigate the convergence properties of the method for 1D to 3D uniform grids for energy dissipation, an error relative to the exact solution, and von Neumann dissipation and dispersion errors. Two dimensional simulations demonstrate the effectiveness of the method in resolving sharp wave fronts.
  • R. Abedi, R. Haber, and P. Clarke. “Effect of material randomnness on dynamic fracture of quasi-brittle materials”, International Journal of Fracture: Special Issue for Integrated Computational Structure-Material Modeling of Deformation & Failure Under Extreme Conditions An IUTAM Symposium, 2017. Accepted.
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    We propose an asynchronous Spacetime Discontinuous Galerkin (aSDG) method combined with a novel rate-dependent interfacial damage model as a means to simulate crack nucleation and propagation in quasi-brittle materials. Damage acts in the new model to smoothly transition the aSDG jump conditions on fracture surfaces between Riemann solutions for bonded and debonded conditions. We use the aSDG method’s powerful adaptive meshing capabilities to ensure solution accuracy without resorting to crack-tip enrichment functions and extend those capabilities to support fracture nucleation, extension and intersection. Precise alignment of inter-element boundaries with aw orientations and crack-propagation directions ensures mesh-independent crack-path predictions. We demonstrate these capabilities in a study of crack-path convergence as adaptive error tolerances tend to zero.The fracture response of quasi-brittle materials is highly sensitive to the presence and properties of microstructural defects. We propose two approaches to modeling these inhomogeneities. In the first, we represent defects explicitly as crack-like features in the analysis domain’s geometry with random distributions of size, location, and orientation. In the second, we model microscopic flaws implicitly, with probabilistic distributions of strength and orientation, to drive nucleation of macroscopic fractures. Crack-path oscillation, microcracking, and crack branching make numerical simulation of dynamic fracture particularly challenging. We present numerical examples that explore the influence of model parameters and inhomogeneities on fracture patterns and the aSDG model’s ability to capture complex fracture patterns and interactions.
  • R. Abedi, “A comparative and parametric study of dynamic cohesive and linear elastic fracture mechanics models”, International Journal of Solids and Structures 102–103: 163–175, 2016.
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    In cohesive fracture mechanics (CFM), fundamental nondimensional parameters are the ratios of space- time domain geometries and loadings to corresponding intrinsic scales implied by the cohesive fracture traction–separation relations (TSRs). One of these parameters is the nondimensional load-to-strength parameter which is the ratio of the applied loads, expressed in stress form, to an intrinsic strength scale implied by a TSR. Herein the radii of stress singularity from asymptotic Linear Elastic Fracture Mechanics (LEFM) solutions are derived to normalize cohesive process zone (CPZ) sizes from CFM. By approximating these nondimensional CPZ sizes, a simple small-scale yielding (SSY) indicator is derived for dynamic fracture which in turn is shown to be proportional to the square of the load-to-strength parameter. Thus, the load-to-strength parameter serves two purposes. First, increasing this ratio is shown to correspond to more ductile response for families of cohesive fracture self-similar solutions. Second being related to SSY condition, it is used to evaluate the validity of an LEFM model. Numerical results compare characteristic differences between these groups of CFM solutions, investigate the accuracy of the proposed SSY indicator, demonstrate LEFM solutions underestimate crack length and speed even when the SSY condition is satisfied, and study the evolution of the CPZ size.
  • R. Pal, R. Abedi, A. Madhukar, and R.B. Haber, “Adaptive spacetime discontinuous Galerkin method for hyperbolic advection–diffusion with a non-negativity constraint”, International Journal for Numerical Methods in Engineering, 105(13): 963-989, 2016.
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    Applications where the diffusive and advective time scales are of similar order give rise to advection–diffusion phenomena that are inconsistent with the predictions of parabolic Fickian diffusion models. Non-Fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in Fickian models. In this work, we implement a modified, frame-invariant form of Cattaneo’s hyperbolic diffusion relation within a spacetime discontinuous Galerkin advection–diffusion model. An h-adaptive spacetime meshing procedure supports an asynchronous, patch-by-patch solution procedure with linear computational complexity in the number of spacetime elements. This localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non-negative concentration solution. In contrast to some previous methods, we do not modify the numerical fluxes to enforce non-negative concentrations. Thus, the element-wise conservation properties that are intrinsic to discontinuous Galerkin models are defined with respect to physically meaningful Riemann fluxes on the element boundaries. We present numerical examples that demonstrate the effectiveness of the proposed model, and we explore the distinct features of hyperbolic advection–diffusion response in subcritical and supercritical flows. Copyright © 2015 John Wiley & Sons, Ltd.
  • S.T. Miller and R. Abedi, “Riemann solutions for spacetime discontinuous Galerkin methods”, Journal of Computational and Applied Mathematics, 270: 564 – 570, 2014.
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    Spacetime discontinuous Galerkin nite element methods (cf. [1,2,3]) rely on `target uxes’ on element boundaries that are computed via local one-dimensional Riemann solutions in the direction normal to element face. In this work, we demonstrate a generalized solution procedure for linearized hyperbolic systems based on diago- nalisation of the governing system of partial di erential equations. We show that source terms do not in uence the Riemann solution in the spacetime setting. We provide details for implementation of coordinate transformations and Riemann so- lutions. Exact Riemann solutions for some linear systems of equations are provided as examples.
  • R. Abedi and R.B. Haber, “Riemann solutions and spacetime discontinuous Galerkin method for linear elastodynamic contact”, Computer Methods in Applied Mechanics and Engineering, 270: 150 – 177, 2014.
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    We derive Riemann solutions for stick, slip and separation contact modes in a linear elastic material with an isotropic Coulomb friction relation and explore their numerical implementation. The Riemann solutions preserve the characteristic structure of the underlying elastodynamic system and imply dynamic contact conditions that are distinct from the quasi-static conditions used in some numerical models.Nonphysical discontinuities in the standard Coulomb model at stick-slip transitions can cause contact-mode chatter in numerical simulations. We restate the Coulomb relation to remove these artificial discontinuities and eliminate the need for algorithmic remedies. Discontinuous response at abrupt separation-to-contact transitions is physically reasonable, and we propose a regularization scheme to address this case. We implement the Riemann contact solutions within an adaptive spacetime discontinuous Galerkin (SDG) code and report numerical results that demonstrate the model’s efficacy.
  • R. Abedi and R.B. Haber, “Spacetime dimensional analysis and self-similar solutions of linear elastodynamics and cohesive dynamic fracture”, International Journal of Solids and Structures, 48(13):2076 – ­­2087, 2011.
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    We present a dimensional analysis and self-similar solutions for linear elastodynamics with extensions to dynamic fracture models based on cohesive traction–separation relations. We formulate the problem using differential forms in spacetime and show that the scaling rules expressed in terms of forms are simpler and more uniform than those obtained for tensor representations of the solution. In the extension to cohesive elastodynamic fracture, we identify and study the influence of certain intrinsic cohesive scales on dynamic fracture behavior and describe a fundamental set of nondimensional groups that uniquely identifies families of self-similar solutions. We present numerical studies of the influence of selected nondimensional parameters on dynamic fracture response to verify the dimensional analysis, including the identification of the fundamental set for cohesive fracture mechanics. We show that distinct values of a widely-used nondimensional quantity can produce self-similar solutions. Therefore, this quantity is not fundamental, and it cannot parameterize dynamic, cohesive-fracture response.
  • R. Abedi, M.A. Hawker, K. Matous, and R.B. Haber, “An adaptive spacetime discontinuous Galerkin method for cohesive damage models of elastodynamic fracture”, International Journal for Numerical Methods in Engineering, 81(10):1207 – 1241, 2010.
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    This paper describes an adaptive numerical framework for cohesive fracture models based on a spacetime discontinuous Galerkin (SDG) method for elastodynamics with element-wise momentum balance. Discontinuous basis functions and jump conditions written with respect to target traction values simplify the implementation of cohesive traction–separation laws in the SDG framework; no special cohesive elements or other algorithmic devices are required. We use unstructured spacetime grids in a h-adaptive implementation to adjust simultaneously the spatial and temporal resolutions. Two independent error indicators drive the adaptive refinement. One is a dissipation-based indicator that controls the accuracy of the solution in the bulk material; the second ensures the accuracy of the discrete rendering of the cohesive law. Applications of the SDG cohesive model to elastodynamic fracture demonstrate the effectiveness of the proposed method and reveal a new solution feature: an unexpected quasi-singular structure in the velocity response. Numerical examples demonstrate the use of adaptive analysis methods in resolving this structure, as well as its importance in reliable predictions of fracture kinetics.
  • R. Abedi, R.B. HaberS. Thite, and J. Erickson, “An h–adaptive spacetime discontinuous Galerkin method for linearized elastodynamics”, Revue Européenne de Mécanique Numérique, special issue on adaptive analysis (ed.),     15(6):619 – 642, 2006 (Invited paper).
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    We present an h-adaptive version of the spacetime-discontinuous Galerkin (SDG) finite element method for linearized elastodynamics (Abedi et al., 2006). The adaptive version inherits key properties of the basic SDG formulation, including element-wise balance of linear and angular momentum, complexity that is linear in the number of elements and oscillation-free shock capturing. Unstructured spacetime grids allow simultaneous adaptation in space and time. A localized patch-by-patch solution process limits the cost of reanalysis when the error indicator calls for more refinement. Numerical examples demonstrate the method’s performance and shock-capturing capabilities.
  • R. Abedi, B. Petracovici, and R.B. Haber, “A spacetime discontinuous Galerkin method for linearized elastodynamics with element–wise momentum balance”, Computer Methods in Applied Mechanics and Engineering, 195:3247 – 3273, 2006.
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    We present a new spacetime discontinuous Galerkin finite element method for linearized elastodynamics that delivers exact balance of linear and angular momentum over every spacetime element. The method is formulated for use with fully unstructured spacetime grids and uses displacement basis functions that are discontinuous across all inter-element boundaries. We introduce a new spacetime formulation of continuum elastodynamics that uses differential forms and the exterior calculus on manifolds to generate a system of spacetime field equations and jump conditions. Then we invoke a Bubnov-Galerkin weighted residuals procedure to formulate the finite element method. We describe an implementation on patch-wise causal meshes that features linear complexity in the number of elements and special per-pixel accurate visualization. Numerical examples confirm an a priori error estimate and demonstrate the method’s shock-capturing capabilities.


Articles in Conference Proceedings

  • R. Abedi, “An adaptive time domain approach to characterize dispersive elastodynamic media” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2017 IMECE, Tampa, Florida, USA – November 5-8, 2017, paper no. 70805 (7 pages), 2017.
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     A time domain approach is presented to compute the transmission and reflection coefficients of a unit cell. The solution of a wave scattering problem to an ultra-short incident wave enables the derivation of these scattering parameters with only one time domain solution. The adaptive operations of a spacetime discontinuous Galerkin method and several or its unique properties, such as linear solution complexity and local / asynchronous solution features, enable accurate computation of scattering parameters. An inverse parameter retrieval method, from the equivalent material impedance and wave speed to dispersive elastic constitutive parameters, is uniquely solved by using the continuity of the wavenumber.
  • P.L. Clarke, R. Abedi, B. Bahmani, K.A. Acton, and S.C. Baxter, “Effect of the spatial inhomogeneity of fracture strength on fracture pattern for quasi-brittle materials” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2017 IMECE, Tampa, Florida, USA – November 5-8, 2017, paper no. 71515 (9 pages), 2017.
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     The response of quasi-brittle materials is greatly influenced by their microstructural architecture and variations. To model such statistical variability, Statistical Volume Elements (SVEs) are used to derive a scalar fracture strength for domains populated with microcracks. By employing the moving window approach the probability density function and covariance function of the scalar fracture strength field are obtained. The Karhunen-Loève method is used to generate realizations of fracture strength that are consistent with the SVE-derived statistics. The effect of homogenization scheme, through the size of SVE, on fracture pattern is studied by using an asynchronous spacetime discontinuous Galerkin (aSDG) finite element method, where cracks are exactly tracked by the method’s adaptive operations. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • K.A. Acton, S.C. Baxter, B. Bahmani, P.L. Clarke, and R. Abedi, “Mesoscale models characterizing material property fields used as a basis for predicting fracture patterns in quasi-brittle materials” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2017 IMECE, Tampa, Florida, USA – November 5-8, 2017, paper no. 71500 (6 pages), 2017.
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     To accurately predict fracture patterns in quasi-brittle materials, it is necessary to accurately characterize heterogeneity in the properties of a material microstructure. This heterogeneity influences crack propagation at weaker points. Also, inherent randomness in localized material properties creates variability in crack propagation in a population of nominally identical material samples. In order to account for heterogeneity in the strength properties of a material at a small scale (or “microscale”), a mesoscale model is developed at an intermediate scale, smaller than the size of the overall structure. A central challenge of characterizing material behavior at a scale below the representative volume element (RVE), is that the stress/strain relationship is dependent upon boundary conditions imposed. To mitigate error associated with boundary condition effects, statistical volume elements (SVE) are characterized using a Voronoi tessellation based partitioning method. A moving window approach is used in which partitioned Voronoi SVE are analysed using finite element analysis (FEA) to determine a limiting stress criterion for each window. Results are obtained for hydrostatic, pure and simple shear uniform strain conditions. A method is developed to use superposition of results obtained to approximate SVE behavior under other loading conditions. These results are used to determine a set of strength parameters for mesoscale material property fields. These random fields are then used as a basis for input in to a fracture model to predict fracture patterns in quasi-brittle materials. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • S. Mudaliar, P.L. Clarke, and S. Mudaliar, “Radiative transfer in turbulent ow using spacetime discontinuous Galerkin finite element method” In: Proceeding XXXIInd International Union of Radio Science General Assembly & Scientific Symposium, URSI 2017 GASS, Palais des congres, Montreal, Canada – August 19-26th, 2017, , paper no. 2965 (4 pages).
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    The radiative transfer equation for a problem that involves scattering, absorption and radiation is solved using spacetime discontinuous Galerkin (SDG) method. The strength of finite element method to handle scattering problems in heterogeneous media with complex geometries is well known. Adaptive operations in spacetime facilitates very accurate and efficient solution algorithm. We investigated the accuracy of the SDG method by using the method of manufactured solutions. For the case of harmonic phase functions we illustrate how the L2 norm error decreases with the choice of high order polynomial and more refined element size. Key merits of the use of SDG for our problem enamates from its linear solution cost, and the ability to obtain the solution for a wide frequency spectrum in one time domain simulation.
  • R. Abedi and S. Mudaliar, “A spacetime adaptive approach to characterize complex dispersive media” In: Proceeding XXXIInd International Union of Radio Science General Assembly & Scientific Symposium, URSI 2017 GASS, Palais des congres, Montreal, Canada – August 19-26th, 2017, paper no. 2439 (4 pages).
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    We present a time domain approach that can obtain reflection and transmission coefficients of a material for a wide range of frequencies. The advanced method of spacetime discontinuous Galerkin method is used to obtain the time domain response of a unit cell to an incident wave. Adaptive operations in space and time permits very efficient and accurate tracking of wave fronts. By Fourier analysis and inversion of the obtained transmission and reflection coefficients in the frequency domain, we obtain equivalent impedance, wave speed, permittivity, and permeability of the unit cell for the given frequencies. The linear solution cost of the SDG method, its powerful adaptive operations, and derivation of the entire spectrum with one time domain simulation are attractive attributes of the proposed method.
  • R. Abedi and S. Mudaliar, “Error analysis and comparison of Riemann and average fluxes for a spacetime discontinuous Galerkin electromagnetic formulation” In: Proceeding XXXIInd International Union of Radio Science General Assembly & Scientific Symposium, URSI 2017 GASS, Palais des congres, Montreal, Canada – August 19-26th, 2017, paper no. 2480 (4 pages).
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    We present a time domain discontinuous Galerkin (TDDG) method for electromagnetics problem that directly discretizes space and time by unstructured grids satisfying a specific causality constraint. This enables a local and asynchronous solution procedure. We show that the numerical method is dissipative, thus ensuring its stability. Numerical results show the convergence rate of 2p + 1 for energy dissipation. We also investigate the choice of Riemann versus average numerical fluxes for noncausal faces and demonstrate that while the more dissipative nature of Riemann   fluxes may render it unsuitable for low order elements, it provides a cleaner solution for high order elements.
  • R. Abedi and S. Mudaliar, “An h-adaptive Time Domain Discontinuous Galerkin Method for Electromagnetics” In: Proceeding 2017 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, San Diego, California, USA – July 9-14, paper no. 1795 (2 pages), 2017.
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    We present an h-adaptive time domain discontinuous Galerkin (TDDG) method for electromagnetics problem in which space and time are directly discretized by unstructured grids that satisfy a specific causality constraint. This enables a local and asynchronous solution procedure with arbitrary high and per element spacetime orders of elements. Our numerical results demonstrate that by using energy dissipation as an error indicator and local adaptive operations in spacetime we can significantly improve the efficiency of the method relative to nonadaptive solutions.
  • R. Abedi, R.B. Haber, and A. Elbanna, “Mixed-mode dynamic crack propagation in rocks with contact-separation mode transitions” In: Proceeding 51th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA – June 25-28, ARMA 17-0679 (12 pages), 2017.
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    We propose an interfacial contact/damage model for simulating dynamic fracture in rocks. An interfacial damage parameter, D, models the evolution of damage on fracture interfaces, while relative contact and contact–stick fractions model contact–separation and stick–slip transitions. The damage rate is determined by an effective stress, written as a scalar function of the normal and tangential components of the Riemann traction solution for assumed bonded conditions. We propose alternative definitions of the effective stress that generate failure criteria that resemble the Tresca and Mohr–Coulomb criteria for compressive stress states, and we compare their compressive strengths and fracture angles under a compressive loading. We adopt a stochastic Weibull model for crack-nucleation in which cracks nucleate at points where the effective stress exceeds the probabilistic fracture strength. We implement the nucleation model with an h-adaptive asynchronous spacetime discontinuous Galerkin (aSDG) method that captures accurately the complex fracture patterns that arise under dynamic loading conditions. Numerical examples illustrate the effects on fracture response of varying the stochastic nucleation parameters and the alternative definitions of the effective stress. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • P.L. Clarke and R. Abedi, “Fracture modeling of rocks based on random field generation and simulation of inhomogeneous domains” In: Proceeding 51th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA – June 25-28, ARMA 17-0643 (11 pages), 2017.
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    Realistic fracture simulations in rock as a heterogeneous brittle material with significant inherent randomness, require the use of models that incorporate its inhomogeneities and statistical variability. Since brittle materials do not match ductile materials in dissipating energy in the bulk, their fracture response is highly dependent on the stochastic microscale distribution and strength of defects. The high dependence of their fracture progress on microstructural defects results in wide scatter in their ultimate strength and the so-called size effect. Our approach for incorporating randomness in rocks is based on the modeling of stochastic volume elements (SVEs). Although representative volume elements (RVEs) are more commonly used in solid mechanics, SVEs are more appropriate for fracture analysis since they ensure that the material randomness is maintained. They still average microscale features similar to RVEs, and provide a more economical solution approach than those methods that explicitly model all microcracks in rock.  To create a random field for macroscopic fracture strength field, we first generate several realizations of rock with a prescribed crack density and distribution. SVEs are then constructed with their centers at known spatial position on these random realizations. Next, by using a moving window approach, where the SVE traverses the known positions in these random realizations, we obtain first and second moments of the target random field. Point-wise probability distribution function and spatial covariance function are derived and used to generate consistent realizations of random fields based on the Karhunen-Loève (KL) method. Finally, such realizations will be used for the analysis of dynamic stimulation of a wellbore in a tight formation. A powerful and mesh adaptive spacetime discontinuous Galerkin finite element method is used for dynamic fracture simulations. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • R. Abedi and P.L. Clarke, “Simulation of refracture and contact mode transitions in tight formations” In: Proceeding 51th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA – June 25-28, ARMA 17-0642 (11 pages), 2017.
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     Transitions between separation and contact modes are prevalent in rock mechanics. For stimulating tight hydrocarbon reservoirs, the transfer of hydraulic load from hydraulically loaded to in-situ cracks, removal of hydraulic load, re-fracturing, and application of cyclic loading are all examples involving contact and separation mode reversal. We propose an interfacial damage model that incorporates all contact and separation modes by combining their corresponding dynamically consistent Riemann solutions. Instead of commonly used penalty method and Lagrange multiplier approach we propose a new regularization scheme—based on the interface displacement and separation velocity jumps—that smoothens contact-separation mode transitions, remedies ill-conditioning that may arise by using penalty methods, and provides a tunable maximum penetration. In addition, we propose an aperture-based regularization approach that enables smooth transfer of hydraulic load to in-situ cracks. Numerical results, obtained by an h-adaptive spacetime discontinuous Galerkin method, demonstrate accurate modeling of contact mode transitions and intersection of cracks in hydraulic fracturing.
  • R. Abedi, O. Omidi, P.L. Clarke, “A numerical study on the effect of loading and randomness on fracture patterns in a tight formation” In: Proceeding 51th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA – June 25-28, ARMA 17-0642 (11 pages), 2017.
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    Hydraulic fracturing has been the most common approach to stimulate tight formations. The geometry of the wellbore and the time history of the hydraulic loading play important roles in induced fracture patterns. For example, generating multiple perforations in a wellbore is nowadays attracting more attention in oil industry to enhance gas recovery. The increased number of fractures can potentially enhance the yield of a reservoir by increasing the regions affected by hydraulic fractures. We use an h-adaptive spacetime discontinuous Galerkin method and an interfacial damage model to study the conditions for which the induced hydraulic fractures become effective and propagate in rock. Our results show that as the loading rate decreases, only a few of these fractures will propagate. As the loading rate increases, more perforations become active, until ultimately all result in crack propagation. Moreover, higher loading rates affect larger zones for each of the initial perforations by dynamic fracture features such as microcracking and crack bifurcation. Our study mainly focuses on stimulation techniques that induce fully dynamic loading on rocks; for example, high explosives detonate and sends a shock wave in rock. Given the limitations of hydraulic fracturing technique, we study the effectiveness of a hybrid approach where initial perforations similar to those for hydraulic fracturing are used as seeds of crack propagation under dynamic loading. Finally for very high rates of loading, we demonstrate that a stochastic approach for crack nucleation predicts more realistic fracture patterns than conventional approaches that assume a macroscopically uniform fracture strength for rock. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • R. Abedi, O. Omidi, and P.L. Clarke, “Numerical simulation of rock dynamic fracturing and failure including microscale material randomness” In: Proceeding 50th U.S. Rock Mechanics/Geomechanics Symposium, Houston, Texas, USA – June 26-29, 2016, ARMA 16-0531, 2016.
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     Realistic fracture simulations in rock as a heterogeneous brittle material with significant inherent randomness, requires the use of models that incorporate its inhomogeneities and statistical variability. Dynamic crack growth in rocks is generally associated with complex features such as crack path oscillations, microcracking and crack branching. We employ two approaches to address rock inhomogeneities for dynamic fracture simulations. First we model fractures explicitly with random size, location and orientation as natural pre-existing crack-like defects. Second, we use a probabilistic nucleation technique based on the Weibull model to implicitly incorporate creation of new cracks during the analysis. Both approaches can be used for the simulation of rocks for which the natural fractures are oriented in a specific angle, as in sedimentary rocks. We use the Spacetime Discontinuous Galerkin (SDG) method to efficiently and accurately capture complex fracture patterns observed in dynamic rock fracture. Specifically we employ a novel crack path tracking method, offered by the SDG method’s powerful adaptive operations, to accurately model crack path oscillations, microcracking, and crack bifurcation. Our approach is applicable to rock fracture as well as problems where an induced major crack propagates and intersects natural fractures. Incorporation of macro-micro crack interactions can provide a more accurate estimation in hydrocarbon recovery in tight formations. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • P.L. Clarke, O. Omidi, and R. Abedi, “Modeling crack connectivity of induced fractures in a naturally fractured formation” In: Proceeding 50th U.S. Rock Mechanics/Geomechanics Symposium, Houston, Texas, USA – June 26-29, 2016, ARMA 16-0532, 2016.
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    Advancing fractures during the hydraulic fracturing process can produce complicated growth pattern when propagating in a pre-exiting natural fracture network. A proper network representation for simulating pressure-driven and natural fracture interactions is crucial as these occurrences may result in significant diversion of fracture paths which potentially generate difficulties in proppant transport and ineffectiveness of the treatment. In this study fracture network and propagation patterns are modeled with a simplicial complex network representation and geometrical information are analyzed with a graph theoretic approach. In conjunction with graph theoretic algorithms, a disjoint-set data structure is employed to track fracture connectivity, dynamic hydraulic load advancing in the fracture network and load transfers between independent sets of fractures. This permits imposing independent loading conditions for arbitrary sets of fracture sets. The procedure is implemented in a spacetime discontinuous Galerkin finite element scheme, whose efficiency and accuracy are very important for the type for fracture simulations considered herein. In addition, the SDG method’s powerful mesh adaptive operations enable direct tracking of arbitrary crack propagation patterns. Numerical results, of the dynamics problem solution, from various crack configurations and loading conditions will be presented which can have applications in the stability analysis of natural faults close to hydraulic fracturing reservoirs. For all case studies, the rock matrix domain is subject to confinement (compressive) stress conditions on the boundary; the simplicial complex network is capable to incorporate the connectivity of the main crack with natural fissures and microcracks that are generated due to dynamic loading. (Acknowledgments: The authors gratefully acknowledge partial support for this work via the U.S. National Science Foundation (NSF), CMMI – Mechanics of Materials and Structures (MoMS) program grant number 1538332.)
  • O. Omidi, R. Abedi, and S. Enayatpour, “Well stimulation in tight formations: a dynamic approach” In: Proceeding 50th U.S. Rock Mechanics/Geomechanics Symposium, Houston, Texas, USA – June 26-29, 2016, ARMA 16-0150, 2016.
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    Hydraulic fracturing is widely employed for well stimulation. Different techniques have been utilized in practice to optimize fracking in the last five decades. However, it has some disadvantages including a lack of control over the direction of fracture propagation, the high treatment cost along with environmental issues. Producing multiple fractures by dynamic stimulation techniques seems to be more promising in naturally fractured reservoirs, since it is an effective way for connecting a pre-existing fracture network to a wellbore. In this study, applying high rate loadings we investigate fracturing in rocks due to explosives and propellants as two common methods for dynamic stimulation of a well. An interfacial damage model implemented in a Spacetime Discontinuous Galerkin finite element framework is utilized to simulate fracturing in rocks. A powerful dynamic mesh adaptivity scheme is implemented to track arbitrary crack paths and align them with element boundaries. High explosives produce shockwaves causing extreme compressive stresses, which results in crushing and compacting the rock around the wellbore. Propellants can generate a pressure pulse producing a fracturing behavior that loads the rock in tension. The main advantage of this later approach is to create multiple fractures and consequently prepare the well for an effective hydraulic fracturing with much lower cost as a re-fracturing solution. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.
  • O. Omidi., R. Abedi, and S. Enayatpour, “An Adaptive Meshing Approach to Capture Hydraulic Fracturing” In: Proceeding 49th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, California, USA – June 28-July 1, 2015, ARMA 15-0572, 2015.
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    Hydraulic fracturing is widely employed to stimulate oil and gas reservoirs to increase the productivity of these naturally fissured rock domains. Different numerical techniques are available to examine how hydraulic fractures propagate. They are mainly categorized into continuum and interface-based methods. Cohesive models are among the most effective class of interfacial approaches representing crack surfaces as sharp material interfaces. In lieu of a traditional cohesive model, we have formulated and employed an interfacial damage model that incorporates the processes of nucleation, growth and coalescence on the fracture surfaces. Utilizing a dynamic adaptive meshing, we employed a Spacetime Discontinuous Galerkin (SDG) finite element method to simulate hydraulic fracture propagation. Our SDG implementation adaptively aligns the element boundaries with crack-path trajectories that are obtained as a part of the solution according to a crack growth criterion. Thus, this model does not suffer the mesh-dependent effects encountered in most other numerical fracture models. Furthermore, no discontinuous features are introduced within the elements as opposed to XFEM and generalized finite element methods. Adaptive mesh refinement in an area allows free nucleation, growth and branching of cracks oriented arbitrarily in the domain without any mesh bias whereas a coarse mesh can be used in other regions of the domain to utilize an efficient implementation. Presenting numerical examples, we performed a sensitivity analysis of some input variables such as the magnitude of in-situ stress components, number and orientation of induced fractures is performed to demonstrate the effectiveness of our approach in resolving hydraulic fracturing.
  • R. Abedi, O. Omidi. and P.L. Clarke, “Spacetime Discontinuous Galerkin FEM: Spectral Response” In: Journal of Physics: Conference Series, Proceeding 22nd International Conference on Spectral Line Shapes  (ICSLS22), Tullahoma, TN, USA – June 1-6, 2014, 548(1), 012065, 2014.
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    Materials in nature demonstrate certain spectral shapes in terms of their material properties. Since successful experimental demonstrations in 2000, metamaterials have provided a means to engineer materials with desired spectral shapes for their material properties. Computational tools are employed in two different aspects for metamaterial modeling: 1. Mircoscale unit cell analysis to derive and possibly optimize material’s spectral response; 2. macroscale to analyze their interaction with conventional material. We compare two different approaches of Time-Domain (TD) and Frequency Domain (FD) methods for metamaterial applications. Finally, we discuss advantages of the TD method of Spacetime Discontinuous Galerkin finite element method (FEM) for spectral analysis of metamaterials.
  • R. Abedi, M.A. Hawker, and R.B. Haber, “Spacetime discontinuous Galerkin models for multi–scale elastodynamic fracture: recent progress” In E.O.I. de Navarra, D.R.J. Owen and B. Suárez, editors, Computational Plasticity – Fundamentals and Applications, COMPLAS IX, Proceeding Ninth International Conference on Computational Plasticity, 412 – 415, Barcelona, Spain, September 4-7, 2007. International Center for Numerical Methods in Engineering.
    View/Hide Abstract
    We review recent progress in the development of spacetime discontinuous Galerkin (SDG) finite element methods for modeling dynamic fracture with cohesive damage models. After reviewing the formulation and implementation of an adaptive SDG cohesive model, we describe new investigations of crack-tip kinetics and singular structure in the near-tip velocity field. We discuss prospects for tracking solution-dependent crack paths and for nucleating cohesive interfaces using an extended set of spacetime meshing operations.
  • R. Abedi, S.H. Chung, J. Erickson, Y. Fan, M. Garland, D. Guoy, R.B. Haber, J. Sullivan, S. Thite, and Y. Zhou,   “Space–time   meshing with adaptive refinement and coarsening” In Proceedings of the Twentieth Annual Symposium on Computational Geometry, SCG ’04, pages 300-309, New York, NY, USA, June 9-11 2004. ACM.
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    We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T, our method constructs a tetrahedral mesh of the spacetime domain Ω × [0, T] in constant running time per tetrahedron in \mathbb{R}^3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.
Book Chapters
  • R. Abedi, S.H. Chung, M.A. Hawker, J. Palaniappan, and R.B. Haber, “Modeling Evolving Discontinuities with spacetime discontinuous Galerkin methods” In A Combescure, R De Borst, and T Belytschko, editors, IUTAM symposium on discretization methods for evolving discontinuities, Proceedings of the IUTAM Symposium held Lyon, France, September 4-7, 2006, volume 5 of IUTAM Bookseries,  59 – 87. IUTAM, Springer, 2007.
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    We review recent progress in applying spacetime discontinuous Galerkin (SDG) finite element methods to problems whose solutions exhibit various types of moving discontinuities. SDG models and related solution methods offer a number of attractive features, including element-wise satisfaction of the governing balance laws, linear computational complexity in the number of spacetime elements, and a computational structure that readily supports parallel implementations. We describe the use of new unstructured spacetime meshing procedures in discretizing evolving discontinuities. Specifically, we show bow h-adaptive spacetime meshing can be used to capture weak shocks in linear elastodynamics, how the SDG framework provides a convenient setting for implementing cohesive models for dynamic fracture, and how more advanced spacetime meshing procedures can deliver sharp representations of discontinuous solution features by tracking the trajectories of contact discontinuities in compressible gas dynamics.
Conference Abstracts and Presentations
  • R. Abedi, “An adaptive time domain approach to characterize dispersive elastodynamic media”, International Mechanical Engineering Congress & Exposition AMSE 2017 IMECETampa, Florida, USA – November 5-8, 2017.
    Abstract
  • P.L. Clarke, R. Abedi, B. Bahmani, K.A. Acton, and S.C. Baxter, “Effect of the spatial inhomogeneity of fracture strength on fracture pattern for quasi-brittle materials”, International Mechanical Engineering Congress & Exposition AMSE 2017 IMECETampa, Florida, USA – November 5-8, 2017.
    Abstract
  • K.A. Acton, S.C. Baxter, B. Bahmani, P.L. Clarke, and R. Abedi, “Mesoscale models characterizing material property fields used as a basis for predicting fracture patterns in quasi-brittle materials”, International Mechanical Engineering Congress & Exposition AMSE 2017 IMECETampa, Florida, USA – November 5-8, 2017.
    Abstract
  • R. Abedi and S. Mudaliar, “A spacetime adaptive approach to characterize complex dispersive media” XXXIInd International Union of Radio Science General Assembly & Scientific Symposium, URSI 2017 GASS, Palais des congres, Montreal, Canada – August 19-26th, 2017.
  • R. Abedi and S. Mudaliar, “Error analysis and comparison of Riemann and average fluxes for a spacetime discontinuous Galerkin electromagnetic formulation” XXXIInd International Union of Radio Science General Assembly & Scientific Symposium, URSI 2017 GASS, Palais des congres, Montreal, Canada – August 19-26th, 2017.
  • R. Abedi and R.B. Haber, “A Rate-Dependent Interfacial Damage Model for Multiscale Dynamic Fracture”, 14th U.S. National Congress on Computational Mechanics (USNCCM14), Montreal, Canada – July 17-20th, 2017.
    Abstract
  • R. Abedi, O. Omidi, P.L. Clarke, and R.B. Haber, “Effect of loading rate and in-situ stress anisotropy on fracture patterns in a tight formation”, 14th U.S. National Congress on Computational Mechanics (USNCCM14), Montreal, Canada – July 17-20th, 2017.
    Abstract
  • P.L. Clarke, R. Abedi, K.A. Acton, S.C. Baxter, and R.B. Haber, “A stochastic approach for modelling dynamic fracture of quasi-brittle materials”, 14th U.S. National Congress on Computational Mechanics (USNCCM14), Montreal, Canada – July 17-20th, 2017.
    Abstract
  • R.B. Haber, A. Madhukar, R. Abedi, and V. Kindratenko, “Barrier-Free Parallel–Adaptive Scheme for Asynchronous Spacetime Discontinuous Galerkin Methods”, 14th U.S. National Congress on Computational Mechanics (USNCCM14), Montreal, Canada – July 17-20th, 2017.
    Abstract
  • K.A. Acton, S.C. Baxter, B. Bahmani, R. Abedi, “Developing Mesoscale Probabilistic Characterizations of the Elastic and Inelastic Properties of Random Composites using Statistical Volume Elements”, 14th U.S. National Congress on Computational Mechanics (USNCCM14), Montreal, Canada – July 17-20th, 2017.
  • R. Abedi and S. Mudaliar, “An h-adaptive Time Domain Discontinuous Galerkin Method for Electromagnetics”, 2017 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS-URSI 2017), San Diego, California, USA – July 9-14, 2017.
  • R. Abedi, R.B. Haber, and A. Elbanna, “Mixed-mode dynamic crack propagation in rocks with contact-separation mode transitions”, 51th U.S. Rock Mechanics/Geomechanics Symposium (ARMA 2017), San Francisco, California, USA – June 25-28, 2017.
    Abstract
  • P.L. Clarke and R. Abedi, “Fracture modeling of rocks based on random field generation and simulation of inhomogeneous domains”, 51th U.S. Rock Mechanics/Geomechanics Symposium (ARMA 2017), San Francisco, California, USA – June 25-28, 2017.
    Abstract
  • R. Abedi and P.L. Clarke, “Simulation of refracture and contact mode transitions in tight formations”, 51th U.S. Rock Mechanics/Geomechanics Symposium (ARMA 2017), San Francisco, California, USA – June 25-28, 2017.
    Abstract
  • R. Abedi, O. Omidi, and P.L. Clarke, “A numerical study on the effect of loading and randomness on fracture patterns in a tight formation”, 51th U.S. Rock Mechanics/Geomechanics Symposium (ARMA 2017), San Francisco, California, USA – June 25-28, 2017.
    Abstract
  • R.B. Haber, R. Abedi, P.L. Clarke, and A. Madhukar, “Dynamic Fracture in Quasi-brittle Materials with Random Defects”, V International Conference on Computational Modeling of Fracture and Failure of Materials and Structures (CFRAC 2017), Nantes, France – June 14-16, 2017 (ketnote talk).
    Abstract
  • R. Abedi and R.B. Haber, “A rate-dependent interfacial damage model: Constitutive equation and fracture simulation”, Engineering Mechanics Institute Conference June 4-7,  (EMI 2017), San Diego, California, USA – June 4-7, 2017.
    Abstract
  • P.L. Clarke, R. Abedi, B. Bahmani, “Dynamic fracture simulation of inhomogeneous rock”, Engineering Mechanics Institute Conference June 4-7,  (EMI 2017), San Diego, California, USA – June 4-7, 2017.
    Abstract
  • P.L. Clarke, R. Abedi, K.A. Acton, and S.C. Baxter, “A comparative study on characterization, stochastic realization, and fracture simulation of quasi-brittle materials”, Engineering Mechanics Institute Conference June 4-7,  (EMI 2017), San Diego, California, USA – June 4-7, 2017.
    Abstract
  • R. Abedi, “Spacetime discontinuous Galerkin method for wave propagation simulation in complex media”, International Mechanical Engineering Congress & Exposition AMSE 2016 IMECE, Phoenix, Arizona, USA – November 13-17, 2016.
    Abstract
  • P.L. Clarke, R. Abedi, and O. Omidi, “Homogenization and simulation of solids with random microstructure”, International Mechanical Engineering Congress & Exposition AMSE 2016 IMECE, Phoenix, Arizona, USA – November 13-17, 2016.
    Abstract
  • R. Abedi, S. Mudaliar, “Formulation of a discontinuous Galerkin method for unstructured causal grids in spacetime and linear dispersive electromagnetic media”, Society of Industrial and Applied Mathematics, Annual Meeting 16 (SIAM AN16), Boston, Massachusetts – USA, July 11-15, 2016.
    Abstract
  • R. Abedi, S. Mudaliar, “Spacetime Discontinuous Galerkin Finite Element Method for Time Domain Electromagnetics”, 2016 IEEE International Symposium on Antennas and Propagation/USNC-URSI National Radio Science meeting (AP-S/URSI 2016), Fajardo – Puerto Rico, June 26-July 1, 2016.
    Abstract
  • R. Abedi, O. Omidi, P.L. Clarke, and S. Enayatpour, “Numerical simulation of rock dynamic fracturing and failure including microscale material randomness”, 50th US Rock Mechanics/Geomechanics Symposium (ARMA 2016), Houston, Texas – USA, June 26-29, 2016.
    Abstract
  • P.L. Clarke, O. Omidi, and R. Abedi, “Modeling crack connectivity of induced fractures in a naturally fractured formation”, 50th US Rock Mechanics/Geomechanics Symposium (ARMA 2016), Houston, Texas – USA, June 26-29, 2016.
    Abstract
  • O. Omidi, R. Abedi, and S. Enayatpour, “Well stimulation in tight formations: a dynamic approach”, 50th US Rock Mechanics / Geomechanics Symposium (ARMA 2016), Houston, Texas – USA, June 26-29, 2016.
    Abstract
  • R. Abedi, P.L. Clarke, O. Omidi, and P. Kumar, “Fracture analysis of a quasi-brittle material based on a random field representation of micro-cracked domain”, Probabilistic Mechanics & Reliability Conference (PMC 2016), Nashville, Tennessee – USA, May 22-25, 2016.
    Abstract
  • R. Abedi, O. Omidi, R.B. Haber, and A. Elbanna, “An interfacial model for mode-I and mode-II dynamic crack propagation in rocks with stick–slip contact transitions”, Engineering Mechanics Institute Conference (EMI 2016), Nashville, Tennessee – USA, May 22-25, 2016.
    Abstract
  • P.L. Clarke, R. Abedi, and O. Omidi, “An approach to track crack connectivity for hydraulic fracturing using graph and disjoint-set data structures”, Engineering Mechanics Institute Conference (EMI 2016), Nashville, Tennessee – USA, May 22-25, 2016.
    Abstract
  • O. Omidi, R. Abedi, P.L. Clarke, and S. Enayatpour, “Effects of Material Spatial Randomness on Dynamic Fracturing in Rocks”, Engineering Mechanics Institute Conference (EMI 2016), Nashville, Tennessee – USA, May 22-25, 2016.
    Abstract
  • R. B. Haber, R. Abedi, “Spacetime Interfacial Damage Model for Dynamic Fracture in Brittle Materials”, Variational Models in Fracture in Banff International Research Station for Mathematical Innovation and Discovery, Banff, Alberta, Canada, May 8-13, 2016.
    Abstract
  • R. Abedi, S.T. Miller, R.B. Haber, and O. Omidi, “Efficiency of High Order Methods in Space and Time: Study of Elastodynamics Problem Using Spacetime Discontinuous Galerkin Finite Element Method”, 13th U.S. National Congress on Computational Mechanics (USNCCM13), San Diego, California, USA – July 26-30, 2015.
    Abstract
  • R.B. Haber, R. Pal, A. Madhukar, and R. Abedi, “Spacetime Discontinuous Galerkin Method for Hyperbolic Advection–Diffusion with a Non-Negativity Constraint”, 13th U.S. National Congress on Computational Mechanics (USNCCM13), San Diego, California, USA – July 26-30, 2015.
    Abstract
  • O. Omidi, R. Abedi, S. Enayatpour, I. McNamara, R.B. Haber, “Spacetime Discontinuous Galerkin Finite Element Method and an Interfacial Damage Model for Hydraulic and Compressive Fracture Simulations”, 13th U.S. National Congress on Computational Mechanics (USNCCM13), San Diego, California, USA – July 26-30, 2015.
    Abstract
  • O. Omidi, R. Abedi, and S. Enayatpour, “An adaptive meshing approach to capture hydraulic fracturing”, 49th US Rock Mechanics / Geomechanics Symposium (ARMA 2015), San Francisco, California – USA, June 28-July 1, 2015.
    Abstract
  • R. Abedi, K. Marwah, I. McNamara, O. Omidi, and R. Haber, “A probabilistic approach for dynamic fracture and fragmentation study of brittle materials”, Pan-American Congress of Applied Mechanics XV (PACAM XV), Urbana, Illinois – USA, May 18-21, 2015.
    Abstract
  • O. Omidi, R. Abedi, S. Enayatpour, I. McNamara, and R.B. Haber, “Dynamic fracture and contact in rocks using an interfacial damage model”, Pan-American Congress of Applied Mechanics XV (PACAM XV), Urbana, Illinois – USA, May 18-21, 2015.
    Abstract
  • I. McNamara, A. Madhukar, R.B. Haber, A. Elbanna, and R. Abedi,  “Spacetime Simulation of Seismic Response”, Pan-American Congress of Applied Mechanics XV (PACAM XV), Urbana, Illinois – USA, May 18-21, 2015.
    Abstract
  • R. Pal, A. Madhukar, R. Abedi, and R.B. Haber, “Spacetime discontinuous Galerkin method for hyperbolic advection–diffusion with a non-negativity constraint”, Pan-American Congress of Applied Mechanics XV (PACAM XV), Urbana, Illinois – USA, May 18-21, 2015.
    Abstract
  • P.L. Clarke, R. Abedi, “Topology Optimization of Lithium-Ion Battery Electrode Micro-structure Morphology for Reduction of Damage Accumulation and Longevity of Battery Life”, COMSOL Conference 2014 Boston, Boston, Massachusetts – USA, October 8-10, 2014.
    Presentation
    Abstract
  • R. Abedi, K. Marwah, I. McNamara, R.B. Haber, and O. Allix, “Interfacial delayed-damage model for dynamic fracture and fragmentation”, 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain – July 20-25, 2014.
    Presentation
    Abstract
  • O. Omidi, R. Abedi, K. Marwah, I. McNamara, and R.B. Haber, “Dynamically consistent interfacial damage and contact model for impact problems”, 17th U.S. National Congress on Theoretical and Applied Mechanics, East Lansing, Michigan, USA – June 15-20, 2014.
    Presentation
    Abstract
  • R. Abedi, R. Pal, I. McNamara, R.B. Haber, “A Riemann-solver-free discontinuous Galerkin method for hyperbolic systems”, 17th U.S. National Congress on Theoretical and Applied Mechanics, East Lansing, Michigan, USA – June 15-20, 2014.
    Presentation
    Abstract
  • P.L. Clarke, S.T. Miller, Reza Abedi, “Discontinuous versus Continuous Galerkin Finite Element Methods for dynamic problems”, 17th U.S. National Congress on Theoretical and Applied Mechanics, East Lansing, Michigan, USA – June 15-20, 2014.
    Presentation
    Abstract
  • R. Abedi, O. Omidi, and P.L. Clarke, “Spacetime Discontinuous Galerkin FEM: Spectral Response”, 22nd International Conference on Spectral Line Shapes  (ICSLS22), Tullahoma, TN, USA – June 1-6, 2014.
    Poster
    Conference Proceeding
    Abstract
  • R. Abedi, R.B. Haber, “Adaptive Spacetime Discontinuous Galerkin Model for Wave Propagation in Layered Composite Plates with Defects”, 12th U.S. National Congress on Computational Mechanics (USNCCM12), Raleigh, NC, USA – July 22-25, 2013.
    Presentation
    Abstract
  • R. Abedi, S.T. Miller, “Comparison of Finite Element Methods for Elastodynamics”, 12th U.S. National Congress on Computational Mechanics (USNCCM12), Raleigh, NC, USA – July 22-25, 2013.
    Presentation
    Abstract
  • R. Abedi, R.B. Haber, “Coupling interfacial and bulk response using Riemann solutions in an adaptive spacetime model for elastodynamic contact”, V International Conference on Coupled Problems in Science and Engineering, Coupled Problems 2013, Santa Eulalia, Ibiza, Spain – June 17-19, 2013 (keynote lecture).
    Presentation
    Abstract
  • S.T. Miller, R. Abedi, “Riemann Solutions for Spacetime Discontinuous Galerkin Methods”, 4th International Congress on Computational Engineering and Sciences (FEMTEC 2013), Las Vegas, NV, USA – May 19-24, 2013.
    Presentation
    Abstract
  • R. Abedi, R.B. Haber, “Riemann conditions and rate-dependent interfacial damage model for elastodynamic contact and fracture”, 11th U.S. National Congress on Computational Mechanics (USNCCM11), Minneapolis, MN, USA – July 25-28, 2011.
    Presentation
    Abstract
  • R.B. Haber, R. Abedi, and O. Allix, “Adaptive spacetime models for dynamic fracture”, International Conference on Computational Modeling of Fracture and Failure of Materials and Structures (CFRAC 2011), Barcelona, Spain – June 6-8, 2011 (Plenary Lecture).
    Abstract
  • R. Abedi, R.B. Haber, “Spacetime interfacial damage model for elastodynamic fracture with Riemann contact conditions”, 16th US National Congress of Theoretical and Applied Mechanics (USNCTAM16), State College, PA, USA – June 28-July 2, 2010.
    Presentation
    Abstract
  • R. Abedi, R.B. Haber, and O. Allix, “Spacetime damage-delay cohesive model for elastodynamic fracture with Riemann contact conditions”, IV European Conference on Computational Mechanics (ECCM 2010), Paris, France – May 16-21, 2010.
    Presentation
    Abstract
  • R. Abedi, R.B. Haber, O. Allix, “A multi-scale, delayed-damage cohesive model for dynamic fracture”, 10th U.S. National Congress on Computational Mechanics (USNCCM10), Columbus, OH, USA – July 16-19, 2009.
    Abstract
  • R. Abedi, S.H. Chung, and R.B. Haber, “Spacetime method for tracking elastodynamic fracture with a damage-based cohesive model”, 8th World Congress on Computational Mechanics (WCCM8), Venice, Italy – June 30-July 4, 2008.
    Abstract
  • R. Abedi, R.B. Haber, “A damage-based cohesive model in an adaptive spacetime discontinuous Galerkin method”, Study of dynamic cohesive fracture using a spacetime discontinuous Galerkin method”, 6th International Conference on Shell and Spatial Structures: Spanning Nano to Mega, Ithaca, NY, USA – June 27-30, 2008.
    Abstract
  • R. Abedi and R.B. Haber, “A study of dynamic cohesive fracture using a spacetime discontinuous Galerkin method”, 9th U.S. National Congress on Computational Mechanics (USNCCM9), San Francisco, CA, USA – July 23-26, 2007.
    Abstract
  • R. Abedi, M.A. Hawker, and R.B. Haber, “Spacetime simulation of dynamic cohesive fracture”, Proceedings of the International Conference on Computational Fracture and Failure of Materials, 15, Nantes, France – June 11-13, 2007.
  • R. Abedi, R.B. Haber, “Spacetime discontinuous Galerkin method for nonlinear solid mechanics”, 7th World Congress on Computational Mechanics (WCCM7), Los Angeles, CA, USA – July 16-22, 2006.
    Abstract
  • R. Abedi, S. Thite, R.B. Haber, and J. Erickson, “An h-adaptive spacetime discontinuous Galerkin method for elastodynamics”, 8th U.S. National Congress on Computational Mechanics (USNCCM8), Austin, TX, USA – July 24-28, 2005.
    Abstract
  • M. Hawker, R. Abedi, K. Matous, and R.B. Haber, “An adaptive spacetime discontinuous Galerkin framework for implementing cohesive damage models”, 8th U.S. National Congress on Computational Mechanics (USNCCM8), Austin, TX, USA – July 24-28, 2005.
    Abstract
  • J. Palaniappan, R. Abedi, S. Thite, and R.B. Haber, “A monolithic spacetime discontinuous Galerkin method for fluid–structure interaction problems”, 8th U.S. National Congress on Computational Mechanics (USNCCM8), Austin, TX, USA – July 24-28, 2005.
    Abstract
  • S. Thite, J. Erickson, S.H. Chung, R. Abedi, J. Palaniappan, and R.B. Haber, “Meshing in 2D × Time for front–tracking DG methods”, 8th U.S. National Congress on Computational Mechanics (USNCCM8), Austin, TX, USA – July 24-28, 2005.
  • R. Abedi, S.H. Chung, Y. Fan, S. Thite, J. Erickson, and R.B. Haber, “Adaptive discontinuous Galerkin method for elastodynamics on unstructured spacetime grids”, 21st International Congress of Theoretical and Applied Mechanics (ICTAM21), Warsaw, Poland – August 15-21, 2004.
    Abstract
  • R. Abedi, Y. Fan, M. Hawker, L. Yin, and R.B. Haber, “A spacetime discontinuous Galerkin finite element method for wave propagation and scattering in solids”, 7th U.S. National Congress on Computational Mechanics (USNCCM7), Albuquerque, NM, USA – July 28-30, 2003.
    Abstract
  • R. Abedi, S.H. Chung, J. Erickson, R.B. Haber, J. Sullivan, and L. Yin, “Spacetime meshing with adaptive coarsening and refinement”, 7th U.S. National Congress on Computational Mechanics (USNCCM7), Albuquerque, NM, USA – July 28-30, 2003.
    Abstract
  • R.B. Haber, B. Petracovici, R. Abedi, and R. Jerrard, “A spacetime discontinuous Galerkin method for elastodynamics with element-level momentum balance”, 7th U.S. National Congress on Computational Mechanics (USNCCM7), Albuquerque, NM, USA – July 28-30, 2003.
    Abstract
Theses
  • R. Abedi, Spacetime damage-based cohesive model for elastodynamic fracture with dynamic contact, Ph.D. thesis, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 2010 (academic advisor: R.B. Haber).
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    Dynamic material failure is important in a number of scientific and engineering applications and a variety of numerical methods for its modeling have been proposed. This thesis presents the formulation and implementation of an interfacial-damage, cohesive-fracture model, including contact and friction effects, for dynamic failure of brittle materials. The model is implemented within a spacetime discontinuous Galerkin (SDG) finite element method. An adaptive meshing procedure generates spacetime grids that satisfy a special causality constraint to enable an efficient patch-by-patch, advancing-front solution scheme with O(N) computational complexity. Per-element balance properties, local adaptive operations, and the use of Riemann fluxes provide to the SDG method the extreme accuracy and efficiency required to solve multiscale fracture problems.A dimensional analysis of linear elastodynamics, with extensions to fracture models based on cohesive traction–separation laws, supports the formulation. The problem is formulated and analyzed using differential forms and the exterior calculus in spacetime. The analysis demonstrates that the velocity scalings implied by the spatial and temporal coordinate scalings and by the scalings of the material properties must be identical to obtain a self-similar scaling of an elastodynamic process. The use of differential forms reveals intrinsic structure and relations between the spacetime mechanics fields which are otherwise obscured by conventional tensorial analysis. For example, only four distinct scalings are required to define a scaled elastodynamic process when we work directly with forms, while eight are required when tensorial analysis is used. In the context of dynamic cohesive fracture, the analysis shows that, among the nondimensional variables, the ratio of the stress-loading scale to the cohesive strength is proportional to the ratio of the radius of the singularity-dominant zone from Linear Elastodynamic Fracture Mechanics (LEFM), to the cohesive-process-zone size. These ratios are, in turn, useful indicators of whether the small-scale-yielding caveat of LEFM is satisfied.A novel continuum formulation of the linear elastodynamic contact problem also supports the SDG finite element model. In contrast to previous contact models that invoke quasi-static contact conditions, the proposed model enforces dynamic contact conditions by prescribing momentum flux and compatibility conditions obtained from the local Riemann problems for bonded, separation, contact–stick, and contact–slip modes. This approach preserves the characteristic structure of the underlying equations at the contact interface, a property that is lacking in previous formulations. The fully-bonded and contact–stick conditions are identical, as expected, so the non-penetration and tangential slip constraints are treated exactly in the new continuum formulation. Furthermore, the direction of the tangential contact traction (friction) is shown to be continuous through transitions between contact–stick and contact-slip modes. These favorable properties, which improve the accuracy of and facilitate numerical implementations of the proposed model, are not obtained in many existing models which, for example, replace the non-penetration constraint with a large interfacial stiffness in the normal direction. The transition between separation and contact modes retains its physically discontinuous character, and a regularization of this transition is introduced to facilitate and reduce the cost of numerical implementations. A discretization and numerical implementation within the adaptive SDG framework demonstrate the effectiveness of the new contact model in a numerical setting.A new two-scale cohesive fracture model replaces the usual traction-separation law with a damage model that represents mesoscale processes of void growth and coalescence. The evolution of a single damage parameter D, which represents the debonded area fraction on cohesive interfaces, is governed by an irreversible, time-delay evolution law characterized by a cohesive strength and a relaxation time τ that determines the maximum damage rate. Riemann fluxes for the fully-bonded condition are enforced in the undamaged area fraction (1-D) of the cohesive interface, while the Riemann fluxes for the contact–stick, contact–slip or separation conditions determine the fluxes in the debonded area fraction. These mesoscale Riemann values are averaged to derive macroscopic cohesive conditions. The damage-based cohesive model is implemented within the adaptive SDG finite element framework to produce a numerical model that efficiently and accurately resolves the multi-scale response associated with dynamic fracture and transitions between contact, separation, stick and slip conditions in the event of crack closure. Beyond ensuring solution accuracy, the model uses the SDG scheme’s adaptive meshing capabilities to freely nucleate and extend cohesive interfaces to capture solution-dependent crack paths. The SDG adaptive meshing aligns the boundaries of spacetime elements with crack-path trajectories having arbitrary position and orientation, and two adaptive error indicators ensure the accurate rendering of both the cohesive model and the bulk solution. Thus, the present model does not suffer the limited resolution and mesh-dependent effects encountered in most other numerical fracture models. Numerical results obtained with the proposed model demonstrate crack propagation, microcrack formation and crack branching phenomena.

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