[toggle title=”Journal articles” open=”yes”]

Wang, Hang, Reza Abedi, and Saba Mudaliar. “.”  (2020): 107281.

    • H Wang, R Abedi, S Mudaliar. “Space-Angle Discontinuous Galerkin Method for Radiative Transfer between Concentric Cylinders.” Journal of Quantitative Spectroscopy and Radiative Transfer 107281. 2020.
      PDF PDF[spoiler title=”View/Hide Abstract” icon=”caret”] The integro-differential radiative transfer equation (RTE) for concentric cylinders problem involving scattering, absorption and emission is solved using the discontinuous Galerkin (DG) finite element method (FEM). The space-angle DG method directly solves the cylindrically-symmetric RTE as a three-dimensional problem, where a 1D spatially domain in radial distance r is twice extruded in the cosine of polar angle (μ) and the difference in azimuthal angle () directions. Thus, the method has a higher accuracy than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods. This is reflected by numerically verified convergence rate of  for smooth problems and space-angle polynomial interpolation order of p. The axisymmetric RTE formulation is more complicated than the plane-parallel formulation, for having two independent angle directions (μ and φ) and an extra derivative term with respect to  in the differential equation. This results in a complex characteristic structure in  plane with strong discontinuity lines in radiation intensity I. A method of characteristics is formulated and implemented to verify the DG formulation and demonstrate its accuracy when such strong discontinuities persist in the solution, specifically when there are no scattering and absorption terms. The relaxation of inter-element continuity constraint of continuous FEMs by this DG method implies its superiority in numerically capturing such discontinuities. Finally, a benchmark problem pertained to heat radiation in a gray gas and another one with nonzero phase function demonstrate the effectiveness of the method in modeling black-body and scattering angular integration terms..[/spoiler]
    • R Abedi, AV Amirkhizi. “Use of loss limit approach to zero in scattering-based parameter retrieval of elastic micro-structured media.” International Journal of Solids and Structures 200-201, 34-63 (2020).
      PDF PDF[spoiler title=”View/Hide Abstract” icon=”caret”] The parameter retrieval method based on scattering data is used to derive dynamic constitutive parameters of solids with periodic structure. There are inherent ambiguities in the real part of the retrieved wavenumber and the sign of impedance, though the latter has a one-to-one correspondence with the direction of energy flux. Moreover, for lossless structures there can be multiple solution branches that satisfy passivity and continuity of the wavenumber as a function of frequency, leading to potentially double positive or double negative (in terms of density and modulus) overall descriptions of the micro-structured medium. The continuity of wavenumber for lossy unit cells is used to unambiguously determine their constitutive parameters and by taking the limit when loss approaches zero, one can determine the stable solution branches of lossless micro-structures. The two loss models of nonzero damping and complex modulus are compared in terms of their energy loss characteristics and retrieved parameters. These models are employed in both time and frequency domain calculations. The lossy solutions demonstrate that from double-negative and double-positive solutions, only the latter is the stable solution branch in a pass-band of a lossless 1D unit cell. A 2D photonic crystal example is used to demonstrate that the wavenumber appears to jump, discontinuously, at the transition point between two consecutive stop-bands, hence rendering the existing methods based on continuity of wavenumber ineffective. In contrast, the proposed approach based on taking limit of lossy solutions can successfully be used to determine stable overall properties of such media. Finally, certain features of half- and full-cycle stop-bands are discussed..[/spoiler]
    • JM Garrard,  R Abedi. “Statistical Volume Elements for the Characterization of Angle-Dependent Fracture Strengths in Anisotropic Microcracked Materials.” International Journal of Solids and Structures 200-201, 34-63 (2020).   Garrard, Justin M., and Reza Abedi. “.” ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B, 6(3) (2020).
      PDF PDF[spoiler title=”View/Hide Abstract” icon=”caret”] Statistical volume elements (SVEs) are used to homogenize fracture strength of rock, based on the microcrack statistics of a real-world Yuen-Long marble sample. The small size of SVEs enables maintaining inhomogeneities in fracture properties with lower computational cost compared to methods that explicitly model microcracks at macroscale. Maintaining inhomogeneity is important to capture realistic fracture patterns in rock as a quasi-brittle material. Uniaxial tensile, uniaxial compressive, and shear strengths are derived for arbitrary angle for loading and orientation of a single crack by using the linear elastic fracture mechanics (LEFM) method and incorporating frictional effects. Mesoscopic fracture strength fields are generated for different strengths and angle of loading by traversing the spatial domain with circular SVEs. Increasing the SVE size smoothens the spatial inhomogeneity and angular anisotropy of homogenized strengths. Spatial and angular covariance functions of the random fields are obtained to demonstrate how fracture strength varies in space and by changing the angle of loading. Two isotropic and anisotropic rock domains are studied and shown to have very different single- and two-point statistics. Macroscopic fracture simulations by an asynchronous spacetime discontinuous Galerkin (aSDG) method demonstrate that most macroscopic cracks for the anisotropic domain are aligned with the weakest strength planes.[/spoiler]
    • JM Garrard, R Abedi. “Statistical volume element averaging scheme for fracture of quasi-brittle materials.” Computers and Geotechnics 117: 103229. 2020.
      PDF PDF[spoiler title=”View/Hide Abstract” icon=”caret”] To capture the randomness and inhomogeneity of rock at microscale, a statistical volume element (SVE) averaging approach is proposed. The microcrack statistics of a real-world Yuen-Long marble sample is used to realize 2D microcracked domains. The size effect, i.e. the decrease of the mean and variation of homogenized strength field by increasing SVE size, is analyzed. Increasing the crack density is shown to have a similar effect. While smaller SVEs maintain a greater level of inhomogeneity and are preferred for fracture analysis, it is shown that low density of microcracks pose a lower limit on the SVE size. Beside the actual power-law distribution of microcrack length, by varying the Weibull model shape parameter m other domains are created with different microcrack distribution shapes. Macroscopic fracture simulations, by the asynchronous Spacetime Discontinuous Galerkin (aSDG) method, study the effect of m for a uniaxial tensile problem. By increasing m from 0.5 to 4, the length distribution of microcracks become more uniform; this corresponds to a more uniform and stronger mesoscopic strength field, which results to about 3 and 6 times increase to macroscopic tensile strength and toughness, respectively. However, the more uniform length distribution of microcracks is shown to reduce rock brittleness.[/spoiler]
    • KA Acton, C Sherod, B Bahmani, R Abedi. “Effect of Volume Element Geometry on Convergence to a Representative Volume.” ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering 5.3: 030907. 2019.
      PDF PDF[spoiler title=”View/Hide Abstract” icon=”caret”] To accurately simulate fracture, it is necessary to account for small-scale randomness in the properties of a material. Apparent properties of statistical volume element (SVE) can be characterized below the scale of a representative volume element (RVE). Apparent properties cannot be defined uniquely for an SVE, in the manner that unique effective properties can be defined for an RVE. Both constitutive behavior and material strength properties in SVE must be statistically characterized. The geometrical partitioning method can be critically important in affecting the probability distributions of mesoscale material property parameters. Here, a Voronoi tessellation-based partitioning scheme is applied to generate SVE. Resulting material property distributions are compared with those from SVE generated by square partitioning. The proportional limit stress of the SVE is used to approximate SVE strength. Superposition of elastic results is used to obtain failure strength distributions from boundary conditions at variable angles of loading.[/spoiler]
    • PL Clarke, H Wang, JM Garrard, R Abedi, and Saba Mudaliar. “Space-angle discontinuous Galerkin method for plane-parallel radiative transfer equation.” Journal of Quantitative Spectroscopy and Radiative Transfer 233: 87-98. 2019.
      PDF      [spoiler title=”View/Hide Abstract” icon=”caret”] The radiative transfer equation (RTE) for a plane-parallel problem involving scattering, absorption and radiation is solved using the discontinuous Galerkin (DG) finite element method (FEM). Both space and angle directions are discretized by the DG method. Thus, while the method has a higher accuracy in angle direction than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods, it removes the continuity constraint implied by the form of basis function for continuous FEMs in space and angle. The discrete formulation of the problem is presented for nonzero phase function and a variety of boundary conditions. The numerical results demonstrate a convergence rate when the intensity is interpolated by an order p polynomial in both space and angle. The method is validated against exact solutions, and compared with other space-angle and hybrid FEMs for a few benchmark problems. The appropriateness of the DG formulation for problems with discontinuous solution is demonstrated by solving a problem with delta source term, where an in-element averaging of the source term eliminates negative intensity values for high order elements. Finally, a problem with an angular-line source term and a convergence study where the solution order is zero in angle are used to further demonstrate the advantages of the high order space-angle DG formulation; for the convergence study problem, the error was reduced by about 13 binary orders of magnitude, by increasing the order in both space and angle, rather than in space only.[/spoiler]
    • B Bahmani, R Abedi. “Asynchronous Spacetime Discontinuous Galerkin Formulation for a Hyperbolic Time-Delay Bulk Damage Model.” Journal of Engineering Mechanics 145.10: 04019075. 2019.
    • PDF[spoiler title=”View/Hide Abstract” icon=”caret”]A bulk damage formulation is presented for failure analysis of brittle materials under dynamic loading. A time-delay ordinary differential equation (ODE) is used to model damage evolution. The evolution is driven by the difference between a target static damage value and the instantaneous damage value. A damage length scale is introduced from the model’s intrinsic relaxation time and elastic wave speeds. This length scale addresses the mesh sensitivity problem of some existing damage formulations for dynamic fracture, with less computational effort than some other existing remedies. The authors use the asynchronous spacetime discontinuous Galerkin (aSDG) method for the solution of the resulting hyperbolic system of equations. Local and asynchronous solution process, linear complexity of the solution versus the number of elements, local recovery of balance properties, and high spatial and temporal orders of accuracy are some of the main advantages of the aSDG method. Several numerical examples are presented to demonstrate mesh insensitivity of the method and the effect of boundary conditions on dynamic fracture patterns.[/spoiler]
    • R. Abedi, PL. Clarke. “A computational approach to model dynamic contact and fracture mode transitions in rock.” Computers and Geotechnics 109 : 248-271.”2019. 
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]We propose an interfacial contact and fracture model based on Riemann solutions. Instead of penalty method and Lagrange multiplier approach we propose a regularization scheme—based on the interface displacement and separation velocity jumps—that smoothens contact-separation mode transitions. An aperture-based regularization approach smoothens the transfer of hydraulic load to in-situ cracks. A discontinuous Galerkin implementation is used to solve dynamic fracture problems for uniaxial compression and explosive examples. Moreover, we investigate contact–separation mode transitions and hydraulic load transfer in several hydraulic refracture problems and study the effect of loading rate and in-situ cracks in the number and orientation of activated perforations.[/spoiler]
    • B Bahmani, R Abedi, PL Clarke. “A Stochastic Bulk Damage Model Based on Mohr-Coulomb Failure Criterion for Dynamic Rock Fracture”, Applied Sciences, 9, no. 5: 830. 2019.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]We present a stochastic bulk damage model for rock fracture. The decomposition of strain or stress tensor to its negative and positive parts is often used to drive damage and evaluate the effective stress tensor. However, they typically fail to correctly model rock fracture in compression. We propose a damage force model based on the Mohr-Coulomb failure criterion and an effective stress relation that remedy this problem. An evolution equation specifies the rate at which damage tends to its quasi-static limit. The relaxation time of the model introduces an intrinsic length scale for dynamic fracture and addresses the mesh sensitivity problem of earlier damage models. The ordinary differential form of the damage equation makes this remedy quite simple and enables capturing the loading rate sensitivity of strain-stress response. The asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used for macroscopic simulations. To study the effect of rock inhomogeneity, the Karhunen-Loeve method is used to realize random fields for rock cohesion. It is shown that inhomogeneity greatly differentiates fracture patterns from those of a homogeneous rock, including the location of zones with maximum damage. Moreover, as the correlation length of the random field decreases, fracture patterns resemble angled-cracks observed in compressive rock fracture.[/spoiler]
    • B Bahmani, M Yang, A Nagarajan, PL Clarke, S Soghrati, R Abedi. “Automated homogenization-based fracture analysis: Effects of SVE size and boundary condition”, Computer Methods in Applied Mechanics and Engineering 345, 701-727, 2019.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]To model the sample-to-sample variations and the effect of microscale inhomogeneities on fracture response, statistical volume elements (SVEs) are employed to homogenize the elastic and fracture properties of ZrB-SiC, a two-phase particulate composite often used as a thermal coating. In the mesoscale analysis, 2D finite element models are generated using a non-iterative, automated mesh generation algorithm named Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR). The analysis of SVEs under mixed, traction, and minimal kinematic boundary conditions yields their angle-dependent tensile and shear fracture strengths and elastic stiffnesses. This study shows that homogenized fracture strengths are highly dependent on the SVE size and the particular geometric distribution of inclusions (even for similar volume ratios), whereas elastic properties are mainly a function of the volume fraction. Moreover, mean values of strength and bulk modulus, respectively, decrease and remain almost constant as the SVE size increases. For the macroscale analysis, an isotropic, inhomogeneous field of fracture strength is generated from the homogenization of SVEs. The asynchronous Spacetime Discontinuous Galerkin (aSDG) method is subsequently employed for fracture analysis under uniaxial tensile and thermal strain loadings.[/spoiler]
    • R Abedi, R.B. Haber (contributed equally). “Spacetime simulation of dynamic fracture with crack closure and frictional sliding”, Advanced Modeling and Simulation in Engineering Sciences, 5:22, 2018.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]We combine the asynchronous spacetime discontinuous Galerkin (aSDG) method, an interfacial-damage fracture model, and a dynamic contact model to simulate dynamic fracture and crack closure in brittle materials. The contact model enforces specialized Riemann solutions for bonded, separation, slip and stick conditions while preserving elastodynamic characteristic structure across fracture interfaces. Powerful adaptive spacetime meshing tracks dynamic evolution of fracture-surface networks and captures moving solution features. We present numerical examples to demonstrate the model’s ability to reveal fine details of fracture response in problems that range from dynamic crack initiation, growth, closure, and arrest along a pre-defined planar path to fragmentation of rock by an explosively loaded wellbore with stochastic nucleation, free propagation, and coalescence of fracture surfaces.[/spoiler]
    • R Abedi, O Omidi, S Enayatpour. “A mesh adaptive method for dynamic well stimulation”, Computers and Geotechnics, 102, 12-27, 2018.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]We utilize the adaptive meshing features of an asynchronous spacetime discontinuous Galerkin (aSDG) FE method to address rock fracturing in a reservoir stimulation under high rate loadings. Our aSDG implementation adaptively aligns the element boundaries with crack-path trajectories obtained as a part of the solution and no discontinuous features are introduced within the elements. We propose a novel rate-dependent interfacial damage model to represent fracture processes on crack surfaces while the model incorporates various contact modes under transient settings. Several examples are discussed to demonstrate the effectiveness of our approach in resolving dynamic fracturing driven by the high internal fluid pressure.[/spoiler]
    • PL Clarke, R Abedi. “Modeling the connectivity and intersection of hydraulically loaded cracks with in situ fractures in rock”, International Journal for Numerical and Analytical Methods in Geomechanics, 42 (14), 1592-1623, 2018.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]The interaction of an advancing hydraulically loaded crack and in situ fracture network can yield highly complex patterns. We model the connectivity of cells in a finite element domain and in a fracture network by a simplicial complex data structure. The complete adjacency information between cells is determined by one level down facet and one level up cofacet neighborhood information. Combined with a disjoint set data structure, explicit algorithms are derived to efficiently track network connectivity and load transfer between independent fracture sets. We also propose an approach to regularize the application of hydraulic load to newly intersected in situ cracks to smoothen the transition of pressure on intersected cracks from ambient to hydraulic pressure and to avoid the sudden loading of the entire length of these cracks. Numerical results demonstrate the performance of crack connectivity and load transfer models and the effect of regularization model. The results show that as the angle between an incoming hydraulically loaded crack and an in situ crack increases, the effect of in situ crack shifts from slight realignment to diversion/offsetting of the loaded crack. As the angle difference approaches the normal angle, the loaded crack tends to penetrate through the in situ crack. The proposed schemes are also used for transient simulation of 2D reservoirs with multiple perforations surrounded by in situ cracks with and without a bias in the distribution of their orientation. It was shown that from 2 perforations with angles closer to in situ cracks at low loading rates to all perforations at higher loading rates can result in active hydraulic crack propagation. The h‐adaptive method of asynchronous space‐time discontinuous Galerkin method is used to exactly track complex fracture patterns in these dynamic fracture simulations.[/spoiler]
    • KA Acton, SC Baxter, B Bahmani, PL Clarke, R Abedi. “Voronoi tessellation based statistical volume element characterization for use in fracture modeling”, Computer Methods in Applied Mechanics and Engineering 336, 135-155, 2018.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]Accurate characterization of random heterogeneity in a material microstructure is essential to the accurate characterization of complex fracture patterns that result from random crack nucleation and propagation. It is also important to characterize microstructural behavior at intermediate scales, between the length scale of material heterogeneity and the scale of a Representative Volume Element (RVE). The availability of material property data at multiple scales will ultimately allow adjustment of computational cost and level of accuracy with respect to resolution of complex fracture patterns. Statistical Volume Elements (SVE) may be generated at the mesoscale by partitioning an RVE. SVE provide a probabilistic characterization of material heterogeneity, while also presenting a continuum representation of apparent properties. Appropriate definition of an SVE requires modeling choices, such as partitioning methods and size. An essential modeling assumption is the choice of loading condition used to approximate SVE apparent behavior, since the constitutive properties of an SVE are not necessarily invariant with respect to the boundary condition applied. This work develops a Voronoi tessellation based partitioning scheme applied at various length scales, for heterogeneous materials with various contrast ratios. Results show the variability of failure strength for a given SVE as a function of loading direction. Results of the mesoscale material property analysis are implemented in an asynchronous spacetime discontinuous Galerkin (aSDG) finite element based fracture model.[/spoiler]
    • R. Abedi, S. Mudaliar. “An Asynchronous Spacetime Discontinuous Galerkin Finite Element Method for Time Domain Electromagnetics”, Journal of Computational Physics, , 351:121-144, 2017.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • R. Abedi, R. Haber, and P. Clarke. “Effect of random defects on dynamic fracture in quasi-brittle materials”, International Journal of Fracture: Special Issue for Integrated Computational Structure-Material Modeling of Deformation & Failure Under Extreme Conditions An IUTAM Symposium, 208.1-2, 241-268, 2017.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • 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.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • 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.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • S.T. Miller and R. Abedi, “Riemann solutions for spacetime discontinuous Galerkin methods”, Journal of Computational and Applied Mathematics, 270: 564 – 570, 2014.
    • PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • 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.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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. [/spoiler]
    • 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.
      PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • 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.
      PDF [spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
    • R. Abedi, R.B. Haber, S. 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).
      PDF [spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]

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[toggle title=”Articles in Conference Proceedings” open=”yes”]

  • H Wang, R Abedi, and S Mudaliar. A space-angle discontinuous Galerkin method for one-dimensional cylindrical radiative transfer equation with angular decomposition. In 2021 United States National Committee of URSI National Radio Science Meeting (USNC-URSI NRSM), pages 107-108. IEEE, 2021
  • H. Wang, R. Abedi and S. Mudaliar, “A Parallel Space-angle Discontinuous Galerkin Method for Radiative Transfer in Two-dimensional Rectangular Enclosures,” 2020 IEEE USNC-CNC-URSI North American Radio Science Meeting (Joint with AP-S Symposium), 2020, pp. 1-2.
  • Robert B. Haber, Amit Madhukar, Xiao Ma, Ahmed Elbanna, and Reza Abedi. Distributed parallel-adaptive causal spacetime discontinuous Galerkin method with application to earthquake simulation. In 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation (Waves 2019), Vienna, Austria, August 25-30, 2019
  • Reza Abedi and Robert B. Haber. Applications of adaptive spacetime meshing in the asynchronous spacetime discontinuous Galerkin method. In 14th International Conference on Math- ematical and Numerical Aspects of Wave Propagation (Waves 2019), Vienna, Austria, August 25-30, 2019
  • Hang Wang, Reza Abedi, and Saba Mudaliar. A discontinuous Galerkin method for the solution of two dimensional axisymmetric radiative transfer problem. In 2019 USNC-URSI Radio Science Meeting (Joint with AP-S Symposium), Atlanta, Georgia, USA, 2019.
  • J.M. Garrard, R. Abedi, P.L. Clarke, “Statistical volume elements for the characterization of angle-dependent fracture strengths” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018, paper no. 88257 (10 pages), 2018.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”]As a quasi-brittle material, the fracture response of rock is very sensitive to its microstructural defects. Herein, we use statistical volume elements (SVEs) to characterize rock fracture strength at the mesoscale, based on the distribution of microcracks at the microscale. The use of SVEs ensures that the material randomness is maintained upon “averaging” of microscale features. Certain fracture strengths, such as uniaxial tensile strength, uniaxial hydrostatic strength, shear strength, and uniaxial compressive strength, are obtained and characterized for different angles of loading. Thus, a material with anisotropic fracture strength can be characterized. Statistics of the characterized strengths are analyzed, as well as their auto- and crosscorrelation functions of these random fields to shed light on the length scales, relative to the volume element size, at which homogenized properties vary. While crack interaction is not included, the analysis provides insight on the distribution and correlation of different strengths. Finally, the asynchronous spacetime discontinuous Galerkin method is used for macroscopic fracture analyses of two rock domains homogenized by SVEs.[/spoiler]
  • B. Bahmani, M. Yang, A. Nagarajan, P.L. Clarke, S. Soghrati. R. Abedi, “An integrated approach for statistical microscale homogenization to macroscopic dynamic fracture analysis” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018, paper no. 88429 (10 pages), 2018.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”]Maintaining material inhomogeneity and sample-to-sample variations is crucial in fracture analysis, particularly for quasibrittle materials. We use statistical volume elements (SVEs) to homogenize elastic and fracture properties of ZrB2-SiC, a two-phase composite often used for thermal coating. At the mesoscale, a 2D finite element mesh is generated from the microstructure using the Conforming to Interface Structured Adaptive Mesh Refinement (CISAMR), which is a non-iterative algorithm that tracks material interfaces and yields high-quality conforming meshes with adaptive operations. Analyzing the finite element results of the SVEs under three traction loadings, elastic and angle-dependent fracture strengths of SVEs are derived. The results demonstrate the statistical variation and the size effect behavior for elastic bulk modulus and fracture strengths. The homogenized fields are mapped to macroscopic material property fields that are used for fracture simulation of the reconstructed domain under a uniaxial tensile loading by the asynchronous Spacetime Discontinuous Galerkin (aSDG) method.[/spoiler]
  • K. Acton, B. Bahmani, R. Abedi, “Mesoscale material strength characterization for use in Fracture modeling” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018, paper no. 88249 (7 pages), 2018.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”]To accurately simulate fracture, it is necessary to account for small-scale randomness in the properties of a material. Apparent properties of Statistical Volume Elements (SVE), can be characterized below the scale of a Representative Volume Element (RVE). Apparent properties cannot be defined uniquely for an SVE, in the manner that unique effective properties can be defined for an RVE. Both constitutive behavior and material strength properties in SVE must be statistically characterized. The geometrical partitioning method can be critically important in affecting the probability distributions of mesoscale material property parameters. Here, a Voronoi tessellation based partitioning scheme is applied to generate SVE. Resulting material property distributions are compared with those from SVE generated by square partitioning. The proportional limit stress of the SVE is used to approximate SVE strength. Superposition of elastic results is used to obtain failure strength distributions from boundary conditions at variable angles of loading.[/spoiler]
  • B. Bahmani, P.L. Clarke, R. Abedi, “Comparison of interfacial and continuum models for dynamic fragmentation analysis” In: Proceeding International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018, paper no. 88294 (9 pages), 2018.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”]The microstructural design has an essential effect on the fracture response of brittle materials. We present a stochastic bulk damage formulation to model dynamic brittle fracture. This model is compared with a similar interfacial model for homogeneous and heterogeneous materials. The damage models are rate-dependent, and the corresponding damage evolution includes delay effects. The delay effect provides mesh objectivity with much less computational efforts. A stochastic field is defined for material cohesion and fracture strength to involve microstructure effects in the proposed formulations. The statistical fields are constructed through the Karhunen-Loeve (KL) method. An advanced asynchronous Spacetime Discontinuous Galerkin (aSDG) method is used to discretize the final system of coupled equations. Application of the presented formulation is shown through dynamic fracture simulation of rock under a uniaxial compressive load. The final results show that a stochastic bulk damage model produces more realistic results in comparison with a homogenizes model.[/spoiler]
  • P.L. Clarke, H. Wang, J. Garrard, R. Abedi, S. Mudaliar, “A Discontinuous Galerkin Method for the Solution of One Dimensional Radiative Transfer Equation” In: Proceeding 2018 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Boston, Massachusetts, USA – July 8-13, paper no. 2793 (2 pages), 2018.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] The radiative transfer equation (RTE) for a plane-parallel problem involving scattering, absorption and radiation is solved using the discontinuous Galerkin (DG) finite element method (FEM). Both space and angle directions are discretized by the DG method. The problem is formulated for nonzero phase function.  The method is validated against exact solutions, and compared with other space-angle and hybrid FEMs for a few benchmark problems. The performance of the method is also studies for the solution of problems with discontinuous solution.[/spoiler]
  • R. Abedi, P.L. Clarke, “Modeling of rock inhomogeneity and anisotropy by explicit and implicit representation of microcracks” In: Proceeding 52th U.S. Rock Mechanics/Geomechanics Symposium, Seattle, Washington, USA – June 17-20, ARMA 18-1094 (11 pages), 2018.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]  Fracture patterns experienced under a dynamic uniaxial compressive load are highly sensitive to rock microstructural defects due to its brittleness and the absence of macroscopic stress concentration points. We propose two different approaches for modeling rock microstructural defects and inhomogeneity. In the explicit realization approach, microcracks with certain statistics are incorporated in the computational domain. In the implicit realization approach, fracture strength values are sampled using a Weibull probability distribution. We use the Mohr-Coulomb failure criterion to define an effective stress in the context of an interfacial damage model. This model predicts crack propagation at angles Θ = ±(45 – Φ/2) relative to the direction of compressive load, where Φ is the friction angle. By using appropriate models for fracture strength anisotropy, we demonstrate the interaction of rock weakest plane and Φch. Numerical results demonstrate the greater effect of strength anisotropy on fracture pattern when an explicit approach is employed. In addition, the density of fractures increases as the angle of the weakest planes approaches ± Φch. The fracture simulations are performed by an h-adaptive asynchronous spacetime discontinuous Galerkin (aSDG) method that can accommodate crack propagation in any directions. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.[/spoiler]
  • J.M. Garrard, R. Abedi, P.L. Clarke, “Random field realization and fracture simulation of rocks with angular bias for fracture strength” In: Proceeding 52th U.S. Rock Mechanics/Geomechanics Symposium, Seattle, Washington, USA – June 17-20, ARMA 18-1100 (10 pages), 2018.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]  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. The high dependence of their fracture progress on microstructural defects results in wide scatter in their ultimate strength and the so-called size effect. This paper proposes an approach based on statistical volume elements (SVEs) to characterize rock fracture strength at the mesoscale. The use of SVEs ensures that the material randomness is maintained upon averaging of microscale features. Because the fracture strength varies not just spatially, but also by the angle of loading, this work includes angular variability to properly model a heterogeneous rock domain. Two different microcrack distributions, one angularly uniform and one angularly biased towards a specific angle, are used to show that implementing angle into the random field provides the most realistic fracture simulation. An adaptive asynchronous spacetime discontinuous Galerkin (aSDG) finite element method is used to perform the dynamic fracture simulations. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.[/spoiler]
  • B. Bahmani, R. Abedi, P.L. Clarke, “A bulk damage model for modeling dynamic fracture in rock” In: Proceeding 52th U.S. Rock Mechanics/Geomechanics Symposium, Seattle, Washington, USA – June 17-20, ARMA 18-0826 (9 pages), 2018.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] The fracture response of rock, as a quasi-brittle material, is highly sensitive to its microstructural design. We present a statistical damage formulation to model dynamic rock fracture. The damage model is rate-dependent and the corresponding damage evolution is a dynamic equation which introduces a timescale to the problem. The introduced timescale preserves mesh objectivity of the method with much less computational efforts in comparison with other conventional non-local formulations. We define a statistical field for rock cohesion to involve microstructure effects in the proposed formulation. The statistical field is constructed through the Karhunen-Loeve (KL) method. The damage model is coupled with the elastodynamic equation. The final system of coupled equations is discretized by an asynchronous Spacetime Discontinuous Galerkin (aSDG) method. Robustness of the proposed formulation is shown though dynamic fracture simulation of rock under uniaxial compressive load. The numerical investigation indicates the importance of load amplitude and microstructure randomness on failure response of rock. Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. 1538332.[/spoiler]
  • 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.
    PDF [spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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).
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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).
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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).
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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. [/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”] 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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.)[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    [spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]
  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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 [latex]\mathbb{R}^3[/latex] 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.[/spoiler]

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[toggle title=”Book Chapters” open=”yes”]

  • 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.
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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.[/spoiler]

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[toggle title=”Conference Abstracts and Presentations” open=”yes”]

    • Reza Abedi and Alireza V. Amirkhizi. Analysis of higher order scattering modes in parameter retrieval method for the characterization of dispersive media. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Giang Huynh, Reza Abedi, and Robert B Haber. Formulation and stability analysis of unstructured spacetime discontinuous Galerkin method for hyperbolic and parabolic partial differential equations. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Robert B. Haber, Amit Madhukar, Christian Howard, Volodymyr Kindratenko, and Reza Abedi. A scalable distributed architecture for a spacetime parallel-adaptive hyperbolic solver. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Amit Madhukar, Xiao Ma, Robert B Haber, Amit Elbanna, and Reza Abedi. Spacetime-adaptive simulation of earthquake rupture on a branch fault system. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Christian Howard, Amit Madhukar, Robert B. Haber, Jeff Erickson, and Reza Abedi. Adaptive spacetime meshing in 3d x time for causal spacetime discontinuous Galerkin solvers. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Sanne J van den Boom, Reza Abedi, Fred van Keulen, and Alejandro M Aragon. Topology optimization of phononic crystals with smooth boundary descriptions using an enriched finite element method. In 16th U.S. National Congress on Computational Mechanics (USNCCM21), Online, 25-29 July, 2021.
    • Reza Abedi and Robert B Haber. Spacetime adaptive meshing for tracking and capturing dynamic solution features. In International Conference on Spectral High Order Methods (ICOSAHOM2021), Online, 12-16 July, 2021.
    • Reza Abedi, Giang Huynh, and Robert B Haber. The solution of elliptic pdes by using a hyperbolic pde solver. In International Conference on Spectral High Order Methods (ICOSAHOM2021), Online, 12-16 July, 2021.
    • Huynh Giang, Garrard Justin M, and Abedi Reza. Calibration and homogenization of Mohr-Coulomb failure model for fracture analysis of anisotropic rock. In The Engineering Mechanics Institute Conference 2021 (EMI 2021), Online, 25-28 May, 2021.
    • Reza Abedi, Katherine A. Acton, Soheil Soghrati, Robert B Haber, Philip L Clarke, Justin M Garrard, Bahador Bahmani, Ming Yang, and Anand Nagarajan. Elements for realizing random fields for mesoscopic elastic and fracture properties and macroscopic fracture analysis. In The Engineering Mechanics Institute Conference 2021 (EMI 2021), Online, 25-28 May, 2021.
    • Reza Abedi, Giang Huynh Reza Haber, and Alireza Mazaheri. Causal and noncausal spacetime discontinuous Galerkin methods for the solution of parabolic partial differential equations. In 14th World Congress on Computational Mechanics, (WCCM-ECCOMAS), Online, 11-15 January, 2021.
    • Reza Abedi and Alireza Amirkhizi. Use of loss limit approach to zero in scattering-based parameter retrieval of elastic micro-structured media. In 14th World Congress on Computational Mechanics, (WCCM-ECCOMAS), Online, 11-15 January, 2021.
    • Robert Haber, Amit Madhukar, Christian Howard, Volodymyr Kindratenko, and Reza Abedi. A distributed parallel-adaptive causal spacetime discontinuous Galerkin solver. In 14th World Congress on Computational Mechanics, (WCCM-ECCOMAS), Online, 11-15 January, 2021.
    • Amit Madhukar, Xiao Ma, Robert Haber, Ahmed Elbanna, and Reza Abedi. Extreme multiscale earthquake simulation with an adaptive spacetime discontinuous Galerkin method. In 14th World Congress on Computational Mechanics, (WCCM-ECCOMAS), Online, 11-15 January, 2021.
    • Christian Howard, Amit Madhukar, Robert Haber, Jeff Erickson, and Reza Abedi. Causal meshing in up to e3xr for parallel-adaptive spacetime discontinuous Galerkin solvers. In 14th World Congress on Computational Mechanics, (WCCM-ECCOMAS), Online, 11-15 January, 2021.
    • H Wang, R Abedi, and S Mudaliar. A space-angle discontinuous Galerkin method for one-dimensional cylindrical radiative transfer equation with angular decomposition. In 2021 United States National Committee of URSI National Radio Science Meeting (USNC-URSI NRSM), Boulder, CO, USA, Jan 4-9. 2021.
    • H. Wang, R. Abedi and S. Mudaliar. A parallel space-angle discontinuous Galerkin method for radiative
      transfer in two-dimensional rectangular enclosures. In Proceedings of 2020 IEEE USNC-CNC-URSI North American Radio Science Meeting (Joint with AP-S Symposium), Montreal, QC, Canada, July 5-10, 2020.
    • Amit Madhukar, Robert Haber Ahmed Elbanna, and Reza Abedi. Spacetime DNS method for multi-scale earthquake simulation. In 57th Annual Technical Meeting of the Society of Engineering Science (SES), Online, 28-30 September, 2020.
    • Hang Wang, Reza Abedi, and Saba Mudaliar. Space-angle discontinuous Galerkin method for one-dimensional cylindrical radiative transfer equation. In AIAA Science and Technology Forum and Exposition 2020, Orlando, Florida, USA, January 6-9, 2020.
    • R. Abedi, R.B. Haber. Applications of adaptive spacetime meshing in the asynchronous spacetime discontinuous Galerkin method. In 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation (Waves 2019), Vienna, Austria, August 25-30, 2019.
      Abstract
    • R.B. Haber, A. Madhukar, X. Ma, A. Elbanna, R. Abedi. Distributed parallel-adaptive causal spacetime discontinuous Galerkin method with application to earthquake simulation. In 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation (Waves 2019), Vienna, Austria, August 25-30, 2019.
      Abstract
    • R. Abedi, P.L. Clarke, B. Bahmani, J.M. Garrard, K.A. Acton, R.B. Haber. Effects of mesoscale material inhomogeneity on macroscopic dynamic fracture response. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28-Ausust 1, 2019.
      Abstract
    • J.M. Garrard, R. Abedi, R.B. Haber. Explicit and implicit approaches for characterization and fracture analysis of anisotropic rock. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28-Ausust 1, 2019.
      Abstract
    • R.B. Haber, A. Madhukar, X. Ma, A. Elbanna, R. Abedi, V. Kindratenko. Distributed parallel{adaptive implementation of asynchronous spacetime discontinuous Galerkin methods with application to seismic simulation. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28-Ausust 1, 2019.
      Abstract
    • C. Howard, A. Madhukar, J. Erickson, R.B. Haber, R. Abedi. Tent pitching meshes for asynchronous spacetime discontinuous Galerkin solvers in 3dxtime. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28- Ausust 1, 2019.
      Abstract
    • X. Ma, A. Madhukar, R.B. Haber, R. Abedi, A. Elbanna. Modeling earthquake ruptures with high-resolution fault-zone physics: An adaptive asynchronous space-time discontinuous Galerkin approach. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28-Ausust 1, 2019.
      Abstract
    • A. Madhukar, R.B. Haber, R. Abedi, V. Kindratenko. Performance and scalability of parallel{adaptive asynchronous spacetime discontinuous Galerkin methods. In 15th U.S. National Congress on Computational Mechanics (USNCCM15), Austin, Texas, USA, July 28-Ausust 1, 2019.
      Abstract
    • J.M. Garrard, R. Abedi. A statistical volume element averaging scheme for fracture analysis of microcracked rock. In Engineering Mechanics Institute Conference (EMI 2019), Pasadena, California, USA, June 18-21, 2019.
      Abstract
    • R. Abedi, A. Mazaheri, R.B. Haber. Adaptive space-time discontinuous Galerkin methods for solutions of steady and transient partial differential equations. In 2019 North American High Order Methods Conference (NAHOMCon 19), San Diego, California, USA, June 2-5, 2019.
      Abstract
    • K. Acton, B. Bahmani, R. Abedi, “Mesoscale Material Strength Characterization for use in Fracture Modeling”, International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018.
      Abstract
    • B. Bahmani, M. Yang, A. Nagarajan, P.L. Clarke, S. Soghrati, R. Abedi , “An integrated approach for statistical microscale homogenization to macroscopic dynamic fracture analysis”, International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018.
      Abstract
    • B. Bahmani, P.L. Clarke, R. Abedi, “Comparison of interfacial and continuum models for dynamic fragmentation analysis”, International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018.
      Abstract
    • J.M. Garrard, P.L. Clarke, R. Abedi, “Statistical volume elements for the characterization of angle-dependent fracture strengths”, International Mechanical Engineering Congress & Exposition AMSE 2018 IMECE, Pittsburgh, Pennsylvania, USA – November 11-14, 2018.
      Abstract
    • R. Abedi, A. Mazaheri, R.B. Haber, “Adaptive Space-Time Discontinuous Galerkin Method for Unsteady Elliptic and Parabolic PDEs with First-Order Hyperbolic System Approach”, 13th World Congress on Computational Mechanics (WCCM XIII), 2nd Pan-American Congress on Computational Mechanics (PANACM II), New York City, New York – USA, July 22-27, 2018.
      Abstract
    • B. Bahmani, M. Yang, P.L. Clarke, K.A. Acton, A. Nagarajan, S. Soghrati, S.C. Baxter, R.B. Haber, R. Abedi , “Homogenization and Stochastic Fracture Simulation of Quasi-brittle Materials”, 13th World Congress on Computational Mechanics (WCCM XIII), 2nd Pan-American Congress on Computational Mechanics (PANACM II), New York City, New York – USA, July 22-27, 2018.
      Abstract
    • R.B. Haber, A. Madhukar, V. Kindratenko, and R. Abedi, “Parallel–Adaptive Implementation of Asynchronous Spacetime Discontinuous Galerkin Methods”, 13th World Congress on Computational Mechanics (WCCM XIII), 2nd Pan-American Congress on Computational Mechanics (PANACM II), New York City, New York – USA, July 22-27, 2018.
      Abstract
    • A. Madhukar, R.B. Haber, V. Kindratenko, and Reza Abedi, “Shared–Memory Parallel Implementation of High-Order Asynchronous Spacetime Discontinuous Galerkin Methods”, 13th World Congress on Computational Mechanics (WCCM XIII), 2nd Pan-American Congress on Computational Mechanics (PANACM II), New York City, New York – USA, July 22-27, 2018.
      Abstract
    • A. Elbanna, A. Madhukar, R.B. Haber, R. Abedi, “Simulating Earthquake Rupture on Frictional Interfaces Using an Asynchronous Space-time Discontinuous Galerkin Method”, 13th World Congress on Computational Mechanics (WCCM XIII), 2nd Pan-American Congress on Computational Mechanics (PANACM II), New York City, New York – USA, July 22-27, 2018.
      Abstract
    • R. Abedi, P.L. Clarke, and S. Mudaliar, “Random Field Realization and Time Domain Stochastic Simulation of a Complex Electromagnetic Medium”, 2018 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS-URSI 2018), Boston, Massachusetts, USA – July 8-13, 2018.
      Abstract
    • P.L. Clarke, H. Wang, J.M. Garrard, R. Abedi and S. Mudaliar, “A Discontinuous Galerkin Method for the Solution of One Dimensional Radiative Transfer Equation”, 2018 IEEE AP-S Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS-URSI 2018), Boston, Massachusetts, USA – July 8-13, 2018.
      Abstract
    • R. Abedi (in collaboration with S. Mudaliar), “An h-adaptive asynchronous spacetime discontinuous Galerkin method for TD analysis of complex electromagnetic media ”, The Institute for Computational and Experimental Research in Mathematics (ICERM), Computational Aspects of Time Dependent Electromagnetic Wave Problems in Complex Materials, Providence, Rhode Island, USA – June 22-29, 2018. (invited talk)
      Abstract
      Presentation Slides
      Presentation Video
    • R. Abedi, P.L. Clarke, “Modeling of rock inhomogeneity and anisotropy by explicit and implicit representation of microcracks”, 52th U.S. Rock Mechanics / Geomechanics Symposium (ARMA 2018), Seattle, Washington, USA – June 17-20, 2018.
      Abstract
    • B. Bahmani, R. Abedi, P.L. Clarke, “A bulk damage model for modeling dynamic fracture in rock”, 52th U.S. Rock Mechanics / Geomechanics Symposium (ARMA 2018), Seattle, Washington, USA – June 17-20, 2018.
      Abstract
    • J.M. Garrard, P.L. Clarke, R. Abedi, “Random field realization and fracture simulation of rocks with angular bias for fracture strength”, 52th U.S. Rock Mechanics/Geomechanics Symposium (ARMA 2018), Seattle, Washington, USA – June 17-20, 2018.
      Abstract
    • R. Abedi, B. Bahmani, R.B. Haber, “A Unified Model for Tensile- and Shear-driven Dynamic Crack Propagation in Rocks”, 18th US National Congress of Theoretical and Applied Mechanics (USNCTAM18), Chicago, Illinois, USA – June 4-9, 2018.
      Abstract
    • R.B. Haber, R. Abedi, P.L. Clarke, “Modeling Rate Effects in Quasi-Brittle Materials with a Two-term Interfacial Damage Rule”, 18th US National Congress of Theoretical and Applied Mechanics (USNCTAM18), Chicago, Illinois, USA – June 4-9, 2018.
      Abstract
    • R. Abedi, P.L. Clarke, B. Bahmani, M. Yang, A. Noshadravan, A. Nagarajan, S. Soghrati, K.A. Acton, “Mesoscopic characterization of SVEs and macroscopic field realization for fracture and elastic properties”,  Engineering Mechanics Institute Conference,  (EMI 2018), Boston, Massachusetts, USA – May 29-June 1, 2018.
      Abstract
    • B. Bahmani, R. Abedi, P.L. Clarke, K.A. Acton, “Comparison of interfacial and bulk damage models for dynamic brittle fracture”,  Engineering Mechanics Institute Conference,  (EMI 2018), Boston, Massachusetts, USA – May 29-June 1, 2018.
      Abstract
    • J. Garrard, P.L. Clarke, R. Abedi, “Random field realization and fracture simulation of rock with angular-bias in microcrack orientation”,  Engineering Mechanics Institute Conference,  (EMI 2018), Boston, Massachusetts, USA – May 29-June 1, 2018.
      Abstract
    • R. Abedi, “An adaptive time domain approach to characterize dispersive elastodynamic media”, International Mechanical Engineering Congress & Exposition AMSE 2017 IMECE, Tampa, 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 IMECE, Tampa, 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 IMECE, Tampa, Florida, USA – November 5-8, 2017.
      Abstract
    • Mudaliar, P. Clarke, and R. Abedi, “Radiative transfer in turbulent flow using spacetime discontinuous Galerkin finite element method” 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, “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, S.C. Baxter, and B. Bahmani, “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
    • 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
    • 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, 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, 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

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[toggle title=”Theses” open=”yes”]

  • 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).
    PDF[spoiler title=”View/Hide Abstract” icon=”caret”]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. [/spoiler]

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