Reza Abedi

Effect of random defects on dynamic fracture in quasi-brittle materials

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 flaw 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 model 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.

Riemann solutions for spacetime discontinuous Galerkin methods

Abstract Spacetime discontinuous Galerkin finite element methods (cf. Abedi etxa0al. (2013), Abedi etxa0al. (2010), Miller and Haber (2008)) rely on ‘target fluxes’ on element boundaries that are computed via local one-dimensional Riemann solutions in the direction normal to the element face. In this work, we provide details of converting a space–time flux expressed in differential forms into a standard one-dimensional Riemann problem on the element interface. We then demonstrate a generalized solution procedure for linearized hyperbolic systems based on diagonalization of the governing system of partial differential equations. The generalized procedure is particularly useful for the implementation aspects of coupled multi-physics applications. We show that source terms do not influence the Riemann solution in the spacetime setting. We provide details for implementation of coordinate transformations and Riemann solutions. Exact Riemann solutions for some linearized systems of equations are provided as examples, including an exact, semi-analytic Riemann solution for generalized thermoelasticity with one relaxation time.

Error analysis and comparison of Riemann and average fluxes for a spacetime discontinuous Galerkin electromagnetic formulation

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.

A spacetime adaptive approach to characterize complex dispersive media

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.

Radiative transfer in turbulent flow using spacetime discontinuous Galerkin finite element method

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.

An h-adaptive time domain discontinuous galerkin method for electromagnetics

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.

Mesoscale Models Characterizing Material Property Fields Used As a Basis for Predicting Fracture Patterns in Quasi-Brittle Materials

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 ∗Address all correspondence to this author. 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 quasibrittle materials. INTRODUCTION Material inhomogeneities at the microstructural scale greatly influence fracture response. Failure initiates locally where stress concentrations are induced in large part by local heterogeneity. Therefore, fracture models that ignore microstructural inhomogeneity, or employ Representative Volume Elements (RVE) to homogenize material properties, may not accurately capture fracture response. The high sensitivity of brittle fracture to material microstructure not only contributes to the form of failure patterns, but also size effects [1–3] and high response variability for samples with identical geometry and loading specifications [4–6]. In previous work, the Spacetime Discontinuous Galerkin 1 Copyright

Effect of the Spatial Inhomogeneity of Fracture Strength on Fracture Pattern for Quasi-Brittle Materials

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.

An asynchronous spacetime discontinuous Galerkin finite element method for time domain electromagnetics

Abstract 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 Maxwells 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.

Spacetime Discontinuous Galerkin FEM: Spectral Response

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 materials 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.