@article{savadatti_guddati_2012, title={Accurate absorbing boundary conditions for anisotropic elastic media. Part 1: Elliptic anisotropy}, volume={231}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2012.05.033}, abstractNote={With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for heterogeneous anisotropic elastic media, we investigate the accuracy aspects of local ABCs designed for tilted elliptic anisotropy in the frequency domain (time-harmonic case). Such media support both anti-plane and in-plane wavemodes with opposing signs of phase and group velocities (cpxcgx<0) that have long posed a significant challenge to the design of accurate (and stable) local ABCs. By first considering the simpler case of scalar anti-plane waves, we show that it is possible to overcome the challenges posed by cpxcgx<0 by simply utilizing the inevitable reflections occurring at the truncation boundaries. This understanding helps us to explain the ability of a recently developed local ABC – the perfectly matched discrete layer (PMDL) – to handle the challenges posed by cpxcgx<0 without the need of intervening space–time transformations. PMDL is a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation-based local ABCs (rational ABCs); it inherits the straightforward approximation properties of rational ABCs along with the versatility of PML. The approximation properties of PMDL quantified through its reflection matrix is used to derive simple bounds on the PMDL parameters necessary for the accurate absorption of all outgoing anti-plane and in-plane wavemodes – including those with cpxcgx<0. Beyond the previously derived bound on the real parameters of PMDL sufficient for the absorption of outgoing propagating anti-plane wavemodes, we present bounds on the complex parameters of PMDL necessary for the absorption of outgoing propagating and evanescent wavemodes for both anti-plane and coupled in-plane pressure and shear waves. The validity of this work is demonstrated through a series of numerical experiments.}, number={22}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Savadatti, Siddharth and Guddati, Murthy N.}, year={2012}, month={Sep}, pages={7584–7607} }
 @article{savadatti_guddati_2012, title={Accurate absorbing boundary conditions for anisotropic elastic media. Part 2: Untilted non-elliptic anisotropy}, volume={231}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2012.05.039}, abstractNote={With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general elastic media, we investigate the accuracy aspects of local ABCs designed for untilted non-elliptic anisotropy in the frequency domain (time-harmonic analysis). While simple space–time transformations are available to treat the wavemodes with opposing phase and group velocities present in elliptic anisotropic media, no such transformations are known to exist for the case of non-elliptic anisotropy. In this paper, we use the concept of layer groupings along with an unconventional stretching of the finite element mesh to guarantee the accuracy of local ABCs designed to treat all propagating wavemodes, even those with opposing phase and group velocities. The local ABC used here is the perfectly matched discrete layer (PMDL) which is a simple variant of perfectly matched layers (PMLs) that is also equivalent to rational approximation-based local ABCs (rational ABCs); it inherits the straightforward approximation properties of rational ABCs along with the versatility of PML. The approximation properties of PMDL quantified through its reflection matrix allow us to (a) show that it is impossible to design an accurate PMDL with wavenumber-independent parameters, (b) theoretically demonstrate the ability of wavenumber-dependent parameters to ensure accuracy, and finally (c) design a practical though unconventional stretching of the finite element PMDL mesh that facilitates the implementation of wavenumber-dependent parameters. The validity of this work is demonstrated through a series of numerical experiments.}, number={22}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Savadatti, Siddharth and Guddati, Murthy N.}, year={2012}, month={Sep}, pages={7608–7625} }
 @article{savadatti_guddati_2010, title={A finite element alternative to infinite elements}, volume={199}, ISSN={["1879-2138"]}, DOI={10.1016/j.cma.2010.03.018}, abstractNote={In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.}, number={33-36}, journal={COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING}, author={Savadatti, Siddharth and Guddati, Murthy N.}, year={2010}, pages={2204–2223} }
 @article{savadatti_guddati_2010, title={Absorbing boundary conditions for scalar waves in anisotropic media. Part 1: Time harmonic modeling}, volume={229}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2010.05.018}, abstractNote={With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the accuracy aspects of local ABCs designed for the scalar anisotropic wave equation in the frequency domain (time harmonic case). The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL). PMDL is a simple variant of perfectly matched layers (PML) and is equivalent to rational approximation-based local ABCs. Specifically, we derive a sufficient condition for PMDL to accurately absorb wave modes with outgoing group velocities and this condition turns out to be a simple bound on the PMDL parameters. The reflection coefficient derived in this paper clearly reveals that the PMDL absorption is based on group velocities, and not phase velocities, and hence a PMDL can be designed to correctly identify and accurately absorb all outgoing wave modes (even those with opposing signs of phase and group velocities). The validity of the sufficient condition is demonstrated through a series of frequency domain simulations. In part 2 of this paper [S. Savadatti, M.N. Guddati, Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling, J. Comput. Phys. (2010), doi:10.1016/j.jcp.2010.05.017], the accuracy condition presented here is shown to govern both the well-posedness and accuracy aspects of PMDL designed for transient (time-dependent) modeling of scalar waves in anisotropic media.}, number={19}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Savadatti, Siddharth and Guddati, Murthy N.}, year={2010}, month={Sep}, pages={6696–6714} }
 @article{savadatti_guddati_2010, title={Absorbing boundary conditions for scalar waves in anisotropic media. Part 2: Time-dependent modeling}, volume={229}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2010.05.017}, abstractNote={With the ultimate goal of devising effective absorbing boundary conditions (ABCs) for general anisotropic media, we investigate the well-posedness and accuracy aspects of local ABCs designed for the transient modeling of the scalar anisotropic wave equation. The ABC analyzed in this paper is the perfectly matched discrete layers (PMDL), a simple variant of perfectly matched layers (PML) that is also equivalent to rational approximation based ABCs. Specifically, we derive the necessary and sufficient condition for the well-posedness of the initial boundary value problem (IBVP) obtained by coupling an interior and a PMDL ABC. The derivation of the reflection coefficient presented in a companion paper (S. Savadatti, M.N. Guddati, J. Comput. Phys., 2010, doi:10.1016/j.jcp.2010.05.018) has shown that PMDL can correctly identify and accurately absorb outgoing waves with opposing signs of group and phase velocities provided the PMDL layer lengths satisfy a certain bound. Utilizing the well-posedness theory developed by Kreiss for general hyperbolic IBVPs, and the well-posedness conditions for ABCs derived by Trefethen and Halpern for isotropic acoustics, we show that this bound on layer lengths also ensures well-posedness. The time discretized form of PMDL is also shown to be theoretically stable and some instability related to finite precision arithmetic is discussed.}, number={18}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Savadatti, Siddharth and Guddati, Murthy N.}, year={2010}, month={Sep}, pages={6644–6662} }