# Browsing by Subject "Integral equations"

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Item A comprehensive comparison of FFT-accelerated integral equation methods vs. FDTD for bioelectromagnetics(2015-05) Massey, Jackson White; Yılmaz, Ali E.; Biros, GeorgeShow more The performance of two FFT-accelerated integral equation methods--the adaptive integral method (AIM) and GMRES-FFT--and the finite-difference time-domain (FDTD) method are systematically compared for their use in bioelectromagnetic (BioEM) analysis. The comparison involves four steps: (i) A BioEM benchmark is developed. The power absorbed by a human model illuminated by an impressed time-harmonic source is selected as the problem of interest. The benchmark consists of three inhomogeneous models (a multilayered spherical head phantom, an anatomical male, and an anatomical female model), two types of models (pixel or surface based), two types of sources (far or near), and three frequencies in the UHF band (402 MHz, 900 MHz, and 2.45 GHz). (ii) Error and cost measures are identified: The total power absorbed, the power absorbed in different tissues, and the absorbed power density are compared to either analytical results or results from other methods. The peak memory requirement and computation time of the simulations are recorded. (iii) The benchmark problems are solved using each method with optimized parameters. (iv) Plots of results, errors, and computational costs are presented and the tradeoff between increased accuracy and cost is quantified for each method. The data show that when surface-based models can be used AIM generally outperforms GMRES-FFT and FDTD: AIM achieves lower errors at the same computational cost or costs less to achieve the same error. When restricted to pixel-based models, however, FDTD generally outperforms GMRES-FFT and AIM: All three methods yield comparable errors, in most cases FDTD is less costly than GMRES-FFT (especially for anatomical models, far sources, and higher frequencies), and GMRES-FFT is slightly less expensive than AIM. These results suggest that for the type of BioEM analysis represented by the benchmark, AIM should be used whenever surface-based models are available and FDTD should be used if only pixel-based models are available.Show more Item A computational procedure for analysis of fractures in two-dimensional multi-field media(2010-12) Tran, Han Duc; Mear, Mark E.; Rodin, Gregory J.; Ravi-Chandar, Krishnaswa; Landis, Chad M.; Tassoulas, John L.Show more A systematic procedure is followed to develop singularity-reduced integral equations for modeling cracks in two-dimensional, linear multi-field media. The class of media treated is quite general and includes, as special cases, anisotropic elasticity, piezoelectricity and magnetoelectroelasticity. Of particular interest is the development of a pair of weakly-singular, weak-form integral equations (IEs) for "generalized displacement" and "generalized stress"; these serve as the basis for the development of a Symmetric Galerkin Boundary Element Method (SGBEM). The implementation is carried out to allow treatment of general mixed boundary conditions, an arbitrary number of cracks, and multi-region domains (in which regions having different material properties are bonded together). Finally, a procedure for calculation of T-stress, the constant term in the asymptotic series expansion of crack-tip stress field, is developed for anisotropic elastic media. The pair of weak-form boundary IEs that is derived (one for generalized displacement and the other one for generalized stress) are completely regularized in the sense that all kernels that appear are (at most) weakly-singular. This feature allows standard Co elements to be utilized in the SGBEM, and such elements are employed everywhere except at the crack tip. A special crack-tip element is developed to properly model the asymptotic behavior of the relative crack-face displacements. This special element contains "extra" degrees of freedom that allow the generalized stress intensity factors to be directly obtained from the solution of the governing system of discretized equations. It should be noted that while the integral equations contain only weakly-singular kernels (and so are integrable in the usual sense) there remains a need to devise special integration techniques to accurately evaluate these integrals as part of the numerical implementation. Various examples for crack problems are treated to illustrate the accuracy and versatility of the proposed procedure for both unbounded and finite domains and for both single-region and multi-region problems. It is found that highly accurate fracture data can be obtained using relatively course meshes. Finally, this dissertation addresses the development of a numerical procedure to calculate T-stress for crack problems in general anisotropic elastic media. T-stress is obtained from the sum of crack-face displacements which are computed via a (regularized) integral equation of the boundary data. Two approaches for computing the derivative of the sum of crack-face displacements are proposed: one uses numerical differentiation, and the other one uses a weak-form integral equation. Various examples are examined to demonstrate that highly accurate results are obtained by means of both approaches.Show more Item Envelope tracking integral equation based hybrid electromagnetic circuit simulators(2016-12) Subramanian, Vivek, Ph. D.; Yilmaz, Ali E.; Neikirk, Dean; Ling, Hao; Alu, Andrea; Chakraborty, SwagatoShow more This dissertation presents envelope-tracking hybrid field-circuit simulator for efficiently analyzing narrowband scattering from distributed structures loaded with nonlinear devices. The simulator models the interactions of fields with distributed structures and lumped elements by coupling and simultaneously solving the electric field integral equation and Kirchhoff’s equations, respectively. The coupled nonlinear system of equations is iteratively solved by a time marching scheme that represents the fields, voltages, and currents of interest (signals) as a truncated series of harmonic sinusoids (carriers) multiplied with complex-valued time-varying coefficients (envelopes). Unlike time-domain simulators, which sample the signals at a rate proportional to their maximum frequency content, the proposed envelope-tracking simulator samples the envelopes at a rate proportional to their maximum bandwidth; thus, it requires significantly fewer time steps when solving narrowband problems. Moreover, the envelope-tracking simulator is generally more accurate than its time-domain counterpart because of smaller integration and interpolation errors. Numerical results demonstrate that the proposed simulator improves the tradeoff between accuracy and computational cost, especially when analyzing distributed structures excited by narrowband signals or/and loaded with weakly nonlinear devices. Although the Fourier envelope simulator uses smaller number of time steps, there are other issues relating to the Fourier envelope simulator which are addressed in this thesis: (i) lumped element models that relate voltage envelopes and current envelopes for nonlinear elements are generally unavailable and the approximations used in the simulator to find them are inaccurate for broader band excitations. Higher order interpolation schemes were used in this dissertation to improve the accuracy of these approximations. Numerical results that demonstrate the ability to solve for problems with broader bandwidth of excitation are presented. (ii) As in its timedomain counterpart, adaptive integral method is used to reduce the computational cost of the simulator thus enabling the simulation of larger problems and (iii) Sparse preconditioners are used to improve the convergence of the solution algorithms. Finally, the Fourier envelope method is extended to the analysis of infinitely periodic arrays containing lumped nonlinear loads. Numerical results are presented to highlight the .features of this algorithm.Show more Item Envelope-tracking integral equation methods for band-pass transient scattering analysis(2015-09-28) Kaur, Guneet; Yilmaz, Ali E.; Ling, Hao; Alu, Andrea; Demkowicz, Leszek; Schulz, KarlShow more This dissertation presents envelope-tracking (ET) integral equation methods to efficiently analyze band-pass scattering problems. Unlike the traditional time-domain marching-on-in-time (TD-MOT) schemes, ET-MOT schemes solve for space-time samples of not the current density but its complex envelope. The time step size used in ET-MOT schemes is inversely proportional to the bandwidth of the fields of interest and not their maximum frequency content; thus, ET-MOT schemes can use (much) larger time step sizes for band-pass analysis: the smaller the bandwidth of the fields compared to their maximum frequency content, the larger the time step size in ET-MOT solutions compared to those in the TD-MOT solutions. Despite the reduction in the number of time steps, ET-MOT schemes suffer from high computational costs that also affect time- and frequency-domain integral equation methods. This dissertation presents an FFT-based algorithm, the ET adaptive integral method (ET-AIM), to reduce the computational complexity of ET-MOT schemes. ET-AIM is both theoretically and empirically compared to its time-domain and frequency-domain counterparts, TD-AIM and FD-AIM, respectively. Because the performance of the envelope-tracking methods is a complex function of the bandwidth of interest and because each method has different accuracy-efficiency tradeoff, only limited deductions can be made from theoretical comparison of the methods. Thus, in addition to theoretical comparisons, an empirical approach for comparing the different methods is presented: To perform a fair, meaningful, and generalizable comparison, benchmark problems are identified, an appropriate error norm is defined, and the key parameters of the methods are optimized subject to a constraint on the error norm. Computational costs are measured and compared for all three methods for solving progressively larger benchmark scattering problems for varying frequency bandwidths. This dissertation also proposes an out-of-core algorithm to ameliorate the high memory requirement of FFT-accelerated time-marching methods. The proposed algorithm exchanges the core memory requirement with external storage space requirement without significantly increasing the simulation time. The performance of the proposed methods is demonstrated by solving surface- and volume-integral equations pertinent to scattering problems that involve good conductors and inhomogeneous volumes with complex dielectric properties. For example, numerical results obtained using ET-AIM are presented for analysis of scattering of radar pulses from a PEC missile, a generic aircraft, etc. and antenna radiation near anatomically realistic human body model.Show more Item Fast integral equation solver for variable coefficient elliptic PDEs in complex geometries(2017-10-26) Malhotra, Dhairya; Biros, George; Engquist, Bjorn; van de Geijn, Robert A; Hughes, Thomas J R; Kloeckner, AndreasShow more This dissertation presents new numerical algorithms and related software for the numerical solution of elliptic boundary value problems with variable coefficients on certain classes of geometries. The target applications are problems in electrostatics, fluid mechanics, low-frequency electromagnetic and acoustic scattering. We present discretizations based on integral equation formulations which are founded in potential theory and Green's functions. Advantages of our methods include high-order discretization, optimal algorithmic complexity, mesh-independent convergence rate, high-performance and parallel scalability. First, we present a parallel software framework based on kernel independent fast multipole method (FMM) for computing particle and volume potentials in 3D. Our software is applicable to a wide range of elliptic problems such as Poisson, Stokes and low-frequency Helmholtz. It includes new parallel algorithms and performance optimizations which make our volume FMM one of the fastest constant-coefficient elliptic PDE solver on cubic domains. We show that our method is orders of magnitude faster than other N-body codes and PDE solvers. We have scaled our method to half-trillion unknowns on 229K CPU cores. Second, we develop a high-order, adaptive and scalable solver for volume integral equation (VIE) formulations of variable coefficient elliptic PDEs on cubic domains. We use our volume FMM to compute integrals and use GMRES to solve the discretized linear system. We apply our method to compute incompressible Stokes flow in porous media geometries using a penalty function to enforce no-slip boundary conditions on the solid walls. In our largest run, we achieved 0.66 PFLOP/s on 2K compute nodes of the Stampede system (TACC). Third, we develop novel VIE formulations for problems on geometries that can be smoothly mapped to a cube. We convert problems on non-regular geometries to variable coefficient problems on cubic domains which are then solved efficiently using our volume FMM and GMRES. We show that our solver converges quickly even for highly irregular geometries and that the convergence rates are independent of mesh refinement. Fourth, we present a parallel boundary integral equation solver for simulating the flow of concentrated vesicle suspensions in 3D. Such simulations provide useful insights on the dynamics of blood flow and other complex fluids. We present new algorithmic improvements and performance optimizations which allow us to efficiently simulate highly concentrated vesicle suspensions in parallel.Show more