Browsing by Subject "Fast algorithms"
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Item A multiregion-integral-equation method for wearable and implantable device design and evaluation : modeling, high-performance algorithms, and postprocessing(2018-07-02) Massey, Jackson White; Yilmaz, Ali E.; Biros, George; Kozlov, Mikhail; Neikirk, Dean; Tewfik, AhmedComputational bioelectromagnetics has an essential role during the design phase of wireless devices that operate near, on, or inside the human body. Simulating the electromagnetic interaction of devices with a nearby human body, however, presents significant challenges that manifest in three areas of the simulation process: (i) Modeling challenges arise when simultaneously modeling devices (with complex geometrical features) and bodies (with inhomogeneous tissues); interfacing such models and creating consistent (conformal) discretizations/meshes is difficult, especially when high-fidelity device and anatomical body models are used, because the two types of models are often developed independently and because they exhibit large differences in length scales, model sizes, and material properties. (ii) Solution challenges arise due to the size of the problem, potentially reaching to 10⁸–10⁹ unknowns, as well as the multitude of simulations that are of interest, e.g., to quantify the impact of variations in anatomy, posture, and device position/orientation on the performance of the design. (iii) Postprocessing challenges arise also because of the problem size and because of the increasing demand for more accurate and numerous secondary quantities, e.g., for multiphysics simulations. Calculating such secondary quantities quickly becomes the bottleneck of the simulation process, especially when powerful modeling methods and solution algorithms are developed to address the other challenges. This dissertation presents an integral-equation based approach to address these challenges and enable more advanced designs of body-worn and implanted devices. The dissertation presents a formulation that hybridizes a multiregion surface-integral equation formulation with a volume-integral equation pertinent to scattering from inhomogeneous tissues. The proposed formulation sidesteps many of the modeling challenges by allowing devices to be modeled and meshed independently of the anatomical human body model and mesh, by coupling them using an equivalent surface, and by simultaneously solving the resulting fully-coupled linear system of equations. To address the solution challenges, a scalable fast iterative algorithm that is accelerated by the multiple-grid adaptive integral method and parallelized using a hybrid shared-memory/distributed-memory (OpenMP/MPI) scheme is presented; moreover, a Schur-complement based algorithm is developed to harness the relatively small size of the system of equations corresponding to the equivalent surface and to rapidly perform the multitude of simulations needed during the design process. To accelerate the postprocessing stage, auxiliary-grid-based methods to rapidly compute far-field and near-field distributions are proposed. The dissertation also presents a benchmark suite that can be used to evaluate competitive computational bioelectromagnetics methods empirically and objectively; this suite is used to quantify the performance of the presented methods. Finally, three classes of problems are simulated to demonstrate the utility of the proposed work: a wearable antenna near an anatomical human model, an ingestible device inside an anatomical human model, and an anatomical human model under MRI exposureItem Fast algorithms for frequency domain wave propagation(2012-12) Tsuji, Paul Hikaru; Ying, Lexing; Ghattas, Omar N.; Engquist, Bjorn; Fomel, Sergey; Ren, KuiHigh-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time.Item Fast direct algorithms for elliptic equations via hierarchical matrix compression(2010-08) Schmitz, Phillip Gordon; Ying, Lexing; Gamba, Irene M.; Ghattas, Omar; Gonzalez, Oscar; Ren, KuiWe present a fast direct algorithm for the solution of linear systems arising from elliptic equations. We extend the work of Xia et al. (2009) on combining the multifrontal method with hierarchical matrices. We offer a more geometric interpretation of that approach, extend it in two dimensions to the unstructured mesh case, and detail an adaptive decomposition procedure for selectively refined meshes. Linear time complexity is shown for a quasi-uniform grid and demonstrated via numerical results for the adaptive algorithm. We also provide an extension to three dimensions with proven linear complexity but a more practical variant with slightly worse scaling is also described.Item Numerical methods for simulations and optimization of vesicle flows in microfluidic devices(2019-05) Kabacaoğlu, Gökberk; Biros, George; Ghattas, Omar; Moser, Robert; Shelley, MichaelVesicles are highly deformable particles that are filled with a Newtonian fluid. They resemble biological cells without a nucleus such as red blood cells (RBCs). Vesicle flow simulations can be used to design microfluidic devices for medical diagnoses and drug delivery systems. This dissertation focuses on efficient numerical methods for simulations and optimization of vesicle flows in two dimensions. We consider flows with very low Reynolds numbers and inextensible vesicle membranes that resist bending. Our numerical scheme is based on a boundary integral formulation which is known to be efficient for such flows. This formulation leads to a set of nonlinear integro-differential equations for the vesicle dynamics. Complex interplay between the nonlocal hydrodynamic forces and the membranes’ elasticity determines the vesicles’ motion. Many state-of-the-art numerical schemes can resolve these complex flows. However, simulations remain computationally expensive since high-resolution discretization is needed. The high computational cost limits the use of the simulations for practical purposes such as optimization. Our first attempt to reduce the cost is to use low-resolution discretization. We present a scheme that systematically integrates several correction algorithms that are necessary for stable and accurate low-resolution simulations. We compare the low-resolution simulations with their high-fidelity counterparts. We observe that our scheme enables both fast and statistically accurate simulations. We accelerate vesicle flow simulations further by replacing expensive parts of the numerical scheme with low-cost function approximations. We propose a machine-learning-augmented reduced model that uses several multilayer perceptrons to model different aspects of the flows. Although we train the perceptrons with high-fidelity single-particle simulations for one time step, our method enables us to conduct long-horizon simulations of suspensions with several particles in confined geometries. It is faster than a state-of-the-art numerical scheme having the same number of degrees of freedom and can reproduce several features of the flow accurately. It generalizes as is to other particles like deformable capsules, drops, filaments and rigid bodies. Moreover, we investigate deformability-based sorting of RBCs using a microfluidic device that enables medical diagnoses of diseases such as malaria. Using our numerical scheme we solve a design optimization problem to find optimal designs of the device that provide efficient sorting of cells with arbitrary mechanical properties