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Archive : Numerical Analysis and PDE

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Giordano Tierra, Temple UniversityGiordano Tierra, Temple UniversityEWG336Efficient numerical schemes for energy-based models and their applicability to mixtures of isotropic and nematic flows with anchoring and stretching effects </br></br> The study of interfacial dynamics between two different components has become the key role to understand the behavior of many interesting systems. Indeed, two-phase flows composed of fluids exhibiting different microscopic structures are an important class of engineering materials. The dynamics of these flows are determined by the coupling among three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid) and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due anchoring effects of the liquid crystal molecules on the interface. </br></br> In this talk I will introduce a PDE system to model mixtures composed by isotropic fluids and nematic liquid crystals, taking into account viscous, mixing, nematic, stretching and anchoring effects and reformulating the corresponding stress tensors in order to derive a dissipative energy law. Then, I will present new linear unconditionally energy-stable splitting schemes that allows us to split the computation of the three pairs of unknowns (velocity-pressure, phase field-chemical potential and director vector-equilibrium) in three different steps. The fact of being able to decouple the computations in different linear sub-steps maintaining the discrete energy law is crucial to carry out relevant numerical experiments under a feasible computational cost and assuring the accuracy of the computed results. </br></br> Finally, I will present several numerical simulations in order to show the efficiency of the proposed numerical schemes, the influence of the shape of the nematic molecules (stretching effects) in the dynamics and the importance of the interfacial interactions (anchoring effects) in the equilibrium configurations achieved by the system. </br></br> This contribution is based on joint work with Francisco Guillen-Gonzalez (Universidad de Sevilla, Spain) and Marıa Angeles Rodrıguez-Bellido (Universidad de Sevilla, Spain). 5/4/2017 3:00:00 PM5/4/2017 4:00:00 PMFalse
Yue Yu, Lehigh UniversityYue Yu, Lehigh UniversityEWG 336Title: Multiscale and multiphysics coupling methods with applications in vascular blood flow simulations Abstract: In this work, we consider the partitioned approach for fluid-structure interactions, and we develop new stabilized algorithms. There are two approaches in formulating the discrete systems in simulating fluid-structure interaction (FSI) problems: the monolithic approach, and the partitioned approach. The former is efficient for small problems but does not scale up to realistic sizes, whereas the latter suffers from numerical stability issues. In particular, in vascular blood flow simulations where the mass ratio between the structure and the fluid is relatively small, the partitioned approach gives rise to the so called added-mass effect which renders the simulation unstable. I will present a new numerical method to handle this added-mass effect, by relaxing the exact no-slip boundary condition and introducing proper penalty terms on the fluid-structure interface, which enables the possibility of stable explicit coupling procedure. The optimal parameters are obtained via theoretical analysis, and we numerically verify that stability can be achieved irrespective of the fluid-structure mass ratio. To demonstrate the effectiveness of the proposed techniques in practical computations, I will also discuss two vascular blood flow applications in three-dimensional large scale simulations. The first applicationis obtained for patient-specific cerebral aneurysms. The 3D fractional-order PDEs (FPDEs) are investigated which better describe the viscoelastic behavior of cerebral arterial walls. In the second application, we apply the stabilized FSI method to heart valves, and simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions. Lastly, I will present the progress of our ongoing project on the multiscale coupling of the peridynamic and the classical elastic theories, with an application on simulating the material damage in arterial simulations, say, the aneurysm rupture or the heart valve failure problems. 4/20/2017 3:00:00 PM4/20/2017 4:00:00 PMFalse
Wolfgang Wendland, University of StuttgartWolfgang Wendland, University of StuttgartEWG 336 Some remarks to the Maxwell equations </br></br> After a short introduction, time harmonic electromagnetic fields are considered and corresponding boundary value problems. Then two variational formulations for the cavity problem with Dirichlet boundary conditions are analized: R. Picard's extended Maxwell-Heaviside system and the weak variational formulation with Hodge decomposition of the spatial fields. Finally boundary integral equations based on the Stratton-Chu representation formulas are formulated. With Rumsey's principle and corresponding Hodge decomposition of the boundary fields lead to a Garding inequality for the variational equation. 4/13/2017 3:00:00 PM4/13/2017 4:00:00 PMFalse
David Nicholls, University of Illinois ChicagoDavid Nicholls, University of Illinois ChicagoEWG 336Title: Numerical Solution of Diffraction Problems: A High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation Method Abstract: The rapid and robust simulation of linear waves interacting with layered periodic media is a crucial capability in many areas of scientific and engineering interest. High–Order Perturbation of Surfaces (HOPS) algorithms are interfacial methods which recursively estimate scattering quantities via perturbation in the interface shape heights/slopes. For a single incidence wavelength such methods are the most efficient available in the parameterized setting we consider here. In this talk we describe a generalization of one of these HOPS schemes by incorporating a further expansion in the wavelength about a base configuration which constitutes an "Asymptotic Waveform Evaluation" (AWE). We not only provide a detailed specification of the algorithm, but also verify the scheme and point out its benefits and shortcomings. With numerical experiments we show the remarkable efficiency, fidelity, and high–order accuracy one can achieve with an implementation of this algorithm. 4/6/2017 3:00:00 PM4/6/2017 4:00:00 PMFalse

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