Philip Avery

FETI-DPH: A DUAL-PRIMAL DOMAIN DECOMPOSITION METHOD FOR ACOUSTIC SCATTERING

A dual-primal variant of the FETI-H domain decomposition method is designed for the fast, parallel, iterative solution of large-scale systems of complex equations arising from the discretization of acoustic scattering problems formulated in bounded computational domains. The convergence of this iterative solution method, named here FETI-DPH, is shown to scale with the problem size, the number of subdomains, and the wave number. Its solution time is also shown to scale with the problem size. CPU performance results obtained for the acoustic signature analysis in the mid-frequency regime of mockup submarines reveal that the proposed FETI-DPH solver is significantly faster than the previous generation FETI-H solution algorithm.

Scalable FETI Algorithms for Frictionless Contact Problems

1 Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, and Center of Intelligent Systems and Structures, CISS Institute of Thermomechanics AVCR, 17. listopadu 15, Ostrava-Poruba, 708 33, Czech Republic. {zdenek.dostal,vit.vondrak,david.horak}@vsb.cz 2 Stanford University, Department of Mechanical Engineering and Institute for Computational and Mathematical Engineering, Stanford, CA 94305, USA. {cfarhat,pavery}@stanford.edu

An enhanced FIVER method for multi-material flow problems with second-order convergence rate

Abstract The finite volume (FV) method with exact two-material Riemann problems (FIVER) is an Eulerian computational method for the solution of multi-material flow problems. It is robust in the presence of large density jumps at the fluid–fluid interfaces, and the presence of large structural motions, deformations, and even topological changes at the fluid–structure interfaces. To achieve simplicity in implementation, it approximates each material interface by a surrogate surface which conforms to the control volume boundaries. Unfortunately, this approximation introduces a first-order error of the geometric type in the solution process. In this paper, it is first shown that this error causes the original version of FIVER to be inconsistent in the neighborhood of material interfaces and degrades its global order of spatial accuracy. Then, an enhanced version of FIVER is presented to rectify this issue, restore consistency, and achieve for smooth problems the desired global convergence rate. To this effect, the original definition of a surrogate material interface is retained because of its attractive simplicity. However, the solution at this interface of a two-material Riemann problem is enhanced with a simple reconstruction procedure based on interpolation and extrapolation. Next, the extrapolation component of this procedure is equipped with a limiter in order to achieve nonlinear stability for non-smooth problems. In the one-dimensional inviscid setting, the resulting FIVER method is also shown to be total variation bounded. Focusing on the context of a second-order FV semi-discretization, the nonlinear stability and second-order global convergence rate of this enhanced FIVER method are illustrated for several model multi-fluid and fluid–structure interaction problems. The potential of this computational method for complex multi-material flow problems is also demonstrated with the simulation of the collapse of an air bubble submerged in water and the comparison of the computed results with corresponding experimental data.

An Adaptive Mesh Refinement Concept for Viscous Fluid-Structure Computations Using Eulerian Vertex-Based Finite Volume Methods

Embedded Boundary Methods (EBMs) [1] for the solution of Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) problems are typically formulated in the Eulerian setting, which makes them more attractive than Chimera and Arbitrary Lagrangian-Eulerian methods when the structure undergoes large structural motions and/or deformations. In the presence of viscous flows however, they necessitate Adaptive Mesh Refinement (AMR) because unlike Chimera and ALE methods, they do not track boundary layers [2]. In general, AMR gives rise to non-conforming mesh configurations that can complicate the semi-discretization process. This is the case when this process is carried out using the popular vertex-based finite volume method and dual cells. Perhaps for this reason, most of the literature on AMR in the context of EBMs and the FV method has focused so far on cell-centered schemes, where the treatment of non-conforming mesh configurations is straightforward [1]. Specifically, most if not all local refinement strategies developed in this context generate “hanging” nodes in the refined mesh that can be easily dealt with using cell-centered but not vertex-based methods. In the latter case, flux assembly after a mesh refinement step becomes a problematic issue, due to presence of dual cells. This talk proposes a simple approach for resolving this issue by appropriately managing the construction of dual cells past each refinement step. The talk will recall the motivations for EBMs and vertex-based FV methods, explain the aforementioned AMR issue that arises in their context, present a method for resolving it, and illustrate this method with the application of the EBM known as FIVER (Finite Volume method with Exact two-phase Riemann solvers) [3, 4] to various examples including the prediction of the aerodynamic performance of a Formula 1 car.

An Embedded Boundary Method for Viscous Fluid/Structure Interaction Problems and Application to Flexible Flapping Wings

FIVER is a robust Eulerian finite volume method for the solution of compressible multi-fluid and multi-fluid-structure interaction problems characterized by large density jumps and highly nonlinear structural deformations. Its key components include an embedded boundary method for CFD equipped with suitable surrogate discrete material interfaces, the construction and solution of local, exact, two-phase Riemann problems for semi-discretizing the convective fluxes at the fluid-fluid interface, the construction and solution of local, exact, half Riemann problems for enforcing the normal component of the kinematic transmission condition at the fluid-structure interface, and a conservative algorithm for loads distribution on a finite element representation of the structure. Originally developed for inviscid fluid and fluid-structure interaction problems, FIVER is extended in this paper to viscous multi-material problems. To this effect, its embedded boundary method component is equipped with a ghost fluid scheme for approximating the viscous and source terms of the governing semi-discrete equations of dynamic equilibrium, and its loads distribution algorithm is extended to account for the contribution of the fluid viscous stress tensor. The extended FIVER method accommodates both explicit and implicit time-integration schemes. Its performance for highly nonlinear fluid-structure interaction problems is highlighted with the simulation of thrust generation of flapping flexible wings at low Reynolds numbers.

A Stochastic Projection-Based Hyperreduced Order Model for Model-Form Uncertainties in Vibration Analysis

A feasible, nonparametric, probabilistic approach for quantifying model-form uncertainties associated with a High-Dimensional computational Model (HDM) and/or a corresponding Hyperreduced Projection-based Reduced-Order Model (HPROM) designed for the solution of generalized eigenvalue problems arising in vibration analysis, is presented. It is based on the construction of a Stochastic HPROM (SHPROM) associated with the HDM and its HPROM using three innovative ideas: the substitution of the deterministic Reduced-Order Basis (ROB) with a Stochastic counterpart (SROB) that features a reduced number of hyperparameters; the construction of this SROB on a compact Stiefel manifold in order to guarantee the linear independence of its column vectors and the satisfaction of any applicable constraints; and the formulation and solution of a reduced-order inverse statistical problem to determine the hyperparameters so that the mean value and statistical fluctuations of the eigenvalues predicted in real time using the SHPROM match target values obtained from available data. If the data are experimental data, the proposed approach models and quantifies the model-form uncertainties associated with the HDM, while accounting for the modeling errors introduced by model reduction. If on the other hand the data are high-dimensional numerical data, the proposed approach models and quantifies the model-form uncertainties associated with the HPROM. Consequently, the proposed nonparametric, probabilistic approach for modeling and quantifying model-form uncertainties can also be interpreted as an effective means for extracting fundamental information or knowledge from data that is not captured by a deter-ministic computational model, and incorporating it in this model. Its potential for quantifying model-form uncertainties in eigenvalue computations is demonstrated for what-if? vibration analysis scenarios associated with shape changes for a jet engine nozzle.