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In this work, a class of methods which combines divergence-conforming discretizations with residual-based subgrid modeling for large eddy simulation of turbulent flows is introduced. These methods fall within two frameworks: residual-based variational multiscale methods and residual-based eddy viscosities. These methods utilize variationally-consistent formulations for the fine-scale velocities in order to construct subgrid-scale models based on the coarse-scale residual. The result is an LES methodology that responds naturally to spatially- and temporally-varying turbulence. Numerical results demonstrate that these new methods demonstrate proper behavior for homogeneous turbulence and outperform classical LES models for transitional flows and wall-bounded turbulent flows. Furthermore, the resulting formulations contain no “tunable” parameters, and thus extend generally across various classes of flow.

References: 

J.A Evans, C. Coley, R.M. Aronson, C.L. Wetterer-Nelson, and Y. Bazilevs, "Residual-based large eddy simulation with isogeometric divergence-conforming discretizations," in T. Tezduyar T. (Editor), "Frontiers in Computational Fluid-Structure Interaction and Flow Simulation: Research from Lead Investigators under 40," Birkhäuser/Springer, Cham, Chapter 3, 91-130, 2018.

C. Coley. "Residual-Based Large Eddy Simulation of Turbulent Flows using Divergence-Conforming Discretizations." Ph.D. Dissertation, University of Colorado Boulder, 2017.

T. M. van Opstal, J. Yan, C. Coley, J. A. Evans, T. Kvamsdal, and Y. Bazilevs. "Isogeometric divergence-conforming variational multiscale formulation of incompressible turbulent flows." Computer Methods in Applied Mechanics and Engineering, 316:859-879, 2017.

In this work, we develop a framework for creating higher-order, mixed-element meshes that are both geometrically exact and analysis suitable. The framework generates meshes composed of Bernstein-Bezier tetrahera, hexahedra, wedges, and pyramids. Generally speaking, the framework requires only two things: 1) a suitable underlying linear mesh and 2) a suitable CAD description of the surface. As a result, the framework is easily employed alongside existing mesh generation technologies to create geometrically exact meshes of complicated engineering geometries.

References: 

L. Engvall and J.A. Evans, "Mesh quality metrics for isogeometric Bernstein-Bezier discretizations." Submitted for publication.

L. Engvall. "Geometrically Exact and Analysis Suitable Mesh Generation Using Rational Bernstein–Bezier Elements." Ph.D. Dissertation, University of Colorado Boulder, 2018.

L. Engvall and J.A. Evans, "Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bezier discretizations." Computer Methods in Applied Mechanics and Engineering, 319:83-123, 2017.

L. Engvall and J.A. Evans, "Isogeometric triangular Bernstein-Bezier discretizations: Automatic mesh generation and geometrically exact finite element analysis." Computer Methods in Applied Mechanics and Engineering, 304:378-407, 2016.

In this work, we develop a design space exploration framework which elucidates design parameter sensitivities used to inform initial and early-stage design of structural parts and assemblies. Moreover, the framework enables the visualization of a full system response, including the displacement and stress fields throughout the domain, by providing an approximation to the system solution vector. This is accomplished through a collocation-like approach where various geometries throughout the design space under consideration are sampled. The sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method. A surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion. While the methodology is developed specifically for structural mechanics, it is applicable to any system of parametric partial differential equations.  

References: 

J. Benzaken. "Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis." Ph.D. Dissertation, University of Colorado Boulder, 2018.

J. Benzaken, A. J. Herrema, M.-C. Hsu, J. A. Evans, “A Rapid and Efficient Isogeomtric Design Space Exploration Framework with Application to Structural Mechanics.” Computer Methods in Applied Mechanics and Engineering, 316:1215-1256, 2017.

Nearly 300,000 diseased heart valves are surgically replaced annually. By far the most popular surgical replacements are bioprosthetic heart valves (BHVs), which are fabricated from biologically derived materials with the design goal of mechanical similarity to native valves. BHVs have more natural hemodynamics than older "mechanical" designs, but the durability of a typical BHV remains limited to about 10-15 years. While much effort has gone into constructing BHVs, methods to extend durability remain largely explored. The design of such methods hinges upon a proper understanding of the stresses acting on valve leaflets over the complete cardiac cycle. In this work, led by David Kamensky of the University of Texas at Austin, we introduce a new geometrically flexible technique for computational fluid-structure interaction (FSI) which is ideally suited for the simulation of BHVs. Our new method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term "immersogeometric analysis" to identify this paradigm. Unlike classical variational approaches to FSI such as the augmented Lagrangian-Eulerian (ALE) method, our new approach is able to stably and accurately simulate FSI problems in which the structure experiences large deformation and self-contact. Indeed, the ability to capture such features is critical in the simulation of BHVs. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

References: 

D. Kamensky, J.A. Evans, M.-C. Hsu, and Y. Bazilevs, “Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid-thin structure interation analysis, with application to heart valve modeling.” Computers and Mathematics with Applications, 74:2068-2088, 2017.

D. Kamensky, M.-C. Hsu, Y. Yu, J.A. Evans, M.S. Sacks, and T.J.R. Hughes, “Immersogeometric cardiovascular fluid-structure interaction analysis with divergence-conforming B-splines.” Computer Methods in Applied Mechanics and Engineering, 314:408-472, 2017.

D. Kamensky, J.A. Evans, and M.-C. Hsu,  "Stability and conservation properties of collocated constraints in immersogeometric fluid-thin structure interaction analysis." Communications in Computational Physics, 18:1147-1180, 2015.

D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, and T.J.R. Hughes, "An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves." Computer Methods in Applied Mechanics and Engineering, 284:1005-1053, 2015.