Lecture 1

Towards Shape Optimization for Ventricular Assist Devices Using Parallel Stabilized FEM

Modeling and computational analysis play an increasingly-important role in bioengineering, particularly in the design of ventricular assist devices. Numerical simulation of flow in blood pumps has the potential to shorten the design cycle and give the designers important insights into causes of blood damage and suboptimal performance. A set of modeling techniques will be presented which are based on stabilized space-time finite element formulation of the Navier-Stokes equations, with a shear-slip mesh update used to accommodate the movement of the impeller with respect to a non-axisymmetric housing. The computed global flow characteristics (performance curves) are validated against experimentally-measured data.

This application presents a ripe target for shape optimization and optimal control. In order to assess the influence of the fluid constitutive model on the outcome of shape optimization tasks, a comparison of model problem computations based on the Navier-Stokes equations on one hand, and on a more accurate shear-thinning model on the other, will be presented.

In order to obtain quantitative hemolysis prediction, as required for any realistic shape optimization, cumulative tensor-based measures of strain experienced by individual blood cells are being developed and correlated with available blood damage data. In the first approximation, red blood cells under shear are modeled as deforming droplets, and their deformation is tracked along pathlines of the computed flow field.

More complex description of blood behavior must take into account viscoelastic phenomena, as represented for example by an Oldroyd-B constitutive model. Recent developments in stabilized methods of GLS-type for simulation of Oldroyd-B flows using low-order extra-stress interpolations will be outlined.

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Lecture 2

Computational Mechanics 2.0: Broadly-Defined Inverse Problems and Simplex Space-Time Discretizations

This talk focuses on broader trends and developments in computational mechanics. The first subject is the emergence of broadly-defined inverse problems---parameter identification, optimal control, shape optimization---as the driver for application of numerical methods to engineering problems. It is in this context that computational analysis takes a truly novel form which cannot be duplicated by experimental techniques.

In the second part of the talk, a straightforward method for generating simplex space-time meshes is presented, allowing arbitrary temporal refinement in selected portions of space-time slabs. The method increases the flexibility of space-time discretizations, even in the absence of dedicated space-time mesh generation tools. The resulting tetrahedral (for 2D problems) and pentatope (for 3D problems) meshes are tested in the context of advection-diffusion equation, and are shown to provide reasonable solutions, while enabling varying time refinement in portions of the domain.

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Lecture 3

Finite Element Simulation Techniques for Marine Applications

Moving-boundary flow simulations are an important design and analysis tool in many areas of engineering, including civil engineering, marine and coating industries, and off-shore exploration. Two alternative computational approaches---interface-tracking and interface-capturing---are commonly considered. While interface-capturing offers unmatched flexibility for complex free-surface motion, the interface-tracking approach is very attractive due to its better mass conservation properties at low resolution. This provides motivation for expanding the reach of the interface-tracking methods.

The fundamentals of interface-tracking moving-boundary flow simulations---stabilized discretizations of Navier-Stokes equations, ALE and space-time formulations on moving grids, general mesh update mechanisms based on solid mechanics---are widely known and adopted. Challenges still exist, and we discuss some of the issues that are limiting the success of interface-tracking approach.

The generalized form of the kinematic condition, in the form of an elevation equation, as well as its stabilized GLS formulation, has been derived for cases where surface nodes move along prescribed straight lines. This method was then used to simulate water motion in trapezoidal tanks and channels. A further generalization is proposed for cases where surface nodes move along prescribed curvilinear spines. This allows for robust representation of the kinematic condition in an even wider set geometries, such as cylindrical vessels and channels.

The wetting of solid walls is often represented on the macro scale with a slip boundary condition, at least in the immediate vicinity of the solid-liquid-air interface. A consistent application of a full slip condition at curved boundaries is non-trivial, and the standard approach is seen to induce spurious gravity-driven circulation in some cases. We propose an implementation of the slip condition that uses the concept of BC-free boundary condition. The usefulness of our overall approach is demonstrated in 2D and 3D situations.