Conventional computational methods often create a dilemma for fluid-structure interaction problems. Typically, solids are simulated using a Lagrangian approach with grid that moves with the material, whereas fluids are simulated using an Eulerian approach with a fixed spatial grid, requiring some type of interfacial coupling between the two different perspectives. Here, a fully Eulerian method for simulating structures immersed in a fluid will be presented. By introducing a reference map variable to model finite-deformation constitutive relations in the structures on the same grid as the fluid, the interfacial coupling problem is highly simplified. The method is particularly well suited for simulating soft, highly-deformable materials and many-body contact problems, and several examples will be presented.

This is joint work with Ken Kamrin (MIT).

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Composite materials make up over 50% of recent aircraft constructions. They are manufactured from very thin fibrous layers  (~10^-4 m) and even  thinner resin interfaces (~10^-5 m). To achieve the required strength, a particular layup sequence of orientations of the anisotropic fibrous layers is used. During manufacturing, small localised defects in the form of misaligned fibrous layers can occur in composite materials, adding an additional level of complexity. After FE discretisation the model exhibits multiple scales and large spatial variations in model parameters. Thus the resultant linear system of equations can be very ill-conditioned and extremely large. The limitations of commercially available modelling tools for solving these problems has led us to the implementation of a robust and scalable preconditioner called GenEO for parallel Krylov solvers. I will discuss using the GenEO coarse space as an effective multiscale model for the fine-scale displacement and stress fields. For the coarse space construction, GenEO computes generalised eigenvectors of the local stiffness matrices on the overlapping subdomains and builds an approximate coarse space by combining the smallest energy eigenvectors on each subdomain via a partition of unity.

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This talk concerns applications of differential geometry in numerical optimization. They arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of certain nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in practical applications include the rotation group SO(3) (generation of rigid body motions from sample points), the set of fixed-rank matrices (low-rank models, e.g., in collaborative filtering), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion tensors), and the shape manifold (morphing).

In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Specialized languages for describing convex optimization problems, and associated parsers that automatically transform them to canonical form, have greatly increased the use of convex optimization in applications. These systems allow users to rapidly prototype applications based on solving convex optimization problems, as well as generate code suitable for embedded applications. In this talk I will describe the general methods used in such systems.

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TBA

There are many constructions that yield C*-algebras. For example, we build them from groups, quantum groups, dynamical systems, and graphs. In this talk we look at C*-algebras that arise from a certain type of game. It turns out that the properties of the underlying game gives us very strong information about existence of traces of various types on the game algebra. The recent solution of the Connes Embedding Problem arises from a game whose algebra has a trace but no hyperlinear trace.

Assumed knowledge: Familiarity with tensor products of Hilbert spaces, the algebra of a discrete group, and free products of groups.

In this talk, we will review the notion of non-invertible symmetries and we will study adjoint QCD in two dimensions. It turns out that this theory has a plethora of such symmetries which require deconfinement in the massless case. When a mass or certain quartic interactions are tunrned on, these symmetries are broken and the theory confines. In addition, we will use these symmetries to calculate the string tension for small mass and make some comments about naturalness along the RG flow.

In 2010, Hrushovski--Loeser studied the homotopy type of the Berkovich analytification of a quasi-projective variety over a valued field. In this talk, we explore the extent to which some of their results might hold in a relative setting. More precisely, given a morphism of quasi-projective varieties over a valued field, we ask if we might construct deformation retractions of the analytifications of the source and target which are compatible with the analytification of the morphism and whose images are finite simplicial complexes.