Schedule for: 24w5313 - Homological Perspective on Splines and Finite Elements

Nechako residence

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Equivariant cohomology is an algebraic invariant associated to a group action on a space. GKM theory describes this graded ring in a very special set of commonly arising circumstances. In this talk, we will explain defining properties of equivariant cohomology for these spaces (with their group actions) and how to use GKM theory to calculate it. Along the way, we will explain how to associate labeled graphs, recently interpreted as splines, to these spaces.

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Abstract: Starting with C^1 spaces on macro triangulations (Alfeld, Worsey-Farin splits), we show that they are part of a finite element complex. We then discuss how some of the spaces can be used for problems in fluid flow, electro-magnetics and solid mechanics.

Spline functions are a well-known and widely employed tool, with applications to CAD, CAE, CAGD, numerical solution of PDEs, etc.. The theory about spline spaces has been and still is continuously growing, and led to countless types of of spline spaces [1, 2]: univariate and multivariate, polynomial and generalized, defined on triangulations and on (locally) tensor-product meshes, etc.. All these splines are traditionally functions s : Rm −→ Rn with varying regularities. In this presentation we explore the possibility to extend the concept of spline to operators (functions) s : X −→ R where X is an infinite-dimensional Hilbert space, motivated by the fact that such a tool could be applied both to approximation methods and to the solution of functional differential equations [3, 4]. We will present a couple of constructions for piecewise k-linear operators, as a proof of concept that splines can be defined and used in this more general setting.

Given a scalar-valued discontinuous piecewise polynomial, a “potential reconstruction” is a piecewise polynomial that is trace-continuous, i.e., H1-conforming. It is best obtained via a conforming finite element solution of local homogeneous Dirichlet problems on patches of elements sharing a vertex. Similarly, given a vector-valued discontinuous piecewise polynomial not satisfying the target divergence, a “flux reconstruction” is a piecewise polynomial that is normal-trace-continuous, i.e., H(div)-conforming, and has the target divergence. It is best obtained via local homogeneous Neumann problems on patches of elements, using the mixed finite element method. These concepts are known to lead to guaranteed, locally efficient, and polynomial-degree-robust a posteriori error estimates. We show that they also allow to devise stable local commuting projectors that lead to p-robust equivalence of global-best approximation over the whole computational domain using a conforming finite element space with local- (elementwise-)best approximations without any continuity requirement along the interfaces and without any constraint on the divergence. Therefrom, optimal hp approximation / a priori error estimates under minimal elementwise Sobolev regularity follow.

Recently, Eric Brechner and I have developed an open source Python package for building spline models of points, curves, surfaces, solids, and n-dimensional manifolds called BSpy. This package, based primarily on a single object class and a handful of methods, offers a powerful capability for building and manipulating geometric models in many dimensions. This talk will explain the history and philosophy that have gone into BSpy and will demonstrate some of the surprisingly complex operations that can be performed with coding idioms that are often only a few lines of code long.

Additional breakout rooms: Art 106, Art 108 and Art 114(big room).

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)

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This talk will explain the tool of folded alcove walks, which enjoy a wide range of applications throughout combinatorics, representation theory, number theory, and algebraic geometry. We will survey the construction of flag varieties through this lens, focusing on the problem of understanding intersections of different kinds of Schubert cells. We then highlight a key application in GKM theory.

The talk concerns the degrees of freedom that can be used to determine univocally the fields in high order Whitney finite element spaces. There are indeed two different families of such degrees of freedom, the weights and the moments. Weights and moments coincide in the lower order case, but are rather different as soon as we consider the high order case. I will mainly fo- cus on weights. Thank to their natural geometrical localization on the mesh, weights allow, for instance, to generalize, to the polynomial interpolation of differential k-forms, some fundamental concepts of the polynomial interpo- lation of regular scalar functions or to extend to the high order case graph techniques used in the low order case. I will also discuss about the relation- ship between weights and moments through a particular isomorphism that preserves the matrix of the gradient operator. This is a long term collaboration with Francesca Rapetti, from the University Coˆte d’Azur, France.

The Powell-Sabin 6-refinement has proven to be a convenient splitting technique for constructing smooth splines over a general triangulation. In this talk we use it to define a C^1 spline space of arbitrary degree with optimal polynomial precision and prescribed super-smoothness at split points inside triangles. Instead of traditional interpolation, we use blossoming to establish a set of functionals that characterize the spline space. The associated basis functions have some favorable properties, namely, they form a convex partition of unity and can be naturally represented in the Bernstein-Bezier form.

Additional breakout rooms: Art 106, Art 108 and Art 114(big room).

Please meet up at the parking lot.

Please use the meal card and get the breakfast items from Comma(or any food merchants)

The Wang-Shi split is a cross-cut partition of a triangle and consequently the dimension formula for the space of C^{p-1} splines of degree p does not have an homology term and it only involves the number of crossing lines containing each inner point. The talk will present a proof that the number of cut-lines containing any inner point is less than or equal to p+1 allowing to simplifying the formula to dim spline = dim polynomials + number of cut-lines.

This talk will introduce a novel construction of adaptive and yet structure preserving finite element discretizations with function spaces induced by subdivision. In many applications, for example in geo- physical fluid dynamics, adaptive and structure-preserving methods can be highly beneficial to simulate the long-term evolution of a multi-scale system with several invariants of motion like the total energy. If the discretizations of such systems do not preserve these invariants, the simulation results can differ significantly from the true physical behaviour of the systems. Combining the benefits of structure preservation and adaptive finite elements is notoriously difficult. If no special care is taken, adaptive mesh refinement algorithms of standard finite element approaches usually lose the property of structure preservation. Alternatively, the refinement can be chosen to be conforming, which in turn leads to unnecessary propagation of the refinement because surrounding cells need to be refined as well. On the other hand, IGA tensor-product techniques suffer from mesh topology restrictions. For this reason, we chose to build our function spaces upon subdivision. We extend the work of [1], who introduced vector field subdivision schemes that commute with the standard vector calculus operators like the gradient or the curl. Translating their work to the finite ele- ments realm yields de-Rham-complex-preserving finite elements for scalar functions, vector fields, and density functions. We added adaptivity to their structure-preserving discretization by leveraging the hi- erarchy of the basis functions induced by the subdivision algorithm. By carefully keeping track of the introduced degrees of freedom across the refinement levels, we maintain a discrete de Rham complex and thus enable structure-preserving simulations. Our method was verified by simulating the Maxwell eigenvalue problem, a well-known test case that reproduces the analytical eigenvalues if the chosen finite element spaces constitute a discrete de Rham complex. We show that our discretization indeed yields the correct spectrum and investigate the compu-tational effort and accuracy gains of our method.

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Through the seminal works of Buffa et al., the fruitful integration of the two discretization paradigms of Finite Element Exterior Calculus (FEEC) and Isogeometric Analysis (IGA) was demonstrated already in 2011. The latter evolved over the last nearly 20 years, stemming from the publications of Hughes et al., into a powerful concept for linking Finite Element Methods with Computer-aided design. In fact, the introduction of isogeometric discrete differential forms by Buffa et al. laid the foundation for discretizing de Rham complexes in a structure-preserving manner using B-splines. However, although the FEEC theory was derived in an abstract setting, and while Hilbert sequences play a role in various physical applications, connecting IGA and FEEC often proves challenging or is sometimes not directly clear. This is especially true for Hilbert complexes that also encompass differential operators of higher orders. We present two approaches to obtain well-posed discretizations of a whole class of Hodge–Laplace problems using IGA, while maintaining the inf-sup stability condition. We focus on mixed weak formulations of saddle-point structure and second-order Hilbert complexes. In particular, we go beyond the standard de Rham case and demonstrate that ideas from FEEC and IGA are useful for non-de Rham chains as well. A central tool for describing the underlying settings and for choosing the Finite Element spaces is the Bernstein–Gelfand–Gelfand (BGG) construction discussed by Arnold and Hu in 2021. Our approach allows us to incorporate geometries with curved boundaries, which is not directly possible with classical FEEC approaches, and also provides suitable discretizations in arbitrary dimensions. We show error estimates for both approximation methods and explain their applicability in the field of linear elasticity theory. The theoretical discussions and estimates are further illustrated with various numerical examples perfomed utilizing the GeoPDEs software package.

Kelowna https://www.tourismkelowna.com/blog/post/how-to-use-transit-to-explore-kelownas-hiking-trails/

Please use the meal card and get the breakfast items from Comma(or any food merchants)