32 research outputs found
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Mini-Workshop: Mathematical Foundations of Isogeometric Analysis
Isogeometric Analysis (IgA) is a new paradigm which is designed to merge two so far disjoint disciplines, namely, numerical simulations for partial differential equations (PDEs) and applied geometry. Initiated by the pioneering 2005 paper of one of us organizers (Hughes), this new concept bridges the gap between classical finite element methods and computer aided design concepts.
Traditional approaches are based on modeling complex geometries by computer aided design tools which then need to be converted to a computational mesh to allow for simulations of PDEs. This process has for decades presented a severe bottleneck in performing efficient simulations. For example, for complex fluid dynamics applications, the modeling of the surface and the mesh generation may take several weeks while the PDE simulations require only a few hours.
On the other hand, simulation methods which exactly represent geometric shapes in terms of the basis functions employed for the numerical simulations bridge the gap and allow from the beginning to eliminate geometry errors. This is accomplished by leaving traditional finite element approaches behind and employing instead more general basis functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) for the PDE simulations as well. The combined concept of Isogeometric Analysis (IgA) allows for improved convergence and smoothness properties of the PDE solutions and dramatically faster overall simulations.
In the last few years, this new paradigm has revolutionized the engineering communities and triggered an enormous amount of simulations and publications mainly in this field. However, there are several profound theoretical issues which have not been well understood and which are currently investigated by researchers in Numerical Analysis, Approximation Theory and Applied Geometry
Space-time least-squares isogeometric method and efficient solver for parabolic problems
In this paper, we propose a space-time least-squares isogeometric method to
solve parabolic evolution problems, well suited for high-degree smooth splines
in the space-time domain. We focus on the linear solver and its computational
efficiency: thanks to the proposed formulation and to the tensor-product
construction of space-time splines, we can design a preconditioner whose
application requires the solution of a Sylvester-like equation, which is
performed efficiently by the fast diagonalization method. The preconditioner is
robust w.r.t. spline degree and mesh size. The computational time required for
its application, for a serial execution, is almost proportional to the number
of degrees-of-freedom and independent of the polynomial degree. The proposed
approach is also well-suited for parallelization.Comment: 29 pages, 8 figure