| Date: | January 22, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Douglas Arnold, University of Minnesota |
| Title: | Complexes from complexes |
| Abstract: | The finite element exterior calculus has highlighted the importance of Hilbert complexes to partial differential equations and their numerical solution. The most canonical and most extensively studied example is the de Rham complex, which is what is required for application to Darcy flow, Maxwell's equations, the Hodge Laplacian, and numerous problems. But there are many other important differential complexes as well, with applications to elasticity, plates, incompressible flow, general relativity, and other areas. These complexes are less well known and in many cases their properties not established. In this talk I will discuss a systematic procedure for deriving such complexes and deriving their crucial properties. |
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| Date: | January 23, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Kseniya Ivanova, Universität Zürich |
| Title: | Multi-dimensional shear shallow water flows |
| Abstract: | The mathematical model of shear shallow water flows of constant density is studied. This is a 2D hyperbolic non-conservative system of equations that is mathematically equivalent to the Reynolds averaged model of barotropic turbulent flows. The model has three families of characteristics corresponding to the propagation of surface waves, shear waves and average flow (contact characteristics). The system is non-conservative: for six unknowns (the fluid depth, two components of the depth averaged horizontal velocity, and three independent components of the symmetric Reynolds stress tensor) one has only five conservation laws (conservation of mass, momentum, energy and mathematical entropy). A splitting procedure for solving such a system is proposed allowing us to define a weak solution. Each split subsystem contains only one family of waves (either surface or shear waves) and contact characteristics. The accuracy of such an approach is tested on 2D analytical solutions describing the flow with linear with respect to the space variables velocity, and on the solutions describing 1D roll waves. The capacity of the model to describe the full transition scenario as commonly seen in the formation of roll waves: from uniform flow to 1D roll waves, and, finally, to 2D transverse fingering of the wave proles, is shown. Finally, we model a circular hydraulic jump formed in a convergent radial flow of water. Obtained numerical results are qualitatively similar to those observed experimentally: oscillation of the hydraulic jump and its rotation with formation of a singular point. |
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| Date: | February 19, 2020 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Maciej Paszynski, AGH University of Science and Technology, Krakow, Poland |
| Title: | Supermodeling of a tumor dynamics employing isogeometric analysis solvers with piece-wise constant test functions |
| Abstract: | In this talk, we show that it is possible to obtain reliable numerical prognoses about cancer dynamics by creating the supermodel of cancer, which consists of several coupled instances (the sub-models) of a generic cancer model, developed with isogeometric analysis (IGA). Its integration with real data can be achieved by employing a prediction/correction learning scheme focused on fitting several values of coupling coefficients between submodels, instead of matching scores (even hundreds) of tumor model parameters as it is in the classical data adaptation techniques. We also show how to speed up the tumor simulations by employing the piece-wise constant test functions in IGA framework. Namely, we show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is Petrov-Galerkin's kind. This observation has the following consequences. The numerical integration cost can be reduced because we do not need to evaluate the test functions since they are equal to one. The resulting method is equivalent to a linear combination of the collocations at points and with weights resulting from applied quadrature over the spans defined by supports of the piece-wise constant test functions. |
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