BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexandre Ern (Université Paris-Est\, CERMICS \, ENPC) DTSTART:20210209T060000Z DTEND:20210209T070000Z DTSTAMP:20260404T111006Z UID:CMWebinar/1 DESCRIPTION:Title: Hybrid high-order methods for the wave equation on unfitted meshe s\nby Alexandre Ern (Université Paris-Est\, CERMICS \, ENPC) as part of Australian Seminar on Computational Mathematics\n\n\nAbstract\nWe desig n and analyze an unfitted hybrid high-order (HHO) method for the acoustic wave equation. The wave propagates in a domain where a curved interface se parates subdomains with different material properties. The key feature of the space discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order form ulation in time of the wave equation and its reformulation as a first-orde r system. For explicit time-stepping schemes\, we study the CFL condition and observe that the unfitted approach combined with local cell agglomerat ion leads to a comparable condition as when using fitted meshes.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Kris van der Zee (University of Nottingham) DTSTART:20210223T070000Z DTEND:20210223T080000Z DTSTAMP:20260404T111006Z UID:CMWebinar/2 DESCRIPTION:Title: Minimal Residual Finite Element Methods in Banach Spaces\nby Kris van der Zee (University of Nottingham) as part of Australian Seminar on Computational Mathematics\n\n\nAbstract\nMinimal-residual (MinRes) fini te element methods have attracted significant attention in the recent nume rical analysis literature\, owing to their conceptual simplicity and strik ing stability properties. While these methods include classical least-squa res and optimal Petrov-Galerkin methods\, recent advances centre around th e minimisation of residuals measured in a (discrete) dual norm\, such as t he discontinuous Petrov--Galerkin (DPG) methodology. \n\nIn this talk\, I will first discuss how MinRes methods can be extended to Banach-space sett ings. This general setting allows for a direct discretization of PDEs in n onstandard non-Hilbert settings that are required when facing rough data a nd low-regular solutions. This development gives rise to a class of nonlin ear Petrov--Galerkin methods\, or\, equivalently\, abstract mixed methods with monotone nonlinearity. Discrete stability and quasi-optimal convergen ce follow under a Fortin condition. I will consider applications to PDEs ( linear transport\, advection-diffusion)\, as well as the regularization of rough linear functionals. \n\nSecondly\, I will show how the MinRes frame work can be utilised for model reduction. In particular\, I will present a machine-learning framework to train a provably stable parametric Petrov-G alerkin method on a fixed underlying mesh\, whose aim is to ensure highly accurate quantities of interest regardless of the mesh size. Some recent n umerics will illustrate these ideas.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Johnny Guzman (Brown University) DTSTART:20210323T000000Z DTEND:20210323T010000Z DTSTAMP:20260404T111006Z UID:CMWebinar/3 DESCRIPTION:Title: A discrete elasticity complex on Alfeld splits\nby Johnny Guz man (Brown University) as part of Australian Seminar on Computational Math ematics\n\n\nAbstract\nWe discuss what appears to be the first finite elem ent elasticity sequence in 3d on tetrahedral meshes. We build this sequen ce using two discrete de Rham sequences that have more smoothness than the Whitney forms. To do this we use Alfeld splits that take a tetrahedral de composition and split each tetrahedron into four tetrahedra by adding the barycenter. We provide degrees of freedom for the spaces as well as commu ting projections. This is joint work with Snorre Christiansen\, Kaibo Hu a nd Jay Gopalakrishnan.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Antti Niemi (University of Oulu) DTSTART:20210504T070000Z DTEND:20210504T080000Z DTSTAMP:20260404T111006Z UID:CMWebinar/4 DESCRIPTION:Title: Computational mathematics for structural engineering with some ar ctic twist\nby Antti Niemi (University of Oulu) as part of Australian Seminar on Computational Mathematics\n\n\nAbstract\nTechnical requirements for structural design are based on economic\, environmental\, and social pillars. Assessment of the various kind of criteria requires mathematical models and computational tools suitable for conceptual design at the early design stage as well as for high-fidelity simulation at the component lev el. Performance of lightweight thin-walled structures is known to be very sensitive to uncertainties in geometry\, material properties\, support con ditions and external loading. Therefore\, reliable mathematical models and numerical methods are of utmost importance in their modeling. \n\nThis ta lk shall address some open problems related to the stability analysis of c urved shell structures and structural reliability of lightweight structure s in general. The ideas are based on variational principles of mechanics a nd applied probability theory.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Giancarlo Sangalli (University of Pavia) DTSTART:20210309T060000Z DTEND:20210309T070000Z DTSTAMP:20260404T111006Z UID:CMWebinar/5 DESCRIPTION:Title: Isogeometric Analysis: a high-order method for PDEs\nby Gianc arlo Sangalli (University of Pavia) as part of Australian Seminar on Compu tational Mathematics\n\n\nAbstract\nIsogeometric Analysis was proposed in the seminal work of Hughes\, Cottrell\, and Bazilevs in 2005\, and be seen as a generalisation of the finite element\nmethod that replaces classical $C^0$ finite elements with smooth\nsplines. Doing so\, IGA aims to be ea sily compatible with\ncomputer-aided geometric design systems\, where smo oth splines are used\nto create computational geometric models. In this fr amework\, \nthere has been a successful creation of novel\, robust\, high- order\naccurate numerical methods for solving PDEs.\n\nThe concept of k-re finement (or K-method) was proposed as one of\nthe key features of isogeo metric analysis\, "a new\, more efficient\,\nhigher-order concept"\, in th e original isogeometric article by Hughes and co-workers. The idea of usin g high-degree\nand continuity splines (or NURBS\, etc.) as a basis for a n ew\nhigh-order method appeared very promising from the beginning. The\nk-r efinement leads to several advantages: higher accuracy per\ndegree-of-free dom\, improved spectral accuracy\, the possibility of\nstructure-preservin g smooth discretizations are the most interesting\nfeatures that have been studied actively in the community. At the same\ntime\, the k-refinement b rings significant challenges at the\ncomputational level: using standard f inite element routines\, its\ncomputational cost grows with respect to the degree\, making degree\nraising computationally expensive. After a brief introduction of\nIsogeometric Analysis\, I will discuss ideas from\n[San galli and Tani\, CMAME\, 2018\, arXiv:1712.08565] and following works\, th at allow a computationally\nefficient k-refinement.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Laura De Lorenzis (ETH Zurich) DTSTART:20210608T070000Z DTEND:20210608T080000Z DTSTAMP:20260404T111006Z UID:CMWebinar/6 DESCRIPTION:Title: Unsupervised discovery of constitutive laws\nby Laura De Lore nzis (ETH Zurich) as part of Australian Seminar on Computational Mathemati cs\n\n\nAbstract\nThe speaker and her group recently proposed a new approa ch for data-driven automated discovery of constitutive laws. The approach is unsupervised\, i.e.\, it requires no stress data but only displacement and global force data\, which are realistically available through mechanic al testing and digital image correlation techniques\; it delivers interpre table models\, i.e.\, models that are embodied by parsimonious mathematica l expressions discovered through sparse regression of a large catalogue of candidate functions\; it is one-shot\, i.e.\, discovery only needs one ex periment — but can use more if available. The problem of unsupervised di scovery is solved by enforcing equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity of the solution is achieved by Lp regularization combined with thresholding\, which calls for a non-line ar optimization scheme. The ensuing fully automated algorithm leverages ph ysics-based constraints for the automatic determination of the penalty par ameter in the regularization term. We focus on isotropic hyperelasticity a nd\, using numerically generated data including artificial noise\, we demo nstrate the ability of the approach to accurately discover five hyperelast ic models of different complexity. We also show that\, if a “true” fea ture is missing in the function library\, the proposed approach is able to surrogate it in such a way that the actual response is still accurately p redicted. We finally outline the first steps in the direction of extending the approach to more complex types of constitutive laws.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Carla Manni (University of Rome Tor Vergata) DTSTART:20210420T070000Z DTEND:20210420T080000Z DTSTAMP:20260404T111006Z UID:CMWebinar/7 DESCRIPTION:Title: Spectral analysis of matrices from isogeometric methods\nby C arla Manni (University of Rome Tor Vergata) as part of Australian Seminar on Computational Mathematics\n\n\nAbstract\nWhen discretizing a linear PDE by a linear numerical method\, the computation of the numerical solution reduces\nto solving a linear system. The size of this system grows when we refine the discretization mesh.\nWe are then in the presence of a sequenc e of linear systems with increasing size.\nIt is usually observed in pract ice that the corresponding sequence of discretization matrices enjoys\n an asymptotic spectral distribution. Roughly speaking this means that there exists a function\, say f\,\nsuch that the eigenvalues of the considered s equence of matrices behave like a sampling of f over an\nequispaced grid o n the domain of f\, up to some outliers.\n\nIsogeometric analysis is a wel l-established paradigm for the analysis of problems governed by PDEs.\nIt provides a design-through-analysis connection by exploiting a common repre sentation model. This connection is achieved\nby using the functions adop ted in CAD systems not only to describe the domain geometry\, but also to represent the numerical solution of\nthe differential problem.\nIn its ori ginal formulation IgA is based on (tensor-product) B-splines and their rat ional extension\, the so-called NURBS [2].\n\nIn this talk we review the m ain spectral properties of discretization matrices arising from isogeometr ic methods\, based on \nd-variate NURBS of given degrees and applied to ge neral second-order\nelliptic differential problems defined on a d-dimensio nal domain [4\,5]\, \ndiscussing the differences and the similarities with the FEM case [6]. \nWe also discuss the relation between outliers and con vergence to eigenfunctions of classical differential operators under k-ref inement. \n\nThe provided spectral information can be exploited for desi gning iterative solvers [3] \nwith convergence speed independent of the fi neness parameters and also substantially independent of the degrees of the used NURBS\, [1].\n\nThe talk is based on joint works with C. Garoni\, F. Pelosi\, E. Sande\, H. Speleers\, S. Serra-Capizzano.\n\nReferences\n\n[1 ] N. Collier\, L. Dalcin\, D. Pardo\, V.M. Calo\nThe cost of continuity: P erformance of iterative solvers on isogeometric finite elements\, \nSIAM J ournal on Scientific Computing\, 35 A767-A784\, 2013\n \n[2] J.A. Cottrell \, T.J.R. Hughes\, Y. Bazilevs\,\nIsogeometric Analysis: Toward Integratio n of CAD and FEA\,\nJohn Wiley & Sons\, 2009.\n\n[3] M. Donatelli\, C. Ga roni\, C. Manni\, S. Serra-Capizzano\, H. Speleers\, \nSymbol-based multig rid methods for Galerkin B-spline isogeometric analysis\, SIAM Journal on Numerical Analysis\, 55\, 31-62\, 2017.\n\n[4] C. Garoni\, C. Manni\, F. P elosi\, S. Serra-Capizzano\, H. Speleers\, \nOn the spectrum of stiffness matrices arising from isogeometric analysis\, Numerische Mathematik\, 127\ , 751-799\, 2014.\n\n[5] C. Garoni\, C. Manni\, S. Serra-Capizzano\, H. Sp eleers\, NURBS in isogeometric discretization methods: A spectral analysi s\, \nNumerical Linear Algebra with Application}\, 2020\;27:e2318.\n\n[6] C. Garoni\, H. Speleers\, S-E. Ekstrom\, A. Reali\, S. Serra-Capizzano\, T.J.R. Hughes\, \nSymbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: \nExposition and review\ , Archives of Computational Methods in Engineering\, 26\, 1639-1690\, 201 9.\n\n[7] E. Sande\, C. Manni\, H. Speleers:\nSharp error estimates for spline approximation: explicit constants\, n-widths\, and eigenfunction co nvergence\,\nMathematical Models and Methods in Applied Sciences}\, 29\, 1175--1205\, 2019\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Colin Cotter (Imperial College) DTSTART:20210720T070000Z DTEND:20210720T080000Z DTSTAMP:20260404T111006Z UID:CMWebinar/8 DESCRIPTION:Title: Hybridised implicit solvers for the Gung Ho dynamical core\nb y Colin Cotter (Imperial College) as part of Australian Seminar on Computa tional Mathematics\n\n\nAbstract\nGung Ho is the name of the Met Office pr oject to build a new dynamical core (fluid dynamics component) for their w eather/climate prediction system. Gung Ho is built around compatible finit e element methods as the apparently unique solution to the question of how to find a consistent gridpoint (i.e. non-spectral) discretisation that su pports various essential wave propagation properties at the discrete level on grids with near-equal edge lengths "pseudo-uniform" on the sphere. One downside of this approach versus their current finite difference approach is the non-diagonal mass matrix for the velocity component\, which means that the usual strategy of eliminating velocity to get an elliptic problem for pressure results in a non-sparse matrix. The solution to this\, known for decades\, is to "hybridise" the mixed system by breaking continuity c onstraints to get a discontinuous velocity space\, and to introduce Lagran ge multipliers as trace variables supported on cell facets to enforce cont inuity of the solution. The system can then be eliminated down to a sparse reduced system for the trace variables only. The question then arises of how to efficiently iteratively solve this system when the domain is very t hin (like the Earth's atmosphere). This question can be answered by combin ing various results from (a) the analysis of hybridised mixed finite eleme nt methods and (b) the analysis of additive Schwarz methods. I will briefl y introduce these\, describe a solver algorithm and sketch a proof that it gives iteration counts that are independent of depth in the thin layer li mit\, before illustrating with some numerical results produced using Fired rake and PETSc.\n LOCATION:https://stable.researchseminars.org/talk/CMWebinar/8/ END:VEVENT END:VCALENDAR