Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 07:07:39pm CET

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Session Overview
Z6: RBF, meshless, unfitted
Thursday, 15/July/2021:
12:00pm - 2:00pm

Session Chair: Jens Markus Melenk
Virtual location: Zoom 7

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12:00pm - 12:20pm

Oversampled and collocation RBF-FD methods for solving nonlinear conservation laws

Igor Tominec, Murtazo Nazarov

Uppsala University, Department of Information Technology, Sweden

We present our latest results on solving nonlinear conservation laws using radial basis function generated finite difference (RBF-FD) method. The analytic solutions to nonlinear conservation laws are known to develop shocks (discontinuities). Shocks lead to an occurrence of non-physical oscillations that pollute the numerical solutions (Gibbs phenomenon), and can cause spurious growth of the numerical solution in time. To stabilize the shocks when using the RBF-FD method, we apply a residual based viscosity (RV) filter borrowed from the finite element community. Here the residual of the conservation law gives an approximate location of a shock, and by that enables a selective smearing of the solution. We focus on applying RV to two settings of the RBF-FD method: (i) a collocation setting, (ii) an oversampled setting. A set of numerical experiments in two dimensions show that the solution in both settings converges with high-order when RV is active and the solution is smooth. The experiments also show that RV is an effective stabilization of the RBF-FD method when solving Burger’s equation and the Kurganov-Petrova-Popov rotating wave problem.

12:20pm - 12:40pm

The least-squares RBF-FD method

Igor Tominec1, Elisabeth Larsson1, Alfa Heryudono2

1Uppsala University, Sweden; 2University of Massachusetts Dartmouth, MA, USA

The radial basis function-generated finite difference (RBF-FD) method is based on stencil approximations over scattered nodes. This provides the simplicity of finite difference methods, combined with flexibility with respect to the geometry. RBF-FD methods work well in practice, but it has been hard to provide formal convergence proofs. There is also some sensitivity to node layouts, especially in the vicinity of boundaries with conditions including derivatives. Here, we introduce a least-squares formulation of the method where each stencil is used for several evaluation points such that the overall linear system of equations to solve becomes overdetermined. This improves the stability of the approximations and the performance of the method. Furthermore, it allows us to prove convergence by considering a nearby continuous least-squares problem. We show that high-order convergence results can be achieved for Poisson test problems with mixed boundary conditions in two and three dimensions.

12:40pm - 1:00pm

A high-order unfitted RBF-FD method for solving stationary PDEs

Igor Tominec1, Eva Breznik1, Elisabeth Larsson1, Alfa Heryudono2

1Uppsala University, Department of Information Technology, Sweden; 2University of Massachusetts Dartmouth, Department of Mathematics, USA

Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to a computational domain. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of the computational domain. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of the computational domain. In this talk we present a modification to an RBF-FD method which allows the interpolation points to be placed in a box that encapsulates the computational domain. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving PDEs with mixed boundary conditions show that our method is robust and highly accurate. Compared with the conventional approaches, the approximation error tends to be smaller when stencil sizes are large. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.

1:00pm - 1:20pm

A high order meshless finite difference method for elliptic interface problems

Oleg Davydov

University of Giessen, Germany

Elliptic interface problems are solved by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the mesh-based finite difference or finite element methods with the same order. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to 6 for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately. Results are obtained jointly with Mansour Safarpoor. Full article is published by Applied Numerical Mathematics, online version:

1:20pm - 1:40pm

Exploring the mortar and point-to-point interpolation methods for handling non-conformal interfaces in high-order discontinuous Galerkin methods

Edward Laughton

University of Exeter, United Kingdom

The capability to incorporate moving geometric features within models for complex simulations is a common requirement in many fields, for example rotational and sliding components in the modelling of turbomachinery. The inclusion of this movement results in the introduction of non-conformal interfaces separating the physical mesh along the regions of movement by detaching nodal connections. In this talk we will consider the numerical effects of two methods used to carry information across this interface in the setting of high-order discontinuous Galerkin methods. The first is the so-called mortar approach, where flux integrals along edges are split according to the positioning of the non-conformal grid. The second is a less-documented point-to-point interpolation method, where the interior and exterior quantities for flux evaluations are interpolated from elements lying on the opposing side of the interface. Although the mortar approach has significant advantages in terms of its numerical properties, in that it preserves the local conservation properties of DG methods, in the context of complex 3D meshes it poses notable implementation difficulties which the point-to-point method handles more readily. We explore the numerical properties of each method, focusing not only on observing convergence orders for smooth solutions, but also how each method performs in under-resolved simulations of linear and nonlinear hyperbolic problems, to inform the use of these methods in implicit large-eddy simulations.

1:40pm - 2:00pm

Development and application of an immersed boundary method for high-order flux reconstruction schemes

Jiaqing Kou1,2, Kunal Puri2, Charles Hirsch2, Esteban Ferrer1,3

1ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain; 2NUMECA International S.A., Chaussee de la Hulpe 187, Brussels, B-1170, Belgium; 3Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain

High-order methods for computational fluid dynamics are known to provide improved accuracy with relatively lower cost over low-order schemes, particularly for unsteady flows, which have the potential to form the next generation of industrial software [1]. However, the use of these methods on unstructured grids requires the generation of curved grids, which can be challenging for complex configurations. Immersed boundary method (IBM) delivers an alternative solution to avoid the complexity of mesh generation through handling nonconforming grids and moving boundaries in Cartesian domains [2]. With the capability of local hp-refinement to reduce the computational cost, developing IBM in the context of high-order framework becomes a promising area of research. In this work, we introduce our development in combining IBM with high-order flux reconstruction schemes [3,4]. Volume penalization [5,6] is chosen to handle the IBM treatment due to its good convergence property, robustness and ease of implementation. Theoretical study based on von Neumann analysis, numerical experiments for static and moving geometries [7], as well as applications to fluid-structure interaction are reported.


[1] Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T. and Kroll, N., 2013. High‐order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 72(8), pp.811-845.

[2] Mittal, R. and Iaccarino, G., 2005. Immersed boundary methods. Annu. Rev. Fluid Mech., 37, pp.239-261.

[3] Huynh, H.T., 2007, June. A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In 18th AIAA Computational Fluid Dynamics Conference (p. 4079).

[4] Vincent, P.E., Castonguay, P. and Jameson, A., 2011. A new class of high-order energy stable flux reconstruction schemes. Journal of Scientific Computing, 47(1), pp.50-72.

[5] Angot, P., Bruneau, C.H. and Fabrie, P., 1999. A penalization method to take into account obstacles in incompressible viscous flows. Numerische Mathematik, 81(4), pp.497-520.

[6] Kolomenskiy, D. and Schneider, K., 2009. A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. Journal of Computational Physics, 228(16), pp.5687-5709.

[7] Kou, J., Joshi, S., Hurtado-de-Mendoza, A., Puri, K., Hirsch, C., and Ferrer, E., 2021. Immersed boundary method for high-order flux reconstruction based on volume penalization.

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