08:30
MS51: Space and time adaptation for PDEs: From theory to practice (Part 1)
Chair: Maurizio Falcone
08:30
25 mins
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High order space-time adaptive ADER schemes for compressible multiphase flows
Simone Chiocchetti, Michael Dumbser
Abstract: In this talk, we present a family of high order ADER Discontinuous Galerkin (DG) and Finite Volume (FV) methods with space-time adaptive mesh refinement for hyperbolic nonconservative systems of PDEs, such as those often used for modeling compressible multiphase flows.
The presented explicit schemes achieve arbitrary high order of accuracy in space and time with a single-step integration, as opposed, for example, to multi-stage Runge–Kutta timestepping. At each timestep, following a full-polynomial WENO reconstruction (for schemes of the Finite Volume variety), or directly from the piecewise polynomial data (for DG schemes), a discrete, element-local, space-time predictor solution to the governing equations is computed and subsequently employed for the evaluation of the volume and surface integrals required for the update of the degrees of freedom of the data. Jumps in the nonconservative terms occurring at boundaries are handled within the path-conservative framework of Castro and Parés.
The use of an element-local space-time predictor allows to avoid communications between processors as much as possible, yielding algorithms that are particularly well-suited for applications on massively parallel distributed memory supercomputers. Moreover, since the predictor solution, which is given as a polynomial in space and time, can be evaluated at any intermediate time for a given timestep, time-accurate local timestepping is easily implemented, along with cell-by-cell adaptive mesh refinement.
Oscillation occurring at shocks and strong gradients are stabilized by the nonlinear WENO reconstruction for high order ADER-FV schemes, while ADER-DG methods employ an a-posteriori subcell Finite Volume limiting strategy, that is, at each timestep and in each control volume, one checks that the solution obtained from the unlimited DG scheme not show spurious oscillations or other violations of physical constraints (for example positivity of density and pressure) and wherever the tests yield negative results, a new solution is recomputed locally on a refined subcell grid with a more robust ADER-WENO of MUSCL Finite Volume scheme.
Finally, an application of the discussed family of methods is shown, by giving a number of numerical results concerning a first-order hyperbolic formulation of two-phase, one-velocity, one-pressure,
compressible flow with surface tension.
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08:55
25 mins
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A-priori versus a-posteriori limiting in high order AMR finite volume schemes
Matteo Semplice, Raphaël Loubère
Abstract: The numerical computation of solutions to hyperbolic equations with high order accurate schemes has to deal with the nonregular features of the solutions being approximated. This difficulty is typically overcome with some form of limiting, either a-priori inside the reconstruction procedure, or a-posteriori after the timestep has been computed.
In this talk I will consider both approaches by comparing two finite volume schemes employing Adaptive Mesh Refinement, both using the numerical entropy production to refine/coarsen the mesh. The first scheme is based on a CWENO reconstruction, while the other one employs central unlimited reconstruction polynomials and the MOOD a-posteriori limiting procedure. This latter consists in detecting the oscillations or non-physical features of the computed solutions and to locally recompute the timestep with a more robust scheme.
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09:20
25 mins
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Mesh adaptivity in the framework of the Cartesian Grid Finite Element Method - cgFEM
Juan José Ródenas, Enrique Nadal, José Albelda, Manuel Tur
Abstract: To avoid the difficulties associated to the creation and refinement of Finite Element (FE) meshes for arbitrary domains, in the Immersed Boundary Methods (IBMs) the domain's geometry is embedded in another domain whose meshing is simple or even trivial. The Cartesian grid Finite Element Method (cgFEM)[1] is an IBM where the embedding domain is a cuboid that is meshed using Cartesian grids. In cgFEM, mesh refinement is obtained by splitting each 'parent' element into 8 'children' elements and the use of Multi-Point Constraints (MPCs) to enforce C0 continuity between elements of different refinement levels. In fact, mesh refinement represents a key ingredient in the basis of cgFEM. In this work we will show how mesh refinement is essential not only for h-adaptive analyses in cg-FEM but also for accurate representation of the domain's geometry given by NURBS or T-spline models. The use of h-refinement has also allowed for the development of methodologies by means of which cgFEM is able to automatically generate FE models form 3D medical images creating h-adapted Cartesian meshes where the boundaries of the different tissues are implicitly represented, avoiding the standard and time-consuming segmentation process. Other uses of h-refinement in cgFEM include the development of an h-adaptive topology optimization strategy for accurate boundary representation, the development of efficient fictional contact algorithms, muli-grid solvers, etc. The hierarchical data structure of cgFEM allows for simple and un-expensive data transfer between 'parent' and 'children' elements that leads to highly efficient algorithms based on mesh refinement, even in the Matlab implementation of cgFEM.
Keywords: cgFEM, mesh refinement, element splitting, Cartesian grids, topology optimization, patient-specific analysis
Acknowledgments: With the support of Ministerio de Economía, Industria y Competitividad of Spain (DPI2017-89816-R) and the Generalitat Valenciana (PROMETEO/2016/007)
[1] E Nadal, JJ Ródenas, J Albelda, M Tur, JE Tarancón, FJ Fuenmayor. Efficient Finite Element Methodology Based on Cartesian Grids: Application to Structural Shape Optimization. Abstract and Applied Analysis 2013, Article ID 953786, 19 pages (2013).
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