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Compatible Reconstruction of Vectors

Compatible Reconstruction of Vectors. Blair Perot Dept. of Mechanical & Industrial Engineering. Compatible Spatial Discretizations for Partial Differential Equations May 14, 2004. Compatible Discretizations. Tangential components (Edge Elements). Normal components

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Compatible Reconstruction of Vectors

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  1. Compatible Reconstruction of Vectors Blair Perot Dept. of Mechanical & Industrial Engineering Compatible Spatial Discretizations for Partial Differential Equations May 14, 2004

  2. Compatible Discretizations Tangential components (Edge Elements) Normal components (Face Elements) • Vector Components are Primary Temperature Gradient Electric Field Vorticity Heat Flux Magnetic Flux Velocity Flux

  3. Why Vector Components • Physics • Mathematics • Numerics Measurements Continuity Requirements Boundary Conditions Differential Forms Gauss/Stokes Theorems Absence of Spurious Modes Mimetic Properties Unknowns should contain Geometry/Orientation Information

  4. So Why Vector Reconstruction ? • Convection • Adaptation • Formulation of Local Conservation Laws • (momentum, kinetic energy, vorticity/circulation) • Construction of Hodge star operators • Nonlinear constitutive relations

  5. M-adaptation Convection/ Adaptation

  6. Conservation • Wish to have discrete analogs of vector laws. • Conservation of Linear Momentum • Conservation of Kinetic Energy Component Equations Linear Momentum 3-Form ?

  7. T1 T2 Hodge Star Operators • Discrete Hodge Star • Interpolate / Integrate • Least Squares Have (tangential) Need (normal)

  8. Notation Compatible/Mimetic Discretization T1 T2 de Rham-like complex Exact Connectivity Matrices Transposes

  9. Dual Meshes T1 T1 T2 T2 Centroid (center of gravity) Circumcenter (Voronio) T1 T1 T2 T2 FE (smeared) Median

  10. FE Reconstruction • 2D: Raviart-Thomas • 3D: Nedelec Face Edge • Compute Coefficients in the Interpolation • Compute Integrals No dual – because FE is an average over all duals. Quadrature rule is a way of weighting the duals. (which is how you can get other methods)

  11. SOM Reconstruction • Shashkov, et al. • Reconstruct local node values • then interpolate • Arbitrary Polygons • When Incompressible and Simplex = FE interpolation S R L N Discrete Hodge Stars

  12. Voronio Reconstruction Diagonal Hodge star operator (due to local orthogonality) • CoVolume method (when simplices). • Used in ‘meshless’ methods (material science) • Locally Conservative (N.S. momemtum and KE). Discrete Maximum Principal in 3D for Delaunay mesh (not true for FE).

  13. Staggered Mesh Reconstruction Dilitation = constant Face normal velocity is constant • Conserves Momentum and Kinetic Energy. • Arbitrary mesh connectivity. • No locally orthogonality between mesh and dual. • Hodge is now sparse sym pos def matrix.

  14. T1 T2 Vector Reconstruction Expand the Hodge star operation Nonlinear Constitutive Relations are no problem

  15. T1 T2 Other Methods • CVFEM • Linear in elements (sharp dual) • Local conservation • Classic FEM • Linear in element (spread dual) • Discontinuous Galerkin / Finite Volume • Reconstruct in the Voronio Cell Methods Differ in: Interpolation Assumptions Integration Assumptions

  16. Uf Staggered Mesh Reconstruction Interpolate Integrate X has same sparsity pattern as D Symmetric Pos. Def. sparse discrete Hodge star operator

  17. Staggered Mesh Conservation Exact Geometric Identities Time Derivative in N.S.

  18. Conservation Properties • Voronoi Method • Conserves KE • Rotational Form -- Conserves Vorticity • Divergence Form – Conserves Momentum • Cartesian Mesh – Conserves Both • Staggered Mesh Method • Conserves KE • Divergence Form – Conserves Momentum

  19. Incompressible Flow • Define a discrete vector potential • So always. • CT of the momentum equation eliminates pressure (except on the boundaries where it is an explicit BC) • Resulting system is: • Symmetric pos def (rather than indefinite) • Exactly incompressible • Fewer unknowns

  20. Conclusions • Physical PDE systems can be discretized (made finite) exactly. Only constitutive equations require numerical (and physical) approximation. • Vector reconstruction is useful for: convection, adaptation, conservation, Hodge star construction, nonlinear material properties. • Hodge star operators have internal structure that is useful and related to interpolation/integration. www.ecs.umass.edu/mie/faculty/perot

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