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Compatible Reconstruction of Vectors for Enhanced PDE Discretization Techniques

This document discusses advanced vector reconstruction methods for optimal spatial discretization of partial differential equations (PDEs). It explores the use of compatible spatial discretizations, focusing on both tangential and normal components to ensure continuity, boundary condition adherence, and the absence of spurious modes. Emphasizing conservation laws and the construction of Hodge star operators, it presents methods applicable to various geometries, ensuring accurate representation of physical phenomena like convection and energy conservation. Key insights into discrete Hodge stars and local conservation principles are included.

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Compatible Reconstruction of Vectors for Enhanced PDE Discretization Techniques

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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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