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

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

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

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

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Convection/ Adaptation M-adaptation

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

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Hodge Star Operators Have (tangential) Need (normal) Have (tangential) Need (normal) n Discrete Hodge Star u Interpolate / Integrate u Least Squares T1T1 T2T2

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Compatible/Mimetic Discretization de Rham-like complex T1T1 T2T2 Notation Exact Connectivity Matrices Transposes Exact Connectivity Matrices Transposes

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Dual Meshes Circumcenter (Voronio) T1T1 T2T2 T1T1 T2T2 T1T1 T2T2 T1T1 T2T2 Centroid (center of gravity) FE (smeared) Median

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FE Reconstruction n 2D: Raviart-Thomas n 3D: Nedelec n Compute Coefficients in the Interpolation n 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) 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) Face Edge

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

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

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

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Vector Reconstruction Expand the Hodge star operation T1T1 T2T2 Nonlinear Constitutive Relations are no problem

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Other Methods T1T1 T2T2 Methods Differ in: Interpolation Assumptions Integration Assumptions Methods Differ in: Interpolation Assumptions Integration Assumptions n CVFEM u Linear in elements (sharp dual) u Local conservation n Classic FEM u Linear in element (spread dual) n Discontinuous Galerkin / Finite Volume u Reconstruct in the Voronio Cell

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

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

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Conservation Properties n Voronoi Method u Conserves KE u Rotational Form -- Conserves Vorticity u Divergence Form – Conserves Momentum u Cartesian Mesh – Conserves Both n Staggered Mesh Method u Conserves KE u Divergence Form – Conserves Momentum

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

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Conclusions n Physical PDE systems can be discretized (made finite) exactly. Only constitutive equations require numerical (and physical) approximation. n Vector reconstruction is useful for: convection, adaptation, conservation, Hodge star construction, nonlinear material properties. n Hodge star operators have internalstructure that is useful and related to interpolation/integration.

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