Uhaina Uhaina A brief ief intr introduction duction Code Uhaina - - PowerPoint PPT Presentation

A brief ief intr introduction duction

Uhaina Uhaina

Code Uhaina

Actors EPOC, IMB, INRIA, I3M Issues Simulation of tsunami propagation at regional scale. Simulation wave shoaling and wave transformation in the surf and swash zones. Wave overtopping and coastal flooding during storm event or tsunami. Operational and open source community code. Realisation New numerical method for the resolution of the Green-Naghdi equations. Parallel implementation using the finite elements library AeroSol.

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Architecture and developement tools

Using Aerosol finite elements library

Uhaina Pre-processing Post-processing AeroSol Computation

Tools C++ GForge Git / SVN Spack CMake - CTest Doxygen ci.inria Parallel test framework.

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Code Uhaina

Inputs Mesh Bathymetry Initial conditions Boundary conditions. Output Submersion areas Elevation field Breaking zone etc...

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Feature

AeroSol Finite elements, both continuous and discontinuous. Unstructured and hybrid meshes to handle complex geometry. Efficient parallel computation. Specific feature Well-balanced formulation. Wet/Dry zones handling Positivity preserving limiter [Zhang et al. 2010] Shock limiter [Guermond et al. 2011] Efficient discretization of dispersive terms [Lannes et al. 2015]

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Aerosol library

Time iterator

Mesh / connectivity Assembly of the matrix System inversion

Spatial discretization

Geometry Quadrature formulas Finite elements functions Model Numerical flux Boundary condition

Operators

MUMPS UMFPACK PaStiX PETSc

Linear solver

Redistribution Memory allocation Communications

PaMPA

Residual computation System inversion Loop over elements Resolution Residual computation Local operations 5 / 17

Well balanced shallow water equations

Hyperbolic system ∂W ∂t + ∇ · F(W, zb) = S(W, zb) W = η q

• ,
• F(W, zb) =
• q

uq + 1

2g(η2 − 2ηzb)

• S(W, zb) =
• −gη ∂zb

∂x

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Discontinuous Galerkin method

Hyperbolic system ∂W ∂t + ∇ · F(W, zb) = S(W, zb) Domain Ω of Rd Mesh Mh such that Ω = ∪e∈MhΩe Pk polynomial with : deg(Pk) ≤ k Approximation space V k

h =

• v ∈ [L2(Ωe)]d+2,

v|e ∈ [Pk(Ωe)]d+2, ∀Ωe ∈ Mh

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Well balanced shallow water equations

Hyperbolic system ∂W ∂t + ∇ · F(W, zb) = S(W, zb) ∀vh ∈ V k

h

Wh approximation of W in V k

h

∀vh ∈ V k

h

• Ωe∈Mh
• Ωe

∂tWhvh dΩ −

• Ωe∈Mh
• Ωe
• F(Wh, zh) · ∇vh dΩ

+

• σ∈Γ
• σ

v−

h hb(W− h , Wb h, z− h , zb h , n) dσ

+

• σ∈Σi
• σ

[ [ vh ] ]H(W−

h , W+ h , z− h , z+ h , n) dσ

=

• Ωe∈Mh
• Ωe
• S(Wh, zh)vh dΩ

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Positivity preserving limiter

Zhang & Shu quadrature for P1 and P2 cases

hmin = min

x∈SZhang

h(x) If hmin ≤ ǫ : ˜ U(x) = Θ(U(x) − ¯ U) + ¯ U, ∀x ∈ SGauss (1) With Θ = min

• 1,

ǫ − ¯ h hmin − ¯ h

• (2)

Zhang, X., Xia, Y., & Shu, C. (2012). Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes. Journal of Scientific Computing. 9 / 17

Shock limiter

E = 1 2

• hu2 + g(η2 + z2)
• D = ∂E

∂t + ∇ · f(E) ≤ 0 D is always zero except in discontinuities. ν = βνmax min

• 1.0, δx |D|

ǫN

• With :

µmax = δx max |λ| Neglecting face integral :

• Ωe∈Mh
• Ωe

ν∇vh · ∇Wh dΩ

Guermond et al (2011), Entropy viscosity method for nonlinear conservation laws. Journal of Computational Physics. 10 / 17

Shock limiter

Entropy viscosity on simple breaking wave.

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Test case : Well balanced property

Well balanced property and dry/wet interface

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Test case : Subcritical flow over a bump

Convergence with second and third order spatial discretization

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Test case : Carrier and Greenspan periodic

Moving Wet/Dry interface with second order spatial discretization

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Test case : Synolakis

Results for Synolakis test case at different time with third order spatial discretization.

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Demo

Simple demonstration of parallel computation

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Thank you for your attention.

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