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High Order NonHigh Order Non--Oscillatory Methods for Wave Propagation:Oscillatory Methods for Wave Propagation:Algorithms and ApplicationsAlgorithms and Applications
77thth April 2005April 2005
Trento, ItalyTrento, Italy
The ADERThe ADER--DG method for the simulation of DG method for the simulation of elastic wave propagation on unstructured elastic wave propagation on unstructured
meshesmeshes
Martin Käser, Michael Dumbser
DICA, University of Trento
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Motivation
Why very high order schemes for elastic waves ?
• physical models of the subsurface become more and more detailed, accurate computations of resulting synthetic seismograms become essential
model validation
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Motivation
Why very high order schemes for elastic waves ?
• physical models of the subsurface become more and more detailed, accurate computations of resulting synthetic seismograms become essential
model validation
• understanding of misfit only possible with highly accurate numerical schemes
separation of errors
PARAMETERIZATIONMODELING OF PROCESS
ERROR
NUMERICAL APPROXIMATION
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Motivation
Why very high order schemes for elastic waves ?
• physical models of the subsurface become more and more detailed, accurate computations of resulting synthetic seismograms become essential
model validation
• understanding of misfit only possible with highly accurate numerical schemes
separation of errors
• for wave propagation problems time accuracy is as important as space accuracy (amplitude and phase errors depend on time and space accuracy respectively)
space AND time accuracy
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Motivation
Why very high order schemes for elastic waves ?
• physical models of the subsurface become more and more detailed, accurate computations of resulting synthetic seismograms become essential
model validation
• understanding of misfit only possible with highly accurate numerical schemes
separation of errors
• for wave propagation problems time accuracy is as important as space accuracy (amplitude and phase errors depend on time and space accuracy respectively)
space AND time accuracy
• highly accurate numerical methods from the CFD community are applicable
CFD developments
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Elastic Wave Equations
theory of linear elasticity(definition of strain, Hooke´s law)
+Newton´s law
(acceleration through forces caused by stress)
=velocity-stress formulation
(non-conservative hyperbolic system)
In a heterogeneous medium:space-dependent material coefficientsLame constants: λ = λ(x,y,z),
µ = µ(x,y,z), density ρ = ρ(x,y,z)
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Eigenstructure of the Hyperbolic System
Compact vector-matrix notation gives
with the vector of unknowns and Jacobian matrices
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Eigenstructure (continued)
Eigenvalues give wave speeds of P- and S-waves
with
P-wave S-wave
Eigenvectors are given through
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Discontinuous Galerkin (DG) Approach (Reed & Hill, 1973; Cockburn & Shu, 1989, 1991)
Ideas: • incorporation of well-established concept of numerical fluxes across element interfaces into the finite element framework
• approximation of unknown function by piecewise polynomials of arbitrary high order
• time evolution of polynomial coefficients instead of only cell averages=> no polynomial reconstruction necessary !
• computation of many integrals on reference element => increase of computational efficiency
• well-suited for parallelization due to local character of the scheme (only direct neighbours necessary)
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
DG versus Finite Volumes
Advantage of DG:
• no WENO approach with expensive oscillation indicators and non-linear weights
• no reconstruction => pays off especially for high orders on unstructured meshes in 3-D
• no involved stencil selection on unstructured meshes
Example: set of 9 stencils for quadratic WENO reconstruction
Sectoral search algorithm
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
DG – The Numerical Scheme (Reed & Hill, 1973; Cockburn & Shu, 1989, 1991)
The governing system of equations is given by
Multiplication with a test function Φk and integration over a triangle T (m) gives
Integration by parts yields
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
DG – The Numerical Scheme (continued)The exact numerical flux in non-conservative form is
with
Splitting the boundary integral into the flux contributions gives
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
DG – The Numerical Scheme (continued)• transformation into the reference element gives
with and
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
DG – The Numerical Scheme (continued)• transformation into the reference element gives
where the following integrals on the reference element can be pre-computed
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
ADER-DG – Time Integration
• time integration over one time step from level n to n+1 gives the final scheme
where at local time level t = tn = 0
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
ADER-DG – Time Integration
• in the reference element we have the governing equation
• for a linear system the k-th time derivative is expressed by space derivatives
• Taylor series expansion around up gives
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
ADER-DG – Time Integration
• projecting this approximation onto the basis functions gives evolution of degrees of freedom from level t = tn to tn+1= tn+∆t
• analytical time integration gives
where we define
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Boundary ConditionsMost important: 1) open, outflow, absorbing, non-reflective boundaries
2) free surface boundaries
Open boundaries:
• only outgoing flux is considered, incoming flux is set to zero
Free surface boundaries:
• only outgoing flux is considered, incoming flux is set to zero
with
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Numerical Convergence Analysis
Regular mesh and its refinement
Irregular mesh and its refinement
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Initial Condition
Material parameters
• plane P- and S-waves travel along the diagonal with different speeds and in different directions
Wave speeds
At simulation timewe get
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application 1: Global Seismology (2-D)
H. Igel
• Staggered FD meshes (spherical structure)mesh spacing decreases towards the centerhanging nodes due to successive re-coarsening
• FV meshes (unstructured)Voronoi meshes are difficult to adapt to geometrical features
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Global Seismology with ADER-DG using PREM
• mesh spacing proportional to P-wave velocity
(Dziewonski & Anderson, 1981)
• decreasing mesh spacing towards the center is avoided
• local CFL criterion is respected in all cells in a globally optimal sense
• no interpolation at hanging nodes
• no grid staggering
• triangular mesh adapts nicelyto geometry
Animation: σxy u v
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Synthetic Seismograms using PREM
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
• Subdomains produced byMETIS for a calculationwith 32 processors
• METIS decomposes themesh into similarly sizedsubdomains.
• Communication betweensubdomains is low
partnmesh <meshfile> <nparts>
Amdahl´s Law
• Parallel performance still has tobe analysed carefully
Parallelization using METIS and MPI
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application 2: Strong Material Contrasts (LeVeque, 2002)
• Elastic medium withlarge contrast in materialproperties
Outside:ρ = 1, λ = 2, µ = 1, cp = 2 Inside:ρ = 1, λ = 200, µ = 100, cp =20
Movie: σxx
Order: 4 Mesh: 200 x 100 CPU time: 100%Order: 6 Mesh: 100 x 50 CPU time: 50%
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application in 3-D: Mesh Generation • Tetraheral meshes give an extreme flexibility needed for complex geometries• METIS can handle large 3-D tetrahedral meshes easily
• Surface topography is included easily(3 362 868 cells decomposed in less than 10 sec)
Movies: 3D2Dx-slice
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application in 3-D: First Results for PREM
velocity component v
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application in 3-D: Trento (Val d‘Adige)
• surface topography given byDEM with 50m resolution(J.Yebrin)
• test run on3 262 657 tetrahedral cellswith 16 processors
• influence of topography onamplitudes andwave pattern
Doss San Rocco
Dosso di S. Agata
Doss Trento
• more detailed geological structureof the subsurface (?)
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Application in 3-D: Trento (Val d‘Adige)
• surface topography given byDEM with 50m resolution(J.Yebrin)
• test run on3 262 657 tetrahedral cellswith 16 processors
• influence of topography onamplitudes andwave pattern
• more detailed geological structureof the subsurface (?)
![Page 32: The ADER DG method for the simulation of elastic wave ...kaeserm/index.php_files/SOURCES/... · OscillOscillatory Methods foatory Methods for Wave Propagation: r Wave Propagation:](https://reader034.vdocuments.mx/reader034/viewer/2022042312/5edbd7a6ad6a402d666640d5/html5/thumbnails/32.jpg)
Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Anisotropic Material• Hooke´s Law in the most general, triclinic case gives
then the corresponding Jacobian matrix with anisotropy is
• eigenvectors are difficult to compute, but eigenvalues are available
=> numerical flux uses
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Anisotropic Material (aligned along coordinate axis)
total velocity
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Anisotropic Material (aligned along space diagonal)
total velocity
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
J. Wassermann(SeismologicalObservatory FUR)
2970m
3500m
350m
300m
1500m1500m
Outlook: Merapi-Project• Seismic monitoring of vulcanoesstructural changes, source mechanisms
• strong, free surface topography
• measured data available
• internal structure, i.e. velocityand density model available
J. Ripperger
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Martin Käser ADER-DG for Elastic Waves on unstructured meshes Trento, April 2005
Outlook:
• Large scale simulations (High Performance computers in München and Stuttgart)
• Incorporation of viscoelasticity (attenuation)
• Reservoir models
• Solutions for 3D Visualization (Problem: extremely large data sets!)
• Code library for SPICE(synthetic seismogramms)
• Comparison with - other schemes (performance)- analytical data (Lamb´s Problem)- measured data
• Maxwell equations (Georadar modeling)