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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

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


Motivation



model validation

• understanding of misfit only possible with highly accurate numerical schemes

separation of errors

PARAMETERIZATIONMODELING OF PROCESS

ERROR

NUMERICAL APPROXIMATION


Motivation



model validation



• 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


Motivation



model validation



• 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


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)


Eigenstructure of the Hyperbolic System

Compact vector-matrix notation gives

with the vector of unknowns and Jacobian matrices


Eigenstructure (continued)

Eigenvalues give wave speeds of P- and S-waves

with

P-wave S-wave

Eigenvectors are given through


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)


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


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


DG – The Numerical Scheme (continued)The exact numerical flux in non-conservative form is

with

Splitting the boundary integral into the flux contributions gives


DG – The Numerical Scheme (continued)• transformation into the reference element gives

with and


DG – The Numerical Scheme (continued)• transformation into the reference element gives

where the following integrals on the reference element can be pre-computed


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



• 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



• 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


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


Numerical Convergence Analysis

Regular mesh and its refinement

Irregular mesh and its refinement


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


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


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


Synthetic Seismograms using PREM


• 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


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%


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


Application in 3-D: First Results for PREM

velocity component v


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 (?)


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 (?)


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


Anisotropic Material (aligned along coordinate axis)

total velocity


Anisotropic Material (aligned along space diagonal)

total velocity


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


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)

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