An asymptotic model for computing the scattering of waves by small heterogeneities in the time domain

An asymptotic model for computing the scattering of waves by small heterogeneities in the time domain Simon Marmorat EPI POems INRIA Saclay / CEA List Joint work with Patrick Joly (EPI POems INRIA/ENSTA/CNRS) Xavier Claeys (LJLL, Paris) Simon MARMORAT - Junior Seminar November, 19th 2013

Motivation and context 2 «Non-destructive testing (NDT) is a wide group of analysis techniques used in science and industry to evaluate the properties of a material, component or system without causing damage.» Wikipedia Applications: Medicine (ultrasonography, EEG, )! Geophysics (ground exploration)! Industrial maintenance (planes, nuclear plants, )

Non destructive testing 3 Study ultrasonic wave propagation with applications to Non-Destructive Testing. Emitter / receiver From the knowledge of the incident and the scattered waves, recover information (shape, size, physical properties, ) on the object of interest.

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t)

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t) Prove the equation has a unique solution Equation is said to be well-posed

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t) Prove the equation has a unique solution Usually no analytical (no nice formula) solution Equation is said to be well-posed When the equation or the geometry! are not simplistic How to represent the phenomenon anyway?! Thanks to simulation! Represent by a finite number of points, compute an approximation of the unknown on each points.

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t) Prove the equation has a unique solution Usually no analytical (no nice formula) solution Equation is said to be well-posed When the equation or the geometry! are not simplistic Mesh How to represent the phenomenon anyway?! Thanks to simulation! Represent by a finite number of points, compute an approximation of the unknown on each points.

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t) Prove the equation has a unique solution Usually no analytical (no nice formula) solution Equation is said to be well-posed When the equation or the geometry! are not simplistic Mesh Since equation time dependent, we also discretize time How to represent the phenomenon anyway?! Thanks to simulation! Represent by a finite number of points, compute an approximation of the unknown on each points. t

What is Numerical Analysis? 4 Equation modeling some physical phenomenon occurring on, with unknown u(x, t) Prove the equation has a unique solution Usually no analytical (no nice formula) solution Equation is said to be well-posed When the equation or the geometry! are not simplistic Mesh Since equation time dependent, we also discretize time How to represent the phenomenon anyway?! Thanks to simulation! Represent by a finite number of points, compute an approximation of the unknown on each points. Leads to a matrix system: solvable by a computer! t

What is Numerical Analysis? 5 Key parameter: the number of points on the mesh number of degrees of freedom n Mesh

What is Numerical Analysis? 5 Key parameter: the number of points on the mesh number of degrees of freedom n More degrees of freedom Better approximation Higher computational cost inverting a matrix of size has complexity O(n 3 ) n Mesh

PhD motivations 6 Non destructive testing for ultrasonic waves in media containing small heterogeneities (for instance concrete with small aggregate in it). Propagation of waves inside smooth media disturbed by small defects with respect to the wavelength. Classical simulation tools struggles to deal with multi scale problems.

PhD motivations 7 Simulation of an elastic wave propagating inside a media disturbed by small inclusions Finite element method Simulation 35 time (µs) CPU time 3527 (s) Degrees of freedom 465132 t (µs) 0.0013

PhD motivations 8 Homogeneous medium Medium with defects

PhD motivations 8 Homogeneous medium Medium with defects about 90 000 degrees of freedom about 450 000 degrees of freedom

PhD motivations 9 We deal with time domain waves propagating in a smooth media with defects of characteristic size central wavelength! Assumption: " <<

PhD motivations 9 We deal with time domain waves propagating in a smooth media with defects of characteristic size central wavelength! Assumption: " << How to choose x to reach satisfying accuracy?

PhD motivations 9 We deal with time domain waves propagating in a smooth media with defects of characteristic size central wavelength! Assumption: " << How to choose x to reach satisfying accuracy? Homogeneous medium Medium with defects x ' x ' "

A model problem 10 Wave acoustic equation: look for u " (x, t) such that x 2 R d, t > 0.! "

A model problem 10 Wave acoustic equation: look for u " (x, t) such that x 2 R d, t > 0.! " rf = 0 B @ @f @x 1. @f @x d 1 C A r ~g = dx i=1 @g i @x i

A model problem 10 Wave acoustic equation: look for u " (x, t) such that x 2 R d, t > 0. The physical parameters ", µ " admit a jump on the defect boundary.! " Construction of an approximate model: defect taken into account via auxiliary sources.

MATCHED ASYMPTOTIC METHOD Simon MARMORAT - Junior Seminar November, 19th 2013

Matched asymptotics 12 Look for the solution u " of the following model problem: (P " ) " @ 2 u " @t 2 u " (x, 0) = @u " @t r (µ " ru " )=f, (x, 0) = 0, x 2 R d, t > 0, x 2 R d.! " Asymptotic behavior of u " as "! 0?

Matched asymptotics 12 Look for the solution u " of the following model problem: (P " ) " @ 2 u " @t 2 u " (x, 0) = @u " @t r (µ " ru " )=f, (x, 0) = 0, x 2 R d, t > 0, x 2 R d. Asymptotic behavior of as? We would like something like u " "! 0 u " (x, t) =! " 1X n=0 and compute the u n (independent of " ). " n u n (x, t)

Matched asymptotics 12 Look for the solution u " of the following model problem: (P " ) " @ 2 u " @t 2 u " (x, 0) = @u " @t r (µ " ru " )=f, (x, 0) = 0, x 2 R d, t > 0, x 2 R d.! " Asymptotic behavior of as? u " "! 0 No uniform development of u " in function of "! Need to distinguish a far-field zone and a near-field zone inside which development holds.

Matched asymptotics 13 Look for the solution u " of the following model problem: (P " ) " @ 2 u " @t 2 u " (x, 0) = @u " @t r (µ " ru " )=f, (x, 0) = 0, x 2 R d, t > 0, x 2 R d.! "

Matched asymptotics 13 Look for the solution u " of the following model problem: (P " ) " @ 2 u " @t 2 u " (x, 0) = @u " @t r (µ " ru " )=f, (x, 0) = 0, x 2 R d, t > 0, x 2 R d. We denote! " " (x) = (x) if x 2!" + (x) if x 2 R d \! " µ " (x) = µ (x) if x 2!" µ + (x) if x 2 R d \! "

Matched asymptotics 14 Model problem " @ 2 u " @t 2 r (µ " ru " )=f,! "

Matched asymptotics 14 Model problem " @ 2 u " @t 2 r (µ " ru " )=f, p "! 2 " p " Near field zone Matching zone Far field zone

Ansatz and total field 15 Select a cutoff function,! rp scale it "(r) =, "! and represent the total field as the sum (r) u " (x, t) = " ( x ) U N " (x, t)+(1 " ( x )) u F " (x, t)

Ansatz and total field 15 Select a cutoff function,! rp scale it "(r) =, "! and represent the total field as the sum (r) u " (x, t) = " ( x ) U N " (x, t)+(1 " ( x )) u F " (x, t) = when x apple p " 2

Ansatz and total field 15 Select a cutoff function,! rp scale it "(r) =, "! and represent the total field as the sum (r) u " (x, t) = " ( x ) U N " (x, t)+(1 " ( x )) u F " (x, t) = when x p "

Matching principle 16 Far field equations + @2 u 0 @t 2 r µ + ru 0 = f, Limit field + @2 u n @t 2 r µ + ru n =0, n 1. Near field equations r (µru n )=0, n =0, 1 r (µru n )= @2 U n 2 @t 2, n 2

Matching principle 16 Far field equations + @2 u 0 @t 2 r µ + ru 0 = f, Limit field Near field equations r (µru n )=0, n =0, 1 + @2 u n @t 2 r µ + ru n =0, n 1. NX Matching principle: NX The partial sums " n u n and " n U n must coïncide in the matching zone, n=0 n=0 r (µru n )= @2 U n 2 @t 2, n 2 8N 0.

TOWARDS AN ASYMPTOTIC METHOD Simon MARMORAT - Junior Seminar November, 19th 2013

An asymptotic model 18 From the initial model problem (P " ) " @ 2 u " @t 2 r (µ " ru " )=f, x 2 R d, t > 0,

An asymptotic model 18 From the initial model problem (P " ) " @ 2 u " @t 2 r (µ " ru " )=f, x 2 R d, t > 0, seeing ", µ " as perturbations of +, µ + we derive the following approximate model:

An asymptotic model 18 From the initial model problem (P " ) " @ 2 u " @t 2 r (µ " ru " )=f, x 2 R d, t > 0, seeing ", µ " as perturbations of +, µ + we derive the following approximate model: auxiliary source ( e P " ) + @2 u " @t 2 r µ + ru " = f + S" N v ", v " = S N "? u", x 2 R d, t > 0, coupling the auxiliary source with the unknown

An asymptotic model 18 From the initial model problem " - wave operator, not suited for discretization (P " ) " @ 2 u " @t 2 r (µ " ru " )=f, x 2 R d, t > 0, seeing ", µ " as perturbations of +, µ + we derive the following approximate model: ( e P " ) + @2 u " @t 2 r µ + ru " = f + S" N v ", v " = S N "? u", " - independent wave operator, well suited for discretization S" N is an operator involving the first N near-field functions N is the order of approximation of the method: ksol (P )" sol ( e P )" kapplec" N x 2 R d, t > 0,

1D Numerical example 19 Physical parameters: + =1 = 10 µ = 10 µ + =1 " =0.04

1D Numerical example 19 Physical parameters: + =1 = 10 µ = 10 µ + =1 " =0.04 Computation of a reference solution via classical numerical method x =0.01

1D Numerical example 19 Physical parameters: + =1 = 10 µ = 10 µ + =1 " =0.04 Computation of a reference solution via classical numerical method x =0.01 Computation of an approximate solution via asymptotic numerical method x =0.04

1D Numerical example 20 Total field Diffracted field CPU time (s) Reference solution h ref =0.001 15 Asymptotic solution h as =0.01 1 Difference Relative L 2 error < 1%

Numerical example 21 Case of a unique defect Number of degrees of freedom 90 000 CPU time 5 minutes!

Numerical example 22 Case of a network of defects Number of degrees of freedom 90 000 CPU time 5 minutes!

Conclusion and prospects 23 Efficient and accurate numerical method for computing the scattering of waves by small defects. Rigorous error estimates.! To be done: Adapt the method to the 3D case. Extension to defects of arbitrary shape. Extension to the elastodynamic wave equation.

Thank you for your attention! Simon MARMORAT - Junior Seminar November, 19th 2013