Model Order Reduction for Wave Equations Rob F. Remis and J orn T. - - PowerPoint PPT Presentation

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Model Order Reduction for Wave Equations Rob F. Remis and J orn T. - - PowerPoint PPT Presentation

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Introduction Wave equation Model Order Reduction for Wave Equations Rob F. Remis and J orn T. Zimmerling DCSE Fall School, Delft, November 4 8, 2019 1 Introduction Wave equation Introduction Rob F. Remis Fac. of Electrical

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Introduction Wave equation

Model Order Reduction for Wave Equations

Rob F. Remis and J¨

  • rn T. Zimmerling

DCSE Fall School, Delft, November 4 – 8, 2019 1

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Introduction Wave equation

Introduction

Rob F. Remis

  • Fac. of Electrical Engineering, Mathematics and Computer Science

Delft University of Technology r.f.remis@tudelft.nl

J¨

  • rn T. Zimmerling

Department of Mathematics University of Michigan j.zimmerl@umich.edu

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Introduction Wave equation

Introduction

Lecture 1: basic wave equations Lecture 2: Discretization and symmetry Lecture 3: Symmetry and Krylov model-order reduction Lecture 4: Capita selecta

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Introduction Wave equation

Introduction

Reduced-order modeling (ROM) is a vast research area We focus on some ROM techniques for wave field problems Reduced-order model(ing) – ROM: Replace a large-scale system by a much smaller one Before you start, you should have good reasons to believe that a significant reduction can be achieved ROM for wave fields (hyperbolic problems) is hard, in general

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Introduction Wave equation

Introduction

Possible ROM scenarios: Solve a forward problem efficiently Replace a large-scale model in a design/imaging/inversion process Replace a large-scale system, while preserving

input-output characteristics and essential system properties (e.g. stability, passivity)

  • f the unreduced system – structure preserving ROM

Apply ROM to directly solve an imaging/inversion problem – data driven imaging

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Introduction Wave equation

Introduction - Example 1

500 1000 1500

y-direction [m]

500 1000 1500 2000 2500 3000

x-direction [m]

Receiver Source PML 1500 2000 2500 3000 3500 4000 4500 5000 5500

Speed [m/s]

1000 2000 3000 4000 5000 6000 7000 8000 9000 Normalized Time

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 Response [a.u.] ×10-5 Comparison ROM 2000 2500 3000 3500 4000 4500 5000

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Introduction Wave equation

Introduction - Example 2

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GRE TSE Without pad With pad

b+

1 = b+ 1;rom(p),

p = pad design parameter vector

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Introduction Wave equation

The Wave Equation

The wave equation

instantaneous reacting material

∇2u − 1 c2 ∂ttu = −q

u = u(x, t): wave field quantity of interest ∇2: Laplacian ∂tt: double derivative with respect to time q = q(x, t): source c(x): wave speed profile c(x) = c0 for a homogeneous medium

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Introduction Wave equation

The Wave Equation

Source has a bounded support in space Solve wave equation for given initial and boundary conditions The wave equation

material exhibiting relaxation

∇2u − 1 c2 ∂tt(u + χ ∗ u) = −q χ(x, t) is called the relaxation function of the material This function must be causal χ(x, t) = 0 for t < 0 and x ∈ R3

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Introduction Wave equation

The Wave Equation

Applying a one-sided Laplace transform gives Helmholtz’s equation ∇2 ˆ u − ˆ γ2 ˆ u = −ˆ q with a propagation coefficient ˆ γ = ˆ γ0 = s/c0 instanteneous reacting and homogeneous ˆ γ = s/c(x) instanteneous reacting and inhomogeneous ˆ γ2 = ˆ γ2

0[1 + ˆ

χ(x, s)] with relaxation and inhomogeneous

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Introduction Wave equation

Maxwell’s equations

Maxwell’s equations −∇ × H + Jc + ∂tD = −Jext and ∇ × E + ∂tB = 0 Jext [A/m2]: external electric-current source (antenna) E: electric field strength [V/m] H: magnetic field strength [A/m]

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Introduction Wave equation

Maxwell’s equations

Jc [A/m2]: electric conduction current D [C/m2]: electric flux density B [T = Vs/m2]: magnetic flux density How matter reacts to the presence of an EM field is described by the constitutive relations Jc = Jc(E), D = D(E), and B = B(H)

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Introduction Wave equation

Maxwell’s equations

Maxwell’s equations in one dimension ∂yHx + Jc;z + ∂tDz = −Jext

z

and ∂yEz + ∂tBx = 0 Vacuum Instantaneously reacting material Matter exhibiting relaxation

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Introduction Wave equation

Maxwell’s equations

Vacuum Jc;z = 0, Dz = ε0Ez, and Bz = µ0Hz Parameters of vacuum

permittivity of vacuum ε0 permeability of vacuum µ0

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Introduction Wave equation

Maxwell’s equations

Instantaneously reacting material Jc;z = σ(x)Ez, Dz = ε(x)Ez, and Bz = µ(x)Hz Medium parameters

conductivity σ permittivity ε permeability µ

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Introduction Wave equation

Maxwell’s equations

Material exhibiting relaxation Jc;z = 0, Dz = ε0Ez+ε0 t

τ=0

χe(x, t−τ)Ez(x, τ) dτ, Bz = µ0Hz χe relaxation function of the material

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Introduction Wave equation

Maxwell’s equations

Model that is often used (for gold at optical frequencies, for example) χe(t) = (ε∞ − 1)δ(t) + ˜ χe(t) with ˜ χe(t) = ∆ε ω2

p

  • ω2

p − δ2 p

exp(−δpt) sin

  • ω2

p − δ2 p

  • U(t)

U(t): Heaviside unit step function ωp: plasma frequency δp: damping coefficient ∆ε = εs − ε∞

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