A Multiphysics Framework Using hp-finite Elements for Electromagnetics Applications

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1 Univ. Carlos III of Madrid A Multiphysics Framework Using hp-finite Elements for Electromagnetics Applications D. Pardo, L. E. García-Castillo, M. J. Nam, C. Torres-Verdín Team: D. Pardo, M. J. Nam, V. Calo, L.E. García-Castillo, I. Garay, M. Paszynski, P. Matuszyk, L. Demkowicz, C. Torres-Verdín March 13th, 2009

2 overview 1. Motivation and Objectives: Joint Multiphysics Inversion 2. Simulation of Forward Problems Parallel Self-Adaptive Goal-Oriented hp-finite Element Method Electromagnetic Applications Sonic Applications 3. Inversion Library (Work in Progress) h-adaptive Newton s Method Implementation 4. Conclusions and Future Work

3 motivation and objectives Seismic Measurements Figure from the USGS Science Center for Coastal and Marine Geology 2

4 motivation and objectives Marine Controlled-Source Electromagnetics (CSEM) Figure from the UCSD Institute of Oceanography 3

5 motivation and objectives Multiphysics Logging Measurements OBJECTIVES: To determine payzones (porosity), amount of oil/gas (saturation), and ability to extract oil/gas (permeability). 4

6 motivation and objectives Main Objective: To Solve a Multiphysics Inverse Problem Software to solve the DIRECT problem is essential in order to solve the INVERSE problem. 5

7 simulation of forward problems (hp-fem) The h-finite Element Method 1. Convergence limited by the polynomial degree, and large material contrasts. 2. Optimal h-grids do NOT converge exponentially in real applications. 3. They may lock (100% error). The p-finite Element Method 1. Exponential convergence feasible for analytical ( nice ) solutions. 2. Optimal p-grids do NOT converge exponentially in real applications. 3. If initial h-grid is not adequate, the p-method will fail miserably. The hp-finite Element Method 1. Exponential convergence feasible for ALL solutions. 2. Optimal hp-grids DO converge exponentially in real applications. 3. If initial hp-grid is not adequate, results will still be great. 6

8 simulation of forward problems (hp-fem) Energy norm based fully automatic hp-adaptive strategy Coarse grids Fine grids (hp) (h/2, p + 1) global hp-refinement global hp-refinement SOL. METHOD ON FINE GRIDS: A TWO GRID SOLVER 7

9 simulation of forward problems (hp-fem) Motivation (Goal-Oriented Adaptivity) Test Problem Infinite Domain Resistivity = 1 Ohm m Frequency = 2 Mhz 50 meters Transmitter (T) Receiver (R) 8

10 simulation of forward problems (hp-fem) Motivation (Goal-Oriented Adaptivity) Infinite Domain Resistivity = 1 Ohm m Frequency = 2 Mhz Solution decays exponentially. E(T) E(R) 1060 Results using energy-norm adaptivity: Transmitter (T) 50 meters Receiver (R) Energy-norm error: 0.001% Relative error in the quantity of interest > %. Goal-oriented adaptivity is needed. Becker-Rannacher (1995,1996), Rannacher-Stuttmeier (1997), Cirak-Ramm (1998), Paraschivoiu-Patera (1998), Peraire-Patera (1998), Prudhomme-Oden (1999, 2001), Heuveline-Rannacher (2003), Solin-Demkowicz (2004). 9

11 simulation of forward problems (hp-fem) Motivation (Goal-Oriented Adaptivity) Infinite Domain Resistivity = 1 Ohm m Frequency = 2 Mhz Solution decays exponentially. E(T) E(R) 1060 Results using energy-norm adaptivity: Transmitter (T) 50 meters Receiver (R) Energy-norm error: 0.001% Relative error in the quantity of interest > %. Goal-oriented adaptivity is needed. Becker-Rannacher (1995,1996), Rannacher-Stuttmeier (1997), Cirak-Ramm (1998), Paraschivoiu-Patera (1998), Peraire-Patera (1998), Prudhomme-Oden (1999, 2001), Heuveline-Rannacher (2003), Solin-Demkowicz (2004). 10

12 simulation of forward problems (hp-fem) Motivation (Goal-Oriented Adaptivity) Transmitter (T) Infinite Domain Resistivity = 1 Ohm m Frequency = 2 Mhz 50 meters Receiver (R) Distance in z axis from transmitter to receiver (in m) Analytical solution Numerical solution Distance in z axis from transmitter to receiver (in m) Analytical solution Numerical solution Amplitude (V/m) Goal-oriented adaptivity is needed Phase (degrees) 11

13 simulation of forward problems (hp-fem) Mathematical Formulation (Goal-Oriented Adaptivity) We consider the following problem (in variational form): { Find L(Ψ), where Ψ V such that : b(ψ, ξ) = f(ξ) ξ V. We define residual r e (ξ) = b(e, ξ). We seek for solution G of: { Find G V V such that : G(r e ) = L(e). This is necessarily solved if we find the solution of the dual problem: { Find G V such that : Notice that L(e) = b(e, G). b(ψ, G) = L(Ψ) Ψ V. 12

14 simulation of forward problems (hp-fem) Algorithm for Goal-Oriented Adaptivity Solve DIRECT and DUAL problems on Grid hp. Solve DIRECT and DUAL problems on Grid h/2, p + 1. Compute e = Ψ h/2,p+1 Ψ hp, and ǫ = G h/2,p+1 G hp. Represent the error as: L(e) = b(e, ǫ) K b K(e, ǫ). Apply the fully automatic hp-adaptive algorithm. Solve DIRECT and DUAL problems on Grid hp. Solve DIRECT and DUAL problems on Grid h/2, p

15 simulation of forward problems (hp-fem) Axisymmetric Logging-While-Drilling (LWD) Simulation ENERGY-NORM HP-ADAPTIVITY Dhp90: A Fully automatic hp-adaptive Finite Element code p=8 p=7 p=6 p=5 p=4 p=3 p=2 p=

16 simulation of forward problems (hp-fem) Axisymmetric Logging-While-Drilling (LWD) Simulation GOAL-ORIENTED HP-ADAPTIVITY (Quadrilateral Elements) Dhp90: A Fully automatic hp-adaptive Finite Element code p=8 p=7 p=6 p=5 p=4 p=3 p=2 p=

17 simulation of forward problems (hp-fem) Axisymmetric Logging-While-Drilling (LWD) Simulation GOAL-ORIENTED HP-ADAPTIVITY (Zoom towards first receiver) Dhp90: A Fully automatic hp-adaptive Finite Element code p=8 p=7 p=6 p=5 p=4 p=3 p=2 p= E

simulation of forward problems (hp-fem) Non-Orthogonal System of Coordinates ζ 3 ζ 2 7 ζ 1 Fourier Series Expansion in ζ 2 DC Problems: σ u = f u(ζ 1, ζ 2, ζ 3 ) = σ(ζ 1, ζ 2, ζ 3 ) = f(ζ 1, ζ 2, ζ 3

18 simulation of forward problems (hp-fem) Non-Orthogonal System of Coordinates ζ 3 ζ 2 7 ζ 1 Fourier Series Expansion in ζ 2 DC Problems: σ u = f u(ζ 1, ζ 2, ζ 3 ) = σ(ζ 1, ζ 2, ζ 3 ) = f(ζ 1, ζ 2, ζ 3 ) = l= l= m= m= n= n= u l (ζ 1, ζ 3 )e jlζ 2 σ m (ζ 1, ζ 3 )e jmζ 2 f n (ζ 1, ζ 3 )e jnζ 2 Fourier modes e jlζ 2 are orthogonal high-order basis functions that are (almost) invariant with respect to the gradient operator. 17

19 simulation of forward problems (hp-fem) De Rham diagram De Rham diagram is critical to the theory of FE discretizations of multi-physics problems. IR W Q V id Π L 2 0 Π curl Π div P IR W p Q p V p W p 1 0. This diagram relates two exact sequences of spaces, on both continuous and discrete levels, and corresponding interpolation operators. 18

20 simulation of forward problems (hp-fem) Tri-Axial Induction Tool 19

21 simulation of forward problems (hp-fem) Tri-Axial Induction Tools in Deviated Wells (0, 30, and 60 degrees) 6 5 Real part of Hzz at 20 khz vertical 30 degrees 60 degrees 100 ohm m 6 5 Real part of Hxx at 20 khz vertical 30 degrees 60 degrees 100 ohm m 6 5 Real part of Hyy at 20 khz vertical 30 degrees 60 degrees 100 ohm m ohm m ohm m ohm m Depth (m) ohm m Depth (m) ohm m Depth (m) ohm m ohm m 2 1 ohm m 2 1 ohm m ohm m 4 20 ohm m 4 20 ohm m Re(Hzz) field (A/m) x 10 6 Re(Hxx) field (A/m) Re(Hyy) field (A/m) Triaxial tools are more sensitive to dip angle effects 20

22 simulation of forward problems (hp-fem) Tri-Axial Induction Tools in Deviated Wells (0, 30, and 60 degrees) 6 5 Imaginary part of Hzz at 20 khz 100 ohm m vertical 30 degrees 60 degrees 6 5 Imaginary part of Hxx at 20 khz 100 ohm m vertical 30 degrees 60 degrees 6 5 Imaginary part of Hyy at 20 khz 100 ohm m vertical 30 degrees 60 degrees ohm m ohm m ohm m Depth (m) ohm m Depth (m) ohm m Depth (m) ohm m ohm m 2 1 ohm m 2 1 ohm m ohm m 4 20 ohm m 4 20 ohm m Im(Hzz) field (A/m) x 10 5 Im(Hxx) field (A/m) Im(Hyy) field (A/m) Triaxial tools are more sensitive to dip angle effects 21

23 simulation of forward problems (hp-fem) DLL Tool 22

24 simulation of forward problems (hp-fem) Groningen Effect 30 Comparison of Groningen Effects on LLd at 100 Hz 30 Comparison of Groningen Effects on LLd at 100 Hz 15 Casing 15 Casing Relative Depth z (m) 0 The earth surface is at z = 2 km or z = 200 m B: at ρ = 1 m, 1 m below the surface Relative Depth z (m) 0 The earth surface is at z = 2 km B: 1 m below the surface, at ρ = 1 m or ρ = 200 m Surface: at z = 2 km Surface: at z = 200 m DC Apparent Resistivity (Ω m) 60 B at ρ = 1 m B at ρ = 200 m DC Apparent Resistivity (Ω m) As we place the current return electrode B farther from the logging instrument, the Groningen effect diminishes 23

25 simulation of forward problems (hp-fem) DC DLL in Deviated Wells 4 2 LLd: Iso LLd: Ani LLs: Iso LLs: Ani Anisotropic Formation (Vertical Well) 4 2 Anisotropic Formation (60 degree Deviated Well) LLd: 60, Iso LLd: 60, Ani LLs: 60, Iso LLs: 60, Ani 0 Resistivity of Formation 0 Resistivity of Formation Relative Depth (m) Vertical Resistivity Vertical Resistivity Relative Depth (m) Vertical Resistivity Vertical Resistivity Apparent Resistivity (Ω m) Apparent Resistivity (Ω m) Anisotropy is better identified when using deviated wells 24

26 simulation of forward problems (hp-fem) Final hp-grid and solution 8 KHz, acoustics, open borehole setting (no logging instrument). 25

27 new library for inverse problems Notation: Variational Formulation (DC) B(u, v; σ) =< v, σ u > L 2 (Ω) (bilinear u, v) F i (v) =< v, f i > L 2 (Ω) + < v, g i > L 2 ( Ω) (linear v) L i (u) =< l i, u > L 2 (Ω) + < h i, u > L 2 ( Ω) (linear u) Direct Problem (homogeneous Dirichlet BC s): { Find û i V such that : Dual (Adjoint) Problem: { B(û i, v; σ) = F i (v) v V Find ˆv i V such that : B(u, ˆv i ; σ) = L i (u) u V 26

28 new library for inverse problems Notation: Variational Formulation (AC) B(E, F; σ) =< F, µ 1 E > L 2 (Ω) < F, (ω 2 ǫ jωσ)e > L 2 (Ω) F i (F) = jω < F, J imp i > L 2 (Ω) +jω < F, J imp S,i > L 2 ( Ω) L i (E) =< J adj i, E > L 2 (Ω) + < J adj S,i, E > L 2 ( Ω) Direct Problem (homogeneous Dirichlet BC s): { Find Ê i W such that : Dual (Adjoint) Problem: { B(Ê i, F; σ) = F i (F) F W Find ˆF i W such that : B(E, ˆF i ; σ) = L i (E) E W 27

29 new library for inverse problems Constrained Nonlinear Optimization Problem Cost Functional: { Find σ > 0 such that it minimizes C β (σ), where : C β (σ) = W m (L(ûσ) M) 2 l 2 + β R(σ σ 0 ) 2 L 2, where M i denotes the i-th measurement, M = (M 1,..., M n ) L i is the i-th quantity of interest, L = (L 1,..., L n ) n M 2 l 2 = M 2 i ; R(σ σ 0 ) 2 L 2 = (R(σ σ 0 )) 2 i=1 β is the relaxation parameter, σ 0 is given, W m are weights Main objective (inversion problem): Find ˆσ = min σ>0 C β(σ) 28

30 new library for inverse problems Solving a Constrained Nonlinear Optimization Problem We select the following deterministic iterative method: σ (n+1) = σ (n) + α (n) δσ (n) How to find a search direction δσ (n)? We will employ a change of coordinates and a truncated Taylor s series expansion. How to determine the step size α (n)? Either with a fixed size or using an approximation for computing L(σ (n) + α (n) δσ (n) ). How to guarantee that the nonlinear constraints will be satisfied? Imposing the Karush-Kuhn-Tucker (KKT) conditions or with a penalization method, or via a change of variables. 29

31 new library for inverse problems Change of coordinates: Search Direction Method h(s) = σ => Find ŝ = min h(s)>0 C β(s) Taylor s series expansion: A) C β (s + δs) C β (s) + δs C β (s) + 0.5δs 2 H Cβ (s) B) L(s + δs) L(s) + δs L(s), R(s + δs) = R(s) + δs R(s) Expansion A) leads to the Newton-Raphson method. Expansion B) leads to the Gauss-Newton method. Expansion A) with H Cβ = I leads to the steepest descent method. Higher-order expansions require from higher-order derivatives. 30

32 new library for inverse problems Computation of Jacobian Matrix Using the Fréchet Derivative: ( ) L i (û i ) ûi = B, ˆv i, h(s) +B s j s j ( ) ( ) ûi ûi L i = B, ˆv i, h(s) s j s j ( ) ˆvi F i = B s j ( û i, ˆv ) i, h(s) s j ( û i, ˆv ) i, h(s) s j +B ( û i, ˆv i, h(s) ) s j Therefore, we conclude: Jacobian Matrix = L i(û i ) s j = B ( û i, ˆv i, h(s) ) s j 31

33 new library for inverse problems Jacobian Function: One TX, one RX 32

34 new library for inverse problems Jacobian Function: One TX, one RX 33

35 new library for inverse problems Jacobian Function: One TX, one RX 34

36 new library for inverse problems Computation of Hessian Matrix Following a similar argument as for the Jacobian matrix, we obtain: 2 ( L i (û i ) ûi = B, ˆv i, h(s) ) ( B û i, ˆv i, h(s) ) ( ) B û i, ˆv i, 2 h(s) s j s k s j s k s j s k s j s k How do we compute û i and ˆv i? s j s j Find û ( ) i ûi such that : B, v i, h(s) s j s j Find ˆv i s j such that : B ( ˆvi s j, u i, h(s) ) = B = B ( û i, v i, h(s) ) s j ( ˆv i, u i, h(s) ) s j We can compute the Hessian matrix EXACTLY by just solving our original problem for different right-hand-sides, and performing additional integrations. v i u i 35

37 new library for inverse problems Hessian Function: One TX, one RX 36

38 new library for inverse problems Hessian Function: One TX, two RXs 37

39 new library for inverse problems Hessian Function: One TX, three RXs 38

40 new library for inverse problems Algorithms implemented within the inverse library JOINT MULTI PHYSICS INVERSION LIBRARY CONSTRAINED OPTIMIZATION SEARCH DIRECTIONS STEP SIZE 1. KKT Conditions 1. Steepest Descent 2. Penalization Method 2. Gauss Newton 3. Change of coordinates 3. Newton Raphson 1. Uniform step size 2. Variable step size (multiple algorithms) The inverse library is composed of multiple algorithms for imposing constraints, and finding search directions and corresponding step sizes. Jacobian and Hessian matrices are computed exactly by simply solving the dual (adjoint) formulation and performing additional integrations. The inverse library is compatible with multi-physics problems. 39

41 conclusions We have described an efficient numerical method for solving PDE s based on a self-adaptive goal-oriented hp refinement strategy. We are developing a multiphysics version of the code. We are building a new module for the resolution of inverse problems. Our main objective is to create a software infrastructure enabling solution of challenging multiphysics inverse problems with applications to geophysics (hydrocarbon detection and monitoring, etc.), aeronautics and medicine. To achieve this objective, we need Ph.D. students, post-doctoral fellows, experienced researchers, and collaborators in different areas (inversion, solvers, etc). 40

Multiphysics, Inversion, and Petroleum

CIC biogune Research Line II: Multiphysics, Inversion, and Petroleum Research Professor at BCAM Team: D. Pardo, I. Garay, I. Andonegui, J. Álvarez Collaborators: P. de la Hoz, M. Paszynski, L.E. García-Castillo,