David Pardo Extended Abstract ADMOS09

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International Conference on Adaptive Modeling and Simulation

ADMOS 2009

Ph. Bouillard and P. Dez

CIMNE, Barcelona, 2009

A HP FOURIER-FINITE-ELEMENT FRAMEWORKWITH MULTIPHYSICS APPLICATIONS

DAVID PARDO, PAWEL MATUSZYK, MYUNG JIN NAM

Basque Center for Applied Mathematics (BCAM), Spaine-mail: [email protected], web page: http://www.bcamath.org/pardo

Department of Petroleum and Geosystems EngineeringThe University of Texas at Austin, Austin, TX, USA

Key words: Fourier Finite Element Method, Multiphysics, Goal oriented Adaptivity

Abstract. We describe the design of our multiphysics high-order Fourier-Finite Element soft-

ware, and we illustrate its performance by simulating petroleum engineering applications.

1 INTRODUCTION

In order to characterize a reservoir, it is customary to measure different physical phenom-

ena both on the surface of the earth as well as at different logging positions. On the surface,

oil companies typically acquire seismic and possibly marine controlled-source electromagnetic

(CSEM) measurements (Fig. 1 left ). With these measurements, experts on the field study

the expected profitability of the reservoir, possibly with the help of a numerical software based

on the inversion of single-physics measurements (see, e.g. [2]). After performing this first as-

sessment of the reservoir, various wells may be drilled into the subsurface at several locations.

Subsequently, logging instruments based on different physics (such as electromagnetism, acous-

tics and nuclear) are introduced into each borehole, and logging measurements are acquired at

various locations along the trajectory of each well. In this context, more accurate characteri-

zation of reservoirs can be obtained by employing multiphysics simulators for both direct and

inverse problems.

Multiphysics simulators are also needed in other industries and sciences such as the medicine

(multiphysics measurements are utilized for the accurate characterization of tumors), aeronau-tics (multiphysics optimization is needed to design aerodynamic aircraft with low radar cross

section), nano-sciences, material sciences, etc.

Motivated by all these applications, in here we present a framework for solving multiphysics

problems based on the use of a hp-Fourier Finite Element Method (hp-FFEM), which is the firstexisting multiphysics simulator based on this method.

The work presented here is a continuation of [6], where we described a hp-FFEM, and [5],where we described a goal-oriented self-adaptive hp-refinement strategy.

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David Pardo, Pawel Matuszyk, Myung Jin Nam

Figure 1: (a) Marine controlled-source electromagnetics (CSEM) scenario, and (b) logging instrument in a devi-

ated well.

2 METHODOLOGY

Our simulator is based on a high-order Fourier-Finite Element framework suitable for the

computer simulation of a large variety of 2D and 3D multiphysics problems, with special em-

phasis in electromagnetic (EM) applications. The resulting software incorporates a self-adaptive

goal-oriented hp-refinement strategy, where element sizes h and polynomial orders of approx-imation p vary locally throughout the entire computational grid. The software is suitable forboth shared and distributed memory parallel machines. It incorporates LagrangeH1 andNedelecH(curl) elements, and the implementation of Raviart-ThomasH(div) and

L

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elements is currently under development. The framework can be employed to simulate ef-ficiently both multiphysics as well as single-physics problems, and it works efficiently in both

sequential and parallel machines.

hp Fourier-Finite Element Method (hp-FFEM). This method is based on a new geometry-based formulation for simulating 3D resistivity borehole measurements employing a mix of 2D

and 1D algorithms. In so doing, we utilize a 2D self-adaptive goal-oriented hp-adaptive strategy(where h indicates the element size, and p the polynomial order of approximation) combinedwith a Fourier series expansion in a non-orthogonal system of coordinates. This combina-

tion naturally generates a spatial domain decomposition that is used as building block for the

construction of an efficient iterative solver, thereby making unnecessary the use of algebraic

domain-partitioning algorithms. Moreover, the 2D self-adaptive refinement strategy enables

accurate simulations of problems that include high material contrasts (occurring, for example,when simulating a metallic mandrel in an oil-based mud).

Fourier Method. The Fourier transform (or Fourier series expansion in the case of finite size

domains) is selected in a spatial dimension where material coefficients are as smooth as possible.

Thus, the use of a high-order method is justified. We note that the Fourier dimension may not

coincide with a Cartesian direction, and a possibly non-orthogonal change of coordinates may

precede the selection of an appropriate dimension suitable for the Fourier method, as shown

in [6].

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A Self-Adaptive hp Goal-Oriented Algorithm. To determine an optimal distribution of el-

ement size h and polynomial order of approximation p, we employ a goal-oriented self-adaptiverefinement strategy based on the iterative scheme described in [5]. At each step, given an arbi-trary hp-grid, we first perform a global and uniform hp-refinement to obtain the h/2, p+ 1-grid.Second, we approximate the error function in the hp-grid by evaluating the difference betweenthe solutions associated to the hp-and h/2, p + 1-grids. If the error exceeds a user-prescribedtolerance error, then we employ the error function to guide optimal refinements over the hp-grid, and we iterate the process. Once the prescribed tolerance error has been met, we deliver

the h/2, p + 1-grid as the ultimate solution of the problem. This two-dimensional refinementstrategy has been proved to be efficient, robust, and highly accurate for both EM and sonic

problems [3, 6].

Rham diagram. Our implementation employs multiphysics finite elements of variable order

compatible with the so-called de Rham diagram [1] (Fig. 2).

IR H1 H(curl)

H(div)

L2 0

id

curl

div

P

IR Wp Qp

Vp

Wp1 0 .

Figure 2: The de Rham diagram is composed of two exact sequences: one at the continuous level (top), and the

second one at the discrete finite element space level (bottom). Equipped with the projection based interpolationoperators , curl, anddiv, the de Rham diagram commutes.

A Perfectly Matched Layer (PML). A PML is utilized to efficiently truncate the com-

putational domain. For details, see [4], where we demonstrate the robustness of the PML in

presence of anisotropic materials at different frequencies and with high contrast in conductivity

(for example, in cased wells).

Solvers of Linear Equations. To ensure efficiency of the forward simulations, we need fast

solvers of linear equations. We employ both direct (Gauss factorization) and iterative solvers

(see Fig. 3 left panel ).

Parallel Computations. To ensure fast computations, the above method has been imple-

mented in parallel computers for faster execution in multiple core processors as well as in

parallel distributed memory machines.

3 NUMERICAL RESULTS

Left panel of Fig. 3 displays a comparison between a direct solver (MUMPS) and our itera-

tive solver for a logging-while-drilling application. Right panel of Fig. 3 displays the solution

for the marine CSEM problem described in Fig. 1 (left panel). We observe fast convergence as

we increase the number of Fourier modes.

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0 5 10 15 20 250

50

100

150

200

250

Time(sec)

Number of unknowns = 12896 * Number of Fourier Modes

Number of Fourier Modes

Direct SolverIterative Solver

0 2000 4000 6000 8000 1000010

16

1015

1014

1013

1012

1011

1010

AmplitudeofElectricField(V/(Am

2)

Horizontal Distance between TX and RX (m)

With Oil, 0.75 Hz

1 MODE5 MODES

9 MODESEXACT

Figure 3: LEFT PANEL: CPU time used by the direct and iterative solvers, respectively, as a function of the

number of Fourier modes for a particular logging-while-drilling (LWD) application. For the simulations, we have

employed a machine equipped with a 2.0 GHz processor and 8 GB of RAM. RIGHT PANEL: Amplitude of the

electric field as a function of the horizontal distance between transmitter and receivers. Different curves indicate

different numbers of Fourier modes: (a) 1 mode (dotted pink), (b) 5 modes (blue +), (c) 9 modes (black circles),

and (d) exact solution (red solid line). Operating frequency: 0.75 Hz.

REFERENCES

[1] L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz. The Rham diagram for hp finite

element spaces. Computers and Mathematics with Applications, 39(7):2938, 2000.

[2] J. Gunning and M. E. Glinsky. Detection of reservoir quality using Bayesian seismic inversion.

Geophysics, 72:R37R49, 2007.

[3] C. Michler, L. Demkowicz, J. Kurtz, and D. Pardo. Improving the performance of perfectly matched

layers by means ofhp-adaptivity. Numerical Methods for Partial Differential Equations, 23(4):832

858, 2007.

[4] D. Pardo, L. Demkowicz, C. Torres-Verdn, and C. Michler. PML enhanced with a self-adaptive

goal-oriented hp finite-element method and applications to through-casing borehole resistivity mea-

surements. SIAM Journal on Scientific Computing., 30:29482964, 2008.

[5] D. Pardo, L. Demkowicz, C. Torres-Verdn, and L. Tabarovsky. A goal-oriented hp-adaptive finiteelement method with electromagnetic applications. Part I: electrostatics. International Journal for

Numerical Methods in Engineering, 65:12691309, 2006.

[6] D. Pardo, C. Torres-Verdn, M. J. Nam, M. Paszynski, and V. M. Calo. Fourier series expansion in

a non-orthogonal system of coordinates for simulation of 3D alternating current borehole resistivity

measurements. Computer Methods in Applied Mechanics and Engineering, 197:38363849, 2008.

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David Pardo Extended Abstract ADMOS09

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