A self-adaptive goal-oriented hp finite element method with electromagnetic applications. Part II: Electrodynamics

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1 Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx A self-adaptive goal-oriented finite element method with electromagnetic applications. Part II: Electrodynamics D. Pardo a,b, *, L. Demkowicz a, C. Torres-Verdín a,b, M. Paszynski a,1 a Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, T 7871, United States b Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, T 7871, United States Received 1 November 005; accepted 30 October 006 Abstract We present the formulation, implementation, and applications of a self-adaptive, goal-oriented, -Finite Element (FE) Method for Electromagnetic (EM) problems. The algorithm delivers (without any user interaction) a sequence of optimal -grids. This sequence of grids minimizes the error in a prescribed quantity of interest with respect to the problem size, and it converges exponentially in terms of the relative error in a user-prescribed quantity of interest against the CPU time, including problems involving high material contrasts, boundary layers, and/or several singularities. The goal-oriented refinement strategy is an extension of a fully automatic, energy-norm based, -adaptive algorithm. We illustrate the efficiency of the method with D numerical simulations of Maxwell s equations using both H 1 -conforming (continuous) elements and H(curl)-conforming (Nédélec edge) elements. Applications include alternate current (AC) resistivity logging instruments in a borehole environment with steel casing for the assessment of rock formation properties behind casing. Logging instruments, steel casing, and rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. For the presented challenging class of problems, the self-adaptive goal-oriented -FEM delivers results with 5 7 digits of accuracy in the quantityof-interest. Ó 007 Elsevier B.V. All rights reserved. eywords: -finite elements; Exponential convergence; Goal-oriented adaptivity; Computational electromagnetism; Maxwell s equations; Through casing resistivity tools (TCRT) 1. Introduction During the last decades, different algorithms have been designed and implemented to generate optimal grids in the solution of relevant engineering problems. Among those algorithms, a self-adaptive, energy-norm based, - Finite Element (FE) refinement strategy has been developed at the Institute for Computational Engineering and Sciences (ICES) of The University of Texas at Austin. * Corresponding author. Address: Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, T 7871, United States. Tel.: ; fax: address: dzubiaur@gmail.com (D. Pardo). 1 Department of Computer Science, AGH University of Science and Technology, Cracow, Poland. The strategy produces automatically a sequence of meshes that delivers exponential convergence rates in terms of the energy-norm error against the number of unknowns (as well as the CPU time), independently of the number and type of singularities in the problem. Thus, it provides high accuracy approximations of solutions corresponding to a variety of engineering applications. Furthermore, the self-adaptive strategy is problem independent, and it can be applied to FE discretizations of H 1 -, H(curl)-, and H(div)-spaces, as well as to nonlinear problems (see [7,18] for details). The self-adaptive strategy iterates along the following steps. A given (coarse) conforming mesh is first globally refined in both h and p to yield a fine mesh, i.e. each element is broken into eight new elements, and the discretization order of approximation p is raised uniformly by one /$ - see front matter Ó 007 Elsevier B.V. All rights reserved. doi: /j.cma Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

2 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx Subsequently, the problem of interest is solved on the fine mesh. By subtracting the coarse grid solution from the fine grid solution, we obtain a representation of the error function over the coarse grid, which allows for determining optimal refinements by maximizing the decrease of the projection based interpolation error [6] averaged by the added number of unknowns. The final fine grid solution is provided to the user as the final result. Notice that each fine grid problem may contain typically about 0 30 times more unknowns than the corresponding coarse grid problem. A two-grid solver may be used to efficiently approximate the fine grid solution (see [14,13] for details). Its use is critical for large-scale 3D computations. The energy-norm is a quantity of limited relevance for most engineering applications, especially when a particular objective is pursued, for instance, to simulate the electromagnetic (EM) response of geophysical resistivity logging instruments in a borehole environment. In these instruments, the amplitude of the measurement (for example, the electric field) is typically several orders of magnitude smaller at the receiver antennas than at the transmitter antennas. Thus, small relative errors of the solution in the energy-norm do not imply small relative errors of the solution at the receiver antennas. Indeed, it is not uncommon to construct energy-norm based adaptive grids delivering a relative error in the energy-norm below 1%, while the solution at the receiver antennas still exhibits a relative error above 1000% (see [1]). Consequently, a self-adaptive strategy is needed to approximate a specific feature of the solution. Refinement strategies of this type are called goal-oriented adaptive algorithms [10,17], and are based on minimizing the error of a prescribed quantity of interest mathematically expressed in terms of a linear functional (see [,9,11,10,17,19] for details). This paper formulates, implements, and studies (both theoretically and numerically) a self-adaptive goal-oriented algorithm intended to solve electrodynamic problems. The algorithm is an extension of the fully automatic (energy-norm based) -adaptive strategy described in [7,18], and a continuation of concepts presented in [15,0] for elliptic problems. The organization of this document is as follows: in Section, we introduce Maxwell s equations, governing the EM phenomena. We also derive four corresponding variational formulations for axisymmetric problems. A selfadaptive goal-oriented -algorithm for electrodynamic problems is described in Section 3. The corresponding details of implementation are discussed in the same section. In Section 4, we illustrate the efficiency of the method with axisymmetric D numerical simulations of TCRT. Finally, in Section 5 we draw the main conclusions, and outline future lines of research.. Maxwell s equations In this section, we first introduce the time-harmonic Maxwell s equations in the frequency domain. They form a set of first-order Partial Differential Equations (PDE s). Then, we describe boundary conditions needed for the simulation of our logging applications of interest. Finally, we derive a variational formulation in terms of either the electric or the magnetic field, and we reduce the dimension of the computational problem by considering axial symmetry..1. Time-harmonic Maxwell s equations Assuming p a time-harmonic dependence of the form e jxt, where j ¼ ffiffiffiffiffiffi 1 is the imaginary unit, t denotes time, and x 6¼ 0 is the angular frequency, Maxwell s equations can be written as 8 $H ¼ðrþjxÞEþJ imp Ampere s Law; >< $E ¼ jxlh M imp Faraday s Law; $ ðeþ¼q Gauss Law of Electricity; and >: $ ðlhþ¼0 Gauss Law of Magnetism: ð:1þ Here H and E denote the magnetic and electric field, respectively, J imp is a prescribed, impressed electric current density, M imp is a prescribed, impressed magnetic current density, tensors ; l; and r stand for dielectric permittivity, magnetic permeability, and electrical conductivity of the medium, respectively, and q denotes the electric charge distribution. We assume detðlþ 6¼ 0, and detðr þ jxþ 6¼ 0. The equations described in (.1) are to be understood in the distributional sense, i.e. they are satisfied in the classical sense in subdomains of regular material data, and they also imply appropriate interface conditions across material interfaces. Energy considerations lead to the assumption that both electric field E and magnetic field H must be square integrable. According to Eqs. (.1) and (.1) 4, M imp must be divergence free. Maxwell s equations are not independent. Taking the divergence of Faraday s Law yields Gauss Law of magnetism. By taking the divergence of Ampere s Law, and by utilizing Gauss Electric Law we arrive at the so called continuity equation $ ðreþþjxq þ $ J imp ¼ 0: ð:þ.. Boundary conditions (BC s) There exists a variety of BC s that can be incorporated into Maxwell s equations. In the following, we describe those BC s that are of interest for the logging applications discussed in this paper. At this point, we are considering general 3D domains. A discussion on boundary terms corresponding to the axisymmetry condition is included in Section Perfect electric conductor (PEC) Maxwell s equations are to be satisfied in the whole space minus domains occupied by a PEC. A PEC is an ide- Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

3 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx 3 alization of a highly conductive media. Inside a region where r ¼ diagðr c ; r c ; r c Þ with r c!1, the corresponding electric field converges to zero by applying Ampere s law. Faraday s law implies that the tangential component of the electric field E must remain continuous across material interfaces in the absence of impressed magnetic surface currents. Consequently, the tangential component of the electric field must vanish along the PEC boundary, i.e., ^n E ¼ 0; ð:3þ where ^n is the unit normal (outward) vector. Since the electric field vanishes inside a PEC, Faraday s law implies that the magnetic field should also vanish inside a PEC in the absence of magnetic currents. The same Faraday s law implies that the normal component of the magnetic field premultiplied by the permeability tensor must remain continuous across material interfaces. Therefore, the normal component of lh must vanish along the PEC boundary, i.e., ^n ðlhþ ¼0: ð:4þ The tangential component of the magnetic field (surface current) and normal component of the electric field (surface charge density) need not be zero, and may be determined a posteriori. Notice that the effect of a PEC may be reproduced by imposing either a homogeneous essential BC in terms of the electric field, or a homogeneous natural BC in terms of the magnetic field.... Source antennas Antennas are modeled by prescribing an impressed volume electric current J imp or an impressed volume magnetic current M imp. Using the equivalence principle (see, for example, [8]), we can replace the original impressed electric volume current J imp with an equivalent electric surface current J imp S ¼½^n HŠ S ð:5þ defined on an arbitrary surface S enclosing the support of J imp, where ½^n HŠ S denotes the jump of ^n H across S in the case of an interface condition, or simply ^n H on S in the case of a boundary condition. Similarly, an impressed magnetic volume current M imp can be replaced by the equivalent magnetic surface current M imp S ¼ ½^n EŠ S ð:6þ defined on an arbitrary surface S enclosing the support of M imp. This result is true under the physical consideration that impressed volume current J imp and re should remain finite, i.e., hj imp ; wi, hre; wi < 1 for every test function w. See [16] for details...3. Closure of the domain We consider a bounded computational domain. A variety of BC s can be imposed on the boundary o such that the difference between the solution of such a problem and the solution of the original problem defined over R 3 is small. For example, it is possible to use an infinite element technique (as described in [3]). Also, since the EM fields and their derivatives decay exponentially in the presence of lossy media (non-zero conductivity), we may simply impose a homogeneous essential or natural BC on the boundary of a sufficiently large computational domain. In the field of geophysical logging applications, it is customary to impose a homogeneous essential BC on the boundary of a large computational domain (for example, 0 m in each direction from a MHz source antenna in the presence of a resistive media). We will follow the same approach. According to the boundary conditions discussed above, we will represent the boundary C ¼ o as the disjoint union of C E, where M imp C E ¼ ½^n EŠ CE ðwith M imp C E possibly zeroþ; and ð:7þ, where J imp ¼½^n HŠ CH ðwith J imp possibly zeroþ: ð:8þ.3. Variational formulation From Maxwell s equations and the BC s described above, we derive the corresponding standard variational formulation in terms of the electric or magnetic field as follows. First, we notice from Faraday s law that $ E ðl ðþþ 3 if and only if M imp ðl ðþþ 3. Since our objective is to find a solution E H ðcurl; Þ¼ ff ðl ðþþ 3 : $ F ðl ðþþ 3 g, we shall assume in the case of the electric field formulation (E-formulation) derived below that M imp ðl ðþþ 3. If the prescribed M imp Hðcurl; Þ 0 ðl ðþþ 3, we may still solve Maxwell s equations with H ðcurl; Þ-conforming finite elements for the magnetic field by using the H-formulation (Section.3.), or simply by prescribing an equivalent source fm imp ðl ðþþ 3 such that M imp f M imp does not radiate outside the antenna [1]. Similarly, for the H-formulation, we shall assume that J imp ðl ðþþ 3. If the prescribed J imp Hðcurl; Þ 0 ðl ðþþ 3, we may still solve Maxwell s equations with H ðcurl; Þ-conforming finite elements for the electric field by using the E-formulation (Section.3.1), or simply by prescribing an equivalent source e J imp ðl ðþþ 3 such that J imp e J imp does not radiate outside the antenna E-formulation First, we multiply the inverse of the magnetic permeability tensor by Faraday s law. Then, we multiply the resulting Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

4 4 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx equation by $ F, where F H CE ðcurl; Þ ¼fF Hðcurl; Þ : ðn FÞj CE ¼ 0g is an arbitrary test function, and the symbol - denotes complex conjugate. Finally, integrating over the domain, we arrive at the identity ðl 1 $ EÞð$ FÞdV ¼ jx H ð$ FÞdV ðl 1 M imp Þð$ FÞdV : ð:9þ By integrating R H ð$ FÞdV by parts, and by applying Ampere s law, we obtain H ð$ FÞdV ¼ ð$ HÞFdV ^n H F t ds C N ¼ ðr þ jxþe FdV þ J imp FdV ^n H F t ds; ð:10þ where F t ¼ F ðf^nþ^n is the tangential component of vector F on,and^n is the unit outward normal (with respect if o) vector. Substitution of (.10) into (.9), together with Eq. (.5) yields the following variational formulation: Find E E CE þ H CE ðcurl; Þ such that : ðl 1 $ EÞð$ FÞdV ðk EÞFdV ¼ jx J imp FdV þ jx ðl 1 M imp Þð$ FÞdV J imp F t ds 8F H CE ðcurl; Þ; ð:11þ where k ¼ x jxr, and E CE is a lift (typically E CE ¼ 0) of the essential boundary condition data E CE (denoted with the same symbol) related to M imp C E by (.7). Conversely, we can derive (.1), (.3), and (.5) from the variational problem (.11)..3.. H-formulation First, we multiply ðr þ jxþ 1 by Ampere s law. Then, we multiply the resulting equation by $ F, where F H CH ðcurl; Þ ¼fFHðcurl; Þ : ðn FÞj CH ¼ 0g is an arbitrary test function. Finally, by integrating over the domain, we arrive at the identity jx ðk Þ 1 ð$ HÞ ð$ FÞdV ¼ E ð$ FÞdV jx ðk Þ 1 J imp ð$ FÞdV : ð:1þ By integrating R E ð$ FÞdV by parts, and by applying Faraday s law, we obtain E ð$ FÞdV ¼ ¼ jx ðlhþfdv ð$ EÞFdV ^n E CE F t ds C E M imp FdV ^n E CE F t ds: C E ð:13þ Substitution of (.13) into (.1), together with Eq. (.6) yields the following variational formulation: Find H H CH þ H CH ðcurl; Þ such that : h i ðr þ jxþ 1 ð$ HÞ ð$ FÞdV þ jx ðlhþfdv ¼ M imp FdV þ M imp C E F t ds C E h i þ ðr þ jxþ 1 J imp ð$ FÞdV 8F H CH ðcurl; Þ; ð:14þ where H CH is a lift (typically H CH ¼ 0) of the essential boundary condition data H CH (denoted with the same symbol) related to J imp by (.8)..4. Cylindrical coordinates and axisymmetric problems We consider a cylindrical coordinate system ðq; /; zþ. For the geophysical logging applications considered in this article, we assume that both the logging instrument and the rock formation properties are axisymmetric (invariant with respect to the azimuthal component /) around the axis of the borehole. Under this assumption, we obtain that for any vector field A ¼ ^qa q þ ^/A / þ ^za z $ A ¼ ^q oa / oz þ ^/ oa q oz oa z oq N q;z þ ^z 1 oðqa / Þ q oq : ð:15þ We also assume that tensors r, l, and, describing the material properties of the medium, can be represented as 3 N qq 0 N qz 6 7 N ¼ 4 0 N // 0 5 N zq 0 N zz 3 3 N qq 0 N qz ¼ þ 4 0 N // 0 5 ¼ N q;z þ N / ; ð:16þ N zq 0 N zz fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} where N denotes either r, l, or. According to this decomposition in terms of the meridian and azimuthal components (N q;z, and N /, respectively), we define the meridian material properties r q;z, l q;z, and q;z, and the azimuthal material properties r /, l /, and /. We also define k q;z ¼ x q;z jxr q;z, and k / ¼ x / jxr /. In order to simplify the notation, we introduce for every vector quantity F ¼ðF q ; F / ; F z Þ, its meridian part F q;z ¼ Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl. N /

5 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx 5 ðf q ; 0; F z Þ, and its azimuthal part F / ¼ð0; F / ; 0Þ, so that F ¼ F q;z þ F / E-formulation Next, we consider the space of all test functions F H CE ðcurl; Þ such that F ¼ F / ¼ð0; F / ; 0Þ. According to (.15) $ F ¼ ^q of / oz þ ^z 1 q oðqf / Þ : ð:17þ oq The variational formulation (.11) reduces to a formulation in terms of the scalar field E / only, namely, Find E / E /;CE þ eh 1 C E ðþ such that : ðl 1 q;z $ E /Þð$ F / ÞdV ðk /E / ÞF / dv ¼ jx J imp / F / dv þ jx J imp /; F / ds ðl 1 q;z Mimp q;z ÞF / dv 8F / eh 1 C E ðþ; ð:18þ where eh 1 C E ðþ ¼fE / : E / H CE ðcurl; Þg ¼ E / L ðþ : 1 q E / þ oe / oq L ðþ; oe / oz L ðþ; E / j CE ¼ 0g. Similarly, for a test function F ¼ F q;z, the variational problem (.11) simplifies to Find ðe q ;E z ÞE CE þ eh CE ðcurl;þ such that : ðl 1 / $ E q;zþð$ F q;z ÞdV ð k q;z E q;z ÞF q;z dv ¼ jx J imp q F q þ J imp z F z dv þ jx J imp q; F q þ J imp z; F z ds C H ðl 1 / Mimp / ÞF q;z dv 8ðF q ;F z Þ eh CE ðcurl;þ; where eh CE ðcurl; Þ ¼ ðe q ; E z ÞðL ðþþ : ð$ E q;z Þj / ð:19þ q L ðþ; ð^n E q;z Þj CE ¼ 0 : In summary, problem (.11) decouples into a system of two simpler problems described by (.18) and (.19). Remark 1. If jr /;q jþjr q;/ jþjr /;z jþjr z;/ j 6¼ 0, then r cannot be decomposed as the sum of its meridian and azimuthal components and, as a result, we cannot replace problem (.11) with two simpler problems of lower dimension. Similarly, conditions jl /;q jþjl q;/ jþ jl /;z jþ jl z;/ j¼0, and j /;q jþj q;/ jþj /;z jþj z;/ j¼0 are essential in order to derive the formulations (.18) and (.19). Remark. It has been shown in [1, Lemma 4.9] that the space eh 1 C E ðþ can also be expressed as eh 1 C E ðþ ¼ E / L ðþ : 1 E q / L ðþ; r ðq;zþ E / L ðþ; E / j CE ¼ 0g..4.. H-formulation Using the same decomposition of test functions (i.e., F ¼ F /,andf ¼ F q;z ) for variational problem (.14), we arrive at the following two decoupled variational problems in terms of H / (.0), andh q;z (.1), respectively Find H / H /;CH þ eh 1 ðþ such that : h i ðr q;z þ jx q;z Þ 1 $ H / ð$ F / ÞdV þ jx ðl / H / ÞF / dv ¼ M imp / F / dv þ M imp /;C E F / ds C E h i þ ðr q;z þ jx q;z Þ 1 J imp q;z $ F / dv 8F / eh 1 ðþ; ð:0þ where eh 1 ðþ ¼fE / : E / H CH ðcurl; Þg ¼ E / L ðþ : 1 q E / þ oe / oq L ðþ; oe / oz L ðþ; E / j CH ¼ 0 : Find H ¼ðH q ; H z ÞH CH þ eh CH ðcurl; Þ such that : ½ðr / þ jx / Þ 1 $ H q;z Šð$ F q;z ÞdV þ jx ðl q;z H q;z ÞF q;z dv ¼ M imp q F q þ M imp z F z dv þ M imp q;c E F q þ M imp z;c E F z ds C E þ ½ðr / þ jx / Þ 1 J imp / Šð$ F q;zþdv 8ðF q ; F z Þ eh CH ðcurl; Þ; where eh CH ðcurl; Þ ¼ ðe q ; E z ÞðL ðþþ ; ð$ E q;z Þj / ð:1þ ¼ oe q oz oe z oq L ðþ; ð^n EÞj CH ¼ 0 : From the formulation of problems (.18) (.1), we remark the following: Physically, the solution of problems (.19), and (.0) correspond to the TE / -mode (i.e. E / ¼ 0), whereas the solution of problems (.18) and (.1) correspond to the TM / -mode (i.e. H / ¼ 0). The axis of symmetry is not a boundary of the original 3D problem, and therefore, a boundary condition (BC) is not needed to solve this problem. Nevertheless, formulations of problems (.18) (.1) require the use of spaces eh 1 C E ðþ; eh 1 ðþ and eh CE ðcurl; Þ; eh CH ðcurl; Þ described above. The two former spaces involve the singu- Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

6 6 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx lar weight 1, which implicitly requires a homogeneous q essential boundary condition along the axis of symmetry. The two latter spaces are two-dimensional H ðcurlþ spaces with no BC on the axis of symmetry. Thus, no BC is used to solve the problem. From the computational point of view, the effect of having no BC can be achieved by artificially adding a homogeneous natural BC. 3. Self-adaptive goal-oriented -FEM We are interested in solving the variational problems.11 and.14 (or alternatively, (.18) (.0) and (.1)), that we state here in terms of sesquilinear form b, and antilinear form f Find E E D þ V; ð3:þ bðe; FÞ ¼f ðfþ 8F V; where E D is a lift of the essential BC. V is a Hilbert space. f V 0 is an antilinear and continuous functional on V. b is a sesquilinear form. We have 8 ðl 1 $ EÞð$ FÞdV k E FdV E-Form; fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} >< a bðe;fþ¼ E ðe;fþ ðk 1 $ EÞð$ FÞdV >: fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} a H ðe;fþ c E ðe;fþ le FdV fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} c H ðe;fþ H-Form; ð3:3þ where sesquilinear forms a E, a H, c E and c H are Hermitian, continuous and semi-positive definite. We define an energy inner product on V as 8 ðl 1 $ EÞð$ FÞdV þ jk je FdV E-Form; fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} >< a ðe;fþ :¼ E ðe;fþ ðjk j 1 $ EÞð$ FÞdV >: fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ~a H ðe;fþ þ ~c E ðe;fþ le FdV fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} c H ðe;fþ ; H-Form; ð3:4þ with the corresponding (energy) norm denoted by kek. The symbol jk j refers to a matrix of the same dimensions as k with its entries being the absolute value of the entries corresponding to k. At this point, we have assumed that all entries in the tensors included in Eq. (3.4) are non-negative Representation of the error in the quantity of interest Given an -FE subspace V V, we discretize (3.) as follows: Find E E D þ V ; ð3:5þ bðe ; F Þ¼fðF Þ 8F V : The objective of goal-oriented adaptivity is to construct an optimal -grid, in the sense that it minimizes the problem size needed to achieve a given tolerance error for a given quantity of interest L, with L denoting a linear and continuous functional. By recalling the linearity of L, we have Error of interest ¼ LðEÞ LðE Þ¼LðE E Þ¼LðeÞ; ð3:6þ where e ¼ E E denotes the error function. By defining the residual r V 0 as r ðfþ ¼f ðfþ bðe ; FÞ ¼ bðe E ; FÞ ¼bðe; FÞ, we seek the solution of the dual problem ( Find W V; ð3:7þ bðf; WÞ ¼LðFÞ 8F V: Problem (3.7) has a unique solution in V. The solution W, is usually referred to as the influence function. By discretizing (3.7) via, for example, V V, we obtain ( Find W V ; ð3:8þ bðf ; W Þ¼LðF Þ 8F V : The definition of the dual problem plus the Galerkin orthogonality for the original problem imply the final representation formula for the error in the quantity of interest, namely LðeÞ ¼bðe; WÞ ¼bðe; W F Þ¼ fflfflfflffl{zfflfflfflffl} ~ bðe; Þ: At this point, F V is arbitrary, and ~ bðe; Þ ¼bðe; Þ denotes the bilinear form corresponding to the original sesquilinear form. Notice that, in practice, the dual problem is solved not for W but for its complex conjugate W utilizing the bilinear form and not the sesquilinear form. The linear system of equations is factorized only once, and the extra cost of solving (3.8) reduces to only one backward and one forward substitution (if a direct solver is used). Once the error in the quantity of interest has been determined in terms of the bilinear form ~ b, we wish to obtain a sharp upper bound for jlðeþj that depends upon the mesh parameters (element size h and order of approximation p) only locally. Then, a self-adaptive algorithm will be defined with the intent to minimize this bound. First, using a procedure similar to the one described in [7], we approximate E and W with fine grid functions Eh ;pþ1; Wh ;pþ1, which have been obtained by solving the corresponding linear system of equations associated with the FE subspace Vh ;pþ1. In the remainder of this article, E and W will denote the fine grid solutions of the direct and dual problems (E ¼ Eh ;pþ1, andw ¼ Wh ;pþ1, respectively), and we will restrict ourselves to discrete FE spaces only. Next, we bound the error in the quantity of interest by a sum of element contributions. Let b denote a contribution from element to the sesquilinear form b. It then follows that: jlðeþj ¼ jbðe; Þj 6 jb ðe; Þj; ð3:9þ Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

7 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx 7 where summation over indicates summation over elements. Note again that e denotes now the difference between the fine and coarse grid solutions. Similarly is the difference between the fine grid approximation of the influence function and an arbitrary function F V. 3.. Projection based interpolation operator Once we have a representation formula for the error in the quantity of interest in terms of the sum of element contributions given by (3.9), we wish to express this upper bound in terms of local quantities, i.e. in terms of quantities that do not vary globally when we modify the grid locally. For this purpose, we introduce the idea of projection-based interpolation operators. First, in order to simplify the notation, we define the following three spaces of admissible solutions: V ¼ H D ðcurl; Þ, V D ¼ eh D ðcurl; Þ, and, V 1D ¼ eh 1 D ðþ. The corresponding -Finite Element spaces will be denoted by V, V D 1D, and V, respectively. At this point, we introduce three projection-based interpolation operators that have been defined in [6,5], and used in [7,18] for the construction of the fully automatic energynorm based -adaptive algorithm: P curl;3d : V! V, P curl;d : V D! V D,and, P 1D : V 1D! V 1D. We shall also consider three Galerkin projection operators: P curl;3d : V! V, P curl;d : V D! V D, and, P 1D : V 1D! V 1D. To further simplify the notation, we will utilize the unique symbol P curl to denote all projection based interpolation operators mentioned above. Depending upon the problem formulation (and corresponding space of admissible solutions), P curl should be understood as P curl;3d for problems (.11) and (.14), P curl;d for problems (.19) and (.1),orP 1D for problems (.18) and (.0). Similarly, we will use the unique symbol P curl to denote either P curl;3d, P curl;d,orp 1D. We denote E ¼ P curl E. Eq. (3.9) then becomes jlðeþj 6 jb ðe; Þj ¼ b ðe P curl E; Þþb ðp curl E Pcurl E; Þ: ð3:30þ Given an element, we conjecture that jb ðp curl E P E; Þj will be negligible compared to jb ðe P curl E; Þj. Under this assumption, we conclude that jlðeþj jb ðe P curl E; Þj: ð3:31þ In particular, for ¼ W P curl W, we have jlðeþj b E P curl E; W Pcurl W : ð3:3þ By applying Cauchy Schwartz s inequality, we obtain the next upper bound for jlðeþj jlðeþj k~ek k~k ; ð3:33þ where ~e ¼ E P curl E, ~ ¼ W Pcurl W,andkk denotes energy-norm kkrestricted to element Fully automatic goal-oriented -refinement algorithm We describe an self-adaptive algorithm that utilizes the main ideas of the fully automatic (energy-norm based) -adaptive algorithm described in [7,18]. We start by recalling the main objective of the self-adaptive (energynorm based) -refinement strategy, which consists of solving the following maximization problem: 8 Find an optimal -grid ~ in the following sense : >< E P P curl E E P curl E b >: f ¼ arg max ; DN b ð3:34þ where E ¼ Eh ;pþ1 is the fine grid solution, and DN > 0 is the increment in the number of unknowns from grid to grid c. Similarly, for goal-oriented -adaptivity, we propose the following algorithm based on estimate (3.33): 8 Find an optimal -grid ~ in the following sense : ~ P E Pcurl E W Pcurl W ¼ arg max 6 DN >< b 4 >: where E P curl E b W P curl W b DN ; ð3:35þ E ¼ Eh ;pþ1 and W ¼ Wh ;pþ1 are the fine grid solutions corresponding to the direct and dual problems, and DN > 0 is the increment in the number of unknowns from grid to grid c. Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

8 8 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx The implementation of the goal-oriented -adaptive algorithm is based on the optimization procedure used for the energy-norm -adaptivity [7,18], which utilizes a multi-step approach (first optimization of edges, and then optimization of interior degrees of freedom). The subspace associated with an optimal FE grid is always contained in the subspace associated with the FE fine grid computed during the previous step Implementation details We outline shortly the main implementation details needed to extend the fully automatic (energy-norm based) -adaptive algorithm [7,18] to a fully automatic goal-oriented -adaptive algorithm: (1) First, the solution W of the dual problem on the fine grid is necessary. This goal can be attained either by using a direct (frontal) solver or an iterative (twogrid) solver (see [1]). () Subsequently, we treat both solutions as satisfying two different partial differential equations (PDE s). We select the functions E and W as the solutions of the system of two PDE s. (3) We proceed to redefine the evaluation of the error. The energy-norm error evaluation of a two-dimensional function is replaced with the product ke P curl EkkW Pcurl Wk. (4) After these simple modifications, the energy-norm based self-adaptive algorithm may now be utilized as a self-adaptive goal-oriented algorithm. 4. Numerical results In this section, we first describe a model problem corresponding to an alternating current (AC) through-casing resistivity tool (TCRT) in a borehole environment. Then, we utilize this problem to illustrate the main characteristics of the self-adaptive goal-oriented -FEM described in this article, namely: the advantages of using goal-oriented -adaptivity as opposed to energy-norm -adaptivity, the advantages and disadvantages of using the azimuthal formulation (with continuous elements) as opposed to the meridian formulation (with edge elements), and the final -grids, accuracy, and performance delivered by the self-adaptive goal-oriented -FEM Modeling of an AC through-casing resistivity tool (TCRT) in a borehole environment We consider a TCRT instrument composed of the following axisymmetric materials (all dimensions are given in cm): one transmitter and two receiver coils defined on (1) C1 ¼fðq; /; zþ : 0:08 m < q < 0:1 m; 0:0105 m < z < 0:00975 mg, () C ¼fðq; /; zþ : 0:08 m < q < 0:1 m; 0: m < z < 0:00665 mg, and, (3) C3 ¼fðq; /; zþ : 0:08 m < q < 0:1 m; 0: m < z < 0:00865 mg, respectively. The transmitter antenna is composed of a toroidal coil with an impressed volume magnetic current I M ¼ 1=ðpa Þ A, where a is the radius of the toroidal coil. The corresponding magnetic far-field solution in homogeneous and isotropic medium is given by: H ¼ ^/ðr þ jxþjk e jkd 4pd 1 j q kd d : ð4:36þ This TCRT instrument moves along the vertical direction (z-axis) in a subsurface borehole environment composed of: a borehole mud with resistivity 0.1 m defined on (1) BH ¼fðq; /; zþ : q < 0:1 mg, a casing, with resistivity 10 6 m defined on (1) CS ¼fðq; /; zþ : 0:1 m6 q 6 0:117 mg, and, three formation materials of resistivities 100 m, 10,000 m, and 1 m, defined on (1) M1 ¼fðq; /; zþ : q P 0:117 m; ðz < 0:5 morz > 1mÞg, () M ¼fðq; /; zþ : q P 0:117 m; 0:5 m6 z < 0mg, and, (3) M3 ¼fðq; /; zþ : q P 0:117 m; 0m6 z 6 1mg, respectively. Fig. 1 shows the geometry of the described logging instrument and borehole environment. From the engineering point of view, the main objective of these simulations is to compute the (normalized) difference of the vertical component of the electric field at the receiving coils. More precisely, we are interested in computing the quantity of interest Q given by QðEÞ¼ 1 j C1 j " E z ðlþdl 1 #, E z ðlþdl dðc 1 ;C Þ; C1 j C j C ð4:37þ where dðc 1 ; C Þ is the distance between the two receiver antennas. From the numerical point of view, solution of this TCRT problem is rather challenging for the following reasons: It incorporates three singularities (of different strength) located at points where three different materials meet. Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

9 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx m 0.0 m 0.65 m 0.5 m 0.5 m 0.1m 0.1 Ohm-m Ohm-m 100 Ohm- m 1 Ohm-m Ohm-m The resistivity between adjacent materials varies by as much as ten orders of magnitude. The quantity of interest Q is typically several orders of magnitude smaller than the electric field measured at the receiver coils (see [15] for details). Thus, it is necessary to approximate the solution at the receiver coils very accurately, in order to obtain an acceptable error level in the quantity of interest. Due to the presence of steel casing, we need to either consider a long computation domain in the vertical direction (3000 m or more) or to close the computational domain with an absorbing-type boundary condition (BC). Here, we consider a 4000 m domain in the vertical direction, and we impose a homogeneous essential BC. 4.. Numerical comparison between energy-norm and goal-oriented -adaptivity The approximate straight lines obtained in Figs. and 3 (in an algebraic vs. logarithmic scale) indicate the exponential convergence of both, the energy-norm -adapz 100 Ohm- m Fig. 1. D cross section of the geometry of a TCRT, composed of one transmitter and two receiver antennas (coils), a borehole, a m thick steel casing, and four horizontal layers in the rock formation (with different resistivities). We consider the TCRT problem described in Section 4.1 operating at 10 khz. In Fig., we display the convergence history obtained using the self-adaptive energy-norm algorithm [7]. The corresponding convergence history using the self-adaptive goal-oriented -algorithm described in this paper is shown in Fig. 3. From these figures, we make the following observations: a 10 4 Continuous Elements b 10 4 Edge Elements Upper bound for L(e) / L(u) Energynorm error L(e) / L(u) L(e) / L(u) Relative Error in % Relative Error in % Number of Unknowns N (scale N 1/3 ) Number of Unknowns N (scale N 1/3 ) Fig.. Convergence history for the TCRT problem operating at 10 khz using energy-norm -adaptivity. The solid curves describe the relative energynorm error, while dashed curves describe the relative error in the quantity of interest. Left panel: convergence history using formulation (.0) with continuous (standard) elements. Right panel: convergence history using formulation (.19) with edge elements. Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

10 10 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx Frequency: 10 hz L(e) / L(u) (Cont. Elements) Upper bound for L(e) / L(u) (Cont. Elements) L(e) / L(u) (Edge Elements) Upper bound for L(e) / L(u) (Edge Elements) Frequency: 10 Hz L(e) / L(u) (Cont. Elements) Upper bound for L(e) / L(u) (Cont. Elements) L(e) / L(u) (Edge Elements) Upper bound for L(e) / L(u) (Edge Elements) Relative Error in % Relative Error in % /3 Number of Unknowns N (scale N ) Fig. 3. Convergence history for the TCRT problem operating at 10 khz using goal-oriented -adaptivity. The solid curves describe the use of formulation (.0) with continuous elements. The dashed curves describe the use of formulation (.19) with edge elements. Light grey curves describe the relative error in the quantity of interest. Black curves describe the upper bound estimate (3.33). tivity (in terms of the relative energy-norm error vs. the number of unknowns) and the goal-oriented -adaptivity (in terms of upper bound (3.33) vs. the number of unknowns). The final slow-down in convergence associated with edge elements is explained in Section 4.3. Exponential convergence in terms of the energy-norm does not imply an acceptable convergence in terms of the quantity of interest. Indeed, the final grid produced by the energy-norm adaptive algorithm exhibits a relative energy-norm error below %, while the relative error in the quantity of interest remains large (over 00%). The self-adaptive goal-oriented -FEM delivers exponential convergence rates in terms of estimate (3.33), which is an upper bound estimate for the error in the quantity of interest. Thus, the relative error in the quantity of interest rapidly approaches 0 (as the number of unknowns increases), even though the relative error in the quantity of interest may not decrease monotonically with the problem size /3 Number of Unknowns N (scale N ) Fig. 4. Convergence history for the TCRT problem operating at 10 Hz using goal-oriented -adaptivity. Solid curves indicate the use of formulation (.0) with continuous elements. Dashed curves indicate the use of formulation (.19) with edge elements. Light grey curves display the relative error in the quantity of interest. Black curves display upper bound estimate (3.33). for the case of simulating the TCRT problem considered in this paper: The edge element formulation may entail a sudden slowdown in convergence due to round-off errors. As the frequency and/or finite element size decreases, the ratio between the L - and curl-contributions to the bilinear form b becomes smaller. In the limit, the L -contribution is not measurable with the computer (due to finite precision arithmetic), and the corresponding system of linear equations becomes singular (gradients become ill-defined). The slow-down in convergence observed in Figs. and 3(right panel) correspond to the pre-limiting case, where the -grid contains elements for which the Table 1 Quantity of interest for continuous elements Quantity of interest Real part Imag. part Final coarse grid e e 09 Final fine grid e e Numerical comparison between continuous and edge finite elements At this point, we describe the main advantages and disadvantages of using edge elements with formulation (.19) as opposed to continuous elements with formulation (.0), Table Quantity of interest for edge elements Quantity of interest Real part Imag. part Final coarse grid e e 09 Final fine grid e e 09 Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

11 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx 11 Fig. 5. TCRT problem operating at 10 Hz. Amplification (1.83 m 1.13 m) of the final goal-oriented -grid in the vicinity of the two receiving antennas. Two circles indicate the location of the receiving antennas. This final goal-oriented -grid uses continuous elements and entails an error in the quantity of interest below 0.1%. Different colors indicate different polynomial orders of approximation, ranging from 1 to 8. L -contribution is orders of magnitude smaller than the corresponding curl-contribution. As we decrease the frequency, this slow-down in convergence occurs earlier, as shown in Fig. 4. The latter stability problem can be overcome by explicitly re-enforcing the divergence-free condition (Gauss law) into the variational formulation at the expense of introducing a new variable (Lagrange multiplier). See [4] for details. Continuous finite elements converge at a faster rate than edge elements (see Figs. 3 and 4). This difference may be explained by the fact that an edge element of order p contains, for p > 1, more unknowns (p þ p) than a continuous element of order p (the number of unknowns is equal to p þ p þ 1). For higher-order elements (p > 1), the number of unknowns associated with nodes located within the interior of an element is larger for edge elements than for continuous elements. Thus, for a fixed number of unknowns, a well-designed direct solver (performing static condensation on the interior of each element) should solve a problem using edge elements faster than using continuous elements. 3 3 We do not display a convergence curve comparison against the CPU time due to the lack of an efficient implementation for edge elements at this time. In summary, it is difficult to assess what type of elements (continuous or edge) are more suitable for simulating axisymmetric resistivity logging applications. In any case, by comparing results obtained from both the continuous and the edge element formulations we can verify the results, and perhaps utilize this comparison for constructing a posteriori error estimates Final grids, accuracy, and performance delivered by the self-adaptive goal-oriented -FEM In the following, we display the final results obtained for the TCRT problem described in Section 4.1 operating at 10 khz. Tables 1 and display the final value of the quantity of interest Q. By using the fine grids as an error estimate for the coarse grids, we realize that the final coarse grids entail a relative error in the quantity of interest below 0.005%. By comparing fine grid results for continuous and edge elements, we estimate that both fine grids entail a relative error in the quantity of interest below %. The final -grids delivering a relative error in the quantity of interest below 0:1% for the TCRT problem operating at 10 Hz are displayed in Figs. 5 and 6 (continuous elements) (edge elements). Table 3 displays the total CPU time associated with the self-adaptive goal-oriented -FEM to solve the TCRT problem for 80 different vertical positions (spaced at 3 cm intervals) of the logging instrument along the z-axis. For this performance analysis we use continuous elements and consider different antennas (a toroidal coil, a ring of vertical dipoles, a ring of horizontal dipoles, and a ring Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

12 1 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx Fig. 6. TCRT problem operating at 10 Hz. Amplification (1.83 m 1.13 m) of the final goal-oriented -grid in the vicinity of the two receiving antennas. Two circles indicate the location of the receiving antennas. This final goal-oriented -grid uses edge elements and entails an error in the quantity of interest below 0.1%. Different colors indicate different polynomial orders of approximation, ranging from 1 to 8. Table 3 TCRT problem 80 Vertical positions Electrical resistivity of casing 10 6 m 10 5 m Toroid (10 khz) Ring of vert. dipoles (10 khz) Ring of horiz. dipoles (10 khz) Electrodes (0 Hz) Performance of the self-adaptive goal-oriented -FEM. of electrodes) and different casing resistivities, using 10 6 m and 10 6 m, respectively. These results were obtained in an IBM Power 4 (1.3 GHz) processor. The corresponding final logs are shown in Fig Summary and conclusions We have developed and successfully tested a self-adaptive goal-oriented -FE algorithm that produces a sequence of -grids delivering exponential convergence rates in terms of a user-prescribed quantity of interest vs. the number of unknowns (as well as CPU time). The self-adaptive strategy combines the use of: a projection based interpolation operator, which is independent of the material properties of the problem, and thus, it allows for a simple and problem-independent implementation, with Position of the receiver antenna in the zaxis (in m) Continuous Elements (10 hz) Toroid Vert. dipoles (ring) Hor iz. dipoles (ring) Amplitude of first difference of E (V/m ) z Fig. 7. TCRT problem operating at 10 khz. Final log for different types of antennas. Results obtained using the self-adaptive goal-oriented - FEM. evaluation of element-based energy norms that incorporate material properties. This is essential for obtaining a sharp stable upper bound of the quantity of interest that Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.

13 D. Pardo et al. / Comput. Methods Appl. Mech. Engrg. xxx (007) xxx xxx 13 we can use for minimization of the error. In the examples presented above, the ratio between the upper bound used for minimization and the quantity of interest typically ranges from 10 up to 100. Notice that if we had not included the material properties in the evaluation of the energy-norms, then this ratio would have become as large as 10 11, since the material properties vary by as much as ten orders of magnitude. Numerical results illustrate the performance and highaccuracy simulations obtained with the goal-oriented - FEM for problems that include several singularities and large material contrasts. These results illustrate the necessity and importance of using goal-oriented algorithms for approximating an arbitrary quantity of interest of the solution. A comparison of continuous vs. edge element discretizations for solving axisymmetric Maxwell s equations reveals the advantages and disadvantages of each type of discretization. For the model problem considered in this paper, the use of edge elements provides a better initial approximation of the solution at low frequencies, but the rate of convergence is roughly half to that obtained with continuous elements. Since the influence function used by the self-adaptive goal-oriented -adaptive algorithm is approximated via Finite Elements, the numerical method presented in this article is problem independent, and therefore can be applied to FE discretizations of H 1 -, HðcurlÞ-, and HðdivÞ-spaces. Acknowledgements This work was financially supported by Baker Atlas and The University of Texas at Austin s Joint Industry Research Consortium on Formation Evaluation sponsored by Baker Atlas, BP, ConocoPhillips, ENI E&P, ExxonMobil, Halliburton Energy Services, Mexican Institute for Petroleum, Occidental Oil and Gas Corporation, Petrobras, Precision Energy Services, Schlumberger, Shell International E&P, Statoil, and Total. We would also like to acknowledge the expertise and technical advise received from L. Tabarovsky, A. Bespalov, T. Wang, and other members of the Science Department of Baker-Atlas. References [1] F. Assous, C. Ciarlet, S. Labrunie, Theoretical tools to solve the axisymmetric Maxwell equations, Math. Methods Appl. Sci. 5 (00) [] R. Becker, R. Rannacher, Weighted a posteriori error control in FE methods, in: Hans Georg Bock et al. 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Suttmeier, A posteriori error control in finite element methods via duality techniques: application to perfect plasticity, Comput. Mech. 1 (1998) [0] P. Solin, L. Demkowicz, Goal-oriented -adaptivity for elliptic problems, Comput. Methods Appl. Mech. Engrg. 193 (004) [1] J. Van Bladel, Singular Electromagnetic Fields and Sources, Oxford University Press, New York, Please cite this article in press as: D. Pardo et al., A self-adaptive goal-oriented finite element method..., Comput. Methods Appl.