Non-planar sensing skins for structural health monitoring based on electrical resistance tomography

Electrical resistance tomography (ERT) -based distributed surface sensing systems, or sensing skins, offer alternative sensing techniques for structural health monitoring, providing capabilities for distributed sensing of, for example, damage, strain and temperature. Currently, however, the computational techniques utilized for sensing skins are limited to planar surfaces. In this paper, to overcome this limitation, we generalize the ERT-based surface sensing to non-planar surfaces covering arbitrarily shaped three-dimensional structures; We construct a framework in which we reformulate the image reconstruction problem of ERT using techniques of Riemannian geometry, and solve the resulting problem numerically. We test this framework in series of numerical and experimental studies. The results demonstrate that the feasibility of the proposed formulation and the applicability of ERT-based sensing skins for non-planar geometries.

A component of SHM is a sensor network consisting of variety of sensors utilizing a variety of techniques, that continuously monitors the condition of the infrastructure [ ]. While the sensing techniques have advanced signi cantly over the past twenty years, utilization of SHM to real-life infrastructure is still relatively rare. Many factors contribute to the slow adaptation of SHM for infrastructure, including the high cost of implementing and maintaining, as well as di culty of the interpretation of measurements. The interpretation of the measurements is especially challenging when a large number of discrete sensors are used without the utilization of a model-based interpretation approach. Distributed sensors and sensing systems may o er an alternative that at times can be more cost e ective. Especially, distributed sensors that are model-based and provide direct visualization of the data can overcome many of the limitations of discrete sensors. An example of such system is an electrical resistance tomography (ERT) -based sensing skin [ ].
ERT based sensing skin is a distributed surface sensing system that uses a layer of electrically conductive material (such as colloidal metallic paint [ , ] or carbon nanotube lm [ , ]) which is In addition to SHM, ERT-based sensing systems have been applied to robotics, where the sensing skin is used for detecting and localizing touch via pressure sensing [ , , ]. In publication [ ], an ERT-based touch sensor made of conductive fabric was wound around an arti cial arm. The winding did not cause wrinkles to the fabric, but since the fabric was bent, the geometry was non-planar. The computational model used in the study, however, assumed a planar geometry. Although earlier studies have indicated that at least certain sensing skin materials are very sensitive to stretching and bending [ , ], neglecting these e ects by the use of planar approximation did not cause signi cant reconstruction artifacts in [ ]. Nevertheless. it is not guaranteed that the planar approximation works with all materials, especially when aiming at quantitative imaging [ ]. Even more importantly, in many potential SHM applications, the planar approximation of the sensor is impossible, because of the nontrivial topology of the surface. This is the case for example with all the geometries considered in the numerical and experimental studies of this paper ( Figure ).
Another application, very similar to SHM with sensing skin is the use of ERT with self-sensing materials [ , ]. Recently, ERT imaging was applied to self-sensing composite tubes for damage detection, and the structure was non-planar [ ]. In this case, the D structure of the target material was modeled as in other D ERT applications [ , , ]. While in the self-sensing applications, the structures -and thus also sensors -are inherently three-dimensional, in sensing skin applications the thickness of a sensor is several orders of magnitudes lower than its other dimensions. Clearly, this type of sensor can be modeled as a surface in three-dimensional space, and a full three-dimensional model would be unnecessarily complicated, making the computations more complex and more prone to numerical errors.
In this paper, we formulate the problem of imaging a thin, electrically conductive surface material -sensing skin -applied on an arbitrarily shaped three-dimensional object by modeling it as a twodimensional surface in the three-dimensional space, or, mathematically as a manifold. The mathematical framework of the formulation is referred to as di erential geometry. Since the formulation and its mathematical proofs are very technical, their details are left to an extended, technical version of this paper, published with open access in arXiv [arxiv-viite]. The focus of the journal paper is in the numerical and experimental evaluation of this approach. In numerical and experimental studies, we evaluate the approach in cases of three non-planar geometries. In these studies, we consider two target applications; crack detection and imaging of di usive processes (such as heat conduction on solid materials).
In ERT imaging, the conductivity of the target is reconstructed from the voltage and current data obtained through a set of electrodes placed on the surface of the target. Typically, the target is treated as a three-dimensional or as a planar two-dimensional domain. However, in order to reconstruct the conductivity of an arbitrary shaped sensing skin, we consider the target as an arbitrary surface in three-dimensions.
In this section, we rst write a model that describes the ERT measurements given the surface conductivity; this is referred to as the forward model of ERT, and it is approximated numerically using the nite element method (FEM). The inverse problem of ERT is to reconstruct the conductivity given the current/potential measurements. The inverse problem is ill-posed in the sense that the "conventional" solutions to this problem are non-unique and extremely intolerant to measurement noise and modelling errors. For this reason, the solutions of the inverse problem require a priori information on the conductivity, or regularization of the problem [ ]. In this paper, we formulate the inverse problem as a regularized least-squares problem, where the data delity term utilizes the FEM approximation of the forward model.

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Consider a measurement setup in which the measurement data is obtained by sequentially setting each electrode to a known potential, grounding others, and measuring the electric current caused by potential di erence. We note that many of the existing ERT measurement systems operate the other way round -using current excitations and potential measurements. However, for a such system, the formulation of both the forward and inverse problem are analogous with the formulation written in this section. The choice of using potential excitations and current measurements is made, because the commercial measurement device employed for the experiments (Section ) uses this procedure.
The above described measurement setting constitutes the following forward problem: solve the electric current ( ) through each electrode , given the spatially distributed conductivity ( ) (where = ( , , ) is the spatial variable) and a set of electric potentials corresponding to an excitation . We model this relation using the complete electrode model (CEM) [ ] which consist of a partial di erential equation and a set of boundary conditions, where ⊂ ℝ is a surface with boundary , is the part of the representing the edge of the 'th electrode, is contact resistance, −ˆis an inward unit normal of (i.e. a vector tangent to , pointing inwards), and is the number of electrodes. In addition, the currents are required to satisfy Kirchho 's law = = . We write˜for the in nitesimal length elements of the one-dimensional boundary . By calling a surface, we mean that we can parametrize = ( ( , ), ( , ), ( , )) locally for some ( , ) ∈ ⊂ ℝ and some ∈ ⊂ . This means that the functions and di erential operators in ( . ) are two-dimensional, and can be formally de ned through Riemannian geometry.
Formally, we equip manifold M with a Riemannian metric . Metric de nes a product on tangent vectors analogous to a dot product in vector spaces (see Figure ) and it consequently de nes the Figure : An illustration of how the shortest path between two points in the non-planar two-dimensional model di ers from the shortest path between these points in the three-dimensional model. Essentially the Riemannian metric determines how the distance is de ned in the domain. divergence and the gradient operators; where : → ℝ (e.g. = ∇ ),ˆ: → ℝ (e.g.ˆ= ). The maps generalize directional derivatives to ; technically : ( ) → ( ), where ( ) is a collection of di erentiable functions on and = , , form a local basis for the tangent plane. In this basis, | | is the determinant of the matrix formed from the components of . Furthermore,˜in ( . ) is the Riemannian volume measure of a curve (length in ℝ ) in ( , ). Since in practice, M is an embbed manifold, we de ne as the pullback of the standard dot product in ℝ to and as the pullback of to [ , ].
. We approximate ( . ) with a Galerkin nite element method, as described in detail in the technical Appendix . Indeed, by introducing test function ( , ), we can write ( . ) in a variational form We write for the in nitesimal area elements of the two-dimensional surface . Notation-wise, the variational form ( . ) is almost the same as the one written for the D electrical [ ]. However, the functions in ( . ) are de ned on only the surface ⊂ ℝ , the di erential operators according ( . ), and inner products are de ned with respect to the Riemannian metrics and .
We can now concatenate the simulated measurements to form a vector ( ) = ( ( ), ..., ( )) . Further, we denote the vector containing the corresponding measured data by . The typical approach to solve the inverse problem of EIT is to solve a conductivity that minimizes the sum of a so-called data term, ( ( ) − ) , and a regularization functional ( ). In sensing skin applications, however, we may improve the reconstruction quality by utilizing measurements where = { ( ) ∈ ( ) | min ≤ ( ) ≤ max , ∀ ∈ }, ( ) is a nite dimensional function space on , and is a matrix for which is so-called data precision matrix. The matrix accounts for the magnitude of noise in the measurements. Furthermore, the lower constraint min > comes from the natural, physics-based limit for the positivity of the conductivity and the upper constraint max restricts the conductivity from above whenever the maximum conductivity is known. In cases where the maximum conductivity is unknown, we set max to an arbitrary large number.
Note that the regularization function ( ) in ( . ) is chosen depending on the information that is available about the conductivity prior to the measurements. In the numerical and experimental cases of the following sections, we consider two choices of regularization functionals. We note, however, that the non-planar ERT scheme proposed in this paper is not restricted to any particular choices of regularization. Although the above modeling error correction method based on the discrepancy term is highly approximative, it has shown to be useful in several cases with real data [ ], and is thus used also in this paper. A more advanced formulation of the inverse problem for detecting complex crack patterns in the presence of inhomogeneous background was proposed in [ ]. We note that, if needed, this computational method would also be directly applicable to the non-planar ERT model described above. We evaluate the proposed ERT imaging scheme with numerical simulation studies using two non-planar geometries; one resembles a pipe segment ( rst column in Figure ) and one resembles a pressure vessel (second column in Figure ). The gures also illustrate the locations of the electrodes. We note that majority of them are internal electrodes, in the sense that they are surrounded by the sensing skin. Such a setting is chosen in order to improve the sensitivity of ERT measurements; the use of internal electrodes improves the quality of ERT reconstructions from the case where all electrodes are in the perimeter of the sensing skin even in planar geometries [ ] -in non-planar imaging the e ect is presumably even stronger. Furthermore, we consider two target applications; crack detection and imaging of di usive processes (such as distributed temperature sensing [ ] or strain measurement [ , ]).
Both geometries are used to study crack detection (Cases and ). In each geometry, we consider ve stages of cracking. In the rst stage, stage , the sensing skin is intact and the conductivity is homogeneous. Measurements simulated in this stage are used as the reference measurements ref and utilized for computing the homogeneous background estimate ( . ). In the subsequent stages, to simulate evolving crack pattern, we lower the conductivity at the locations that correspond to the cracks. The di usive process imaging is studied in Case , where the geometry is same as in Case . Here, the conductivity distribution is spatially smooth, and it evolves in the di usive manner, mimicking an application where the surface temperature distribution is monitored using a sensing skin. page of .
The rst column in Figure shows the pipe segment geometry. The radius of the pipe segment is . m and it consists of three . m long straight cylindrical sections connected by two curved sections that both turn degrees to from an "S"-shaped geometry. The three straight sections each have eight symmetrically placed electrodes on them and the two curved sections both have four electrodes on their convex side. These electrodes are square-shaped with . m side length. The second column in Figure shows the geometry of a pressure vessel. The diameter of the pressure vessel is m and the length of the cylindrical middle section is .
m. The radius of curvature for the spherical top section is . m. Furthermore, the chamber has cylindrical extensions. One of the extensions is attached to the top section of the chamber. The radius of this extension is .
m. The other two extensions are attached on the cylindrical section. The radius of the larger horizontal extension is . m and the radius of the smaller diagonal extension is . m. On each extension, electrodes are placed radially. Furthermore, the cylindrical section of the chamber has four layers of radially placed electrodes. The topmost and bottommost layers have electrodes each, and the two layers in between have and electrodes. The total number of electrodes is . The inner electrodes on the main chamber are square-shaped with side length of .
m. The other are rectangular with side lengths of . m and . m. The FE mesh that we use in the data simulation for the pipe segment geometry has nodes and elements, and the FE mesh for the pressure vessel has nodes and elements. In each simulation, we initially set the surface conductivity to ( ) = S and use it to generate the reference measurements (stage ). Subsequently, we generate measurements from stages of varying conductivities, each stage being a continuation of the previous one (stages -). When simulating cracks (Cases and ), stages -consists of spatially narrow areas of low conductivity, ( crack ) = − S (top rows in Figures and ). When simulating the spatially smooth distribution (Case ), the minimum conductivity is set to . S in a single point on the curved surface, and it gradually increases to background value S as function of space. To mimic the di usive process, the size of the area with lowered conductivity is increased between consecutive stages from to (top row in Figure ).
. We reconstruct the conductivity by solving the minimization problem of ( . ). In the crack detection problems in Cases and , we utilize total variation (TV) regularization [ ] TV regularization penalizes the magnitude of the spatial gradient of in norm and is often suitable for cases where the conductivity features sharp edges on relatively homogeneous background. TV regularization is shown to be feasible in ERT based crack detection [ ].
In Case , we utilize Gaussian smoothness regularization , ∈ ℝ are the locations of the nodes and in the FE mesh, = and = . . This is often a feasible choice of regularization functional in cases of di usive phenomena, because it promotes spatial smoothness of the conductivity distribution.
In all the studies, the matrix is diagonal with [ ] , = and the minimum conductivity is  In Case , we set max = ∞, that is, the conductivity distribution is not constrained from above. The meshes used in the image reconstruction are sparser than those used when simulating the data. For example, the mesh for the pipe segment has nodes and elements while the mesh for the pressure vessel mesh has nodes and elements. To solve the minimzation problem ( . ), we utilize the recently published iterative Relaxed Inexact Gauss-Newton (RIPGN) algorithm [ ]. RIPGN is a Gauss-Newton variant; it linearizes the non-linear operator ( ) of ( . ) at each iterate, nds an approximate solution to the associated proximal problem using primal dual proximal splitting (the algorithm of Chambolle and Pock [ ]), and interpolates between this solution and the one computed at the previous iteration step. After computing each iterate, we check the convergence of the algorithm by comparing the value of the objective function in ( . ) at the current iterate to the value objective function at the previous iterates. Furthermore, we limit the maximum amount of computed iterations to .
The reason for applying the RIPGN method to optimization in this paper is that it was shown to shown to be very e ective both in D and planar D ERT [ ]. We note, however, that standard Gauss-Newton and Newton methods based on smoothing the minimum and maximum constraints and the TV functional [ , ] could be utilized as well. All the code used in the study was written in Julia ( . . ). Computations were done on AMD Ryzen X CPU with GB of RAM (DDR , MHz, CL ). Parts of the RIPGN algorithm utilize CUDA code. CUDA code was run on Nvidia RTX Ti GPU.
. . . : The results of Case are illustrated in Figure . The top row shows the (true) simulated conductivity, and the reconstructed conductivity is depicted in the bottom row. Each column corresponds to a di erent cracking stage.
In the rst stage (Figure , column ), a crack forms at the middle section of the pipe segment. The reconstruction captures the shape of this crack quite accurately and only a small artifact is visible near the crack. The conductivity value at the crack is − S, which equals to min .

Reconstruction Simulation
Stage .
Stage . Stage . Stage . In the second stage ( Figure , column ), two new cracks appear in the pipe segment, on the side opposite to the crack in stage . The reconstruction shows these cracks clearly: The locations and lengths of the cracks are somewhat correct. The orientation of the upper crack is slightly biased, but this bias is insigni cant from practical point of view.
In the third stage ( Figure , column ) the rst crack (state ) is lengthened upwards and further extended to two branches, forming a "Y"-shaped crack. The reconstructed surface conductivity traces the "Y"-shape of the crack well. The junction of the branches is slightly dislocated, but the size of the crack is again well recovered. In the nal stage (Figure , column ) the two small cracks of stage are inter-connected, forming a single crack extending from top to the mid section of the pipe segment. Again, the crack is well tracked by the ERT reconstruction, yet a couple of very small defects appear next to it. Note that the cracks in the reconstructed conductivity are thicker than the simulated ones since the inversion mesh is sparser.
. . : Figure shows the simulated and reconstructed conductivity in each cracking stage in Case where the geometry corresponds to a part of a pressure vessel. The reconstructions in Case trace the evolution of the crack pattern well. In all stages of cracking, the reconstruction quality is similar to that in Case , although a few more de ciencies are present. This small reduction in quality compared to Case is, however, expected. The surface area of the pressure vessel is thirteen times larger than the surface area of the pipe segment in Case and the geometry is far more complex. Overall, the results of Case further con rm the feasibility of the non-planar D ERT to crack detection applications.  and seem to be related to the type of regularization that is used. The simulation clearly demonstrates that ERT imaging of di usive phenomena is achievable also in non-planar geometry. In rst stage (the st column in Figure ), a spatially smooth region of low conductivity appears at the middle section of the pipe segment. In the subsequent stages (columns -in Figure ), the surface area of this region increases and the value within the region decreases further. Each reconstruction re ects the corresponding stage clearly and the de ciencies in these reconstructions are apparent only at the last two stages. These de ciencies, however, look similar to what is observed in D and planar D ERT studies [ ], and seem to be related to the type of regularization that is used. The simulation clearly demonstrates that ERT imaging of di usive phenomena is achievable also in non-planar geometry. .
For the experimental validation of the non-planar sensing skin, we used a setup where the outer surface of a hollow plastic cube was covered with conductive paint. We refer to the experimental test case as Case . The paint was a : mixture of graphite powder (manufactured by Cretacolor, www.cretacolor.com) and black coating paint (RUBBERcomp, manufactured by Maston, www.maston.fi). Figure ). Each side of the cube had eight electrodes. On the vertical sides, ve of these electrodes were inner electrodes, and the remaining three were shared with the adjacent sides. On the top side, this con guration was four and four. In total, the number of electrodes was . The electrodes were square-shaped, and the side length of an electrode was .

Side length of the cube was . m and bottom of the cube was open (last column in
m. The electrodes shared by two cube sides were bent along the edges.
We measured the reference data in the initial stage in which the sensing skin was intact. Subsequently, we simulated the cracking of the underlying structure by cutting the surface of the paint layer with a knife. We generated four di erent stages of cracking and carried out the ERT measurements corresponding to each of these stages. The same approach to "physically simulating" di erent stages of cracking has been used previously in cases on planar geometries, e.g, in [ , ]. Based on these studies, Reconstruction Photo Stage .
Stage . Stage . Stage . the quality of ERT reconstructions is similar in cases where a sensing skin is damaged with knife and where real crack patterns of the same complexity are monitored on the surface of a, e.g., a concrete beam.
We measured the data with an ERT device manufactured by Rocsole Ltd. (www.rocsole.com). This ERT device samples the currents with MHz frequency, and computes the current amplitudes from the samples using discrete Fourier transform. The device outputs the amplitudes for the excitation potentials and for the measured electric currents. The device selects the amplitude for the excitation potentials automatically. Furthermore, we used the kHz excitation frequency, and to reduce the measurement noise, the current amplitudes that we used in the reconstructions were one-minute time averages.
Similarly to Cases and , we use TV regularization to for the crack reconstructions (see Section . ). Furthermore, the parameters are the same in the numerical cases. Figure c shows the FE mesh used in the inversion. This mesh has nodes and elements. .
The top row in Figure shows a photo of the sensing skin at each stage and the bottom row shows the corresponding reconstructions (Case ). We highlight the crack made at each stage with light teal color and the cracks made at the previous stages are darkened; the cracks are very thin (less than mm in thickness) and would otherwise be indistinguishable from the background. In the rst stage ( Figure , column ), we created a diagonal crack on one the vertical sides of the cube. Reconstruction shows this crack accurately, although a small gap is visible in the reconstruction; the actual crack is fully connected. In the second stage (Figure , column ), we extended the rst crack so that it reaches the top side of the cube. The reconstruction shows the location and size of this crack quite accurately, although the curved extension of the crack is wider than the initial crack at stage .
In the third stage ( Figure , column ), we created an additional crack on the adjacent side of the cube. This crack is clearly visible in the reconstruction. In the nal stage (Figure , column ), we extended the crack made on the third stage so that it reaches through the top side to the adjacent side. This extended crack is correctly located by ERT, although the reconstruction shows a blocky area in the corner of the cube. This reconstruction artifact is an expected one, since the electrodes are quite far from the cube corners, and therefore the ERT measurements are less sensitive to conductivity variations in these areas. Note also that the cracks in the reconstructed conductivities are thicker than the actual cracks made on the physical sensing skin. This is, again, partly caused by the sparsity of the nite element mesh, and partly a result of limited sensitivity of ERT to thickness of the cracks [ ].
One goal of the structural health monitoring research is to develop cost-e ective sensor technologies. ERT based sensing skins have been proposed as a cost-e ective distributed surface sensing systems for SHM. In the previous studies, the sensing skins sensors have been planar. To extend the usability of the ERT-based sensing skin to more complex structures it is necessary consider non-planar sensing skins and computational models.
In this paper, we formulated the computational model for ERT in the case of non-planar surface sensing. We gave a brief outline of the numerical scheme to reconstruct the non-planar surface conductivity of the sensing skin. In this scheme, we modelled the relationship between the measured electric currents and the known electric potentials on a surface of an arbitrary object in D, and we used this model to formulate a minimization problem that yields the conductivity as the solution. Furthermore, we studied the feasibility of the scheme with three sets of numerical simulations and one set of experimental data.
In the synthetic cases, we acquired highly accurate reconstructions, and we observed only minor artifacts in the reconstructed conductivity. These artifacts were similar to what has been observed in previous planar sensing skins studies. With the measurement data, the reconstruction quality was slightly worse than in synthetic cases but su cient for most practical applications. Furthermore, we noted that the reconstructions from the measurement data could be improved, for example, by using a model for inhomogeneous background conductivity or by using a di erent electrode arrangement.
Overall, the reformulation of ERT imaging problem by using non-planar surface model proved to be viable; we did not observe any loss of reconstruction quality that could be related to nonplanarity of the sensing skins. We conclude that with the proposed approach, ERT-based sensing skin is viable in monitoring complicated non-planar surfaces. In the future, non-planar sensing skins should allow monitoring of complex industrial structures such as those in aerospace, civil and mechanical engineering.
Similarly to Euclidean spaces [ ], we derive a nite element (FE) approximation for ( . ). Although ( . ) looks identical to the Euclidean counterpart, the de nitions of the operators in ( . ) are more involved, containing calculations based on the Riemannian metric .
The FE approximation relies on the weak formulation of ( . ). The well-posedness of this weak formulation has been previously shown for ( , ) (i.e. the potential measurement setup) [ ] in Euclidean spaces, however for ( , ) (i.e. the current measurement setup), no previous work exists; we will show the well-posedness of the weak formulation for ( , ) in the manifold setting, which also extends to the Euclidean setting.
Initially, we take as an arbitrary metric on . However, to see how to compute the FE approximation through integration in ℝ , we need to x . In this case, to properly account for the shape of in ℝ , we take as the metric induced on by the natural metric on ℝ [ , ]. Namely, for tangents , on the tangent plane at a point (illustrated in Figure ), it is de ned by ( , ) :=˜( ( ), ( )), where : → ℝ is the inclusion map ( ) := and˜= ( ) + ( ) + ( ) is the Euclidean metric in ℝ .
The solutions ( , ) of ( . ) comprise a twice continuously di erentiable function ∈ := ( ) and a vector ∈ ℝ with components , = , . . . , . We denote ( , ) ∈ := ( ) ⊕ ℝ . We will show that the nite element approximation of ( . ), however, satis es the weak formulation, where is bilinear and is linear. The space where ( ) is a Hilbert space of twice weakly di erentiable functions. We de ne it as the completion of ∞ ( ) with respect to the norm · ( ) [ , Chapter ]. It corresponds to the common space (Ω) also used with planar CEM [ ]. The natural norm for this space is [ , , ] ( . ) where the inner products inducing the individual norms are In the following lemmas, we assume that the model ( . ) has at least two electrodes, i.e. ≥ . Lemma . . Suppose that > is constant on ∀ , the part of corresponding to electrode . Then the PDE ( . ) admits a weak formulation ( . ), where the bilinear operator : × → ℝ and the linear operator : → ℝ are given by Proof. Suppose that ( , ) solves ( . ). We need to show that it solves ( . ). So let ( , ) ∈ be arbitrary. We de ne := ∇ . Applying ∫ · to ( . a) we get where is the Riemannian volume corresponding to the metric on . Denoting by˜the Riemannian volume on , using the product rule, the divergence theorem on Riemannian manifolds [ , Appendix A], and ( . b) to replace ∇ ,ˆ , we obtain The equations ( . b) and ( . c) both hold for each = , . . . , and de ne the vectors , ∈ ℝ . By multiplying each component of by / , where is a component of a test vector ∈ ℝ , integrating over , and summing over = , . . . , , we get Subtracting ( . ) from ( . ) and plugging in gives Finally, since assume is constant, by subtracting ∫ ( − )/˜we get ( . ).
The next lemma shows that the weak formulation ( . ) is well-posed, meaning that the solution ( , ) exists and is unique, and is continuous, leading eventually to the invertibility of the linear system of the FE approximation. For the simplicity, we assume that the boundary of is ∞ . However, the arguments that we use in the following proofs should extend to domains with boundaries of lesser smoothness. Now, if we were solving for ( , ) instead of ( , ), we could follow the treatment in [ ] by replacing relevant theorems on Sobolev spaces by their (compact Riemannian) manifold counterparts. However, no well-posedness proof for the weak formulation of ( , ) exists. To prove the wellposedness for ( , ), we show that the conditions of the Banach-Nesča-Babuška theorem (BNB) hold for and that is continuous. The Euclidean case will follow as long as the domain for ( ) is bounded.
Follows directly from Young's inequality.
To see that ( . a) holds, start by denoting = Plugging (˜,˜) into ( . ) and simplifying gives This nishes the proof.

Now since sup
For the next lemma, we will replace and by their nite element approximations = and = − = (˜), where we allow to be an arbitrary FE basis function. For ∈ ℝ , we x basis vectors ∈ ℝ so that we can utilize Kirchho 's law to eliminate one of the components: we choose vectors ∈ ℝ such that the components of are ( ) = , ( ) + = − , and otherwise ( ) = . This xes the value of the so that = − = . Indeed, due to the Kirchho ' law, we only have − unknown currents. Note also that no longer appear in the lemma, since the value is easy to determine.
The vector = (s , s ), where (s ) = , ( s ) =˜contains the coe cients of the nite element approximations for and . Furthermore, the problem is well-posed, Galerkin orthogonality holds and for the exact solution (ˆ,ˆ) and some constant C > we have Proof. Since is continuous and since BNB holds for , by applying the Cea's lemma [ , Lemma ], we see that the problem is well-posed also in and the Galerkin orthogonality and the solution ( . ) hold for ( , ).
[ ] A. Voss, Imaging moisture ows in cement-based materials using electrical capacitance tomography, PhD thesis, University of Eastern Finland, .