Curl-curl Equation Research Articles

An efficient cascadic multigrid method with regularization technique for 3D electromagnetic finite-element anisotropic modeling

Efficient forward modeling of electromagnetic (EM) fields is the basis of interpretation and inversion of the EM data, which plays an important role in practical geophysical exploration. A novel extrapolation cascadic multigrid (EXCMG) method is developed to solve large linear systems encountered in geophysical EM modeling. The original curl-curl equation with arbitrary anisotropic conductivity is regularized by including the gradient of a scaled divergence correction term, and a linear edge element method is used to discretize the equation. We develop a novel approach to address the issue of edge unknowns in 3D edge element discretizations on nonuniform rectilinear grids. Inspired by the original EXCMG method for nodal elements, we introduce a new prolongation operator that treats edge unknowns as defined on the midpoints of edges. This operator aims to provide an accurate approximation of the finite-element solution on the refined grid. Using a good initial guess significantly reduces the number of iterations required by the preconditioned biconjugate gradient stabilized solver, which is used as a smoother for the EXCMG algorithm. Numerical experiments are carried out to validate the accuracy and efficiency of our EXCMG method, including models from magnetotellurics and controlled-source EM modeling. The testing results indicate that EXCMG is more efficient than traditional Krylov-subspace iterative solvers, the algebraic multigrid solver, and those depending on the auxiliary-space Maxwell solver, especially for large-scale problems where the number of unknowns exceeds 10 million. The EXCMG method can also be applied to solve other large-scale forward modeling problems encountered in geophysics.

A Highly Efficient and Accurate Divergence-Free Spectral Method for the Curl-Curl Equation in Two and Three Dimensions

A Highly Efficient and Accurate Divergence-Free Spectral Method for the Curl-Curl Equation in Two and Three Dimensions

Hybrid CPU-GPU solution to regularized divergence-free curl-curl equations for electromagnetic inversion problems

The Curl-Curl equation is the foundation of time-harmonic electromagnetic (EM) problems in geophysics. The efficiency of its solution is key to EM simulations, accounting for over 95% of the computation cost in geophysical inversions for magnetotelluric or controlled-source EM problems. However, most published EM inversion codes are still central processing unit (CPU)-based and cannot utilize recent computational developments on the graphic processing units (GPUs). Based on a previously proposed divergence-free algorithm developed on CPUs, this study demonstrates the current limits of the CPU-based inversion procedure. To exploit the high throughput capability of GPUs, we propose a hybrid CPU-GPU framework to solve forward and adjoint problems required for EM inversions. The large sparse linear systems arising from the staggered-grid finite difference approximation of the Curl-Curl equation are solved with a mixed-precision Krylov subspace solver implemented on a GPU. The algorithm is then tested in EM forward and adjoint calculations, with real-world three-dimensional numerical examples. Test results show promising 30× kernel-level speed-ups over the conventional CPU algorithm. This approach may further take the complex frequency domain EM inversions onto the next, practical stage on small affordable GPU platforms.

A divergence-free vector finite-element method for efficient 3D magnetotelluric forward modeling

For large-scale magnetotelluric (MT) forward modeling using vector finite-element methods, iterative solvers are commonly applied. However, as frequency decreases close to zero, the iterative solution process struggles to converge. This is mainly caused by the fact that the weak conductivity term in the curl-curl equation governing the electromagnetic (EM) diffusion in the earth fades during the iterative solution for EM fields. This can lead to violation of the divergence-free condition for current and false jumps in the calculated fields. We develop a regularization technique for vector finite-element MT forward modeling, in which a scaled grad-div operator for electrical fields is included in the curl-curl equation to enforce the divergence-free condition explicitly. Because of its use of basis functions, the direct edge element discretization of the scaled term can lead to zero coefficients. To address this, an equivalent substitute of the scaled grad-div operator based on potential representation of electrical fields is used, discretized with node elements. For one specific element, this is finally reduced to a scaled gradient of the surface integral of current across all the surfaces of the element, resulting in better connectivity of the linear system of equations. The correctness of our algorithm is verified with two synthetic models and an inversion model from real data. The numerical performance is compared to the traditional iterative divergence correction technique (and without the application of the divergence correction for one case) for each of the three models. The results indicate that our algorithm is generally more efficient and stable compared to the traditional technique for all models at all periods considered, with significant improvement at long periods.

Iterative solution methods for 3D controlled-source electromagnetic forward modelling of geophysical exploration scenarios

We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.

Forward calculation of 3D controlled-source electromagnetic responses based on joint application of secondary field and coupled potential formulations

In the traditional framework of 3D controlled-source electromagnetic (CSEM) modeling, either the field formulation or the coupled potential formulation is implemented. We have developed an improved algorithm that can calculate the 3D CSEM responses based on a joint application of the secondary field and secondary coupled potential formulations. The vector finite-element method is implemented for the discretization of the curl-curl equation of the secondary field. The vector and nodal elements are implemented simultaneously for the discretization of the coupled potential formulation. We use the stiffness matrix resulting from the coupled potential formulation to construct an efficient preconditioner, which is used for the solution of the linear system of equations obtained based on the secondary field formulation. The linear system of the equations could be solved efficiently by the quasiminimum residual method within several hundred iterations with the constructed preconditioner. We compare different modeling schemes to calculate the CSEM responses for a horizontally layered earth model excited by an electric dipole and find that the iteration process of the improved algorithm converges faster on average than other modeling schemes for the frequency band considered. Thus, the least computation time can be obtained over the frequency band for the improved algorithm compared with other modeling schemes, with the same level of accuracy reached. We further verify the efficiency of the algorithm through a small-scale 3D earth model excited by a magnetic dipole. Finally, we illustrate the performance of the algorithm with two 3D subsurface models, including a land-based model and a marine disk model. By comparison with published work, it is illustrated that our algorithm can efficiently calculate accurate land-based and marine CSEM responses with the new constructed preconditioner implemented.

Extension of the regularization technique to controlled-source electromagnetic modeling in general anisotropic conductivity media

Traditional Krylov subspace methods can be slow for electromagnetic modeling, particularly at lower frequencies. One of the commonly used remedies is to apply the divergence correction iteratively during the solution process, which requires solving an additional divergence correction equation. Alternatively, we have developed an efficient regularization technique to carry out the forward modeling of the anisotropic effect for 3D controlled-source electromagnetic data. Inside this scheme, we explicitly include the gradient of a scaled divergence correction term into the original curl-curl equation for general anisotropic conductivity structure. This inclusion leads to a significantly better-conditioned linear system of equations without changing the solution of the original system and avoids the solution of an additional equation. The correctness of the developed algorithm is examined based on a vertical transverse isotropic layered model with the semianalytical solution available. Then, we use a tilted transverse isotropic ocean canonical reservoir model and a general anisotropic 3D model to investigate the stability and efficiency of the algorithm. Numerical experiments demonstrate that the developed technique is 0.49–2.9 times faster than the standard algorithm at the considered frequencies.

Three-dimensional controlled-source electromagnetic forward modeling by edge-based finite element with a divergence correction

We have developed an edge-based finite-element (FE) modeling algorithm with a divergence correction for calculating the controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field to eliminate the source singularity. The choice of the edge-based FE method enables us to properly handle the discontinuity of the normal component of electric fields across conductivity boundaries. Although we can solve the resulting complex-symmetric linear system of equations efficiently by a quasiminimal residual (QMR) method preconditioned with an incomplete Cholesky decomposition for the high-frequency band, the iterative solution process encounters a common problem in the field formulation and does not converge within a practically feasible number of iterations for low frequencies. To overcome this difficulty and to accelerate the iterative solution process in general, we combine a divergence correction technique with the secondary field solution using the QMR solver. We have found that applying the divergence correction intermittently during the iterative solution process ensures the calculation of sufficiently accurate electric and magnetic fields and can significantly speed up the solution process by more than an order of magnitude. We have tested the efficiency and accuracy of our algorithm with 1D and 3D models, and we have found that the divergence correction technique is able to guide the electric field to satisfy the boundary conditions across conductivity interfaces. Although there is a computational overhead required for applying the divergence correction, that cost is significantly offset by the substantial gains in the solution accuracy and speed-up. The work makes the field-based curl-curl formulation using edge elements an efficient and practical method for CSEM simulations.

Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics

In this paper we study the asymptotic behaviour of the quasilinearcurl-curlequation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in Gangl and Sturm (ESAIM: COCV26(2020) 106) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems inH1is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.

3D finite difference modeling of controlled-source electromagnetic response in frequency domain based on a modified curl-curl equation

3D controlled-source electromagnetic (CSEM) modeling involves the solution of the Maxwell's curl-curl equations, which have abundant null space. This is commonly handled by adding one additional iteration of divergence correction after each certain iterations of the solution process of the curl-curl equations (CC-DC). Its efficiency typically deteriorates as lower frequencies are used. In this paper, we propose a 3D finite difference CSEM modeling algorithm based on a modified version of the curl-curl equations with scaled grad-div operator (CCGD) in frequency domain. In this approach, we integrate the divergence correction term in the original system with its importance controlled by scaling factors (the same values as model resistivities are used in this paper) and avoid the application of the iterative divergence correction. The accuracy of the proposed approach is verified using a layered model due to the existence of analytical solution for the simple model. Then based on two more complex synthetic models, we examine the numerical performance of the CCGD approach, and compare it with the traditional CC-DC approach in terms of computing time and iteration number. The results indicate that the proposed CCGD approach is efficient and stable over different frequencies.

An Efficient and Robust Mixed-Gauge Scheme Based on Current-Splitting for Nonlinear Magnetostatic Field Computation

In the edge element discretization for magnetostatic problems, a gauge scheme, such as tree gauge, Lagrangian multiplier (LM) gauge, and auto gauge, is usually adopted to handle the singular curl-curl equation in terms of the magnetic vector potential. However, tree-gauge and LM-gauge schemes are not very efficient as a nonlinear problem with many increments has to be resolved. In order to overcome those difficulties, a mixed-gauge scheme based on current-splitting is proposed in this article. First, we make an improvement for LM-gauged formulation and adapt it to calculate the initial increment with a specially designed CG solver; then split the discrete source current density into a compatible part and incompatible part based on discrete Helmholtz decomposition; auto gauge cooperates with the compatible part to calculate the following increments. The mixed-gauge scheme combines the advantages of LM-gauge and auto-gauge schemes, which avoids the step to determine the source term, and at the same time, has high computational efficiency. The mixed-gauge scheme is tested in some examples against some other gauge schemes and found to be efficient and robust, where others possibly converge slowly or fail to produce a reliable solution.

Highly Parallelized Contour Integral Method for Computing Resonant Modes of Lossy Cavities

In this article, we address an efficient solver of the Maxwell eigenvalue problem for lossy cavity resonators. The curl-curl equation for the electric field is discretized using curved tetrahedral incomplete quadratic finite elements, resulting in a nonlinear eigenvalue formulation. The eigenvalue problem is efficiently solved using a contour integral method (CIM). This method enables an accurate computation of all eigenvalues within a predefined region and is implemented in a highly parallelized framework to enhance the performance of the algorithm. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

3D MT modeling using the T–Ω method in general anisotropic media

We developed an algorithm to simulate magnetotelluric responses for three-dimensional general electrical anisotropic structures with finite element (FE) method. From the Maxwell's equations, the curl-curl equation in terms of the electric vector potential and the magnetic scalar potential is obtained by decomposing the magnetic field into the incident field and the resultant field. For this method, the only unknown variable in the air is the magnetic scalar potential. Galerkin weighted residual method is adopted to generate the variational equations. Then we made a comparison with the edge-based FE method which is in terms of electric fields to validate the accuracy of this method. Later, two types of anisotropic models are studied and the results are analyzed in detail. Finally, based on these analyses, two main meaningful conclusions are obtained.

Efficient Computation of the Matrix Entries in a Hierarchical-Hexahedral Finite-Element Solution of Curl-Curl Equation

This paper presents an efficient computation of the matrix entries in a hierarchical finite element solution of curl-curl equation on large hexahedral elements. The method applies a product to sum approach to convert a weighting basis product into a sum of appropriate polynomials. This conversion reduces the number of required triple integrals from $O(D^{2})$ to $O(D)$ , where $D$ is the degree of freedom in the hexahedral element. The efficiency of the method is verified through some numerical examples.

Existence of Cylindrically Symmetric Ground States to a Nonlinear Curl-Curl Equation with Non-Constant Coefficients

We consider the nonlinear curl-curl problem \nabla\times\nabla\times U + V(x) U=f(x, |U|^2)U in \mathbb R^3 related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient V and subcritical cylindrically symmetric nonlinearity f . The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions [J. Funct. Anal. 41 (1981)(2), 236–275], new rearrangement type inequalities by Brock [Proc. Indian Acad. Sci. Math. Sci. 110 (2000), 157–204] and the recent extension of the Nehari-manifold technique from Szulkin and Weth [Handbook of Nonconvex Analysis and Applications (2010), pp. 597–632].

Stochastic Modeling and Regularity of the Nonlinear Elliptic curl--curl Equation

This paper addresses the nonlinear elliptic curl-curl equation with uncertainties in the material law. It is frequently employed in the numerical evaluation of magnetostatic fields, where the uncertainty is ascribed to the so-called B-H curve. A truncated Karhunen-Lo\`eve approximation of the stochastic B-H curve is presented and analyzed with regard to monotonicity constraints. A stochastic non-linear curl-curl formulation is introduced and numerically approximated by a finite element and collocation method in the deterministic and stochastic variable, respectively. The stochastic regularity is analyzed by a higher order sensitivity analysis. It is shown that, unlike to linear and several nonlinear elliptic problems, the solution is not analytic with respect to the random variables and an algebraic decay of the stochastic error is obtained. Numerical results for both the Karhunen-Lo\`eve expansion and the stochastic curl-curl equation are given for illustration.

Algebraic Two-Level Convergence Theory for Singular Systems

We consider the algebraic convergence theory that gives its theoretical foundations to classical algebraic multigrid methods. All the main results constitutive of the approach are properly extended to singular compatible systems, including the recent sharp convergence estimates for both symmetric and nonsymmetric systems. In fact, all results carry over smoothly to the singular case, which therefore does not require a specific treatment (except a proper handling of issues associated with singular coarse grid matrices). Regarding problems with a low-dimensional null space, the presented results thus mainly confirm what has been observed often at a more practical level in many previous works. As illustrated by the discussion of the application to the curl-curl equation, the potential impact is greater for problems with large-dimensional null space. Indeed, it turns out that the design of multilevel methods can then actually be easier for the singular case than it is for nearby regularized problems.

Three-dimensional magnetotellurics modeling using edgebased finite-element unstructured meshes

Three-dimensional forward modeling is a challenge for geometrically complex magnetotellurics (MT) problems. We present a new edge-based finite-element algorithm using an unstructured mesh for accurately and efficiently simulating 3D MT responses. The electric field curl-curl equation in the frequency domain was used to deduce the H (curl) variation weak form of the MT forward problem, the Galerkin rule was used to derive a linear finite-element equation on the linear-edge tetrahedroid space, and, finally, a BI-CGSTAB solver was used to estimate the unknown electric fields. A local mesh refinement technique in the neighbor of the measuring MT stations was used to greatly improve the accuracies of the numerical solutions. Four synthetic models validated the powerful performance of our algorithms. We believe that our method will effectively contribute to processing more complex MT studies.

Local Fourier Analysis of Multigrid for the Curl-Curl Equation

We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk, and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. This procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows several well known properties to be recovered in Fourier space, such as the commutation properties of the classical Nedelec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to two- and three-dimensional problems in $H$(curl) and $H$(div) and to other types of regular meshes.

Equilibrated residual error estimator for edge elements

Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. A cheap computation of such functions via equilibration is well known for scalar equations of second order. We simplify and modify the equilibration such that it can be applied to the curl-curl equation and edge elements. The construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions. For this reason, Raviart-Thomas elements are extended in the spirit of distributions.