Goal-oriented Adaptive Finite Element Research Articles

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.

An Efficient Goal-Oriented Adaptive Finite Element Method for Accurate Simulation of Complex Electromagnetic Radiation Problems

A goal-oriented adaptive frequency domain finite element method for solving electromagnetic radiation problems including complex structures is presented in this paper. Compared with the traditional adaptive finite element method, the goal-oriented method can flexibly control the refined regions according to the parameters of interest; therefore it has better convergence and has made significant progress in scattering problems and eigenvalue problems. To simulate complex antennas, this paper proposes an error indicator with high accuracy and low computational cost, and it uses the adjoint problem to weight element residuals without additional degrees of freedom. Moreover, high-quality mesh refinement algorithms adapted to this indicator are developed using a suitable point insertion strategy for multiscale structures. By simulating two practical antennas, comparisons with the traditional goal-oriented FEM and the well-developed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -adaptive finite element method in commercial software demonstrate the accuracy and efficiency of the proposed method.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

Analyzing the effect of dipping anisotropy on 3D marine controlled-source electromagnetic response based on a goal-oriented adaptive finite element method

This paper presents a goal-oriented adaptive edge-based finite element (FE) algorithm for modelling marine controlled-source electromagnetic (CSEM) responses in three-dimensional (3-D) dipping anisotropic conductive media. We adopt the secondary field formulation for quasi-static Maxwell's equations to avoid the source singularity. The algorithm is implemented on unstructured tetrahedral grids, which allow for the simulation of complex model geometries, including tilted electrical interfaces and bathymetry. We validate the adaptive finite element code against a one-dimensional (1-D) layered model with vertical anisotropy. Further, we compare the 3-D FE numerical solutions with those obtained by a 2-D FE code for a two-dimensional dipping anisotropic model. The modelling results from the new 3-D FE algorithm are in good agreement with both the analytical solutions derived from the 1-D code and the numerical results from the anisotropic 2-D FE algorithm. We then use the algorithm to simulate the marine CSEM responses over different anisotropic 3-D models, including an anisotropic reservoir, anisotropic surrounding sediments, and an anisotropic overburden, and performed a detailed analysis of the effect of VTI, HTI, and TTI anisotropy on 3-D marine CSEM response. The modelling results illustrate that marine CSEM responses are considerably influenced by various anisotropic parameters, but to different extents. The research on different types of anisotropy within different media can help us avoid the misinterpretation of marine CSEM data when processing and interpreting of the measured geophysical electromagnetic data.

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

A parallel adaptive finite-element approach for 3-D realistic controlled-source electromagnetic problems using hierarchical tetrahedral grids

SUMMARY To effectively and efficiently interpret or invert controlled-source electromagnetic (CSEM) data which are recorded in areas with the kind of complex geological environments and arbitrary topography that are typical, 3-D CSEM forward modelling software that can quickly solve large-scale problems, provide accurate electromagnetic responses for complex geo-electrical models and can be easily incorporated into inversion algorithms are required. We have developed a parallel goal-oriented adaptive mesh refinement finite-element approach for frequency-domain 3-D CSEM forward modelling with hierarchical tetrahedral grids that can offer accurate electromagnetic responses for large-scale complex models and that can efficiently serve for inversion. The approach uses the goal-oriented adaptive vector finite element method to solve the total electric field vector equation. The geo-electrical model is discretized by unstructured tetrahedral grids which can deal with complex underground geological models with arbitrary surface topography. Different from previous adaptive finite element software working on unstructured tetrahedral grids, we have utilized a novel mesh refinement technique named the longest edge bisection method to generate hierarchically refined grids. As the refined grids are nested into the coarse grids, the refinement technique can precisely map the electrical parameters of inversion grids onto the forward modelling grids so that the extra numerical errors generated by the inconsistency of electrical parameters between inversion grids and forward modelling grids are eliminated. In addition, we use the parallel domain-decomposition technique to further accelerate the computations, and the flexible generalized minimum residual solver (FGMRES) with an auxiliary Maxwell solver pre-conditioner to solve the final large-scale system of linear equations. In the end, we validate the performance of the proposed scheme using two synthetic models and one realistic model. We demonstrate that accurate electromagnetic fields can be obtained by comparison with the analytic solutions and that the code is highly scalable for large-scale problems with millions or even hundreds of millions of unknowns. For the synthetic 3-D model and the realistic model with complex geometry, our solutions match well with the results calculated by an existing 3-D CSEM forward modelling code. Both synthetic and realistic examples demonstrate that our newly developed code is an effective, efficient forward modelling engine for interpreting CSEM field data acquired in areas of complex geology and topography.

Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates

In this study, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and geometric singularities in bounded domains of Rd. Our AMLMC algorithm is built on the results of the weak convergence rates in the work (Moon et al., 2006) for an adaptive algorithm using isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in a deterministic setting. Designed to suit the geometric nature of the singularities in the solution, our AMLMC algorithm uses a sequence of deterministic, non-uniform auxiliary meshes as a building block. The above-mentioned deterministic adaptive algorithm generates these meshes, corresponding to a geometrically decreasing sequence of tolerances. In particular, for a given realization of the diffusivity coefficient and accuracy level, AMLMC constructs its approximate sample using the first mesh in the hierarchy that satisfies the corresponding bias accuracy constraint. This adaptive approach is particularly useful for the lognormal case treated here, which lacks uniform coercivity and thus produces functional outputs that vary over orders of magnitude when sampled. Furthermore, we discuss iterative solvers and compare their efficiency with direct ones. To reduce computational work, we propose a stopping criterion for the iterative solver with respect to the quantity of interest, the realization of the diffusivity coefficient, and the desired level of AMLMC approximation.From the numerical experiments, based on a Fourier expansion of the diffusivity coefficient field, we observe improvements in efficiency compared with both standard Monte Carlo (MC) and standard MLMC (SMLMC) for a problem with a singularity similar to that at the tip of a slit modeling a crack.

Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.

A posteriori error estimates of goal-oriented adaptive finite element methods for nonlinear reaction–diffusion problems

In this paper, we construct two goal-oriented a posteriori error estimates with finite element approximations in the enriched space for the reaction–diffusion equation with the nonlinear reaction term. The first error estimator is proved to be the rigorous global lower and upper bounds on the error in the quantity of interest, and the second one is computed simply and efficiently. We propose the corresponding goal-oriented adaptive finite element algorithms, and show the effectiveness and the similar performance of those methods with a series of numerical experiments.

A goal-oriented adaptive finite-element approach for 3-D marine controlled-source electromagnetic problems with general electrical anisotropy

SUMMARY We present a goal-oriented adaptive finite-element algorithm for accurately modelling marine controlled-source electromagnetic responses in 3-D media with general electrical anisotropy. We formulate the primal boundary value problem in terms of the total electric field for general applications. Following the goal-oriented adaptivity concept, the dual problem is derived from a functional designed to measure the data errors of interest in light of the data quality. We approximate the solutions to the primal and dual formulations using the edge finite-element method on tetrahedral grids for a flexible treatment of complex geological settings and survey geometries. To control the mesh adaptation, we develop a reliable residual-type a posteriori error estimation that takes account of the volumetric residual and the numerical discontinuity of the normal component of the electrical current density and that of the tangential component of the magnetic field with respect to non-smooth and anisotropic coefficients. We demonstrate the proposed modelling solver on 1-D MCSEM scenarios with varying degrees of electrical anisotropy. The comparison with goal-oriented adaptivity results obtained from other three commonly used error indicators shows that our approach is robust in dealing with both moderate and strong electrical anisotropy. After that, we constructed a 3-D hydrocarbon-bearing reservoir model with slope seafloor topography and tilted transverse isotropy in the background to examine our algorithm for the case of multiple sources. Finally, we implement a sensitivity analysis procedure to evaluate the resolution of the electrical anisotropy. The quantitative results indicate limitations and preferences of conventional MCSEM data in resolving anisotropic models, providing fundamental insights for inversion based data interpretation.

Multilevel correction goal-oriented adaptive finite element method for semilinear elliptic equations

In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations.

Adaptive modelling of variably saturated seepage problems

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

Weighted Goal-oriented Adaptive Finite-element for 3D Transient EM Modeling

The unstructured finite-element method has been widely used in 3D time-domain electromagnetic (EM) modeling due to its flexibility for modeling rugged topography and complex underground structures. However, how to generate high-quality grids becomes the key to high-accuracy EM responses. We have developed a weighted goal-oriented adaptive finite-element method based on hybrid posterior error estimation in combination with unstructured vector finite-element method and Backward Euler scheme to create an effective mesh. By introducing a weighting factor and adjusting the relative weights of the hybrid posterior errors, the numerical accuracy and convergence rate are greatly improved. To handle the huge difference of EM responses at different time channels, we introduce another weighting factor defined by the exponential power of time to achieve a synchronous refinement of shallow and deep meshes. The numerical experiments on a homogenous half-space model show that our algorithm performs better than the traditional adaptive method both from the accuracy and convergence. Further, we also test the effectiveness of our algorithm for modeling different abnormal bodies under a topographic earth.

Adaptive finite element modelling of three-dimensional magnetotelluric fields in general anisotropic media

We present a goal-oriented adaptive finite element (FE) modelling algorithm for 3-D magnetotelluric fields in generally anisotropic conductivity media. The model consists of a background layered structure, containing anisotropic blocks. Each block and layer might be anisotropic by assigning to them 3×3 conductivity tensors. The second-order partial differential equations are solved using the adaptive finite element method (FEM). The computational domain is subdivided into unstructured tetrahedral elements, which allow for complex geometries including bathymetry and dipping interfaces. The grid refinement process is guided by a global posteriori error estimator and is performed iteratively.The system of linear FE equations for electric field E is solved with a direct solver MUMPS. Then the magnetic field H can be found, in which the required derivatives are computed numerically using cubic spline interpolation. The 3-D FE algorithm has been validated by comparisons with both the 3-D finite-difference solution and 2-D FE results. Two model types are used to demonstrate the effects of anisotropy upon 3-D magnetotelluric responses: horizontal and dipping anisotropy. Finally, a 3D sea hill model is modelled to study the effect of oblique interfaces and the dipping anisotropy.

3-D direct current resistivity anisotropic modelling by goal-oriented adaptive finite element methods

3-D direct current resistivity anisotropic modelling by goal-oriented adaptive finite element methods

Adaptive unified continuum FEM modeling of a 3D FSI benchmark problem.

In this paper, we address a 3D fluid-structure interaction benchmark problem that represents important characteristics of biomedical modeling. We present a goal-oriented adaptive finite element methodology for incompressible fluid-structure interaction based on a streamline diffusion-type stabilization of the balance equations for mass and momentum for the entire continuum in the domain, which is implemented in the Unicorn/FEniCS software framework. A phase marker function and its corresponding transport equation are introduced to select the constitutive law, where the mesh tracks the discontinuous fluid-structure interface. This results in a unified simulation method for fluids and structures. We present detailed results for the benchmark problem compared with experiments, together with a mesh convergence study.

A goal-oriented adaptive finite-element method for 3D scattered airborne electromagnetic method modeling

We have developed a goal-oriented adaptive unstructured finite-element method based on the scattered field for 3D frequency-domain airborne electromagnetic (AEM) modeling. To guarantee the EM field divergence free within each element and the continuity conditions at electrical material interfaces, we have used the edge-based shape functions to approximate the electrical field. The posterior error for finite-element adaptive meshing procedure is estimated from the continuity of the normal component of the current density, whereas the influence functions are estimated by solving a dual forward problem. Because the imaginary part of the scattered current is discontinuous and the real part is continuous, we use the latter to estimate the posterior error. For the multisources and multifrequencies problem in AEM, we calculate the weighted posterior error for each element by considering only those transmitter-receiver pairs that do not adhere to our convergence criteria. Finally, we add a minimum volume constraint to improve the stability of the adaptive procedure. To check the accuracy, we compared our adaptive results with the semianalytical solutions for AEM systems over a half-space model. To test the effectiveness of our algorithm for multiple sources and multiple frequencies of AEM, we analyzed meshes for separate frequencies and for combined frequencies. Finally, we calculated the AEM responses over a hill model with and without embedded abnormal bodies to prove the feasibility of our algorithm for AEM variable topography modeling.

MARE2DEM: a 2-D inversion code for controlled-source electromagnetic and magnetotelluric data

This work presents MARE2DEM, a freely available code for 2-D anisotropic inversion of magnetotelluric (MT) data and frequency-domain controlled-source electromagnetic (CSEM) data from onshore and offshore surveys. MARE2DEM parametrizes the inverse model using a grid of arbitrarily shaped polygons, where unstructured triangular or quadrilateral grids are typically used due to their ease of construction. Unstructured grids provide significantly more geometric flexibility and parameter efficiency than the structured rectangular grids commonly used by most other inversion codes. Transmitter and receiver components located on topographic slopes can be tilted parallel to the boundary so that the simulated electromagnetic fields accurately reproduce the real survey geometry. The forward solution is implemented with a goal-oriented adaptive finite-element method that automatically generates and refines unstructured triangular element grids that conform to the inversion parameter grid, ensuring accurate responses as the model conductivity changes. This dual-grid approach is significantly more efficient than the conventional use of a single grid for both the forward and inverse meshes since the more detailed finite-element meshes required for accurate responses do not increase the memory requirements of the inverse problem. Forward solutions are computed in parallel with a highly efficient scaling by partitioning the data into smaller independent modeling tasks consisting of subsets of the input frequencies, transmitters and receivers. Non-linear inversion is carried out with a new Occam inversion approach that requires fewer forward calls. Dense matrix operations are optimized for memory and parallel scalability using the ScaLAPACK parallel library. Free parameters can be bounded using a new non-linear transformation that leaves the transformed parameters nearly the same as the original parameters within the bounds, thereby reducing non-linear smoothing effects. Data balancing normalization weights for the joint inversion of two or more data sets encourages the inversion to fit each data type equally well. A synthetic joint inversion of marine CSEM and MT data illustrates the algorithm's performance and parallel scaling on up to 480 processing cores. CSEM inversion of data from the Middle America Trench offshore Nicaragua demonstrates a real world application. The source code and MATLAB interface tools are freely available at http://mare2dem.ucsd.edu.

NCP Function--Based Dual Weighted Residual Error Estimators for Signorini's Problem

In this paper, we consider goal-oriented adaptive finite element methods for Signorini's problem. The basis is a mixed formulation, which is reformulated as nonlinear variational equality using a nonlinear complementarity function. For a general discretization, we derive error identities w.r.t. a possible nonlinear quantity of interest in the displacement as well as in the contact forces, which are included as Lagrange multiplier, using the dual weighted residual method. Afterwards, a numerical approximation of the error identities is introduced. We exemplify the results for a low order mixed discretization of Signorini's problem. The theorectical findings and the numerical approximation scheme are finally substantiated by some numerical examples.

A goal-oriented adaptive finite element method for 3D resistivity modeling using dual-error weighting approach

A goal-oriented adaptive finite element (FE) method for solving 3D direct current (DC) resistivity modeling problem is presented. The model domain is subdivided into unstructured tetrahedral elements that allow for efficient local mesh refinement and flexible description of complex models. The elements that affect the solution at each receiver location are adaptively refined according to a goal-oriented posteriori error estimator using dual-error weighting approach. The FE method with adapting mesh can easily handle such structures at almost any level of complexity. The method is demonstrated on two synthetic resistivity models with analytical solutions and available results from integral equation method, so the errors can be quantified. The applicability of the numerical method is illustrated on a resistivity model with a topographic ridge. Numerical examples show that this method is flexible and accurate for geometrically complex situations.