Goal-oriented Adaptive Mesh Refinement Research Articles

An improved goal-oriented adaptive finite-element method for 3-D direct current resistivity anisotropic forward modeling using nested tetrahedra

We developed a novel adaptive finite element method (FEM) to address the problem of 3-D direct current (DC) resistivity forward modeling with complex surface topography and arbitrary conductivity anisotropy. The tetrahedra-based FEM and secondary virtual potential algorithm are first used to handle arbitrary complex geo-models. Then, to ensure the accuracy of the simulation solution, an improved goal-oriented adaptive mesh refinement (AMR) algorithm is proposed to realize an optimized mesh density distribution. To avoid the drawback of the traditional goal-oriented AMR algorithm for the DC forward modeling problem, we incorporate a volume-based weighting factor into the posterior error estimation procedure to further optimize the density distribution of the forward modeling grid. In addition, instead of traditional open source mesh generation software, we propose using the longest-edge bisection (LEB) algorithm to perform the mesh refinement process, which can preserve the topological structure between different-level meshes. Finally, the comprehensive test using a two-layered model and two complex 3-D models demonstrate the capability of our newly developed code to obtain highly accurate solutions even on relatively coarse initial grids. By incorporating the volume factor, our novel AMR algorithm achieves a more uniform and reasonable mesh density distribution during these experiments. The LEB refinement technique can generate a series of nested tetrahedral elements and provide fewer tetrahedral elements compared to the traditional Delaunay-based AMR method. The proposed 3-D DC forward modeling method has been implemented into an open source C++ code, which will contribute to the advancement of the 3-D DC resistivity imaging field.

DC3DPAFEM: An efficient and accurate 3-D direct current resistivity anisotropic forward modeling software for complex geological settings

Nowadays, there is a growing trend that direct current (DC) field surveys are shifting towards challenging areas characterized by mountainous topography and electrical anisotropy. Given these complex geological settings, there is an urgent need for 3-D DC forward modeling software capable of effectively addressing large-scale problems and delivering accurate modeling results to interpret field data. However, most open-source software packages face certain limitations, such as the high numerical cost to handle complex surface topography, the lack of consideration for anisotropic conductivity, the absence of mesh refinement techniques to guarantee accuracy in forward modeling, and the lack of parallel computing techniques to solve large-scale problems. In this study, we develop an efficient and highly accurate 3-D DC anisotropic forward modeling software, namely DC3DPAFEM, using the adaptive finite element algorithm based on the unstructured tetrahedral mesh. Firstly, we construct a strong compatible boundary value problem (BVP) for 3-D anisotropic DC problems by adopting a specialized secondary potential approach to handle the surface topography efficiently. Then, we develop a goal-oriented adaptive mesh refinement (AMR) technique to ensure accurate forward modeling results, even with a coarse initial mesh. To ensure time and memory efficiency, we employ a robust conjugate gradient (CG) algorithm preconditioned by the algebraic multigrid (AMG) solver to solve the large-scale linear system of equations resulting from complex geological structures. We aim to investigate the performance of the AMG scheme in anisotropic DC cases. Furthermore, we incorporate the domain decomposition technique into the iterative solution scheme for further efficiency gains. This technique significantly improves computing efficiency for large-scale problems in parallel clusters. Finally, we conduct comprehensive performance tests for DC3DPAFEM using a two-layer anisotropic model and a 3-D complex model with undulating terrain. The results of both examples validate the accuracy of DC3DPAFEM, as they closely align with the analytical solutions and the solutions obtained from the existing 3-D DC forward modeling code. Compared to traditional direct solver MUMPS and ILU-preconditioned iterative solvers, DC3DPAFEM exhibits highly scalable performance for large-scale problems, offering significant advantages in terms of memory and time consumption. Overall, DC3DPAFEM demonstrates substantial advances in efficiency, accuracy, and practicality through a series of numerical examples. This open-source code provides an efficient and available tool for developing a 3-D DC inversion method that can deal with large-scale problems involving intricate topography and anisotropic media.

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.

An adaptive multilevel Monte Carlo algorithm for the stochastic drift–diffusion–Poisson system

We present an adaptive multilevel Monte Carlo algorithm for solving the stochastic drift–diffusion–Poisson system with non-zero recombination rate. The a-posteriori error is estimated to enable goal-oriented adaptive mesh refinement for the spatial dimensions, while the a-priori error is estimated to guarantee linear convergence of the H1 error. In the adaptive mesh refinement, efficient estimation of the error indicator gives rise to better error control. For the stochastic dimensions, we use the multilevel Monte Carlo method to solve this system of stochastic partial differential equations. Finally, the advantage of the technique developed here compared to uniform mesh refinement is discussed using a realistic numerical example.

Goal-Oriented Adaptive Mesh Refinement for Discontinuous Petrov--Galerkin Methods

In this article, a new unified duality theory is developed for Petrov-Galerkin finite element methods. This novel theory is then used to motivate goal-oriented adaptive mesh refinement strategies for use with discontinuous Petrov-Galerkin (DPG) methods. The focus of this article is mainly on broken ultraweak variational formulations of stationary boundary value problems, however, many of the ideas presented within are general enough that they be extended to any such well-posed variational formulation. The proposed goal-oriented adaptive mesh refinement procedures require the construction of refinement indicators for both a primal problem and a dual problem. In the DPG context, the primal problem is simply the system of linear equations coming from a standard DPG method and the dual problem is a similar system of equations, coming from a new method which is dual to DPG. This new method has the same coefficient matrix as the associated DPG method but has a different load. We refer to this new finite element method as a DPG* method. A thorough analysis of DPG* methods, as stand-alone finite element methods, is not given here but will be provided in subsequent articles. For DPG methods, the current theory of a posteriori error estimation is reviewed and the reliability estimate in [13, Theorem 2.1] is improved on. For DPG* methods, three different classes of refinement indicators are derived and several contributions are made towards rigorous a posteriori error estimation. At the closure of the article, results of numerical experiments with Poisson's boundary value problem in a three-dimensional domain are provided. These results clearly demonstrate the utility of the goal-oriented adaptive mesh refinement strategies for quantities of interest with either interior or boundary terms.

Adaptive and iterative methods for simulations of nanopores with the PNP–Stokes equations

We present a 3D finite element solver for the nonlinear Poisson–Nernst–Planck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule dynamics inside nanopore sensors. The source code is released online at github.com/mitschabaude/nanopores.We add to existing numerical approaches by deploying goal-oriented adaptive mesh refinement. To reduce the computation overhead of mesh adaptivity, our error estimator uses the much cheaper Poisson–Boltzmann equation as a simplified model, which is justified on heuristic grounds but shown to work well in practice. To address the nonlinearity in the full PNP–Stokes system, three different linearization schemes are proposed and investigated, with two segregated iterative approaches both outperforming a naive application of Newton's method. Numerical experiments are reported on a real-world nanopore sensor geometry.We also investigate two different models for the interaction of target molecules with the nanopore sensor through the PNP–Stokes equations. In one model, the molecule is of finite size and is explicitly built into the geometry; while in the other, the molecule is located at a single point and only modeled implicitly – after solution of the system – which is computationally favorable. We compare the resulting force profiles of the electric and velocity fields acting on the molecule, and conclude that the point-size model fails to capture important physical effects such as the dependence of charge selectivity of the sensor on the molecule radius.

SU2: An Open-Source Suite for Multiphysics Simulation and Design

This paper presents the main objectives and a description of the SU2 suite, including the novel software architecture and open-source software engineering strategy. SU2 is a computational analysis and design package that has been developed to solve multiphysics analysis and optimization tasks using unstructured mesh topologies. Its unique architecture is well suited for extensibility to treat partial-differential-equation-based problems not initially envisioned. The common framework adopted enables the rapid implementation of new physics packages that can be tightly coupled to form a powerful ensemble of analysis tools to address complex problems facing many engineering communities. The framework is demonstrated on a number, solving both the flow and adjoint systems of equations to provide a high-fidelity predictive capability and sensitivity information that can be used for optimal shape design using a gradient-based framework, goal-oriented adaptive mesh refinement, or uncertainty quantification.

Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method

We have investigated the use of the adaptive high-order finite-element method (FEM) for geoelectromagnetic modeling. Because high-order FEM is challenging from the numerical and computational points of view, most published finite-element studies in geoelectromagnetics use the lowest order formulation. Solution of the resulting large system of linear equations poses the main practical challenge. We have developed a fully parallel and distributed robust and scalable linear solver based on the optimal block-diagonal and auxiliary space preconditioners. The solver was found to be efficient for high finite element orders, unstructured and nonconforming locally refined meshes, a wide range of frequencies, large conductivity contrasts, and number of degrees of freedom (DoFs). Furthermore, the presented linear solver is in essence algebraic; i.e., it acts on the matrix-vector level and thus requires no information about the discretization, boundary conditions, or physical source used, making it readily efficient for a wide range of electromagnetic modeling problems. To get accurate solutions at reduced computational cost, we have also implemented goal-oriented adaptive mesh refinement. The numerical tests indicated that if highly accurate modeling results were required, the high-order FEM in combination with the goal-oriented local mesh refinement required less computational time and DoFs than the lowest order adaptive FEM.

Goal-oriented error estimation and adaptivity for fluid–structure interaction using exact linearized adjoints

We develop duality-based a posteriori error estimates for functional outputs of solutions of steady fluid–structure-interaction problems. The crucial complication in obtaining these estimates pertains to the derivation of the coupled dual (exact linearized adjoint) problem owing to the free-boundary character of fluid–structure interaction. We present two approaches to derive the dual problem. In the domain-map linearization approach, the fluid subproblem is first transformed to a fixed reference domain, after which one essentially linearizes with respect to the domain transformation map. In the shape-linearization approach, fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape-derivative techniques. We show that the dual problems correspond to coupled fluid–structure problems with nonstandard coupling conditions. Furthermore, we present numerical experiments that demonstrate the consistency of the dual-based error estimates and their usefulness in goal-oriented adaptive mesh-refinement.

Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Standard and goal-oriented adaptive mesh refinement (AMR) techniques are presented for the linear Boltzmann transport equation. A posteriori error estimates are employed to drive the AMR process and are based on angular-moment information rather than on directional information, leading to direction-independent adapted meshes. An error estimate based on a two-mesh approach and a jump-based error indicator are compared for various test problems. In addition to the standard AMR approach, where the global error in the solution is diminished, a goal-oriented AMR procedure is devised and aims at reducing the error in user-specified quantities of interest. The quantities of interest are functionals of the solution and may include, for instance, point-wise flux values or average reaction rates in a subdomain. A high-order (up to order 4) Discontinuous Galerkin technique with standard upwinding is employed for the spatial discretization; the discrete ordinates method is used to treat the angular variable.

Efficient reconstruction of local heat fluxes in pool boiling experiments by goal-oriented adaptive mesh refinement

In this paper, we consider the efficient estimation of local boiling heat fluxes from transient temperature measurements in the heater close to the heater surface. For accurate prediction, heat flux estimation is formulated as a transient three-dimensional (3D) inverse heat conduction problem (IHCP). This inverse problem is ill-posed and cannot be treated straightforwardly by established numerical methods. In order to obtain a regularized stable solution, a large-scale time-dependent PDE-constrained optimization problem has to be solved and an appropriate stopping criterion for the termination of the iterative solution process has to be chosen. Since the boiling heat flux is non-uniformly distributed on the heater surface due to the strong local activity of the boiling process, the use of a fixed uniform spatial discretization is not efficient. Instead, an adaptive mesh refinement strategy can be used to obtain an appropriate discretization which significantly reduces the total computational effort. In this work, we present an automatic algorithm incorporating an adaptive mesh refinement via a heat flux-based a-posteriori error estimation technique. The suggested algorithm can cope with both spatially point-wise or highly resolved temperature observations efficiently. It is applied to real measurement data obtained from two different types of pool boiling experiments. The numerical results show that the computational effort can be reduced significantly for given estimation quality. This adaptive IHCP solution technique can be also viewed as an efficient soft sensor to deduce unmeasurable local boiling heat fluxes.

Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Shape-Linearization Approach

We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems via shape-linearization principles. To derive an appropriate dual (linearized adjoint) problem, we linearize the domain dependence of the very weak form and goal functional of interest using techniques from shape calculus. We show for a Bernoulli-type free-boundary problem that the dual problem corresponds to a Poisson problem with a Robin-type boundary condition involving the curvature. Moreover, we derive a generalization of the dual problem for nonsmooth free boundaries which includes a natural extension of the curvature term. The effectivity of the dual-based error estimate and its usefulness in goal-oriented adaptive mesh refinement is demonstrated by numerical experiments.

Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Domain-Map Linearization Approach

In free-boundary problems, the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems that include both sources of error. The derivation of an appropriate dual problem (linearized adjoint) is, however, nonobvious for free-boundary problems. To derive an appropriate dual problem, we present the domain-map linearization approach. In this approach, the free-boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map. We show for a Bernoulli-type free-boundary problem that this dual problem corresponds to a Poisson problem with a nonlocal Robin-type boundary condition. Furthermore, we present numerical experiments that demonstrate the effectivity of the dual-based error estimate and its usefulness in goal-oriented adaptive mesh refinement.

Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model

The quasicontinuum approximation [E.B. Tadmor, M. Ortiz, R. Phillips, Quasicontinuum analysis of defects in solids, Philos. Mag. A 73(6) (1996) 1529–1563] is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation. In regions of the crystal with a highly nonuniform deformation such as around defects, every atom must be a representative atom to obtain sufficient accuracy, but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy. We present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized Frenkel-Kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy.