Ralph-Uwe Börner

Bio: Ralph-Uwe Börner is an academic researcher from Freiberg University of Mining and Technology. The author has contributed to research in topics: Finite element method & Discretization. The author has an hindex of 8, co-authored 24 publications receiving 529 citations.

Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example

Abstract: SUMMARY We present a new 3-D vector finite element code and demonstrate its strength by modelling a realistic marine CSEM scenario. Unstructured tetrahedral meshes easily allow for the inclusion of arbitrary seafloor bathymetry so that natural environments are mapped into the model in a close-to-reality way. A primary/secondary field approach, an adaptive mesh refinement strategy as well as a higher order polynomial finite element approximation improve the solution accuracy. A convergence study strongly indicates that the use of higher order finite elements is beneficial even if the solution is not globally smooth. The marine CSEM scenario also shows that seafloor topography gives an important response which needs to be reproduced by numerical modelling to avoid the misinterpretation of measurements.

Numerical Modelling in Geo-Electromagnetics: Advances and Challenges

Abstract: During the last decade, tremendous advances have been observed in the broad field of numerical modelling for geo-electromagnetic applications. This trend received support due to increasing industrial needs, mainly caused by hydrocarbon and ore exploration industry. On the other hand, the increasing reliability and accuracy of data acquisition techniques further spurs this development. In this review, we will focus on advances and challenges in numerical modelling in geo-electromagnetics. We review recent developments in the discrete solution of the 3-D induction problem in the time and frequency domains. Particularly, advantages and disadvantages of the common numerical techniques for solving partial differential equations such as the Finite Difference and Finite Element methods will be considered.

Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection

Abstract: SUMMARY We present an efficient numerical method for the simulation of transient electromagnetic fields resulting from magnetic and electric dipole sources in three dimensions The method we propose is based on the Fourier synthesis of frequency domain solutions at a sufficient number of discrete frequencies obtained using a finite element (FE) approximation of the damped vector wave equation obtained after Fourier transforming Maxwell's equations in time We assume the solution to be required only at a few points in the computational domain, whose number is small relative to the number of FE degrees of freedom The mapping which assigns to each frequency the FE approximation at these points of interest is a vector-valued rational function known as the transfer function Its evaluation is approximated using Krylov subspace projection, a standard model reduction technique Computationally, this requires the FE discretization at a small number of reference frequencies and the generation of a sufficiently large Krylov subspace associated with each reference frequency Once a basis of this subspace is available, a sufficiently accurate rational approximation of the transfer function can be evaluated at the remaining frequencies at negligible cost These partial frequency domain solutions are then synthesized to the time evolution at the points of interest using a fast Hankel transform To test the algorithm, responses obtained by2-D and 3-D FE formulations have been calculated for a layered half-space and compared with results obtained analytically, for which we observed a maximum deviation of less than 2 per cent in the case of transient EM modelling We complete our model studies with a number of comparisons with established numerical approaches A first implementation of our new numerical algorithm already gives very good results using much less computational time compared with time stepping methods and comparable times and accuracy compared with the Spectral Lanczos Decomposition Method (SLDM)

Adaptive unstructured grid finite element simulation of two-dimensional magnetotelluric fields for arbitrary surface and seafloor topography

Abstract: SUMMARY We present an adaptive unstructured triangular grid finite element approach for effectively simulating plane-wave diffusive electromagnetic fields in 2-D conductivity structures. The most striking advantage of irregular grids is their potential to incorporate arbitrary geometries including surface and seafloor topography. Adaptive mesh refinement strategies using an a posteriori error estimator yield most efficient numerical solutions since meshes are only refined where required. We demonstrate the robustness of this approach by comparison with analytical solutions and previously published numerical simulations. Maximum errors may systematically be reduced to, for example, 0.8 per cent for the apparent resistivity and 0.2° in the phase. An additional accuracy study of the thickness of the air layer in E-polarization suggests to keep a minimum thickness depending on lateral conductivity contrasts within the earth. Furthermore, we point out the new quality and flexibility of our simulation technique by addressing two marine magnetotelluric applications. In the first case, we discuss topographic effects associated with a synthetic sinusoidal sea bottom model and in the second case, we show a close-to-reality scenario using real bathymetry data from the East Pacific Rise at 17°S.

Three-Dimensional Transient Electromagnetic Modeling Using Rational Krylov Methods

Abstract: A computational method is given for solving the forward modeling problem for transient electromagnetic exploration. Its key features are discretization of the quasi-static Maxwell's equations in space using the first-kind family of curl-conforming Nedelec elements combined with time integration using rational Krylov subspace methods. We show how rational Krylov subspace methods may be used to solve the same problem in the frequency domain followed by a synthesis of the transient solution using the fast Hankel transform, arguing that the pure time-domain is more efficient. We also propose a simple method for selecting the pole parameters of the rational Krylov subspace method which leads to convergence within an a priori determined number of iterations independent of mesh size and conductivity structure. These poles are repeated in a cyclic fashion, which, in combination with direct solvers for the discrete problem, results in significantly faster solution times than previously proposed schemes.

Learning Mesh-Based Simulation with Graph Networks

Abstract: Mesh-based simulations are central to modeling complex physical systems in many disciplines across science and engineering. Mesh representations support powerful numerical integration methods and their resolution can be adapted to strike favorable trade-offs between accuracy and efficiency. However, high-dimensional scientific simulations are very expensive to run, and solvers and parameters must often be tuned individually to each system studied. Here we introduce MeshGraphNets, a framework for learning mesh-based simulations using graph neural networks. Our model can be trained to pass messages on a mesh graph and to adapt the mesh discretization during forward simulation. Our results show it can accurately predict the dynamics of a wide range of physical systems, including aerodynamics, structural mechanics, and cloth. The model's adaptivity supports learning resolution-independent dynamics and can scale to more complex state spaces at test time. Our method is also highly efficient, running 1-2 orders of magnitude faster than the simulation on which it is trained. Our approach broadens the range of problems on which neural network simulators can operate and promises to improve the efficiency of complex, scientific modeling tasks.

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

Abstract: 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.

3D finite-difference frequency-domain modeling of controlled-source electromagnetic data: Direct solution and optimization for high accuracy

Abstract: Three-dimensional modeling of marine controlled-source electromagnetic (CSEM) data is vital to improve the understanding of electromagnetic (EM) responses collected in increasingly complex geologic settings. A modeling tool for simulating 3D marine CSEM surveys, based on a finite-difference discretization of the Helmholtz equation for the electric fields, has been developed. Optimizations for CSEM simulations include the use of a frequency-domain technique, a staggering scheme that reduces inaccuracies especially for horizontal electric-dipole sources located near the seafloor, and a new interpolation technique that provides highly accurate EM field values for receivers located in the immediate vicinity of the seafloor. Source singularities are eliminated through a secondary-field approach, in which the primary fields are computed analytically for a homogeneous or a 1D layered background; the secondary fields are computed using the finite-difference technique. Exploiting recent advances in computer technology and algorithmic developments, the system of finite-difference equations is solved using the MUMPS direct-matrix solver. In combination with the other optimizations, this allows accurate EM field computations for moderately sized models on small computer clusters. The explicit availability of matrix factorizations is advantageous for multisource modeling and makes the algorithm well suited for future use within an inversion scheme. Comparisons of simulated data for (1) 1D models to data generated using a 1D reflectivity technique and (2) 3D models to published 3D data demonstrate the accuracy and benefits of the approach.

A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling

Abstract: SUMMARY We present a parallel goal-oriented adaptive finite element method that can be used to rapidly compute highly accurate solutions for 2.5-D controlled-source electromagnetic (CSEM) and 2-D magnetotelluric (MT) modelling problems. We employ unstructured triangular grids that permit efficient discretization of complex modelling domains such as those containing topography, dipping layers and multiple scale structures. Iterative mesh refinement is guided by a goal-oriented error estimator that considers the relative error in the strike aligned fields and their spatial gradients, resulting in a more efficient mesh refinement than possible with a previous approach based on the absolute errors. Reliable error estimation is accomplished by a dual weighted residual method that is carried out via hierarchical basis computations. Our algorithm is parallelized over frequencies, wavenumbers, transmitters and receivers, where adaptive refinement is performed in parallel on subsets of these parameters. Mesh sharing allows an adapted mesh generated for a particular frequency and wavenumber to be shared with nearby frequencies and wavenumbers, thereby efficiently reducing the parallel load of the adaptive refinement calculations. We demonstrate the performance of our algorithm on a large cluster computer through scaling tests for a complex model that includes strong seafloor topography variations and multiple thin stacked hydrocarbon reservoirs. In tests using up to 800 processors and a realistic suite of CSEM data parameters, our algorithm obtained run-times as short as a few seconds to tens of seconds.