Parallel 3-D marine controlled-source electromagnetic modelling using high-order tetrahedral Nédélec elements

We present a parallel and high-order Nédélec finite element solution for the marine controlled-source electromagnetic (CSEM) forward problem in 3-D media with isotropic conductivity. Our parallel Python code is implemented on unstructured tetrahedral meshes, which support multiple-scale structures and bathymetry for general marine 3-D CSEM modelling applications. Based on a primary/secondary field approach, we solve the diffusive form of Maxwell’s equations in the low-frequency domain. We investigate the accuracy and performance advantages of our new high-order algorithm against a low-order implementation proposed in our previous work. The numerical precision of our high-order method has been successfully verified by comparisons against previously published results that are relevant in terms of scale and geological properties. A convergence study confirms that high-order polynomials offer a better trade-off between accuracy and computation time. However, the optimum choice of the polynomial order depends on both the input model and the required accuracy as revealed by our tests. Also, we extend our adaptive-meshing strategy to high-order tetrahedral elements. Using adapted meshes to both physical parameters and high-order schemes, we are able to achieve a significant reduction in computational cost without sacrificing accuracy in the modelling. Furthermore, we demonstrate the excellent performance and quasi-linear scaling of our implementation in a state-of-the-art high-performance computing architecture.This project has received funding from the European Union's Horizon 2020 programme under the Marie Sklodowska-Curie grant agreement No. 777778. Furthermore, the research leading to these results has received funding from the European Union's Horizon 2020 programme under the ChEESE Project (https://cheese-coe.eu/ ), grant agreement No. 823844. In addition, the authors would also like to thank the support of the Ministerio de Educación y Ciencia (Spain) under Projects TEC2016-80386-P and TIN2016-80957-P. The authors would like to thank the Editors-in-Chief and to both reviewers, Dr. Martin Cuma and Dr. Raphael Rochlitz, for their valuable comments and suggestions which helped to improve the quality of the manuscript. This work benefited from the valuable suggestions, comments, and proofreading of Dr. Otilio Rojas (BSC). Last but not least, Octavio Castillo-Reyes thanks Natalia Gutierrez (BSC) for her support in CSEM modeling with BSIT.Peer ReviewedPostprint (author's final draft

Gradient discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media with discontinuous pressures at the matrix fracture interfaces

We investigate the discretization of Darcy flow through fractured porous media on general meshes. We consider a hybrid dimensional model, invoking a complex network of planar fractures. The model accounts for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrix-fracture interfaces. The numerical analysis is performed in the general framework of gradient discretizations which is extended to the model under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to satisfy the gradient scheme framework, which yields, in particular, convergence. Numerical tests confirm the theoretical results. Gradient Discretization; Darcy Flow, Discrete Fracture Networks, Finite Volum

A Comparison of Numerical Methods used for\ud Finite Element Modelling of Soft Tissue\ud Deformation

Soft tissue deformation is often modelled using incompressible nonlinear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular, the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making up the mesh. We investigate the effect of these choices on the accuracy of the computed solution, using a selection of model problems motivated by typical deformations seen in soft tissue modelling. We set up model problems with discontinuous material properties (as is the case for the breast), steeply changing gradients in the body force (as found in contracting cardiac tissue), and discontinuous first derivatives in the solution at the boundary, caused by a discontinuous applied force (as in the breast during mammography). We find that the choice of pressure basis functions are vital in the presence of a material interface, higher-order schemes do not perform as well as may be expected when there are sharp gradients, and in general that it is important to take the expected regularity of the solution into account when choosing a numerical scheme

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Generalized Finite Element Systems for smooth differential forms and Stokes problem

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8 figures and several comments adde