In Finite Element Methods for solving electromagnetic problems, the use of Nédélec Elements has become very popular. In fact, Nédélec Elements are often said to be a cure to many difficulties that are encountered, particularly eliminating spurious solutions, and are claimed to yield accurate results.  In this paper, we present our first steps in developing a Nédélec Elements code for simulation of geophysical electromagnetic problems and first ideas on how implement the key issues of Edge Elements in an efficient way on HPC platforms

Castillo, Octavio

Cela Espín, José M.

Puente, Josep de la

Puzyrev, Vladimir

English

UPCommons

  
 
 
~ 56 ~ 
HPC and edge elements for geophysical 
electromagnetic problems: an overview 
 
Octavio Castillo, Josep de la Puente, Vladimir Puzyrev, José María Cela 
CASE Department, Barcelona Supercomputing Center 
octavio.castillo@bsc.es 
 
 
Abstract- In Finite Element Methods for solving electromagnetic 
problems, the use of Nédélec Elements has become very popular. 
In fact, Nédélec Elements are often said to be a cure to many 
difficulties that are encountered, particularly eliminating 
spurious solutions, and are claimed to yield accurate results.  In 
this paper, we present our first steps in developing a Nédélec 
Elements code for simulation of geophysical electromagnetic 
problems and first ideas on how implement the key issues of 
Edge Elements in an efficient way on HPC platforms. 
 
I. INTRODUCTION 
Electromagnetic Methods (EM) are an established tool in 
geophysics, finding application in many areas such as 
hydrocarbon and mineral exploration, reservoir monitoring, 
CO2 storage characterization, geothermal reservoir imaging 
and many others. The marine controlled-source 
electromagnetic method (CSEM) has become an important 
technique for reducing ambiguities in data interpretation in the 
offshore environment and a commonplace in the industry. In 
the traditional configuration, the sub-seafloor structure is 
explored by emitting low-frequency signals from a high-
powered electric dipole source towed close to the seafloor. By 
studying the received signal, the subsurface structures could 
be detected at scales of a few tens of meters to depths of 
several kilometers. 
 
On the other hand, in the Finite Element Method for solving 
electromagnetic field problems, the use of Edge-based 
elements (Nédélec elements) has become very popular. In fact, 
Nédélec elements are often said to be a cure to many 
difficulties that are encountered, particularly eliminating 
spurious solutions, and are claimed to yield accurate results. In 
Edge elements, the simulated quantity is interpolated by vector 
shape functions that provide several characteristics important 
for the representation of the electric field. 
 
Based on previous ideas and considering the societal value 
of exploration geophysics, since this process is essential to 
among others, we present the key issues of edge element to 
simulate geophysical electromagnetic problems, namely, we 
present our first steps in developing a Nédélec Elements code 
for simulation of geophysical electromagnetic problems and 
first ideas on how implement the key issues of Edge Elements 
in an efficient way. Strong emphasis is placed on three aspects 
not easily found in the literature:  vector shape functions, 
orientation of geometric entities and the technique to avoid the 
source singularities in CSEM simulations. Finally, we make 
some concluding remarks and show the performance of the r 
code in terms of efficiency and accuracy. 
II. NÉDÉLEC ELEMENTS FORMULATION 
The Nédélec Finite Element Method uses vector basis 
functions defined on the edges of the corresponding elements. 
Similar to the conventional node-based finite element method, 
the modeling domain can be discretized using rectangular, 
tetrahedron, hexahedron or other complex elements [1]. 
Fig. 1 (left) is an illustration of the tetrahedron grid that we 
used with node (red numbers) and edge indexing (blue 
numbers). Following the work of [1], the tangential 
component of the electric field is assigned to the center of 
each edge. The edge basis functions defined by [1] are 
divergence free but not curl free. The vector basis functions 
are also continuous at the element boundaries. As a result, the 
divergence free condition of the electric field in the source free 
region and the continuity conditions are imposed directly in 
the Nédélec Finite Element formulation. Thus, the element 
vector basis functions for the field associated with the edges 
are given by Fig. 1 (right) where Li are defined in [1] and li              
represents the lentgh of edge i. 
In order to guarantee global continuity of tangential 
components, special care has to be taken with regard to the 
edge direction. n short, the edge direction strategy works as 
follows. A global index is assigned to each node of the mesh. 
If an edge connects two nodes ni and nj, we define the 
direction of the edge as going from node ni  to node nj if i<j. 
This gives a unique orientation of each edge. The same 
philosophy is used locally to determine the directions of the 
local edges on each element. 
 
 
 
 
 
 
Fig. 1. Nédélec element description with nodes and edges (left). Vector basis 
functions of linear edge elements right).
 
  
 
 
~ 57 ~ 
III. SUMMARY OF RESULTS 
  The following images are the result of our first steps in 
developing a Nédélec Elements code for simulation of 
geophysical electromagnetic problems. 
 
In geophysical simulations with CSEM, the primary field is 
calculated analytically for a background layered-earth model. 
The secondary field is discretized using edge-elements. The 
primary field computation stage is a classic example of 
embarrassingly parallel workload because no exists 
dependency or communication between those parallel tasks.  
On the other hand, numerical modeling of geophysical  
electromagnetic problems imposes a significant 
computational burden. In solving large-scale models, single 
processor-based methods are limited and often incapable of 
managing the required large memory and computational time 
requirements. Therefore, we are exploring parallel 
environments such as MPI and OpenMP in order to improve 
the performance of these simulations. 
 
# T # E Assembl
y time 
Solver 
time 
h  L2 OL 
128 208 7.66e-03 6.76e-03 3.3e-01 3.969e-01 --- 
512 800 6.75e-03 7.72e-03 2.0e-01 9.831e-02 2.016 
2,048 3136 2.22e-02 4.47e-02 1.1e-01 2.430e-02 2.017 
8,192 12,41
6 
1.82e-01 2.08e-01 5.8e-02 6.001e-03 1.999 
32,768 49,40
8 
4.27e-01 6.68e-01 3.0e-02 1.501e-03 2.004 
131,072 197,1
20 
1.51e00 2.27e00 1.5e-02 3.741e-04 2.002 
524,288 787,4
56 
5.83e00 6.71e00 7.8e-03 9.340e-05 2.002 
2,097,1
52 
3,147
,776 
21.32e0
0 
24.7e00 3.9e-03 2.333e-05 2.001 
 Table 1: Summary of results for the 2D case. Number of elements (T),  
number of edges (E), assembly time (seconds), solver time (seconds), mesh 
spacing (h), L2 error and convergence order. 
 
ACKNOWLEDGMENT 
Authors gratefully acknowledge the support from the 
Mexican National Council for Science and Technology 
(CONACYT). This research was made in the facilities of the 
CASE Department-Barcelona Supercomputing Center. 
 
REFERENCES 
[1] Hongzhu, C. (2014). 3D controlled-source electromagnetic modeling in 
anisotropic medium using edge-based finite element method. Computer & 
Geosciences 73, 164-176. 
[2] Jin, J. (2002). The finite element method in electromagnetics. John Wiley & 
Sons. 
[3] Puzyrev, V., Koldan, J., de la Puente, J., Houzeaux, G., Vázquez, M., & Cela, 
J. M. (2013). A parallel finite-element method for three-dimensional 
controlled-source electromagnetic forward modelling. Geophysical Journal 
International, ggt027. 
[4] Weiss, C. J., & Newman, G. A. (2002). Electromagnetic induction in a fully 3-
D anisotropic earth. Geophysics, 67(4), 1104-1114. 
  
Fig. 2 Vectorial basis functions. Subplot (a) describes the shape function along 
edge 1 – 2, Subplot (b) describes the shape function along edge 2 – 3, Subplot 
(c) describes the shape function along edge 3 – 1 and subplot (c)
 
Fig. 4 Primary field of a z-directed dipole in a homogeneous domain with 384 
elements.  
 
Fig. 3 Ability of vectorial basis functions to interpolate a given field. (a) 
constant field to test, (b) field within the element, (c) error of approximation 
(maximum error=0.5551 - 0.331i e-15)
 


HPC and edge elements for geophysical electromagnetic problems: an overview

https://upcommons.upc.edu/bitstream/2099/16433/1/HPC_DS2105-BoAv11-2.pdf

HPC and edge elements for geophysical electromagnetic problems: an overview

Abstract

Similar works

Full text

Available Versions

UPCommons