Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1556-1561 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Computation Time ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Electromagnetic Modeling ◽  
Direct Inversion ◽  
Conductivity Structure ◽  
The Matrix ◽  
Earth Conductivity

A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

Symmetry properties of the scattering matrix in 3-D electromagnetic modeling using the integral equation method

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1199-1202 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Computation Time ◽  
Integral Equation Method ◽  
Scattering Matrix ◽  
Electromagnetic Modeling ◽  
Symmetry Properties ◽  
Computer Storage ◽  
Earth Conductivity ◽  
Number Of Cells

Modeling large three‐dimensional (3-D) earth conductivity structures continues to pose challenges. Although the theories of electromagnetic modeling are well understood, the basic computational problems are practical, involving the quadratically growing requirements on computer storage and cubically growing computation time with the number of cells required to discretize the modeling body.

scholarly journals Electromagnetic modeling of three‐dimensional bodies in layered earths using integral equations

Geophysics ◽  
1984 ◽  
Vol 49 (1) ◽  
pp. 60-74 ◽  
Author(s):  
Philip E. Wannamaker ◽  
Gerald W. Hohmann ◽  
William A. SanFilipo
Keyword(s):  
Integral Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
Computation Time ◽  
Vector Current ◽  
Hankel Transform ◽  
Transverse Magnetic ◽  
Electromagnetic Modeling ◽  
Method Of Integral Equations

We have developed an algorithm based on the method of integral equations to simulate the electromagnetic responses of three‐dimensional bodies in layered earths. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green’s function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green’s functions over the scattering currents. Efficient evaluation of the tensor Green’s functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green’s functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three‐dimensional (3-D) bodies with responses over two‐dimensional (2-D) bodies of identical cross‐section using plane wave incident fields is the only check available on our solution. Agreement is excellent; however, the length that a 3-D body must have before departures between 2-D transverse electric and corresponding 3-D signatures are insignificant depends strongly on the layering. The 2-D transverse magnetic and corresponding 3-D calculations agree closely regardless of the layered host.

Advanced three-dimensional electromagnetic modeling using a nested integral equation approach

2021 ◽  
Author(s):  
Chaojian Chen ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Region Of Interest ◽  
Electromagnetic Modeling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Uniform Grids ◽  
The Matrix ◽  
Integral Equation Approach ◽  
Matrix Vector

Summary Most of the existing three-dimensional (3-D) electromagnetic (EM) modeling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix-vector multiplications. This in turn requires a laterally-uniform discretization of the modeling domain. However, there is often a need for multi-scale modeling and inversion, for instance, to properly account for the effects of non-uniform distant structures, and at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally-uniform grids leads to excessive computational loads, both in terms of memory and time. To alleviate this problem, we developed an efficient 3-D EM modeling tool based on a multi-nested IE approach. Within this approach, the IE modeling is first performed at a large domain and on a (laterally-uniform) coarse grid, and then the results are refined in the region of interest by performing modeling at a smaller domain and on a (laterally-uniform) denser grid. At the latter stage, the modeling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the matrix-vector multiplications. An important novelty of the paper is a development of a “rim domain” concept which further improves the performance of the multi-nested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

scholarly journals Fast Computation by MLFMM-FFT with NURBS in Large Volumetric Dielectric Structures

Electronics ◽  
2021 ◽  
Vol 10 (13) ◽  
pp. 1560
Author(s):  
Alejandro Pons ◽  
Alvaro Somolinos ◽  
Ivan González ◽  
Felipe Cátedra
Keyword(s):  
Fast Multipole Method ◽  
Near Field ◽  
Three Dimensional ◽  
Solution Process ◽  
Computation Time ◽  
Iterative Solution ◽  
Hexahedral Meshes ◽  
Volumetric Objects ◽  
Computational Resources ◽  
Efficient Memory

A refinement for the computation of the rigorous part of the multi-level fast multipole method (MLFMM) of analyzing volumetric objects is presented. A scheme based on the fast Fourier technique (FFT) is proposed with the objective of reducing the computational resources required to accurately analyze large homogeneous and non-homogeneous dielectric volumes. In order to reduce the memory requirements, the storage of the near-field terms of the method of moments (MoM) matrix is performed only for the positions corresponding to a parallelepiped with the size of the level 1 block of the MLFMM, computed with the vacuum permittivity, taking advantage of the Toeplitz symmetry present in regular hexahedral meshes. The FFT avoids applying the near-field MoM matrix in the iterative solution process. The application of this approach results in huge improvements in terms of memory usage, but also a speeds up the iterative solution process because the use of three-dimensional (3D) FFTs is very efficient for computing convolutions when the number of unknowns of the problems becomes very large as happens in volumetric problems. We also propose a new approach for the numerical treatment of the transition of the dielectric permittivity between different dielectrics or between a dielectric and a free space. To validate the computation technique, the radar cross section (RCS) of several dielectric bodies is computed using the classical MLFMM approach and it is compared with the presented FFT-based-MLFMM solution. The results demonstrate that the efficient memory and computation time usage of the proposed approach.

A new Electron Microscopy tomography: Least squares pseudoimage reconstruction technique

1991 ◽  
Vol 49 ◽  
pp. 538-539
Author(s):  
Nengyu Zhang
Keyword(s):  
Electron Microscopy ◽  
Electron Micrographs ◽  
Three Dimensional ◽  
Matrix Inversion ◽  
Least Square ◽  
Reconstruction Technique ◽  
Reconstruction Problem ◽  
Tilt Axis ◽  
3D Object ◽  
The Matrix

The least square pseudoimage (LSP) Reconstruction technique is based on matrix inversion. Due to the absence of information in electron microscopy, usually in an angle range from 60° to 90°, the matrix is degraded. The degraded matrix is used to compute two-dimensional (2D) projections of the three-dimensional (3D) object along the Z-axis (direction of the electron beam) and tilted around the Y-axis (tilt axis). Applying the pseudoinverse of the degraded matrix to electron micrographs, which are 2D projections of the 3D object, gives the pseudoimage.Since all of the slices which are perpendicular to the Y axis are related to the same degraded matrix, the 3D reconstruction problem can be simplified as series 2D reconstruction problems. All of the 2D pseudoimages are formed by the same pseudoinverse of a degraded matrix. In this case, however, the degraded matrix projects a 2D object onto an ID projection along the Z axis. Each slice is computed from a single line in all of the electron micrographs.

Designing TIMP (tissue inhibitor of metalloproteinases) variants that are selective metalloproteinase inhibitors

2003 ◽  
Vol 70 ◽  
pp. 201-212 ◽  
Author(s):  
Hideaki Nagase ◽  
Keith Brew
Keyword(s):  
Three Dimensional ◽  
Therapeutic Interventions ◽  
Tissue Inhibitors Of Metalloproteinases ◽  
Structural Basis ◽  
Structural Elements ◽  
Ridge Structure ◽  
Extracellular Matrix Components ◽  
Metalloproteinase Inhibitors ◽  
The Matrix ◽  
A Disintegrin And Metalloproteinase

The tissue inhibitors of metalloproteinases (TIMPs) are endogenous inhibitors of the matrix metalloproteinases (MMPs), enzymes that play central roles in the degradation of extracellular matrix components. The balance between MMPs and TIMPs is important in the maintenance of tissues, and its disruption affects tissue homoeostasis. Four related TIMPs (TIMP-1 to TIMP-4) can each form a complex with MMPs in a 1:1 stoichiometry with high affinity, but their inhibitory activities towards different MMPs are not particularly selective. The three-dimensional structures of TIMP-MMP complexes reveal that TIMPs have an extended ridge structure that slots into the active site of MMPs. Mutation of three separate residues in the ridge, at positions 2, 4 and 68 in the amino acid sequence of the N-terminal inhibitory domain of TIMP-1 (N-TIMP-1), separately and in combination has produced N-TIMP-1 variants with higher binding affinity and specificity for individual MMPs. TIMP-3 is unique in that it inhibits not only MMPs, but also several ADAM (a disintegrin and metalloproteinase) and ADAMTS (ADAM with thrombospondin motifs) metalloproteinases. Inhibition of the latter groups of metalloproteinases, as exemplified with ADAMTS-4 (aggrecanase 1), requires additional structural elements in TIMP-3 that have not yet been identified. Knowledge of the structural basis of the inhibitory action of TIMPs will facilitate the design of selective TIMP variants for investigating the biological roles of specific MMPs and for developing therapeutic interventions for MMP-associated diseases.

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