Planar multilayer medium electromagnetic scattering modeling method based on layered medium Green function

Abstract

The invention belongs to the technical field of electromagnetic modeling of a target in a layered medium, particularly provides a planar multilayer medium electromagnetic scattering modeling method based on a layered medium Green function, and is used for solving the problems that the number of unknown quantities of a system is large and the calculation efficiency is low when electromagnetic scattering characteristics of the target are analyzed in the layered medium. The method comprises the following steps: firstly, carrying out mesh generation on a medium target in a planar multilayer medium; then, a layered medium Green function is calculated based on a space list and an interpolation method, a surface integral equation of the target is solved by adopting a moment method, and a matrix equation of target current and magnetic current is constructed; and finally, solving a matrix equation by applying an improved laminated off-diagonal low-rank matrix, and calculating a target scattering field. According to the method, a direct solving calculation method is adopted, and the problem that an ill-conditioned matrix is difficult to converge in a conventional iteration method is effectively solved; especially for a multi-port excitation problem, repeated calculation is not needed, the solving efficiency of a large-scale problem is improved, and calculation resource consumption is reduced.

Description

Technical Field

[0001] The present invention belongs to the technical field of electromagnetic modeling of targets in layered media, relates to a method for solving the scattering field of a medium target in a planar multilayer medium, and specifically provides a planar multilayer medium electromagnetic scattering modeling method based on the layered medium Green's function. Background Art

[0002] The plane layered medium model is a simplified model of a longitudinally layered non-uniform medium. It is widely used in the fields of planar antenna or circuit design, ground penetrating radar and well logging. This model can effectively and comprehensively model the target and its complex environment. Compared with uniform media or free space, the electromagnetic scattering problem in layered media is more complicated. When the traditional moment method is used to solve it directly, a large dense matrix equation will be generated. As the scale of the problem increases, the unknown quantity of the system will increase significantly when the conventional RWG basis function is used for discretization, resulting in a large consumption of computing time and memory resources.

[0003] High-order methods provide better fitting effects than plane triangles by using larger-sized surface units to discretize targets, thereby reducing the number of units and unknown quantities. When dealing with complex electromagnetic targets, grids of different scales are used for division according to different geometric shapes. The resulting system matrix is ​​often ill-conditioned. Conventional iterative methods face the problems of slow convergence speed, low accuracy, or even no convergence when solving such matrices. High-order stacked vector basis functions are highly flexible and can adaptively select the order of basis functions according to the size of grid units, so as to more effectively realize electromagnetic modeling of complex targets; however, in actual engineering applications, it often involves analyzing the single-station electromagnetic scattering characteristics of the target, that is, the problem of multiple right-hand terms; the iterative solution method needs to be re-iterated for each right-hand excitation vector, resulting in low computational efficiency. Summary of the invention

[0004] The purpose of the present invention is to provide a fast calculation method for electromagnetic scattering of medium targets in planar multilayer media based on layered medium Green's function, so as to solve the problems of large number of unknown quantities in the system and low calculation efficiency when analyzing the electromagnetic scattering characteristics of targets in layered media; specifically, the present invention adopts a fast and direct solution method, which can quickly solve problems with multiple right-hand sides.

[0005] To achieve the above object, the technical solution adopted by the present invention is:

[0006] A planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function, characterized by comprising the following steps:

[0007] S01: Meshing the medium target in the planar multilayer medium (PLM) to obtain mesh information;

[0008] S02: Calculate the layered medium Green's function (LMGF) based on the grid information using the space list and interpolation method;

[0009] S03: Based on the Green's function of the layered medium, the moment method is used to solve the surface integral equation of the medium target and construct the matrix equation of the target current and magnetic current;

[0010] S04: The improved stacked non-diagonal low-rank matrix method (M-HODLR) is used to solve the matrix equation to obtain the surface current and surface magnetic current of the dielectric target, and the scattering field of the dielectric target is calculated based on the surface current and surface magnetic current.

[0011] Furthermore, the specific process of step S01 is as follows:

[0012] S011: Use curved triangle mesh to discretize the surface area of ​​the medium target.

[0013] Furthermore, the specific process of step S02 is:

[0014] S021: Read the grid information of the medium target, and set the number of sampling points and interpolation interval required for interpolation;

[0015] S022: Obtain the spatial coordinates of the interpolation node according to the number of sampling points and the interpolation interval, and calculate the LMGF corresponding to the interpolation node position;

[0016] S023: Determine whether the field point and the source point are located in the same dielectric layer. If so, execute two-dimensional space list and interpolation. Otherwise, execute three-dimensional space list and interpolation.

[0017] S024: Calculate the LMGF at the point to be interpolated using the Lagrange interpolation method (LIM).

[0018] Furthermore, the specific process of step S03 is:

[0019] S031: Based on the surface equivalent principle, the surface integral equation of the target is obtained, and the equivalent surface current and equivalent surface magnetic current are used to replace the medium target in PLM;

[0020] S032: Discretize the equivalent surface current and the equivalent surface magnetic current, and select high-order stacked vector basis functions to expand the surface current and surface magnetic current to be solved;

[0021] S033: adopting the Galerkin matching method, selecting the high-order stacked vector basis function in step S032 as the weight function, and constructing the matrix equation of the medium target in the PLM;

[0022] S034: Calculate the element values ​​of the impedance matrix and the right-hand vector in the matrix equation using the LMGF obtained in step S02.

[0023] Furthermore, the specific process of step S04 is as follows:

[0024] S041: According to the spatial position of the basis function, the basis function group is grouped and sorted using the binary tree method, thereby obtaining the stacked structure H-matrix of the impedance matrix in the matrix equation;

[0025] S042: Determine whether the matrix blocks in the non-diagonal position after the division meet the extended compatibility condition, perform ACA low-rank compression on the matrix blocks that meet the condition, and perform downward segmentation on the matrix blocks that do not meet the condition until they are segmented to the leaf layer; directly perform ACA low-rank compression on the non-diagonal matrix blocks of the leaf layer, and perform full-rank filling on the diagonal matrix blocks;

[0026] S043: reaggregating the matrix blocks divided downlink in step S042 into a single matrix block through uplink aggregation to obtain an updated stacked structure H-matrix;

[0027] S044: Use the SMW (Sherman-Morrison-Woodbury) formula to solve the approximate inverse of the impedance matrix and decompose the original matrix of the laminated structure into the form of the product of multiple block diagonal matrices;

[0028] S045: Multiply the inverse matrix of the impedance matrix obtained in step S044 by the right-hand vector to obtain the current coefficient and the magnetic current coefficient, calculate the surface current and the surface magnetic current of the dielectric target from the current coefficient and the magnetic current coefficient, and calculate the scattering field of the dielectric target based on the surface current and the surface magnetic current.

[0029] Based on the above technical solution, the beneficial effects of the present invention are:

[0030] The present invention provides a planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function, which has the following advantages:

[0031] 1) The high-order stacked vector basis function is used to replace the traditional RWG basis function, which effectively reduces the number of unknowns, reduces the dimension of the system matrix, and further improves the solution efficiency;

[0032] 2) The direct solution calculation method is used to effectively solve the problem that the ill-conditioned matrix is ​​difficult to converge in the conventional iterative method; especially for multi-port excitation problems, repeated calculations are not required, which greatly improves the calculation efficiency;

[0033] 3) The M-HODLR method was introduced to analyze the electromagnetic scattering problem of targets in layered media, which significantly reduced the complexity of the solution matrix, improved the efficiency of solving large-scale problems, and reduced the consumption of computing resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is a flow chart of the method for modeling electromagnetic scattering of planar multilayer media based on layered medium Green's function in the present invention.

[0035] Figure 2 Schematic diagram of the surface triangulation grid unit used in the present invention.

[0036] Figure 3 Schematic diagram of the Lagrange interpolation method (LIM) used in the present invention, wherein (a) is a three-dimensional LIM and (b) is a two-dimensional LIM.

[0037] Figure 4 Schematic diagram of the equivalent problem of the target when electromagnetic wave is incident in the present invention, wherein (a) is a schematic diagram of the external equivalent problem, and (b) is a schematic diagram of the internal equivalent problem.

[0038] Figure 5 Schematic diagram of the process of using the binary tree method to divide the basis function group and construct the M-HODLR matrix in the present invention.

[0039] Figure 6 It is a schematic diagram of decomposing the M-HODLR matrix into the product of multiple block diagonal matrices in the present invention.

[0040] Figure 7 Schematic diagram of a dielectric sphere model in a planar multilayer medium in Example 1 of the present invention.

[0041] Figure 8 1 is a comparison diagram of the scattering field calculation results of the dielectric sphere model in the planar multilayer medium in Example 1 of the present invention, wherein (a) is a comparison diagram of the scattering electric field results, and (b) is a comparison diagram of the scattering magnetic field results. DETAILED DESCRIPTION

[0042] In order to make the purpose, technical solution and beneficial effects of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments.

[0043] Example 1

[0044] This embodiment provides a planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function. The specific implementation process is as follows: Figure 1 As shown in FIG. 1 , the main steps include: performing high-order meshing on the target in the layered medium, establishing a layered medium Green's function interpolation table, applying the moment method to construct the matrix equation of the target surface current and surface magnetic current, using a fast direct solution method to solve the constructed matrix equation, and calculating the scattering field of the target; the detailed steps are as follows:

[0045] S01: Meshing of medium targets in planar multilayer media (PLM);

[0046] S02: Calculation of layered medium Green's function (LMGF) based on space list and interpolation method;

[0047] S03: Use the moment method to solve the target surface integral equation and construct the matrix equation of the target current and magnetic flow;

[0048] S04: Apply the improved stacked non-diagonal low-rank matrix method (M-HODLR) to solve the matrix equation and calculate the target scattered field.

[0049] Furthermore, in step S01, the specific process of meshing the medium target in the planar multilayer medium is as follows:

[0050] S011: Use curved triangle mesh to discretize the target surface area.

[0051] Furthermore, the curved triangular mesh unit used in step S01 is as follows: Figure 2 As shown:

[0052] The target surface area is discretized using a surface triangle mesh; the mesh unit is a second-order parametric surface defined by six parameter control points, such as Figure 2 As shown in (a), it can be transformed into a standard triangle in the parametric coordinate system by projection, as Figure 2 As shown in (b); any point r on the surface triangle is expressed in parameter space as:

[0053]

[0054] Among them, r j is the spatial coordinate of the parameter control point, ξ 1 ,ξ 2 ,ξ 3 is the parameter coordinate, satisfying 0<ξ<1,ξ 1 +ξ 2 +ξ 3 =1, is the shape function, and the specific expression is:

[0055]

[0056] Furthermore, in step S02, the specific process of calculating the layered medium Green's function based on the space list and interpolation method is as follows:

[0057] S021: Read the grid information of the target model and set the number of sampling points and interpolation interval required for interpolation;

[0058] S022: Obtain the spatial coordinates of the interpolation node according to the number of sampling points and the interpolation interval, and calculate the LMGF corresponding to the interpolation node position;

[0059] S023: Determine whether the field point and the source point are located in the same dielectric layer. If so, execute two-dimensional space list and interpolation; otherwise, execute three-dimensional space list and interpolation;

[0060] S024: Calculate the LMGF at the point to be interpolated using the Lagrange interpolation method (LIM).

[0061] Furthermore, in step S02, the specific process of calculating the layered medium Green's function based on the space list and interpolation method is as follows:

[0062] After meshing the target in step S011, read the mesh information, set the interpolation interval according to the error requirement, thereby obtaining the spatial coordinates of the interpolation node, calculate the Sommerfeld integral at the interpolation point by direct numerical integration method, and then obtain the LMGF at the node;

[0063] LMGF is the field point coordinates With the source point coordinates function, which can be further expressed as the lateral distance in a plane layered medium When the field point and the source point are located in different dielectric layers, it is necessary to list and interpolate ρ, z, z′ in three-dimensional space. The LMGF at the eight vertices of the hexahedron frame is interpolated to calculate the LMGF at the point to be calculated in the frame, such as Figure 3 As shown in (a); when the two are in the same dielectric layer, LMGF can be expressed as a binary function of ρ and z±z′. It is necessary to list and interpolate ρ and z±z′ in two-dimensional space, and calculate the LMGF at the desired point in the box through the LMGF interpolation at the four vertices of the rectangular box, as shown in Figure 3 As shown in (b); the interpolation method uses the Lagrange interpolation method, and its interpolation basis function is defined as:

[0064]

[0065] Interpolation function L n (x) can be calculated from the interpolation basis function and the function value at the interpolation node:

[0066]

[0067] The calculation time of impedance elements is mainly reflected in the numerical calculation of Sommerfeld integral. This method can replace the point-by-point calculation of Sommerfeld integral with a simpler interpolation calculation, reduce the number of numerical integration calculations while meeting the accuracy requirements, and thus significantly improve the calculation efficiency.

[0068] Furthermore, in step S03, the method of moments is used to solve the target surface integral equation, and the specific process of constructing the matrix equation of the target current and magnetic current is:

[0069] S031: Based on the surface equivalent principle, the surface integral equation of the target is obtained, and the equivalent surface current and equivalent surface magnetic current are used to replace the medium target in PLM;

[0070] S032: Discretize the equivalent surface current and the equivalent surface magnetic current, and select high-order stacked vector basis functions to expand the current and magnetic current to be calculated;

[0071] S033: adopting the Galerkin matching method, selecting the high-order stacked vector basis function in step S032 as the weight function, and constructing the matrix equation of the medium target in the PLM;

[0072] S034: Calculate the element values ​​of the system matrix and the excitation vector using the LMGF obtained in step S024.

[0073] Furthermore, in step S03, the method of moments is used to solve the target surface integral equation, and the specific process of constructing the matrix equation of the target current and magnetic flow is as follows:

[0074] Based on the LMGF and Love field equivalent principle, we can get the following Figure 4 The external equivalence problem shown in (a) is similar to Figure 4 The surface electric field integral equation of the internal equivalent problem shown in (b) is:

[0075]

[0076] Among them, r and r ′ represent the field point coordinates and the source point coordinates respectively, the superscript "o" represents the external equivalent, the superscript "i" represents the internal equivalent, J and M represent the surface current and the surface magnetic current respectively, The operator is defined as follows:

[0077]

[0078] Among them, μ represents magnetic permeability, ε represents dielectric constant, ω represents angular frequency, They represent the electric dyadic Green's function and the magnetic dyadic Green's function in PLM respectively, which are calculated in step S02;

[0079] The surface current and surface magnetic current are expanded by using high-order stacked vector basis functions, and a set of linear equations about the expansion coefficients are obtained by using the Galerkin matching method.

[0080] According to the duality principle, we can get Figure 4 The external equivalence problem shown in (a) is similar to Figure 4 The surface magnetic field integral equation of the internal equivalent problem shown in (b) is:

[0081]

[0082] Similarly, another set of linear equations about the expansion coefficients is obtained, and the PMCHWT (Poggio-Miller-Chang-Harrington-Wu-Tsai) equation of the medium target in PLM is obtained by combining:

[0083]

[0084] Combined with the LMGF obtained in step S02, the impedance matrix (system matrix) and the right-hand side vector (excitation vector) are calculated.

[0085] Furthermore, in step S04, the improved stacked non-diagonal low-rank matrix method (M-HODLR) is applied to solve the matrix equation, and the specific process of calculating the target scattered field is:

[0086] S041: according to the spatial position of the basis function, the basis function group is grouped and sorted using a binary tree method, thereby obtaining the stacked structure H-matrix of the impedance matrix in step S034;

[0087] S042: Determine whether the matrix blocks in the non-diagonal position after the division meet the extended compatibility condition, perform ACA low-rank compression on the matrix blocks that meet the condition, and perform downward segmentation on the matrix blocks that do not meet the condition until they are segmented to the leaf layer; directly perform ACA low-rank compression on the non-diagonal matrix blocks of the leaf layer, and perform full-rank filling on the diagonal matrix blocks;

[0088] S043: reaggregating the matrix blocks divided downlink in step S042 into a single matrix block through uplink aggregation, and finally obtaining an updated stacked structure H-matrix;

[0089] S044: Use the SMW (Sherman-Morrison-Woodbury) formula to solve the approximate inverse of the impedance matrix and decompose the original matrix of the laminated structure into the form of the product of multiple block diagonal matrices;

[0090] S045: Solve the surface current and surface magnetic current, and calculate the target scattering field: multiply the inverse matrix of the impedance matrix obtained in step S044 by the right-hand vector to obtain the current coefficient and the magnetic current coefficient, thereby obtaining the surface current and the surface magnetic current, and calculate the medium target scattering field based on the surface current and the surface magnetic current.

[0091] Furthermore, in step S04, the specific process of applying M-HODLR to solve the matrix equation and calculating the target scattered field is:

[0092] In order to construct the impedance matrix obtained in step S03 into an H-matrix form, the binary tree method is used to group the basis functions, and the basis function groups are continuously divided downward according to the spatial distance until the number of basis functions contained in the child node is less than the preset minimum value, such as Figure 5 As shown; Unlike the matrix equation containing only unknown current coefficients, the PMCHWT equation contains both current and magnetic flow, and the current and magnetic flow basis functions need to be grouped in multiple layers at the same time. Any matrix block in each layer is expressed as a form of mutual coupling between current and magnetic flow:

[0093]

[0094] The traditional HODLR method regards the constructed H-matrix non-diagonal sub-matrix blocks as satisfying the compatibility condition and uses the low-rank compression method for approximation. It is often inefficient and has poor accuracy when processing sub-matrix blocks with short spatial distances and poor low-rank characteristics. The improved M-HODLR method applies the extended compatibility condition:

[0095]

[0096] Among them, m i represents the basis function subgroup, n j represents the weight function subgroup, diam(·) represents the geometric diameter of the basis (weight) function subgroup, dist(·) represents the geometric distance between the basis function subgroup and the weight function subgroup, and η is a preset positive real number;

[0097] The non-diagonal sub-matrix blocks after division are judged, wherein the sub-matrix blocks that meet the extended compatibility conditions are subjected to ACA low-rank compression, and the sub-matrix blocks that do not meet the conditions are subjected to downstream segmentation, that is, the sub-matrix blocks are divided into smaller matrix blocks, and the extended compatibility conditions are judged for the re-divided small matrix blocks. The above operation is repeated until the number of basis functions contained in the sub-matrix block is less than the preset minimum value. At this time, the smallest sub-matrix block is directly subjected to ACA low-rank compression; the matrix after ACA compression can be expressed as:

[0098]

[0099] in, is a low-rank approximation matrix, r is the approximate rank;

[0100] In order to restore the low-rank approximate matrix formed after downlink segmentation to an H-matrix structure, it is necessary to perform uplink aggregation on it. This step can be expressed as:

[0101]

[0102] in, is the low-rank decomposition matrix obtained after upstream aggregation. At this time, the rank of the matrix is ​​still large and contains a large number of zero elements. The rank of the matrix can be further reduced by the QR-SVD recompression method. Reduced to k mn, so far the M-HODLR matrix construction for the PMCHWT equation has been completed;

[0103] like Figure 6 As shown, the M-HODLR matrix can be further decomposed into the following form:

[0104]

[0105] Among them, the subscript l represents the number of layers, Z l represents the l-th layer block diagonal matrix, Represents the block diagonal matrix where the main diagonal is located; then the solution to the matrix equation Z·I=V can be converted into solving the inverse of the block diagonal matrix of each layer:

[0106]

[0107] Block diagonal matrix The inverse matrix of Z can be obtained by LU decomposition, l The inverse matrix of can be obtained by SMW formula:

[0108]

[0109] At this point, the calculation of the coefficient vector is converted into the multiplication of the inverse matrix of the multi-layer block diagonal matrix and the right-hand side vector through the M-HODLR fast and direct solution method. Since it is relatively easy to invert the block diagonal matrix, the calculation efficiency can be significantly improved, especially for problems with multiple right-hand sides. After obtaining the coefficient vector, its scattering field can be obtained through the target equivalent surface current and equivalent surface magnetic current.

[0110] like Figure 8 FIG. 2 shows the target electromagnetic properties calculated by the method of the present invention, wherein (a) is the scattered electric field and (b) is the scattered magnetic field; the dielectric sphere model in the planar multilayer medium is as follows: Figure 7 As shown; the present invention compares the proposed method with the traditional MoM method using CG iteration. From the results, it can be seen that the method of the present invention has high calculation accuracy for the scattering field of the medium target in PLM; at the same time, in this embodiment, the number of face elements required for traditional plane triangle mesh division is 3924, and the number of face elements required for high-order surface triangle mesh in this method is reduced to 776. The traditional CG iteration method takes 348.326 seconds to calculate, while the M-HODLR method takes 63.125 seconds to calculate, and the calculation efficiency is improved by 81.88%.

[0111] The above description is only a specific implementation mode of the present invention. Any feature disclosed in this specification, unless otherwise stated, can be replaced by other alternative features that are equivalent or have similar purposes; all the disclosed features, or all the steps in the methods or processes, except for mutually exclusive features and / or steps, can be combined in any way.

Claims

1. A method for modeling electromagnetic scattering of planar multilayer media based on layered medium Green's function, characterized in that: The following steps are involved: S01: Meshing the medium target in the planar multilayer medium (PLM) to obtain mesh information; S02: Calculate the layered medium Green's function (LMGF) based on the grid information using the space list and interpolation method; S03: Based on the Green's function of the layered medium, the moment method is used to solve the surface integral equation of the medium target and construct the matrix equation of the target current and magnetic flow; S04: The improved stacked non-diagonal low-rank matrix method (M-HODLR) is used to solve the matrix equation to obtain the surface current and surface magnetic current of the dielectric target, and the scattering field of the dielectric target is calculated based on the surface current and surface magnetic current.

2. The planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function according to claim 1 is characterized in that: The specific process of step S01 is: S011: Use a curved triangle mesh to discretize the surface area of ​​the medium target. Furthermore, the specific process of step S02 is: S021: Read the grid information of the medium target, and set the number of sampling points and interpolation interval required for interpolation; S022: Obtain the spatial coordinates of the interpolation node according to the number of sampling points and the interpolation interval, and calculate the LMGF corresponding to the interpolation node position; S023: Determine whether the field point and the source point are located in the same dielectric layer. If so, execute two-dimensional space list and interpolation. Otherwise, execute three-dimensional space list and interpolation. S024: Calculate the LMGF at the point to be interpolated using the Lagrange interpolation method (LIM).

3. The planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function according to claim 1 is characterized in that: The specific process of step S03 is: S031: Based on the surface equivalent principle, the surface integral equation of the target is obtained, and the equivalent surface current and equivalent surface magnetic current are used to replace the medium target in PLM; S032: Discretize the equivalent surface current and the equivalent surface magnetic current, and select high-order stacked vector basis functions to expand the surface current and surface magnetic current to be solved; S033: adopting the Galerkin matching method, selecting the high-order stacked vector basis function in step S032 as the weight function, and constructing the matrix equation of the medium target in the PLM; S034: Calculate the element values ​​of the impedance matrix and the right-hand vector in the matrix equation using the LMGF obtained in step S02.

4. The planar multilayer medium electromagnetic scattering modeling method based on layered medium Green's function according to claim 1 is characterized in that: The specific process of step S04 is: S041: According to the spatial position of the basis function, the basis function group is grouped and sorted using the binary tree method, thereby obtaining the stacked structure H-matrix of the impedance matrix in the matrix equation; S042: Determine whether the matrix blocks in the non-diagonal position after the division meet the extended compatibility condition, perform ACA low-rank compression on the matrix blocks that meet the condition, and perform downward segmentation on the matrix blocks that do not meet the condition until they are segmented to the leaf layer; directly perform ACA low-rank compression on the non-diagonal matrix blocks of the leaf layer, and perform full-rank filling on the diagonal matrix blocks; S043: reaggregating the matrix blocks divided downlink in step S042 into a single matrix block through uplink aggregation to obtain an updated stacked structure H-matrix; S044: Use the SMW (Sherman-Morrison-Woodbury) formula to solve the approximate inverse of the impedance matrix and decompose the original matrix of the laminated structure into the form of the product of multiple block diagonal matrices; S045: Multiply the inverse matrix of the impedance matrix obtained in step S044 by the right-hand vector to obtain the current coefficient and the magnetic current coefficient, calculate the surface current and the surface magnetic current of the dielectric target from the current coefficient and the magnetic current coefficient, and calculate the scattering field of the dielectric target based on the surface current and the surface magnetic current.

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