CN-117148458-B - Mixed precision area decomposition solving method based on electric source electromagnetic method exploration

CN117148458BCN 117148458 BCN117148458 BCN 117148458BCN-117148458-B

Abstract

The invention discloses a mixed precision area decomposition solving method based on electric source electromagnetic method exploration, which comprises the steps of establishing a forward model and carrying out tetrahedral mesh subdivision, carrying out resistivity assignment on any tetrahedral mesh, dividing the tetrahedral mesh into a plurality of mutually non-overlapping subareas, establishing a discrete equation of an electric field control equation for any subarea, coupling the subareas according to boundary conditions, constructing a single-precision interface equation and a double-precision interface equation, carrying out single-precision iterative solution on the single-precision interface equation by adopting a minimum residual method, correcting the single-precision iteration by adopting double-precision iterative residual calculated by adopting the double-precision interface equation when the Nordi residual norm stagnates or reaches preset conditions to obtain corrected parameters, carrying out single-precision iterative solution on the corrected parameters by adopting the single-precision interface equation to obtain the solution of the interface equation, and substituting the solution of the interface equation into corresponding receiving points in the subareas to obtain electromagnetic responses of receiving points.

Inventors

Hui Zhejian
WANG XUBEN

Assignees

成都理工大学

Dates

Publication Date: 20260508
Application Date: 20230831

Claims (10)

1. The mixed precision area decomposition solving method based on the electric source electromagnetic method exploration is characterized by comprising the following steps: Establishing a forward model according to the underground ground electric structure and performing tetrahedral mesh subdivision to obtain node and mesh information of the tetrahedral mesh; carrying out resistivity assignment on any tetrahedral mesh; dividing the tetrahedral mesh into a plurality of mutually non-overlapping subareas; establishing a discrete equation of an electric field control equation for any subarea; coupling the sub-regions according to boundary conditions, and constructing a single-precision interface equation and a double-precision interface equation; Carrying out single-precision iterative solution on the single-precision interface equation by adopting a minimum residual error method; In single-precision iteration, when the Noraddi residual norm stagnates or reaches a preset condition, correcting the single-precision iteration by adopting a double-precision iteration residual calculated by a double-precision interface equation to obtain corrected parameters; Carrying out single-precision iterative solution on the corrected parameters by adopting a single-precision interface equation until convergence conditions are met; substituting the solution of the interface equation into the corresponding receiving points in the sub-region to obtain the electromagnetic response of the receiving points.
2. The method for solving the mixed-precision regional decomposition based on the electric source electromagnetic method according to claim 1, wherein in single-precision iteration, when the norm of the minoxidil residual is stagnated or reaches a preset condition, the single-precision iteration is corrected by adopting a double-precision iteration residual calculated by a double-precision interface equation, so as to obtain corrected parameters, and the method comprises the following steps: Presetting a convergence threshold k of single-precision iteration and a relative change threshold m of residual norms of adjacent iterations; Carrying out single-precision iterative solution on the single-precision interface equation; And if the single-precision iteration residual norm is smaller than the convergence threshold k or the relative change of the adjacent iteration residual norms is smaller than the relative change threshold m of the adjacent iteration residual norms, correcting the single-precision iteration by adopting the double-precision iteration residual calculated by the double-precision interface equation, and obtaining corrected parameters.
3. The method for solving the mixed-precision region decomposition based on electric source electromagnetic method exploration according to claim 1 or 2, wherein in correcting single-precision iterations by adopting double-precision calculation residuals calculated by double-precision interface equations, the method further comprises: Presetting a convergence threshold p of double-precision iteration; If the relative iteration residual norm calculated by the double-precision interface equation is smaller than the convergence threshold p, stopping iteration; and if the relative iteration residual norm calculated by the double-precision interface equation is greater than or equal to the convergence threshold p, carrying out iteration solution on the corrected parameter by using the single-precision interface equation until the convergence condition is met.
4. The method for solving the mixed-precision regional decomposition based on electric source electromagnetic prospecting according to claim 3, wherein the expression of the electric field control equation is: Wherein, the Represents differential arithmetic, E represents electric field strength, ω represents angular frequency, μ represents magnetic permeability, σ represents electrical conductivity, and J s represents applied electrical source strength.
5. The method for solving the mixed-precision regional decomposition based on electrical source electromagnetic prospecting of claim 4, wherein the expression of the discrete equation is: (A p +iωB p )E p ＝-iωS p -λ p Where p denotes the p-th sub-region, A p denotes the stiffness matrix of the p-th sub-region, B p denotes the mass matrix of the p-th sub-region, S p denotes the discrete emission source term of the p-th sub-region, and lambda p denotes the boundary term of the p-th sub-region.
6. The method for solving the mixed-precision regional decomposition based on electric source electromagnetic prospecting according to claim 5, wherein the boundary condition is expressed as: Where n represents the unit normal vector at the interface, μ represents permeability, Λ p represents an unknown value, Γ p represents the internal boundary.
7. The method for solving the mixed-precision regional decomposition based on the electric source electromagnetic method exploration according to claim 1, wherein the expression of the single-precision interface equation is: F s λ s ＝b s Wherein F s represents a coefficient matrix formed by coupling sub-region discrete equation under single precision, lambda s represents an unknown to-be-solved term formed according to boundary term lambda p under single precision, and b s represents a known term formed by coupling sub-region discrete equation source term under single precision; the expression of the double-precision interface equation is as follows: F d λ d ＝b d Wherein F d represents a coefficient matrix formed by coupling sub-region discrete equation under double precision, lambda d represents an unknown to-be-solved term formed according to boundary term lambda p under double precision, and b d represents a known term formed by coupling sub-region discrete equation source term under double precision.
8. The method for solving the mixed-precision regional decomposition based on electric source electromagnetic prospecting according to claim 2, wherein the single-precision iterative residual norm is Arnoldi residual norm in the calculation process of the minimum residual method.
9. The method for solving the mixed-precision regional decomposition based on the electric source electromagnetic method according to claim 1, wherein the double-precision iterative residual error calculated by the double-precision interface equation is a residual error epsilon of a double-precision equation set to be solved, and the expression is: ε=F d λ n -b d Wherein lambda n represents the iteration update quantity of the nth iteration of double precision, F d represents a coefficient matrix formed by coupling sub-region discrete equations under double precision, and b d represents a known term formed by coupling sub-region discrete equation source terms under double precision; The double-precision iteration residual error calculated by the double-precision interface equation corrects the single-precision iteration to obtain corrected parameters, wherein the corrected parameters comprise an iteration update quantity lambda n of the double-precision nth iteration and an iteration error calculated by the double-precision interface equation.
10. The method for solving the mixed-precision regional decomposition based on electric source electromagnetic prospecting according to claim 9, wherein the relative iterative residual norms L calculated by the double-precision interface equation are expressed as

Description

Mixed precision area decomposition solving method based on electric source electromagnetic method exploration Technical Field The invention relates to the technical field of geophysical exploration, in particular to a mixed precision area decomposition solving method based on electric source electromagnetic method exploration. Background As is well known, the controllable source electromagnetic method has the advantages of high transmitting power, strong anti-interference capability and the like, and is widely applied to mineral exploration and engineering geology. Compared with magnetic source emission, the electric source emission has large exploration depth, is sensitive to high-resistance media, has better vertical resolution, and is widely applied to the electromagnetic exploration fields such as semi-aviation electromagnetic method, ocean electromagnetic method and the like. With the increasing complexity of exploration environment and the increasing requirement of exploration precision, the conventional one-dimensional and two-dimensional forward and backward modeling method cannot meet the exploration requirement. Currently, three-dimensional forward and backward methods have become a hotspot and difficulty in research in the field. Three-dimensional forward modeling is the basis of three-dimensional inversion, and it is necessary to study an accurate and efficient three-dimensional forward modeling method. Compared with forward methods such as finite volume and finite difference methods, the finite element method has higher calculation accuracy. At present, a finite element method based on an unstructured tetrahedral mesh can accurately simulate a complex model, can be used for fine detection of a complex terrain area, and is one of the most common methods in electromagnetic forward modeling. For the finite element method, the calculation area needs to be finely divided to obtain an accurate solution, the grid number can reach millions or even tens of millions, the calculation resource consumption is huge, and the calculation efficiency is low. The FETI-DP method (Dual-PRIMAL FINITE ELEMENT TEARING AND Interconnecting method) combining finite element and region decomposition can decompose the solution region into a plurality of sub-regions and couple all the sub-regions through internal boundary conditions to form a reduced interface equation. The method reduces the solving dimension, is easy to calculate in parallel and improves the calculation efficiency. The reduced interface equation solves internal boundary conditions, and the requirement on calculation accuracy is not high. At present, the research in the prior art adopts double precision to calculate, the single precision calculation speed of the computer is twice of that of the double precision, the calculation in the process causes the waste of calculation force, and the calculation efficiency is reduced. Therefore, it is highly desirable to provide a hybrid-precision interface equation solving method that ensures both computational accuracy and can utilize low computational force speeds. Disclosure of Invention Aiming at the problems, the invention aims to provide a mixed precision area decomposition solving method based on electric source electromagnetic method exploration, which adopts the following technical scheme: the mixed precision area decomposition solving method based on the electric source electromagnetic method exploration comprises the following steps: Establishing a forward model according to the underground ground electric structure and performing tetrahedral mesh subdivision to obtain node and mesh information of the tetrahedral mesh; carrying out resistivity assignment on any tetrahedral mesh; dividing the tetrahedral mesh into a plurality of mutually non-overlapping subareas; establishing a discrete equation of an electric field control equation for any subarea; coupling the sub-regions according to boundary conditions, and constructing a single-precision interface equation and a double-precision interface equation; Carrying out single-precision iterative solution on the single-precision interface equation by adopting a minimum residual error method; In single-precision iteration, when the Noraddi residual norm stagnates or reaches a preset condition, correcting the single-precision iteration by adopting a double-precision iteration residual calculated by a double-precision interface equation to obtain corrected parameters; Carrying out single-precision iterative solution on the corrected parameters by adopting a single-precision interface equation until convergence conditions are met; substituting the solution of the interface equation into the corresponding receiving points in the sub-region to obtain the electromagnetic response of the receiving points. Further, in the single-precision iteration, when the norm of the minoxidil residual error stagnates or reaches a preset condition, the single-precision iteration is co