CN108287371A - Background grid Adaptive meshing method in dc resistivity element-free menthod - Google Patents
Background grid Adaptive meshing method in dc resistivity element-free menthod Download PDFInfo
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Abstract
The present invention provides a kind of background grid Adaptive meshing methods in dc resistivity element-free menthod, include the following steps：The medium of determining two dimension earth-electricity model, anomalous body, landform, electrode equal distribution situation, establish dc resistivity element-free menthod computational domain, utilize the discrete earth-electricity model of the node of irregular distribution；Using a small amount of coarse simple initial quadrangle background grid coverage domain, the given controlling value for instructing Adaptive meshing, background grid carries out Adaptive meshing according to the controlling value and the distribution situation of node；N is arranged in each background gridgA Gauss integration point xg, obtain the region element-free menthod equation group；It solves equation group and obtains node electric field value, calculate the apparent resistivity parameter for obtaining observation point.The present invention can be based on arbitrary node and be distributed discrete model, adaptable to arbitrarily complicated earth-electricity model, and the numerical stability and computational efficiency of customary DC resistivity element-free menthod forward modeling are improved using background grid Adaptive meshing method.
Description
Technical field
The present invention relates to a kind of dc resistivity forward modeling methods in exploration geophysics field, more particularly to intricately electric mould
The high-precision of type, high flexibility, high-adaptability and efficiently without unit forward modeling method.
Background technology
Dc resistivity exploration is a kind of important method in geophysical exploration, is widely used in SOLID MINERAL RESOURCES
Exploration, Hydrogeologic Survey, environmental improvement and monitoring, engineering geophysics such as reconnoitre at the fields.The apparent resistivity of measurement and underground
The resistivity of medium has direct relationship, and by manually powering to underground, observing apparent resistivity in earth's surface or well can be with
Resistivity anomaly body distribution in underground is judged.With the development of dc resistivity exploration engineering, it is situated between to complicated landform, underground
The matter complexity form and high-precision of the earth-electricity model of distribution, high-adaptability and flexibility forward modeling method demand it is growing,
Element-free menthod be a kind of emerging method for numerical simulation (Belytschko, et al., 1994；Hadinia and Jafari,
2015) nodal information, is only needed, mesh link information is not depended on, the constraint of grid has been broken away from and there is high flexibility and adaptation
The characteristics of property, is widely studied, simultaneously because it has the characteristics that high-precision using high-precision interpolation method at present in direct current
Application (prosperous English of fiber crops etc., 2017) has been obtained in Resistivity Forward Modelling simulation.In conventional dc resistivity element-free menthod, often exist
Subdivision background grid, background grid seriously affect mould with whether Node distribution is adapted by the way of uniform subdivision on computational domain
Quasi- precision, therefore, in order to ensure that simulation precision is often distributed using thinner background grid.However, element-free menthod is not needing
Unit link information between node often uses the node of Arbitrary distribution, the dense background grid and part that uniform subdivision obtains
Dense Node distribution region is adaptable, but is then extremely mismatched in the region of node sparse distribution, these regions are often
Field value changes small region, it is only necessary to which matched sparse background grid can reach good computational accuracy, these regions
Interior dense background grid distribution considerably increases calculation amount, and simulated cost is multiplied, and limits element-free menthod in direct current
Application in resistance rate forward simulation.
Therefore, it is necessary to design it is a kind of can be according to Node distribution situation Adaptive meshing background grid, numerical stability
The dc resistivity high, computational efficiency is high is without unit forward modeling method.
Invention content
Technical problem solved by the invention is, in view of the deficiencies of the prior art, provides a kind of dc resistivity without list
Background grid Adaptive meshing method in first method gives the simple quadrangle background grid of initial raw and Adaptive meshing control
Value processed carries out background grid Adaptive meshing according to Node distribution situation, obtains the background grid point being adapted with Node distribution
Cloth improves the numerical stability of dc resistivity element-free menthod, ensures simulation precision, and greatly improves the calculating effect of element-free menthod
Rate.
The technical scheme is that：
Step 1, earth-electricity model are established：
First, according to the distribution of two-dimentional earth-electricity model intermediary matter resistivity, the geometric shape and hypsography feelings of anomalous body
Condition establishes element-free menthod region Ω, and sets pole layout position, observation device, observation point position, in element-free menthod region
It is middle two-dimentional earth-electricity model is carried out using the node of one group of Arbitrary distribution it is discrete, according to geology abnormal posture set with geometric shape,
Hypsography form, the artificial part of electrode position arrangement Node distribution situation can arbitrary encryption node, such as topography variation region,
Near current electrode, anomalous body and its peripheral region, and according to forward simulation demand, value variation on the scene is little or far from field
The electrically constant region in source uses sparse Node distribution, so that node reasonable layout, reduces and calculate cost；
Step 2, background grid Adaptive meshing：
Use a small amount of coarse simple initial quadrangle background grid coverage domain, the given control for instructing Adaptive meshing
Value M processed；
Initial quadrangle background grid is numbered, is once operated according to number order：Judge i-th of background net
Whether the number of nodes for including in lattice is more than controlling value M, if it is will according to the midpoint of the four edges of grid by the background grid
The background grid is subdivided into four new background grids, equally does same judgement and operation to four new background grids,
It is not more than controlling value M until newly-generated background grid is satisfied by the number of nodes for including inside it, then stops i-th of background grid
Adaptive meshing obtains the one group new background grid that meets the requirements of i-th of initial background grids after Adaptive meshing；
All initial background grids are carried out after as above operating, met the requirements on element-free menthod computational domain one group of back of the body is obtained
Scape grid, these background grids be according to Node distribution under the guidance of controlling value M Adaptive meshing, they and Node distribution
It is adapted；
Step 3 carries out element-free menthod calculating to earth-electricity model：
The background grid obtained by background grid Adaptive meshing method is numbered, the cloth in each background grid
Set ngA Gauss integration point xg, to each Gauss integration point xgThe local support region of construction one, uses the n for including inside it
Nodal information constructs RPIM shape functions, using this group of shape function to Gauss integration point xgThe field value at place carries out interpolation calculation, uses
Gauss integration calculates corresponding variational problem (2) formula (Xu Shizhe, 1994) of 2.5 dimension dc resistivity boundary value problem (1) formulas；
Wherein, Γ is boundary symbol, ΓSFor ground, ΓTFor cutoff boundary, σ=(x, z) is dielectric conductance rate, U
(λ, x) is wave-number domain current potential, and λ is wave number, I0For electric current, δ is Kronecker delta functions, x=[x, z]TFor appointing on Ω
A bit, A is field source location to meaning,For gradient operator, rAFor the air line distance at any point on point source and boundary, n is outer method
Line unit vector, cos (r, n) are rAWith the included angle cosine of outer normal direction n, K0、K1Respectively the second class single order, zeroth order correct shellfish plug
That function, δ are variation symbol；
The step that element-free menthod shape function is important in element-free menthod is constructed, the present invention uses RPIM shape functions, has
Kronecker delta function properties, to cross point interpolation so that essential boundary can directly apply, and relative to traditional nothing
Elements method shape function, such as Moving Least Squares shape function, RPIM shape functions are theoretical simple, are not related to weight function；
To calculate point xgThe local support region of construction, internal includes n node, and element-free menthod shape letter is constructed using them
Number, then to calculating point xgPlace carries out field and is worth interpolation, according to pertinent literature (Liu and Gu, 2007；Prosperous English of fiber crops etc., 2017)
RPIM shape functions can be written as：
The wherein corresponding RPIM shape functions of support region interior joint are
ΦT(x)=[φ1(x) φ2(x) … φn(x)] (4)
G is RPIM coefficient matrixes
The wherein corresponding coefficient matrix of radial basis function is
The corresponding coefficient matrix of polynomial basis function is
The dc resistivity element-free menthod equation of variational problem (2) formula can be exported using the global weak formula methods of Galerkin：
KU=F (8)
Wherein K is the element-free menthod stiffness coefficient matrix of N × N-dimensional, and N is computational domain interior joint sum, and U is element-free menthod area
The column vector that domain node wave-number domain current potential corresponding N × 1 is tieed up, F are the element-free menthod equation right-hand vector column vector that N × 1 is tieed up, and are being carried on the back
Scape grid ΩeIt is interior, for each point in its local support region ΩqInterior (8) formula can be written as sub- equation group form
KqUq=Fq (9)
Wherein
For the sub- stiffness coefficient matrix of local support region,It itemizes sub- stiffness matrix for volume,To use
Boundary integral item stiffness matrix when third boundary condition, the expression formula of its each element are
Wherein i, j=1,2 ..., n, n are the node total number for including, φ in local support regioniAnd φjRespectively in support region
I-th and j node shape function；Since RPIM shape functions have Kronecker delta function properties, right-hand vector FqIt is each
Element expression is (15) formula；
The sub- equation group of local support region (9) formula of point in all background grids is assembled, D.C. resistance can be obtained
Rate element-free menthod equation (8) formula；
Step 4 calculates apparent resistivity：
The element-free menthod equation obtained in step 3 is solved, node electric field value is obtained, is calculated according to observation device
Obtain the apparent resistivity parameter of observation point；
Advantageous effect：
Background grid seriously affects simulation precision with whether Node distribution is adapted in dc resistivity element-free menthod, conventional
Dc resistivity element-free menthod in using the uniform subdivision of homogeneous background grid method obtain background grid, if intensive in node
When distributed areas background grid number is insufficient to, simulation precision may be caused low, therefore, in order to ensure simulation precision often use compared with
Thin background grid distribution, with the regional area that adaptive model interior joint is most intensive, when node density uneven distribution, part
The intensive grid distribution of uniform subdivision is incompatible with Node distribution in sparse Node distribution region, is added significantly to no unit
The calculation amount of method causes computational efficiency low.The background grid Adaptive meshing method of the present invention gives a small amount of coarse simple four
Side shape initial background grids and Adaptive meshing controlling value M can carry out background grid according to the distribution situation of node and adaptively cut open
Point, the background grid being adapted with Node distribution is obtained, is mutually fitted in node close quarters Adaptive meshing and node dense distribution
The intensive background grid answered so that Gauss integration point improves dc resistivity without unit according to Node distribution situation reasonable layout
The numerical stability of method ensures simulation precision, in adaptable dilute of node sparse region Adaptive meshing and node sparse distribution
Background grid is dredged, Gauss integration point quantity is reasonably greatly decreased, improves dc resistivity element-free menthod forward modelling efficiency.
The present invention can to the arbitrary geometric shape of two dimension, hypsography, resistance parameter complex distribution earth-electricity model into
The numerical stability of direct current element-free menthod can be improved in row dc resistivity forward modelling in the case of node arbitrarily irregular distribution
Property, ensure simulation precision, the computational efficiency of dc resistivity element-free menthod forward modeling greatly improved.
Description of the drawings：
Fig. 1 is the background grid Adaptive meshing effect diagram in the present invention；(a) it is the grid schematic diagram before subdivision,
(b) it is Adaptive meshing effect diagram that self adaptive control value M is 4；
Wherein, 1, point source, 2, initial background grids, 3, field node, 4, the background grid of Adaptive meshing.
Fig. 2 is the schematic diagram of the background grid Adaptive meshing background grid segmentation process in the present invention.
Fig. 3 is dc resistivity element-free menthod forward simulation calculating process schematic diagram；
Wherein, 5, computational domain, 6, support region, 7, Gauss point, 8, node.
Fig. 4 be homogeneous half space point source in electric model use background grid Adaptive meshing method when, (a), (b) and
(c) background grid obtained when being respectively self adaptive control value M=4,8 and 16.
Fig. 5 is analog result of the homogeneous half space point source for different background grid cutting algorithm in electric model Potential analogue
Comparison diagram.
Fig. 6 is complicated earth-electricity model schematic diagram, and (a) is distributed for complicated earth-electricity model resistivity and Node distribution schematic diagram,
(b) for using background grid Adaptive meshing method adaptive when the initial background grids and self adaptive control value M=4 that are used when
The background grid schematic diagram that subdivision obtains.
Fig. 7 is that apparent resistivity analog result, (a)~(c) are observed in complicated earth-electricity model dc resistivity element-free menthod forward modeling
The apparent resistivity analog result of M=4,8 and 16, (d)~(f) are respectively when respectively using background grid Adaptive meshing method
The uniform subdivision of background grid is the apparent resistance of 0.5m × 0.5m, 1m × 1m and 2m × 2m when using background grid uniform subdivision method
Rate analog result.
Specific implementation mode：
Below in conjunction with the drawings and specific embodiments, the present invention is further illustrated.
Dc resistivity observation computational methods of the present invention include the following steps：
The design of step 1, forward modeling earth-electricity model Parameter File：According to the distribution of two-dimentional earth-electricity model intermediary matter resistivity,
Model discrete nodes confidence file is arranged in the geometric shape and hypsography situation of anomalous body, and pole layout, observation dress is arranged
It sets and element-free menthod relevant parameter.
The design of step 2, adaptive background mesh generation file：Use a small amount of coarse simple quadrangle initial background net
Lattice coverage domain, by the vertex point coordinate information of initial background grids and given self adaptive control value input file.
Step 3 carries out adaptive background mesh generation：According to Node distribution situation, under the guidance of self adaptive control value M
Adaptive meshing is carried out to initial background grids.
Step 4 carries out element-free menthod calculating：On computational domain, dc resistivity element-free menthod is carried out according to modelling
Forward modelling obtains dc resistivity element-free menthod equation group.
Step 5 solves the dc resistivity element-free menthod equation group of acquisition, obtains forward model node electric field
Value calculates according to observation device and obtains observation apparent resistivity.
It is that the present invention calculates a homogeneous half space point source for the intricately electric mould of electric model Potential analogue and one below
The example of the high density urethane acrylate anionomer observation of apparent resistivity of type.
As shown in Fig. 1,120m (X one wide are established:- 60~60m), high 60m (Z:0~60m), resistivity 100
The homogeneous half space model of Ω m, there are one the point sources of electric current I=1A to think that underground powers at X=0m, using irregular point
The node discrete model of cloth, as shown in Fig. 1.The minimum interval of Node distribution dense distribution near point source, node is
0.125m, centered on point source, to the gradual sparse distribution of exterior node, the largest interval of node is 4m.First, using background net
Background grid subdivision is that 0.5m × 0.5m and 1m × 1m carries out element-free menthod calculating respectively by the uniform subdivision method of lattice.Then,
Using adaptive background grid cutting algorithm, two initial background grids are only inputted as shown in Fig. 1, to self adaptive control value M
=4,8 and 16 carry out element-free menthod calculating respectively；
Point source Potential analogue result is as shown in Fig. 5, and knot is simulated when background grid is uniformly split into 0.5m × 0.5m
Fruit coincide very well with analytic solutions, and when background grid is uniformly split into 1m × 1m, analog result differs greatly with analytic solutions, obtains
The result of mistake.The analog result obtained when use background grid Adaptive meshing, self adaptive control value M=4,8 and 16
It coincide with analytic solutions fine.The analog result that the two different background grid subdivision methods of comparative analysis obtain, due to node
Density is distributed, when the uniform subdivision of background grid is 0.5m × 0.5m, the node of the density of background grid in node close quarters
Density is adapted, therefore obtains preferable analog result, when the uniform subdivision of background grid is 1m × 1m, the density of background grid
In node close quarters and node density analog result that is incompatible, therefore having there is no.As shown in Fig. 4, work as use
Background grid Adaptive meshing, when self adaptive control value M=4,8 and 16, the background grid distribution of acquisition with Node distribution phase
It adapts to, the analog result that comparison background grid uniform subdivision obtains when being 0.5m × 0.5m and 1m × 1m is it is found that in node unevenness
In the case of even Arbitrary distribution, the numerical stability of dc resistivity element-free menthod can be improved in background grid Adaptive meshing method,
Ensure simulation precision.
Establish 200m (X one wide:- 100~100m), high 80m (Z:0~80m) complexity earth-electricity model, resistivity and section
Point distribution is as shown in attached drawing 6 (a).In ground Table X:2m arranges 59 power supplies and observation electrode at equal intervals within the scope of -58~58m, right
Model carries out high-density electric urethane acrylate anionomer observation of apparent resistivity forward simulation.First, using the uniform subdivision method of background grid
It is that 0.5m × 0.5m, 1m × 1m and 2m × 2m carry out element-free menthod calculating respectively by background grid subdivision.Then, use is adaptive
Background grid subdivision method is answered, two initial background grids, to self adaptive control value M=4,8 are only inputted as shown in attached drawing 6 (b)
Element-free menthod calculating is carried out respectively with 16.
Apparent resistivity analog result is as shown in Fig. 7, and attached drawing 7 (a)~(c) is respectively to use background grid Adaptive meshing
The apparent resistivity analog result of M=4 when method, 8 and 16, attached drawing 7 (d)~(f) are respectively to use the uniform subdivision side of background grid
Background grid subdivision is the apparent resistivity analog result of 0.5m × 0.5m, 1m × 1m and 2m × 2m when method.Comparative analysis attached drawing 7 can
Know, the apparent resistivity isoline of attached drawing 7 (a)~(e) displays is almost the same, however the apparent resistivity isoline of attached drawing 7 (f) display
Figure is substantially the same with attached drawing 7 (a)~(e), but comparison can be seen that certain regional areas have many difference in detail, such as schemes
It is apparent poor that the apparent resistivity isoline of middle-shallow layer (0~10m of depth) region attached drawing 7 (f) display has with attached drawing 7 (a)~(e)
It is different, the region by near field sources and Node distribution it is intensive, illustrate uniform subdivision be 2m × 2m background grid in the region and node
Be distributed it is incompatible, when background grid by uniform subdivision be 0.5m × 0.5m and 1m × 1m when, obtain with use background grid
The consistent analog result of Adaptive meshing method shows thinner background grid subdivision in the region and intensive Node distribution phase
It adapts to, ensure that simulation precision.
Table 1 gives using different background grid cutting algorithm reasons for its use grid sum, Gauss point sum and calculates
Time, comparative analysis it is found that when subdivision method uniform using background grid, reasons for its use grid sum, Gauss point sum and
The time is calculated significantly more than using background grid Adaptive meshing method, in order to ensure that simulation precision, background grid uniformly cut open
Point method needs the intensive subdivision of background grid ensureing the computational accuracy of node close quarters, but considerably increases background grid
Number, to increase Gauss points so that dc resistivity element-free menthod calculation amount is significantly increased.It is adaptive using background grid
Subdivision method can automatically obtain the background grid being adapted with Node distribution so that Gauss point rationally divides according to Node distribution situation
Cloth greatly reduces Gauss point quantity, to greatly improve the computational efficiency of dc resistivity element-free menthod.
Background grid subdivision method reasons for its use grid sum, Gauss point be not total and calculates the time for table 1
The foregoing is only a preferred embodiment of the present invention, is not intended to restrict the invention, for the skill of this field
For art personnel, the invention may be variously modified and varied.All within the spirits and principles of the present invention, any made by repair
Change, equivalent replacement, improvement etc., should all be included in the protection scope of the present invention.
Claims (2)
1. a kind of background grid Adaptive meshing method in dc resistivity element-free menthod, which is characterized in that including following step
Suddenly：
Step 1, earth-electricity model are established：
First, it is built according to the distribution of two-dimentional earth-electricity model intermediary matter resistivity, the geometric shape of anomalous body and hypsography situation
Vertical element-free menthod region Ω, and pole layout position, observation device, observation point position are set, it will in element-free menthod region
Two-dimentional earth-electricity model is discrete using the node progress of one group of Arbitrary distribution, is set and geometric shape, landform according to geology abnormal posture
Rolling shape, electrode position arrange Node distribution situation encryption node, and according to forward simulation demand, value variation on the scene it is little or
Person uses sparse Node distribution far from the electrically constant region of field source；
Step 2, background grid adaptive design：
First, given to instruct Adaptive meshing using a small amount of coarse simple initial quadrangle background grid coverage domain
Controlling value M；
Initial quadrangle background grid is numbered, following operation is carried out according to number order：Judge in i-th of background grid
Including number of nodes whether be more than controlling value M, if it is by the background grid according to the midpoints of the four edges of grid by the back of the body
Scape grid subdivision is four new background grids, equally does same judgement and operation to four new background grids, until
Newly-generated background grid is satisfied by the number of nodes for including inside it and is not more than controlling value M, then it is adaptive to stop i-th of background grid
Subdivision is answered, the one group new background grid that meets the requirements of i-th of initial background grids after Adaptive meshing is obtained；
All initial background grids are carried out after as above operating, met the requirements on element-free menthod computational domain one group of background net is obtained
Lattice, these background grids are that Adaptive meshing obtains under the guidance of controlling value M according to Node distribution, they and Node distribution
It is adapted；
Step 3 carries out element-free menthod calculating to earth-electricity model：
The background grid obtained by background grid Adaptive meshing method is numbered, n is arranged in each background gridg
A Gauss integration point xg, to each Gauss integration point xgThe local support region of construction one, uses n node for including inside it
Information structuring RPIM shape functions, using this group of shape function to Gauss integration point xgThe field value at place carries out interpolation calculation, uses Gauss
Integral and calculating corresponding variational problem of 2.5 dimension dc resistivity boundary value problems as shown in (1) formula；
Wherein, Γ is boundary symbol, and σ=(x, z) is dielectric conductance rate, and Ω is computational domain, and U (λ, x) is wave-number domain current potential, and λ is
Wave number, I0For electric current, δ is Kronecker delta functions, x=[x, z]TFor any point on Ω, A is field source location,
▽ is gradient operator, rAFor the air line distance at any point on point source and boundary, n is exterior normal unit vector, cos (r, n)
For rAWith the included angle cosine of outer normal direction n, K0、K1Respectively the second class single order, zeroth order modified Bessel function, δ are variation symbol；
After all background grids are carried out with element-free menthod calculating as above, the element-free menthod equation group KU=F, wherein K are obtained
For the element-free menthod stiffness coefficient matrix of N × N-dimensional, N is computational domain interior joint sum, and U is element-free menthod Area Node wave-number domain
The column vector that current potential corresponding N × 1 is tieed up, F are the element-free menthod equation right-hand vector column vector that N × 1 is tieed up；
Step 4 calculates apparent resistivity：
The element-free menthod equation obtained in step 3 is solved, node electric field value is obtained, is calculated and is obtained according to observation device
The apparent resistivity parameter of observation point.
2. the background grid Adaptive meshing method in a kind of dc resistivity element-free menthod described in claim 1, feature
It is：For the sub- stiffness coefficient matrix of local support region,It itemizes sub- stiffness matrix for volume,To use
Boundary integral item stiffness matrix when three classes boundary condition, the expression formula of its each element are
Wherein i, j=1,2 ..., n, n are the node total number for including, φ in local support regioniAnd φjRespectively i-th in support region
With the shape function of j node, ΩeFor e-th of background grid,For ΩeWith cutoff boundary Γ∞The sub- boundary overlapped, σ is medium
Conductivity, λ are wave number, rAFor the air line distance at any point on point source and boundary, n is exterior normal unit vector, cos (r, n)
For rAWith the included angle cosine of outer normal direction n, K0、K1Respectively the second class single order, zeroth order modified Bessel function.
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