CN108304651B - The third boundary condition processing method of dc resistivity Finite element method simulation - Google Patents
The third boundary condition processing method of dc resistivity Finite element method simulation Download PDFInfo
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Abstract
The present invention provides third boundary condition processing methods new in a kind of dc resistivity Finite element method simulation, comprising the following steps: establishes dc resistivity Finite Element computational domain, utilizes irregular or regular distribution the discrete earth-electricity model of node；Obtain non-structured Finite Element triangular element；It is calculated in computational domain using Finite Element, linear interpolation is used in unstructured triangular element, the volume subitem in the weak formula integral equation of variational problem met to dc resistivity calculates；Calculating dc resistivity using Moving Least has elements method boundary integral item, using shape function to Gauss integration point xgThe field value at place carries out interpolation calculation；It is solved after volume subitem equation group and boundary integral item equation group are merged, obtains the apparent resistivity parameter of observation point.The present invention handles dc resistivity Finite Element third boundary condition using Moving Least, substantially reduces computational domain area requirement, simplifies modeling, and simulation precision is high.
Description
Technical field
The present invention relates to the boundary processing methods in a kind of dc resistivity Finite element method simulation, in particular to intricately
High-precision, high-adaptability and the efficient dc resistivity Finite Element forward simulation of electric model.
Background technique
Dc resistivity exploration is one of geophysical exploration important method, 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.Forward simulation is the important means in the interpretation of dc resistivity survey data,
Mainly there are finite difference calculus, integral equation method and Finite Element, preceding two numerical value in dc resistivity the Forward Modeling at present
Analogy method is theoretical simple, is easily programmed realization, and computational efficiency is high, but not high to complicated earth-electricity model adaptability.Limited list
First method adaptability and flexibility are high, and the Finite Element for being based particularly on unstructured elements can be to arbitrarily complicated body and landform
Simulation, while Finite Element computational efficiency is higher, simulation precision is high, is answered extensively in dc resistivity forward simulation at present
With (Xu Shizhe, 1994；Ruan Baiyao and Xiong Bin, 2001).In dc resistivity Finite Element forward simulation, first have to establish
Earth-electricity model carries out subdivision using unstructured elements, node irregularly arbitrarily divides for more preferable simulation intricately electric model
Cloth, under other element same case, bigger computational domain needs to cover using more nodes and mesh discretization, models difficulty
It is bigger.When using third boundary condition, it usually needs biggish computational domain range, to guarantee the precision of feature modeling, so that
It is larger to model difficulty.
Therefore, it is necessary to which designing one kind is suitable for smaller range computational domain, the easy and high-precision dc resistivity of modeling
The boundary processing method of Finite Element.
Summary of the invention
Technical problem solved by the invention is in view of the deficiencies of the prior art, it is limited to provide a kind of dc resistivity
Elements method simulates new boundary processing method, can be within the scope of lesser computational domain, so that third boundary condition calculating obtains
Satisfied precision is obtained, substantially reduces computational domain range compared to Finite Element conventional boundary calculation method, simplifies modeling work, mention
The computational efficiency of high dc resistivity Finite Element.
The technical solution of the present invention is as follows:
Step 1, earth-electricity model are established:
Firstly, determining the distribution situation of two dimension earth-electricity model intermediary matter resistivity and the geometric shape and ground of anomalous body
Shape rises and falls, and sets pole layout position, observation device, observation point position, supply current；
The Finite Element region Ω for establishing range small packet nucleus containing earth-electricity model, in Finite Element region
Earth-electricity model is carried out using one group of any irregular or regular distribution node discrete, is set according to geology abnormal posture and geometry
Form, hypsography form, electrode position arrange Node distribution, node can irregular Arbitrary distribution, locally can arbitrarily encrypt
Node；
Step 2, unstructured triangular mesh subdivision:
Earth-electricity model carries out being discretely formed node listing using one group of any irregular or regular distribution node, to node
List carries out unstructured triangular element method subdivision using Delaunay triangle subdivision method, obtains non-structured limited
Elements method triangular element；
The unstructured triangular element of Delaunay is generated using Bowyer-Watson algorithm (Yuan Youwei, 2007).
Bowyer-Watson algorithm is designed based on the empty circumscribed circle property of Delaunay trigonometric ratio, and basic ideas are: setting up an office and collect Pn-1
={ Pi: i=1,2 ..., n-1 } corresponding triangle sets are Tn-1, then in point set Pn-1One new point P of middle insertioniIt obtains a little
Collect Pn={ Pi: i=1,2 ..., n }, in Tn-1In find out and delete comprising point PiTriangle, form cavity, on cavity
Node respectively with PiLink forms new triangle, obtains Pn={ Pi: i=1,2 ..., n } corresponding triangle sets are Tn。
Bowyer-Watson algorithm schematic diagram is as shown in Fig. 2；
The boundary rectangle comprising established all nodes of earth-electricity model is found first, then connects rectangle any bar
Diagonal line, forms two triangles, and to their labels, using them as initial Delaunay triangular element, then according to
Node is inserted by above-mentioned method one by one, will include square after mostly completing insertion and subdivision Delaunay triangular element a little
Triangle other than shape vertex and landform is deleted, and the unstructured elements set of established earth-electricity model is obtained；
Step 3, dc resistivity Finite Element volume subitem calculate:
After subdivision obtains unstructured triangular element in computational domain, dc resistivity is met using Finite Element
The weak formula integral equation of variational problem (1) formula (Xu Shizhe, 1994) in volume subitem calculated
Wherein, Γ is boundary symbol, ΓTFor cutoff boundary symbol, σ is dielectric conductance rate, and σ is the scalar letter of coordinate position
Number, i.e. σ=σ (x), U (λ, x) they are wave-number domain current potential, and λ is wave number, and I is electric current, and δ is Kroneckerdelta function, x=[x,
z]TFor any point on Ω, A is field source location,For gradient operator, rAFor point source on boundary any point it is straight
Linear distance, n are exterior normal unit vector, and cos (r, n) is rAWith the included angle cosine of outer normal direction n, K0、K1Respectively the second class one
Rank, zeroth order modified Bessel function, δ0For variation symbol；
Assuming that any point approximation field value u in triangular unith(x) with conductivityσ's linear change, the structure in triangular element
Make Finite Element shape function Φ=[φ1 φ2 φ3]T, φ1、φ2、φ3Indicate three vertex correspondences of triangular element
Node calculates volume subitem by Finite Element shape function linear interpolation in triangular element, has in each triangular unit:
The substitution of (2) formula itemizes to the volume of variational problem (1) formula and is unfolded to have:
Triangular element ΩeSub- stiffness coefficient matrix are as follows:
Carrying out dc resistivity as described above to all triangular elements has an elements method volume subitem to calculate, assembling
Get up to obtain dc resistivity Finite Element volume subitem equation group stiffness coefficient matrix KV；
Step 4, third boundary condition calculate:
When as shown in Fig. 1, using third boundary condition, D.C. resistance is calculated using Moving Least (MLS)
Rate has elements method boundary integral item, arranges n on cutoff boundary firstgA Gauss integration point xg, to each Gauss integration point xg
A local support region is constructed, calculates MLS shape function using n node for including inside it, using MLS shape function to Gauss product
Branch xgThe field value at place carries out interpolation, and calculating dc resistivity by Gauss integration has elements method boundary integral item；
To calculate point xgConstruct local support region Ωq, internal includes n node, constructs MLS shape function using them, so
Afterwards to calculating point xgPlace carries out field and is worth interpolation, and according to pertinent literature (Liu and Gu, 2007), MLS shape function can at Gauss point
It is written as:
ΦT(x)=[φ1(xg) φ2(xg) … φn(xg)]=pT(xg)A-1(xg)B(xg) (5)
A (x in formulag) and B (xg) be respectively
B (x)=[w (r1)p(x1) w(r2)p(x2) … w(rn)p(xn)] (7)
Wherein w (ri) it is Gauss point xgThe weight at place,Basic function selects pT(xg)=[1,
xg,zg], select cubic spline function as weight function:
WhereinTo uniform distance length, indicate are as follows:
Wherein rwFor round weight function domain ΩwRadius, to each Gauss point by its weight function ΩwWith support region Ωq
It is set as the same；
There is the sub- boundary of triangular element being overlapped with cutoff boundaryOn, utilize (5) formula to the Gauss on cutoff boundary
Point xgThe interpolation as shown in (2) formula is carried out, substitutes into variational problem (1) formula, the change of interpolation calculation is multiplied using minimum movement two
Boundary integral item and it is unfolded to have in point problem:
Wherein Uq=[u1 u2 … un]TFor support region ΩqInside is worth vector, sub- boundary comprising the field of n nodeUpper son
Boundary Stiffness coefficient matrix are as follows:
To all sub- boundaries of triangular element for having with cutoff boundary and being overlappedIt is upper to carry out dc resistivity as described above
There is an elements method boundary integral item to calculate, assembles and obtain dc resistivity Finite Element boundary integral item equation group rigidity
Coefficient matrix KT；
Step 5, equation group stiffness coefficient matrix K that volume is itemizedVWith boundary integral item equation group stiffness coefficient matrix KTIt closes
And obtain dc resistivity Finite Element entirety equation group stiffness coefficient matrix K, Finite Element entirety equation group right-hand vector
Column vector F has Kronecker delta function property, element expression due to Finite Element shape function are as follows:
The potential fields value that entirety equation group KU=F obtains computational domain Ω interior nodes is solved, it can further according to observation device information
The apparent resistivity parameter of observation point is calculated.
The utility model has the advantages that
It in dc resistivity Finite Element forward simulation, first has to establish earth-electricity model, in order to which more preferable simulation is multiple
Miscellaneous earth-electricity model, using unstructured elements carry out subdivision, the irregular Arbitrary distribution of node, under other element same case,
Bigger computational domain needs to cover using more nodes and mesh discretization, and modeling difficulty is bigger.Using third boundary condition
When, it usually needs biggish computational domain range, to guarantee the precision of feature modeling, so that modeling difficulty is larger.The present invention uses
Moving Least Squares interpolation algorithm calculates third boundary condition integral term by Gauss integration, improves boundary integral item meter
Precision is calculated, high-precision analog result is obtained in smaller range computational domain, realizes the purpose for reducing computational domain area requirement.
The present invention can to any geometric shape of two dimension, hypsography, resistance parameter complex distribution earth-electricity model into
Row dc resistivity forward modelling, greatly reduces computational domain area requirement, the small range computational domain in overlay model region
It inside can get high-precision analog result, so that model foundation is more convenient.
Detailed description of the invention:
Fig. 1 is the schematic diagram that the present invention handles boundary using Moving Least；
Wherein, 1, anomalous body boundary, 2, triangular element, 3, field node, 4, Gauss integration point, 5, support region, 6, earth's surface
Γs, 7, cutoff boundary ΓT。
Fig. 2 is the schematic diagram of the unstructured triangular element process of Bowyer-Watson algorithm subdivision；
8, cavity, 9, circumscribed circle, 10, insertion point.
Fig. 3 is that round abnormal body Model carries out high-density electric urethane acrylate anionomer observation of apparent resistivity schematic diagram.
Fig. 4 is the element subdivision schematic diagram of different computational domains, and (a) is the small range computational domain Ω comprising abnormal body Model1
Element subdivision, (b) in small range computational domain Ω1Carry out the computational domain Ω of flared end2Element subdivision.
Fig. 5 is the Forward modelling result that equal half space model carries out high-density electric urethane acrylate anionomer observation of apparent resistivity, (a)
To use conventional DC current Finite Element to computational domain Ω1The analog result of forward simulation acquisition is carried out, is (b) use
Conventional DC current Finite Element is to computational domain Ω2The analog result of forward simulation acquisition is carried out, (c) for using the present invention
Method to computational domain Ω1Carry out the analog result of forward simulation acquisition.
Fig. 6 is the Forward modelling result that round abnormal body Model carries out high-density electric urethane acrylate anionomer observation of apparent resistivity,
It (a) is using conventional DC current Finite Element to computational domain Ω1Carry out forward simulation acquisition analog result, (b) be
Using conventional DC current Finite Element to computational domain Ω2The analog result of forward simulation acquisition is carried out, (c) for using this
The method of invention is to computational domain Ω1Carry out the analog result of forward simulation acquisition.
Specific embodiment:
Below in conjunction with the drawings and specific embodiments, the present invention is further illustrated.
Dc resistivity of the present invention observation calculation method the following steps are included:
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,
The geometric shape and hypsography situation of anomalous body are arranged model discrete nodes confidence file, and pole layout and observation are arranged
Device closes parameter；
Step 2, unstructured triangular mesh subdivision: to model discrete nodes using the unstructured triangular element of progress
Method subdivision obtains non-structured Finite Element triangular element list；
Step 3, dc resistivity Finite Element volume subitem calculate: in computational domain, to all triangular elements, into
Row dc resistivity Finite Element carries out volume subitem and calculates；
Step 4, third boundary condition calculate: calculating dc resistivity using Moving Least on cutoff boundary
There is elements method boundary integral item；
Step 5 solves the potential fields value that dc resistivity finite elements normal equation system obtains computational domain interior nodes, further according to
The apparent resistivity parameter of observation point can be calculated in observation device information.
The following are the examples that the present invention calculates the high density urethane acrylate anionomer observation of apparent resistivity of a round abnormal body Model.
Round exception body Model is as shown in Fig. 3, in electricalresistivityρ1In the horizontal landform homogeneous half space of=100 Ω m,
There are an electricalresistivityρs2=10 Ω m, radius r=5m, the round anomalous body of center of circle buried depth h=15m, in earth's surface X:-58~58m
2m arranges 59 power supplies and observation electrode at equal intervals in range, carries out high-density electric urethane acrylate anionomer observation of apparent resistivity to model
Forward simulation.Firstly, horizontal direction width 140m (X:-80~80m) is established, the packet of vertical direction width 40m (Z:0~-60m)
Small range computational domain Ω containing round abnormal body Model1, using the node discrete model of irregular distribution in computational domain, with suitable
It answers round anomalous body boundary, and the encryption node near electrode, improves simulation precision, node total number 11149, unstructured three
Corner shaped elements sum is 21856, and the unstructured triangular element distribution of the model domain is respectively adopted often as shown in attached drawing 4 (a)
The Finite Element of rule and method of the invention carry out forward simulation to the model domain.Then, in small range computational domain Ω1Base
Flared end is carried out on plinth, establishes horizontal direction width 1560m (X:-780~780m), vertical direction width 760m (Z:0~-760m)
The larger range computational domain Ω comprising round abnormal body Model2, total node number 12919, unstructured triangular element is total
Number is 25523, and the unstructured triangular element distribution of the model domain uses conventional Finite Element as shown in attached drawing 4 (b)
Forward simulation is carried out to the model domain.
Firstly, setting the identical ρ of shoulder-bed resistivity (SBR) for round abnormal body resistivity2=ρ1=100 Ω m, at this time model be
Homogeneous half space model, apparent resistivity analytic solutions are 100 Ω m, and conventional Finite Element and side of the invention is respectively adopted
Method is to small range computational domain Ω1Forward simulation is carried out, shown in analog result such as attached drawing 5 (a) and attached drawing 5 (b), is had using conventional
Limit elements method computational domain Ω2Forward simulation is carried out, shown in analog result such as attached drawing 5 (c).Attached drawing 5 (a) shows have with conventional
Elements method is limited to small range computational domain Ω1The analog result and analytic solutions for carrying out forward simulation acquisition deviate serious, attached drawing 5 (b)
Show that conventional Finite Element can significantly improve simulation precision after expanding computational domain range, obtain good analog result.It is attached
Fig. 5 (c) shows with method of the invention to small range computational domain Ω1The analog result for carrying out forward simulation acquisition is good, with expansion
The analog result that conventional Finite Element obtains after big computational domain range is almost the same, shows the feasible of method of the invention
Property.
Then, ρ is set by round abnormal body resistivity2=10 Ω m, model is round abnormal body Model at this time, depending on
Resistivity observation Forward modelling result is as shown in Fig. 6, attached drawing 6 (a) and attached drawing 6 (b) be respectively conventional Finite Element and
Method of the invention is to computational domain Ω1The analog result of forward simulation is carried out, attached drawing 6 (c) is that conventional Finite Element calculates
Domain Ω2Carry out the analog result of forward simulation.Compare attached drawing 6 (a) and attached drawing 6 (c), both using conventional finite elements method into
Row forward simulation, small-scale computational domain Ω1With large range of computational domain Ω2Analog result difference it is larger.Compare attached drawing 6
(b) and attached drawing 6 (c), using method of the invention to small-scale computational domain Ω1The analog result of forward simulation acquisition is carried out,
With the use large range of computational domain Ω of conventional finite elements method2The analog result for carrying out forward simulation acquisition is consistent.It is right
Than analysis shows, to small-scale computational domain, conventional dc resistivity Finite Element is led since feature modeling precision is inadequate
Cause analog result poor, expanding computational domain can be improved the precision of analog result.Comparative analysis shows to small-scale computational domain,
Method of the invention can get good analog result, realizes and reduces dc resistivity Finite Element to computational domain range
It is required that purpose.
The above is that further detailed description of the invention, and it cannot be said that the present invention in conjunction with specific embodiment
Specific implementation is limited only to this；Belonging to the present invention and for those skilled in the technology concerned, it is being based on skill of the present invention
Under the premise of art scheme thinking, it is made expansion and operating method, data replacement, should all fall in the scope of the present invention it
It is interior.
Claims (2)
1. new third boundary condition processing method in a kind of dc resistivity Finite element method simulation, which is characterized in that packet
Include following steps:
Step 1, earth-electricity model are established:
Firstly, determining that the distribution situation of two dimension earth-electricity model intermediary matter resistivity and the geometric shape and landform of anomalous body rise
Volt, and set pole layout position, observation device, observation point position, supply current；
The Finite Element region Ω for establishing range small packet nucleus containing earth-electricity model, by ground in Finite Element region
Electric model is carried out discrete using one group of any irregular or regular distribution node, is set according to geology abnormal posture and geometric form
State, hypsography form, pole layout location arrangements Node distribution；
Step 2, unstructured triangular mesh subdivision:
Earth-electricity model carries out being discretely formed node listing using one group of any irregular or regular distribution node, to node listing
Unstructured triangular element method subdivision is carried out using Delaunay triangle subdivision method, obtains non-structured finite elements
Method triangular element；
Step 3, dc resistivity Finite Element volume subitem calculate:
After subdivision obtains unstructured triangular element in computational domain, dc resistivity is met using Finite Element change
Volume subitem in point weak formula integral equation of problem (1) formula is calculated
Wherein, Γ is boundary symbol, ΓTFor cutoff boundary symbol, σ is dielectric conductance rate, and σ is the scalar function of coordinate position, i.e.,
σ=σ (x), U (λ, x) are wave-number domain current potential, and λ is wave number, and I is electric current, and δ is Kronecker delta function, x=[x, z]TFor
Any point on Ω, A are field source location,For gradient operator, rAFor any point on point source and boundary straight line away from
From n is exterior normal unit vector, and cos (r, n) is rAWith the included angle cosine of exterior normal n, K0、K1Respectively the second class zeroth order, one
Rank modified Bessel function, δ0For variation symbol；
Finite Element shape function Φ=[φ is constructed in triangular element1 φ2 φ3]T, φ1、φ2、φ3Indicate triangle list
The shape function of the node of three vertex correspondences of member is calculated in triangular element by Finite Element shape function linear interpolation
Volume subitem, obtains the sub- equation group on triangular element；
Dc resistivity Finite Element volume subitem is carried out to all triangular elements to calculate, and assembles acquisition direct current
Resistance rate Finite Element volume subitem equation group；
Step 4, third boundary condition calculate:
Dc resistivity Finite Element boundary integral item is calculated using Moving Least, is arranged on cutoff boundary first
ngA Gauss integration point xg, to each Gauss integration point xgA local support region is constructed, n section for including inside it is used
Point calculates MLS shape function, using this group of shape function to Gauss integration point xgThe field value at place carries out interpolation calculation, assembles D.C. resistance
Rate Finite Element boundary integral item equation group；
Step 5, equation group that volume is itemized and boundary integral item equation group merge, and it is whole to obtain dc resistivity Finite Element
Equation group solves the potential fields value that whole equation group obtains computational domain Ω interior nodes, can calculate further according to observation device information
To the apparent resistivity parameter of observation point.
2. new third boundary condition processing side in dc resistivity Finite element method simulation according to claim 1
Method, it is characterised in that: had using boundary integral item expansion in the variational problem of Moving Least Squares interpolation calculation:
Wherein Uq=[u1 u2 … un]TFor support region ΩqInside is worth vector, Φ=[φ comprising the field of n node1 φ2 …
φn]T, φ1、φ2…φnIndicate support region ΩqInside includes the corresponding Moving Least Squares shape function value of n node, sub- boundaryUpper sub- Boundary Stiffness coefficient matrix are as follows:
To all sub- boundaries of triangular element for having with cutoff boundary and being overlappedUpper progress dc resistivity Finite Element boundary
Integral term calculates, and assembles and obtains dc resistivity Finite Element boundary integral item equation group stiffness coefficient matrix KT。
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| JPH096756A (en) * | 1995-06-20 | 1997-01-10 | Canon Inc | Device and method for processing information |
| CN103698814A (en) * | 2013-12-31 | 2014-04-02 | 中国海洋石油总公司 | Implementation method for mixed absorbing boundary condition applied to variable density acoustic wave equation |
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