A method for estimating shape functions for different thicknesses of a fillet

Abstract

This invention proposes a method for estimating the shape function of a strip at different thicknesses, comprising the following steps: generating an initial shape function dBi, i = 0; and using the shape function dBi... i Obtain the shape function dB for different thicknesses of the inlay. e This refers to the estimated shape function after variations in the thickness of the inlay strip. This invention, by designing a predicted shape function, allows for a single calculation of the basic shape function (dB) using finite element software during magnetic field patching. i The subsequent shape function dB e It can be achieved through the basic shape function dB i The percentage of the change in angle width in the previous cycle f i (R) and the percentage of the next change in angular width f e The product of the ratios (R) is used to calculate the shape function, which effectively reduces the calculation time, shortens the project schedule, and reduces the cost of project implementation.

Description

Technical Field

[0001] This invention belongs to the field of magnetic field padding technology for cyclotron accelerators, and particularly relates to a method for estimating the shape function of strips with different thicknesses. Background Technology

[0002] Magnetic field padding is an essential component of isochronous cyclotron accelerators. Traditional methods typically require at least three padding operations. Each time the insert is cut, the angular width changes (the angular width refers to the angular width of the pole containing the insert; it could be on one side of the pole or on both sides), necessitating a recalculation of the shape function. Thus, completing the entire padding process requires at least three shape function calculations. The reason for requiring at least three shape function calculations is that insert padding must never be "over-padded" but must employ a "gradual approximation" method. This is because during the padding process, only the insert can be cut, not its thickness increased. If only the initial shape function dB is used... i Performing a one-step padding calculation can lead to "over-padding" when the padding amount is large, meaning the insert is cut too much, causing padding failure. Therefore, it's necessary to use smaller padding amounts each time, repeatedly calculate the shape function, and repeatedly send the insert to the machining plant for cutting to prevent "over-padding." However, multiple padding operations require multiple shape function calculations. The shape function is formed as follows... Figure 3 As shown, the horizontal axis represents the accelerator radius, and the vertical axis represents the magnetic field. Figure 3 The initial shape function dB in i It is composed of multiple shape functions with independent peaks superimposed. According to statistics, calculations using finite element software... Figure 2 The shape curve of even one independent peak takes more than two hours to determine. Figure 2 It takes nearly two days to calculate all the shape functions with independent peaks, and this is only the time for one shape function calculation. If there are more than three paddings, it will take several days to calculate the shape functions. This means that the traditional padding method requires high computing resources and a long magnetic field padding time cycle. Summary of the Invention

[0003] To address the problems existing in the prior art, this invention proposes a method for estimating the shape function of inlay strips with different thicknesses, aiming to solve the problem that the traditional magnetic field padding method requires a long magnetic field padding time period.

[0004] To solve its technical problems, the present invention proposes the following technical solutions:

[0005] A method for estimating the shape function of a strip at different thicknesses includes the following steps:

[0006] Step 1: Generate the initial shape function dBi, i = 0;

[0007] The initial shape function dB i That is, successively at the radii R2, R4, ... R of the strip. 2n A triangle with a width of 2dR and a height of h was cut off at a certain point, and its shape function dB was obtained through finite element method simulation. i dB i Contains n curves: the n curves These correspond to radii R2, R4, ... R, respectively. 2n The average change in magnetic field caused by cutting off a triangle with a width of 2dR and a height of h;

[0008] Step 2: Utilize the shape function dB i Obtain the shape function for different thicknesses of the inlay. dB e This is the shape function estimated after the thickness of the inlay changes.

[0009] Step two utilizes the shape function dB i Predicting the shape function of the inlay at different thicknesses

[0010] The specific process is as follows:

[0011] 1) Calculate the initial angle width of the inlay as A. i relative angular width change at (R) Among them, A i (R) is the initial angle width vector of the strip, A i (R)=[A i (R2),A i (R4),…,A i (R 2n )];

[0012] The angular width vector is located at a distance h from the current edge curve of the strip with radius R; f i (R) is the vector of relative angular width change; f i (R)=[f i (R2),f i (R4),…,f i (R 2n The edge curve of the inlay is a polyline composed of 2n+1 points, each with a radius of R1, R2, ..., R. 2n+1 The radial distance between any two adjacent points is dR, and n is the number of cutting triangles;

[0013] 2) Calculate the width of the inlay corner to become Ae The change in relative angular width at (R): Among them, A e (R) is the vector of the changed strip width, A e (R)=[A e (R2),A e (R4),…,A e (R 2n )). The angular width vector at a distance h from the current edge curve of the strip at radius R:

[0014] f e (R) is the vector of relative angular width change: f e (R)=[f e (R2),f e (R4),…,f e (R 2n )];

[0015] 3) Obtain the estimated shape function:

[0016] The angular width at a distance h from the current edge curve of the strip refers to the angular width at a distance h from each fold point on the edge of the strip, and this angular width includes the angular width when the strip is present on one side of the magnetic pole or when the strip is present on both sides of the magnetic pole.

[0017] Advantages and effects of the present invention

[0018] This invention designs a predictive shape function. This allows for the calculation of the basic shape function dB only once using finite element software during magnetic field padding. i The subsequent shape function dB e It can be achieved through the basic shape function dB i The percentage of the change in angle width in the previous cycle f i (R) and the percentage of the next change in angular width f e (R) ratio The product of these factors is used to calculate the shape function, which effectively reduces the calculation time, shortens the project schedule, and lowers the cost of project implementation. Attached Figure Description

[0019] Figure 1 This is a schematic diagram showing the angle width of the inlay when the triangle is not cut.

[0020] Figure 2 A schematic diagram of a triangle being cut at a radius using finite element method software;

[0021] Figure 3 Based on the shape function dB i Schematic diagram;

[0022] Figure 4 A schematic diagram of cutting triangles at even-numbered nodes;

[0023] Figure 5 This is a polyline graph of a stripe after one cutting operation using padding and interpolation algorithms.

[0024] Figure 6 The initial angular width A i (R) and the angular width at a distance h from it. Schematic diagram;

[0025] Figure 7 This is a global schematic diagram illustrating the predicted shape function effect of the present invention;

[0026] Figure 8 for Figure 7 A magnified view of a portion of the image;

[0027] Figure 9 This is a flowchart of the process for this method. Detailed Implementation

[0028] Design principle of the invention

[0029] The invention will be further explained below with reference to the accompanying drawings.

[0030] (i) Reasons for the long repair cycle of traditional magnetic field padding methods

[0031] Step 1: Two conditions are required before the first magnetic field patching: First, establish the basic shape function dB. i Second, the measured magnetic field difference; the basic shape function dB. i like Figure 1-3 As shown, triangles with a width of 2dR and a height of h are cut off at each radius R along the unmachined strip. The shape function dB is obtained through finite element software simulation. i Basic shape function dB i The significance lies in analyzing the change in magnetic field caused by cutting unit triangles on the strip, which serves as one of the bases for the strip padding algorithm. The "magnetic field difference" refers to the difference between the measured average magnetic field at each radius of the magnetic pole containing the strip and the desired isochronous magnetic field; this "magnetic field difference" is the second base for the strip padding algorithm.

[0032] The second step involves processing a portion of the magnetic field and then supplementing that portion. The supplementation of the magnetic field difference at each radius of the inlay is not a single step but rather multiple steps, gradually approximating the target. The method involves multiplying the magnetic field difference at each radius by a percentage. This percentage is determined empirically, for example, 50%. Note that in traditional methods, the magnetic field difference calculation is only performed the first time using the set percentage; subsequent calculations use actual measured magnetic field values ​​from the cutting process at the factory. This difference is then compared to the desired isochronous magnetic field at that radius.

[0033] Step 3: First cutting at the processing plant. The padding algorithm calculates the padding amounts h1-h5 for the first time based on the basic shape function dB. i And the difference in the partial magnetic field at each radius is used to deduce how much can be cut along each broken line of the inlay. Assuming the padding amount calculated by the padding algorithm in the first step is as follows: Figure 5 The broken lines shown below yield h1-h5 respectively;

[0034] Step 4: Second cutting at the processing plant. The second cutting at the processing plant requires the padding algorithm to recalculate the cutting or padding amount. Each padding algorithm calculation relies on two essential conditions: the current shape function and the magnetic field difference between each radius. The magnetic field difference between each radius is obtained by measuring the magnetic field under the current inlay conditions, comparing the current magnetic field with the desired isochronous magnetic field, and then multiplying it by a percentage (e.g., 50%). The reason for recalculating the current shape function is that after the first cutting, the angular width of each fold point on the inlay line changes. Figure 4 As shown, the average magnetic field and the peak height of the shape function change with the change of the angular width; all calculations of the shape function are performed using finite element software.

[0035] Step 5: Second Calculation of Shape Function: ① The second shape function calculation in the finite element software differs from the first. The first calculation cuts along the straight line of the strip, while the second calculation cuts along the broken line of the strip. The reason for cutting along the broken line in the second calculation is that the strip has already been cut once according to the padding algorithm, so the edge of the strip is no longer a straight line but a broken line. ② Regardless of whether the finite element software cuts the triangle along the broken line of the strip for the first, second, or nth time, the thickness h of the triangle cut is the same each time.

[0036] Step 6: After each cut, the inlay strip must be retrieved and the magnetic field remeasured. The shape function also needs to be recalculated. This is because each cut changes the angular width of the magnetic pole (including the inlay strips on either side or one side of the pole). This change in angular width refers to the change in angular width at each fold point on the inlay strip's fold line. A change in angular width results in a change in the magnetic field. If the padding algorithm uses the previous shape function to calculate the padding amount when the magnetic field changes, errors will occur. Firstly, the shape function represents the change in the magnetic field after a triangle is removed. If the current magnetic field is not the actual magnetic field but rather the magnetic field before the change, the resulting shape function is meaningless. Secondly, the current padding algorithm calculates the padding amount based on the difference between the current shape function and the current magnetic field. If the current shape function and the current magnetic field difference do not match, errors will occur. Therefore, the shape function must be recalculated after each cut.

[0037] In summary, the traditional magnetic field patching method has a long cycle time because: First, magnetic field patching cannot be completed in one go; it must be done in multiple stages, each time processing only a portion of the magnetic field and patching only a portion of the magnetic field difference. Therefore, the entire magnetic field difference at a given radius needs to be taken to the processing plant for cutting multiple times, resulting in a long round trip time. Second, before each calculation of the patching amount, the patching algorithm needs to recalculate the shape function dB1, dB2, ... dB. e The reason for recalculating the shape function is that after one or more cuts, the angular width at each fold point of the inlay line changes, and this change in angular width signifies a change in the magnetic field. Since each shape function calculation requires repeatedly cutting triangles at each radius, repeatedly calculating the average magnetic field change caused by each triangle cut, and repeatedly calculating the contribution of the triangle cut at that radius to the surrounding magnetic field, the calculation cycle is very long, taking 1-2 days to calculate one shape function. If three rounds of padding are needed, it will take 5-6 days to complete the shape function calculation, which will prolong the magnetic field padding project.

[0038] (ii) The innovative points of this invention:

[0039] Using estimation methods, in addition to the basic shape function dB i Besides using finite element software for calculation, other shape functions dB1, dB2... dB e The shape functions were estimated using an "estimation" method, rather than calculated using conventional finite element software. The difference between the estimation method and the conventional calculation method is that the estimation method only needs to find a dB value. i and dB e The scaling factor between the two is used; for each estimation, simply multiply this scaling factor by dB. iThis yields a new shape function. Conventional finite element method (FEA) calculations involve cutting triangles at each radius, then calculating the magnetic field change at each triangle and its contribution to the magnetic field changes at other radii. In contrast, the method of this invention requires only one finite element calculation, reducing the computation time by several times compared to multiple calculations.

[0040] (iii) Design principle of the estimation method of this invention

[0041] Estimating shape function It is based on three principles.

[0042] The first rule is that no matter how many times the padding algorithm is used to cut, the change in angle width is approximately equal each time the corresponding shape function is calculated.

[0043] The second pattern is that although the change in angular width is approximately equal in the previous and next cycles, the angular width gradually decreases because the strip is cut thinner and thinner, so the angular width of the magnetic pole also becomes smaller and smaller.

[0044] The third rule is that the angular width ratio f of the next shape function e (R) must be greater than the previous shape function angular width percentage f. i (R), this pattern of change is the shape function dB for the next iteration. e dB of the basic shape function i The changing pattern. From this, the predicted shape function is obtained:

[0045] Based on the above-mentioned inventive principles, this invention designs a method for estimating the shape function of a strip with different thicknesses, such as... Figure 1-9 As shown, it includes the following steps:

[0046] Step 1: Generate the initial shape function dBi, i = 0;

[0047] The initial shape function dB i like Figure 3 As shown, this is achieved by sequentially adjusting the radii of the strips R2, R4, ... R... 2n A triangle with a width of 2dR and a height of h was cut off at a certain point, and its shape function dB was obtained through finite element method simulation. i dB i Contains n curves: the n curves These correspond to radii R2, R4, ... R, respectively. 2n The average change in magnetic field caused by cutting off a triangle with a width of 2dR and a height of h;

[0048] Supplementary Note 1

[0049] Two points to note regarding step one, cutting the triangle: First, the cut must be made at an even-numbered fold point of the strip. Second, the strip's edge should be designed to consist of 2n+1 fold points. When the number of fold points is odd rather than even, cutting the triangle at an even-numbered node will always result in a complete triangle. For example... Figure 4 As shown, if there are 4 fold points, then only half of the second triangle can be cut when cutting the second triangle.

[0050] Step 2: Utilize the shape function dB i Obtain the shape function for different thicknesses of the inlay. dB e This is the shape function estimated after the thickness of the inlay changes.

[0051] The specific process is as follows:

[0052] ① Calculate the initial angle width of the inlay as A i relative angular width change at (R) Among them, A i (R) is the initial angle width vector of the strip, A i (R)=[A i (R2),A i (R4),…,A i (R 2n )];

[0053] The angular width vector is located at a distance h from the current edge curve of the strip with radius R; f i (R) is the vector of relative angular width change; f i (R)=[f i (R2),f i (R4),…,f i (R 2n The edge curve of the inlay is a polyline composed of 2n+1 points, each with a radius of R1, R2, ..., R. 2n+1 The radial distance between any two adjacent points is dR, and n is the number of cutting triangles;

[0054] ② Calculate the corner width of the inlay strip to become A e The change in relative angular width at (R): Among them, A e (R) is the vector of the changed strip width, A e (R)=[A e (R2),A e (R4),…,A e (R 2n )]; The angular width vector at a distance h from the current edge curve of the strip at radius R:

[0055] f e (R) is the vector of relative angular width change: f e (R)=[f e (R2),f e (R4),…,f e (R 2n )];

[0056] Furthermore, the angular width at a distance h from the current edge curve of the strip refers to the angular width at a distance h from each inflection point on the edge of the strip, and this angular width includes the angular width when the strip is present on one side of the magnetic pole or when the strip is present on both sides of the magnetic pole.

[0057] 3) Obtain the estimated shape function:

[0058] Supplementary Note 2

[0059] This application estimates the function. There are four concepts involved: angle width, change in angle width, percentage of change in angle width, and ratio of percentage of change in angle width. Understanding these four concepts is essential for comprehending this application.

[0060] ① Angular width: First, angular width A i (R), A e (R) refers to the fold point of the inlay, which includes two types: one is the fold point when the inlay has not been cut, such as... Figure 4 The folds on the "dashed lines" are folds that were not cut, while the folds are folds after the inlay strip has been cut, such as... Figure 4 The folds on the "solid line" are the folds after the inlay has been cut once. Secondly, the angle width always refers to the even-numbered folds of the inlay; it is the angle width of the even-numbered folds, such as... Figure 4 As shown, this refers to the angle width corresponding to inflection points 2 and 4; with repeated padding of the inlay (the padding amount of the inlay is as follows)... Figure 5 The h1-h5 shown are calculated using a padding algorithm and interpolation method (and are not part of the content protected by this application). The angle width becomes smaller and smaller because the inlay strip is cut thinner and thinner. Figure 5 As shown, compared to when the strip was not cut, the angular width of each of the five fold points corresponding to h1-h5 has become smaller.

[0061] ② The change in angle width is approximate: The change in angle width refers to the change in angle width at a distance h from the current inlay fold point, such as... Figure 6 As shown, assuming points A and B are taken as the two ends of h, this refers to the change in angular width between points A and B. Regarding the change in angular width... As for A i (R) refers to Figure 6 The angular width from point A on the middle side to the corresponding position on the other side of the magnetic pole. It means Figure 6 The angular width from point B to the corresponding position on the other side of the magnetic pole; for the change in angular width In other words, point A is Figure 5 After the strip is cut once, the inflection point corresponding to h2 and h4, point B is... Figure 5 The point at a distance h from h2 and h4. The definition of the change in angular width is: the change in distance h from the current inflection point. Since h is fixed, the change in angular width at each step, regardless of the distance h... still They are all approximately equal.

[0062] ③ The percentage of change in angle width is variable: The percentage of change in angle width refers to the ratio of "change in angle width" to "angle width", as shown in the formula. The numerator is the change in angle width, and the denominator is the angle width itself. Since the denominator decreases over time while the numerator remains approximately constant, the next change in angle width will be greater than the previous change, which is f. e (R)>f i (R).

[0063] ④ The ratio of the change in angle width represents the ratio of the change in shape function: because the problem this application aims to solve is to estimate the shape function using estimation methods. Shape function dB e It only depends on the depth h of each triangle cut, which is also related to the angle width A. i (R), A e (R) is related to the shape function and the angular width A. i (R), A e (R) is related because the shape function is as follows Figure 7 , Figure 8 As shown, it describes the magnetic field changes caused by cutting a triangle at different radii. When the angular width at the same radius is different, the magnetic field changes described by the shape function are also different. Therefore, the shape function is only related to the change in angular width. and angular width A i (R), A e (R) is related, therefore, the ratio of the change in angle width to the total change in angle width. It is the ratio of the shape function change.

[0064] Supplementary Note 3

[0065] ① Figure 8 yes Figure 7The enlarged view shows three solid lines. The top solid line is the shape function curve without cutting the triangle. The two lower solid lines are each accompanied by a dashed line, which represents the function estimated using this invention. The derived shape function shows that the shape function calculated by estimation is very similar to the shape function of cutting triangles using finite element software.

[0066] ② The magnetic field changes significantly at the small radius while the changes at the medium and large radii are very small. This is because the radian corresponding to the angular width accounts for a large proportion on the circumference of the small radius, but a negligible proportion on the circumference of the large radius. Therefore, the change in magnetic field caused by each change in angular width is only very noticeable at the small radius.

[0067] Example 1

[0068] A method for one-time magnetic field padding using shape function estimation includes the following steps:

[0069] Step 1: Measure the average magnetic field at each radius. The difference between the average magnetic field at each radius and the theoretical isochronous average magnetic field is B. shim If the average magnetic field is measured at M radii, then B shim Let B be an M-dimensional vector. shim＝[ B shim1 B shim2 B shim3 ,…B shimM]

[0070] Supplementary Note 1

[0071] In step one, M represents the number of times the magnetic field is measured at different radii of the strip. From the perspective of accurate magnetic field measurement, the larger the value of M is, the better, which means the closer the intervals between the magnetic field measurements are.

[0072] Step 2: Before cutting the insert, the initial shape function dBi is obtained through finite element software simulation calculation;

[0073] Supplementary Note 2

[0074] Step two introduces the initial shape function dBi, where i = 0. The initial shape function dBi is defined using finite element software on an uncut strip (such as...). Figure 1 The shape function obtained by cutting a triangle on the (as shown) is dBi. After a simulated cutting of the inlay using the padding algorithm (the inlay after the first simulated cutting is as shown)... Figure 4 As shown), finite element software in Figure 4 The inlay that has already been cut once (such as the inlay) Figure 4When the triangle is cut again (as shown), the resulting shape function is no longer the initial shape function dBi. The shape function at this point is called dB. k See step five below.

[0075] Step 3: Calculate the magnetic field difference B at each radius. shim Divide into N parts, perform N calculations, and predict the change in magnetic field as B for each calculation. shim / N= [ B shim1 / N,B shim2 / N,B shim3 / N,…B shimM / N ] ;

[0076] Supplementary Note 3:

[0077] ① The uppercase N in step three has a different meaning from the lowercase n that follows. The uppercase N means that the magnetic field difference at a certain radius is divided into N parts. The purpose of dividing it into N parts is to make up for it in N times rather than making up for it all at once, because making up for it all at once has the "risk of over-making up". So each time only 1 / N is made up. The lowercase n is the number of triangles that the finite element software cuts on the entire radius of the strip.

[0078] ② Due to the expected change in magnetic field B shim / N represents the expected change in magnetic field at each radius of the strip, so each calculation of the expected change in magnetic field B shim / N, the B shim / N is not a single data point, but a set of data: B shim / N= [ B shim1 / N,B shim2 / N,B shim3 / N,…B shimM / N ]

[0079] Step 4, k-th calculation: The expected change in magnetic field is B. shim / N; k ranges from 0 to N-1;

[0080] The k-th calculation refers to the calculation when k is greater than 0; when k is greater than 0, all other calculations use the dB value. i The estimated shape function dB k .

[0081] Supplementary Note 4:

[0082] ① The "kth calculation" in step four refers to the kth shape function dB. kThe calculation of the k-th shape function is illustrated in the following example: Assume that the magnetic field difference at each of the M radii is divided into N parts, where N = 3, then the value of k ranges from 0, 1, to 2; the k-th calculation does not include the calculation when k = 0, but refers to the calculation when k is 1 or k is 2; because the shape function when k = 0 is the initial shape function dBi, which has already been calculated in step two.

[0083] ②The shape function dB k This also refers to the shape functions dB1 and dB2 when k is greater than 0 and k takes the value of 1 or 2.

[0084] Step 5: Establish the shape function dB based on the estimation function. k ;

[0085] The shape function based on the estimation function

[0086] Step 6: Obtain the padding amount X calculated in the kth iteration using the padding algorithm. k X k =[X k 2 ,X k 4 ,…X k 2n ], n is the number of times the magnetic field changes are calculated for the cut-off triangle; This represents the padding amount at the 2j-th inflection point during the k-th calculation, where j ranges from 1 to n.

[0087] Supplementary Note 5:

[0088] For step six, calculate the replenishment amount X for the kth time. k The following example illustrates this: When N=3, the first value of k is 1, the second value of k is 2, and when k is 1, assume... Figure 4 If the 5 points are changed to 7 points, then the triangle is cut only at the even-numbered points 2, 4, and 6. In this case, the number of triangles cut is n = 3. Therefore, the padding amount for the kth time is X1 = [X1...]. 2 X1 4 X1 6 Similarly, when k is 2, the amount of the kth replenishment is X2 = [X2] 2 X2 4 X2 6 ];

[0089] Step 7: After completing the above N calculations, the total padding amount at the 2j-th inflection point is the sum of the padding amounts from the N calculations, i.e.: The above calculations include R2, R4, ..., R 2nThe padding amount at n inflection points is given, and the padding amount at the other n+1 inflection points is obtained through linear interpolation.

[0090] Supplementary Note 6:

[0091] The explanation for step seven is as follows:

[0092] ① This represents the total padding amount calculated using the padding algorithm for a given inflection point. Here, j is a variable taking values ​​from 1 to n, where n is the number of triangles cut. Assuming the inlay has 7 inflection points, and triangles are cut at even-numbered inflection points 2, 4, and 6, then n = 3. Since j is a variable taking values ​​from 1 to n, when j equals 1, 2, and 3 respectively, the formula... Calculate the total padding amount at even-numbered inflection points 2, 4, and 6 respectively;

[0093] ② The statement "the padding amounts for the other n+1 inflection points are obtained through linear interpolation" is explained as follows: Assuming there are 7 inflection points, the padding amounts were calculated using a padding algorithm at inflection points 2, 4, and 6. Inflection points 1, 5, and 7 remain; these three inflection points are the so-called "other n+1 inflection points." The padding amounts for inflection points 1, 5, and 7 are calculated using interpolation. Interpolation is an existing technique. For example, if the 5th inflection point is calculated using interpolation, the calculation method is to take the average of the padding amounts for inflection points 2 and 6. The padding amounts for inflection points 1 and 7 are taken from their nearest neighbors, that is, the padding amounts for inflection points 2 and 6.

[0094] Supplementary Note 7:

[0095] Regarding the padding algorithm: The padding algorithm is an existing technology. For details on how to obtain the padding amount through the padding algorithm, please refer to the paper Qin B, Chen DZ, Zhao LC, et al. An improved matrix method for magnet shimmingin compact cyclotrons[J]. Nuclear instruments and methods in physics research, 2010, 620(2-3): P.121-127.

[0096] This article only provides a brief introduction to the padding algorithm as follows:

[0097] ① Two essential conditions for the padding algorithm are: one is the current shape function dB. k One is the difference in magnetic field value (dB) at various radii. Assuming an average magnetic field measurement was performed at m radii, the deviation between this average magnetic field value and the isochronous magnetic field value is given by dB = [dB1, dB2, dB3, ..., dB]. m ];

[0098] ②The representation method of the shim pad amount X: In order to ensure that the finite element software can cut whole triangles rather than half triangles at the entire radius of the shim, the shim edge is set to be composed of 2n + 1 equally spaced folding points. In this way, cutting triangles at the even folding points 2, 4, …, 2n can ensure that the cut triangles are complete rather than half. The shim pad amount X = [x2, x4, x6…x 2n , and this shim pad amount X is a set of data.

[0099] ②The shim equation of the magnetic field is expressed as:

[0100] dB = A·X + ε (1)

[0101] Where A represents the shim matrix, which is a matrix of order m×n. m represents the average magnetic field measurement at m radii, and n is the number of times the finite element software cuts triangles; ε is a random perturbation vector with a mathematical expectation of zero. The matrix representation form of the shim equation is as follows: [[ID=!13]] [[ID=!14]]

[0102] [[ID=!15]] [[ID=!16]] [[ID=!17]]

[0103] The n columns in matrix A are obtained from the shape functions of n cutting triangle calculations at even points on the entire radius of the shim. dB is obtained from magnetic field measurements, and X is the unknown shim to be solved. After obtaining the shim matrix A, the shim equation (1) can be written in the following form to solve:

[0104] X = (A T A) -1 A T dB

[0105] A T is the transpose of A, and (A T A) -1 is to find the inverse matrix of A T A <!

[0106] ④Generally, the spacing of shim points in the radial direction is about 2 cm. The number of elements in the magnetic field error vector dB is greater than the number of elements in the shim amount vector X (m > n). The equation (1) is an overdetermined system of equations, and the shim vector X can be solved by the least squares method. If a relatively small shim point spacing is selected, on the one hand, it will lead to m < n, then the equation (1) is an underdetermined system of equations with infinitely many solutions. That is, there are infinitely many x2, x4, x6 …… Obviously, infinitely many x2, x4, x6 …… have no practical value. On the other hand, it will lead to A becoming a ill-conditioned matrix. Therefore, it must be that the number of elements in the magnetic field error vector dB is greater than the number of elements in the shim amount vector X (m > n), and the shim vector X can be solved by the least squares method.

[0107] It should be noted that there may be some inaccuracies in the translation due to the complexity and专业性 of the patent text. It is recommended to double-check with relevant technical experts or refer to more accurate translations for specific applications.Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention is also intended to include these modifications and variations.

Claims

1. A method for estimating the shape function of a strip of different thicknesses: characterized in that: Includes the following steps: Step 1: Generate the initial shape function dB i , i=0; The initial shape function dB i That is, successively at the radii R2, R4, ... R of the inlay strip. 2n A triangle with a width of 2dR and a height of h was cut off at a certain point, and its shape function dB was obtained through finite element method simulation. i dB i Contains n curves: the n curves These correspond to radii R2, R4, ... R, respectively. 2n The average change in magnetic field caused by cutting off a triangle with a width of 2dR and a height of h; Step 2: Using the initial shape function dB i Obtain the shape function for different thicknesses of the inlay. dB e This is the shape function estimated after the thickness of the inlay changes; Step two utilizes the shape function dB i Predicting the shape function of the inlay at different thicknesses The specific process is as follows: 1) Calculate the initial angle width of the inlay as A. i relative angular width change at (R) Among them, A i (R) is the initial angle width vector of the strip, A i (R)=[A i (R2),A i (R4),…,A i (R 2n )]; The angular width vector is located at a distance h from the current edge curve of the strip with radius R; f i (R) is the vector of relative angular width change; f i (R)=[f i (R2),f i (R4),…,f i (R 2n The edge curve of the inlay is a polyline composed of 2n+1 points, each with a radius of R1, R2, ..., R. 2n+1 The radial distance between any two adjacent points is dR, and n is the number of cutting triangles; 2) Calculate the corner width of the inlay strip to become A e The change in relative angular width at (R): Among them, A e (R) is the vector of the changed strip width, A e (R)=[A e (R2),A e (R4),…,A e (R 2n )]; The angular width vector at a distance h from the current edge curve of the strip at radius R: f e (R) is the vector of relative angular width change: f e (R)=[f e (R2), f e (R4), ..., f e (R 2n )]; 3) Obtain the estimated shape function:

2. The method for estimating the shape function of a strip of different thicknesses according to claim 1, characterized in that: The angular width at a distance h from the current edge curve of the strip with radius R refers to the angular width at a distance h from each inflection point on the edge of the strip, and this angular width includes the angular width when the strip is present on one side of the magnetic pole or when the strip is present on both sides of the magnetic pole.

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