CN114065673A - Bessel integral self-adaptive segmentation method and system in integrated circuit rapid calculation - Google Patents
Bessel integral self-adaptive segmentation method and system in integrated circuit rapid calculation Download PDFInfo
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Abstract
The application provides a Bessel integral self-adaptive segmentation method and a Bessel integral self-adaptive segmentation system in integrated circuit rapid calculation, wherein the method comprises the following steps: determining the order of a Bessel function used in the dyadic Green function; calculating a zero point of the Bezier function by adopting an iterative method according to the order of the Bezier function; determining the range or the number of the zero points; adaptive to the firstmThe sub-intervals are divided, the first one is calculatedmAccumulating the Bessel integrals after the subintervals are divided, and accumulating the accumulation result to the Bessel integral of the whole integral subinterval; if the condition is satisfied, a dyadic Green function of the field generated at the field point by the source point of the current integrated circuit is calculated. The system comprises: the device comprises an order determining module, a zero point determining module and an integral subinterval setting module; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module; the method reduces unnecessary redundant calculation in the Bessel integral self-adaptive segmentation method, and ensuresThe Bessel integral over the interval is guaranteed to be accurate.
Description
Technical Field
The application belongs to the technical field of integrated circuit electromagnetic simulation, and particularly relates to a Bessel integral self-adaptive segmentation method and system in integrated circuit rapid calculation.
Background
When the integrated circuit works, a high-frequency alternating electromagnetic field can be formed on a multilayer layout of the integrated circuit due to the transmission of high-speed signals, and meanwhile, in order to improve the performance of electronic equipment, reduce the volume and reduce the cost, transistors, other components and circuits are integrated on a small semiconductor substrate. In order to realize more functions, the ultra-large scale integrated circuit has a structure from tens of layers to hundreds of layers, each layer of structure is extremely complex, millions or even tens of millions of transistors are integrated, and the ultra-large scale integrated circuit has a multi-scale structure from a centimeter level to the latest nanometer level at present. In order to ensure that the integrated circuit can normally work and realize the function designed in advance, the power integrity and the signal integrity of the integrated circuit need to be ensured firstly, so that the power integrity and the signal integrity of the integrated circuit with a multi-scale structure of tens of layers and hundreds of layers need to be accurately analyzed by adopting an electromagnetic field analysis method, which is a great problem of the electromagnetic field analysis of the ultra-large scale integrated circuit.
A conventional method of analyzing the electromagnetic response of three-dimensional very large scale integrated circuits is a three-dimensional electromagnetic field numerical calculation method, such as a three-dimensional finite element method. The electromagnetic field distribution of the whole calculation region needs to be calculated, and further the electromagnetic response such as the electromagnetic field distribution, the current voltage of the designated port and the like of each layer of the integrated circuit is calculated. However, the characteristic dimensions of the via holes, the wires and the like of the integrated circuit are nano-scale, the dimension of the whole integrated circuit is centimeter-scale, the calculation area determined according to the truncation error is decimeter-scale and meter-scale, and hundreds of millions of grids and unknown quantities can be generated by carrying out uniform grid subdivision on the multi-scale space and then analyzing the space electromagnetic radiation of the multi-scale space, so that the hardware (memory) cost and the CPU time cost are overlarge. Therefore, the electromagnetic response of the three-dimensional large-scale integrated circuit can be calculated by adopting a method combining a finite element method and a moment method. In the three-dimensional large-scale integrated circuit area, a finite element method is adopted; in a large-scale area outside the integrated circuit, a moment method is adopted; the finite element method and the moment method are coupled at the interface of the integrated circuit and the external space. Because the moment method only integrates aiming at the interface, a large number of grid units and unknowns can be reduced, but because the scale range of the integrated circuit is from nano-scale to centimeter-scale, the finite element method directly used for solving the integrated circuit can generate a huge sparse matrix, and because the finite element method and the moment method are coupled, the formed coupling matrix is a dense matrix at the interface, the non-zero element number of the whole sparse matrix and the solving complexity of the sparse matrix are greatly increased, and the calculation time is still long.
The fields generated by the point current sources at any position in space can be calculated based on a green function (or Bessel integral), and the fields generated by the surface current sources at the same position can be calculated by utilizing a Gaussian integral method based on the linear superposition property of the fields generated by the sources, so that the fields generated by the current on the metal plate with the integrated circuit multilayer complex shape at different positions in space can be calculated. However, when the green function method is used to calculate the field generated by the point current source at any position in space, the core and difficulty lies in calculating the bezier integral introduced by the green function, because the bezier function has the characteristics of high oscillation, slow attenuation and the like, so that high-precision calculation for the bezier integral has been a hot problem of research.
Disclosure of Invention
In order to solve the technical problems, the application provides a method and a system for adaptive segmentation of a Bezier integral in rapid calculation of an integrated circuit, so that high-precision calculation of the Bezier integral becomes possible. The method comprises the steps of dividing the whole Bessel integral into integrals of a plurality of subintervals by using a zero point of a Bessel function, then performing self-adaptive segmentation again in each subinterval, determining the most appropriate number of self-adaptive segmentation by using relative errors before and after segmentation as a basic criterion, reducing unnecessary redundant calculation as far as possible, and ensuring that the integral of the subinterval between every two zero points is accurate.
In a first aspect, the present application provides a method for adaptive segmentation of a bezier integral in fast computation of an integrated circuit, including the following steps:
step S1: calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;
step S2: according to the order of one Bezier function, calculating the zero point of the Bezier function by adopting an iteration method; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
step S3: performing self-adaptive integration on an integrand formed by the product of the integral kernel function and the Bessel function, and setting an integral subinterval asm=1；
Step S4: according to the accumulated function formed by the product of the integral kernel function and the Bessel function and the zero point, the first self-adaptive Bessel integralmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
step S5: judging the second of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold, at this time, the Bezier integral of the whole integral subinterval is the final integral result of one Bezier integral of the field dyadic Green' S function of the field generated by the source point of the current integrated circuit at the field point calculated by the self-adaptive segmentation method, and the step is turned to S6; otherwisem=m+1, go to step S4;
step S6: and calculating other Bessel integrals in the dyadic Green function according to the same method of the steps S2-S5, calculating the dyadic Green function of fields generated by any point current source with discrete continuous current on the copper-clad area of any layer of the integrated circuit on other layers based on the Bessel integrals, and further calculating the fields generated by any continuous current on the copper-clad area of any layer of the integrated circuit on other layers.
The electromagnetic parameters comprise conductivity, magnetic permeability and dielectric constant.
The adaptation is tomThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integration result after the sub-interval division comprises the following steps:
step S4.1: setting the number j of segmentation, wherein j is equal to zero before segmentation;
step S4.2: setting j = j +1, and calculating a division point when the division frequency is j
Including a start point and an end point;
step S4.3: using said division point
Will be firstmThe integral of the subinterval is divided into the accumulation of j +1 subinterval integrals of the next level;
step S4.4: integration for each next level subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
；
Step S4.5: when the number of times of division is jmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
；
Step S4.6: if it is satisfied with
If the value is less than the second threshold value, the segmentation is finished to obtain the secondmIntegration results after sub-interval division, i.e.
Otherwise, go to step S4.2.
The division point when the number of division times is j is calculated
The formula is as follows:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
Said using said segmentation points
Will be firstmThe integral of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level, and the formula is as follows:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
The integral value of each next-stage subinterval is calculated by adopting a Gaussian integration method, and the formula is as follows:
wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The time value g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
In a second aspect, the present application provides an adaptive segmentation system for a bezier integral in an integrated circuit fast computation, including: the device comprises an order determining module, a zero point determining module and an integral subinterval setting module; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module;
the order determining module, the zero point determining module and the integral subinterval setting module are arranged on the base; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module, wherein the subinterval segmentation module, the subinterval judgment module and the field calculation module are sequentially connected, and the subinterval judgment module is connected with the subinterval segmentation module;
the order determining module is used for calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;
the zero point determining module is used for calculating the zero point of the Bezier function by adopting an iteration method according to the order of one Bezier function; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
the integral subinterval setting module is used for carrying out self-adaptive integration on an integrand formed by the product of the integral kernel function and the Bessel function, and setting the integral subinterval asm=1；
The subinterval division module is used for adaptively carrying out the first Bessel integral according to an integrand function formed by the product of the integral kernel function and the Bessel function and the zero pointmThe subintervals are divided, Bessel integration is carried out on the subintervals of the next level after the mth subinterval is divided, and the subintervals are accumulated to obtain the first levelmAccumulating the integration result after the sub-interval division,accumulating the accumulated integration result to Bessel integration in the whole interval;
the subinterval judging module is used for judging the second part of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold value, and at the moment, the Bessel integral of the whole integral subinterval is the final integral result of one Bessel integral of a field dyadic Green function of a source point of the current integrated circuit generated at a field point and calculated by a self-adaptive segmentation method, and the final integral result is transferred into the field calculation module; otherwisem=m+1, switching to the subinterval division module;
the field calculation module is used for calculating other Bessel integrals in the dyadic Green function by adopting the same method, calculating the dyadic Green function of fields generated by any point current source with discrete continuous current on the copper-clad region of any layer of the integrated circuit on other layers based on the Bessel integrals, and further calculating the fields generated by any continuous current on the copper-clad region of any layer of the integrated circuit on other layers.
The subinterval segmentation module includes: the device comprises a setting unit, a division point calculating unit, a next-stage sub-integration unit, an integration accumulating unit and a threshold value comparing unit;
the setting unit, the division point calculating unit, the next-stage sub-integration unit, the integration accumulation unit and the threshold value comparing unit are sequentially connected, and the threshold value comparing unit is connected with the division point calculating unit;
the setting unit is used for setting the dividing times j, and j is equal to zero before division;
the division point calculation unit is used for setting j = j +1 and calculating the division point when the division frequency is j
Including a start point and an end point;
the next level sub-integration unit is used for utilizing the division point
Will be firstmThe integral of the subinterval is divided into the accumulation of j +1 subinterval integrals of the next level;
the integral accumulation unit is used for integrating for each next-stage subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
；
The threshold comparison unit is used for comparing j times of divisionmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
If it is satisfied
If the value is less than the second threshold value, the segmentation is finished to obtain the secondmIntegration results after sub-interval division, i.e.
And otherwise, switching to the division point calculation unit.
The division point calculation unit calculates the division point when the division frequency is j
The formula is as follows:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
The next stage of sub-integration unit is used forUsing said division point
Will be firstmThe integral of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level, and the formula is as follows:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
The integral accumulation unit is used for integrating for each next-stage subinterval
And calculating the integral value of each next-stage subinterval by adopting a Gaussian integration method, wherein the formula is as follows:
wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The time value g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
The beneficial technical effects are as follows:
the application provides a Bessel integral self-adaptive segmentation method and system in integrated circuit fast calculation, unnecessary redundant calculation in the Bessel integral self-adaptive segmentation method is reduced as far as possible, integration of subintervals between every two zero points is guaranteed to be accurate, and the integration of the subintervals between all the two zero points is accumulated on the basis of calculation until predefined precision is met.
Drawings
FIG. 1 is a flowchart of an adaptive segmentation method for Bessel integration in rapid computation of an integrated circuit according to an embodiment of the present application;
FIG. 2 is a flow chart illustrating the determination of the zero point distribution of the Bezier function according to the embodiment of the present application;
FIG. 3 is a flowchart of an embodiment of the present applicationmAn integration flow chart after the sub-interval division;
FIG. 4 is a schematic diagram illustrating a principle of a division point according to an embodiment of the present application;
fig. 5 is a schematic block diagram of a bezier integral adaptive segmentation system in an integrated circuit fast computation according to an embodiment of the present application.
The specific implementation mode is as follows:
the present application is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present application is not limited thereby.
The fields generated by the point current sources at any position in space can be calculated based on a green function (or Bessel integral), and the fields generated by the surface current sources at the same position can be calculated by utilizing a Gaussian integral method based on the linear superposition property of the fields generated by the sources, so that the fields generated by the current on the metal plate with the integrated circuit multilayer complex shape at different positions in space can be calculated. However, when the green function method is used to calculate the field generated by the point current source at any position in space, the core and difficulty lies in calculating the bezier integral introduced by the green function, because the bezier function has the characteristics of high oscillation, slow attenuation and the like, so that high-precision calculation for the bezier integral has been a hot problem of research. In the current stage of research, the following method is commonly used for segmentation, and a Bessel function self-adaptive segmentation integration method is provided, wherein the current accumulated integral sum is compared with the integral sum obtained by the previous accumulation, if the error between the current accumulated integral sum and the integral sum obtained by the previous accumulation is smaller than the preset error precision, the current integral sum is taken as an integration result, otherwise, a new segmentation point is additionally arranged at the middle position of every two adjacent current segmentation points, and the new segmentation point is added into the current segmentation point. And when the error is larger than the set error precision, segmenting between zero points according to the following formula to obtain multiplied point distribution, and repeatedly calculating until the precision meets the requirement.
Where j =0 the number of previous subdivisions, j +1=1 the new number of subdivisions, and new1, new2, and the like are sequence numbers before new segment points are not renumbered. The method can obtain higher calculation accuracy, but has the problems that a lot of redundant calculation exists, if the accuracy requirement cannot be met every time, new segmentation points need to be added to all segments, when the accuracy of segmentation integration among some zero points meets the requirement, segmentation and recalculation are still uniformly performed, and the segmentation and recalculation are not necessary, so that the method is mainly researched aiming at the problem of the redundant calculation, and the unnecessary redundant calculation is reduced as much as possible. Therefore, the application provides a Bezier integral self-adaptive segmentation method and system in the rapid calculation of an integrated circuit, so that the high-precision calculation aiming at the Bezier integral becomes possible. The method comprises the steps of dividing the whole Bessel integral into integrals of a plurality of subintervals by using a zero point of a Bessel function, then performing self-adaptive segmentation again in each subinterval, determining the most appropriate number of self-adaptive segmentation by using relative errors before and after segmentation as a basic criterion, reducing unnecessary redundant calculation as far as possible, and ensuring that the integral of the subinterval between every two zero points is accurate. According to the method, the integral sum obtained by current accumulation is not compared with the integral sum obtained by previous accumulation, whether the current accumulation is divided is determined, whether the integral value of the subinterval between every two zero points meets the precision requirement every time, whether the current accumulation is divided further is determined, whether the integral value of the subinterval between every two zero points is accurate is ensured, and the integral of the subinterval between all the two zero points is calculated and accumulated on the basis until the predefined precision is met.
In a first aspect, the present application provides a method for adaptive segmentation of a bezier integral in fast computation of an integrated circuit, as shown in fig. 1, including the following steps:
step S1: calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing Bessel function
For the Bessel function integral, it generally has the following form:
g (r) is a Green function to be integrated, r is a space distance acted by the Green function and is a distance between a source point and a field point of the currently calculated integrated circuit; g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the point current source point on the copper-clad area of the integrated circuit to the field point of other layers,
and calculating a Bessel function used in a dyadic Green function of a current source at any point on a copper-clad region of any layer of the integrated circuit in the field generated by other layers for the v order, wherein v is the order of the Bessel function, and lambda is an integral variable.
Specifically, the whole integrated circuit is sharednLayers, each layer numbered as
The source is atjLayers of electromagnetic parameters of each layer
Thickness of layer
Is located atx T,y T,z T) Is located atx,y,z) The field formed by the field points can be represented by the following Green's function：
Wherein, the nine elements of the dyadic Green function are respectively
The calculation sequence does not affect the final calculation result.
Nine elements of the dyadic Green function, which contains six Bessel integrals
The specific acquisition process refers to patent CN 112989750B.
Wherein the content of the first and second substances,
iis the unit of an imaginary number,i 2=-1；J 0representing a Bessel function of order 0;J 1representing a Bessel function of order 1;
is an integral variable;
expressed as a function of the Bessel integral coefficient,
；x, y, zthe coordinates of the field points are represented,
representing source point coordinates; angular frequency
Represents a frequency;lindicating that the site is at the secondlLayer (A)l=0,1,2,…,n) The number of the layers is 0 to 0n，
Is as followslA z-coordinate of a layer interface;
respectively represent
The number of complex waves in the horizontal and vertical directions of the layer;
respectively representlA layer horizontal dielectric constant, a vertical dielectric constant;
respectively representlHorizontal magnetic conductivity and vertical magnetic conductivity of the layer;
is shown aslThe anisotropy coefficient of the layer;
respectively represent
Integral coefficients of complex wave numbers of the horizontal and vertical layers;A l , B l , C l , D l , E l , F l respectively represent
The undetermined coefficient of a layer,A l , B l the following linear equation is solved:
T1is 2n×2nOf a complex matrix, X1, S1Is of length 2nA complex vector of (a);
C l , D l the following linear equation is solved:
T2is 2n×2nOf a complex matrix, X2, S2Is of length 2nA complex vector of (a);
E l , F l the following linear equation is solved:
T3is 2n×2nOf a complex matrix, X3, S3Is of length 2nA complex vector of (a);
the above e represents the base of the natural logarithm; as described aboveμ 0Represents the permeability of a vacuum; as described aboveμ 1Denotes the permeability of layer 1; as described aboveμ l Is shown aslThe permeability of the layer; as described aboveμ l-1To representl-1 magnetic permeability of the layer; as described aboveμ n Is shown asnThe permeability of the layer; as described aboveμ n-1To representn-1 magnetic permeability of the layer; in the above-mentioned formula, the compound of formula,t 1=z 1-z 0is the thickness of the layer 1 integrated circuit board,t l-1=z l-1-z l-2is as followsl-1 layer of integrated circuit board thickness,t l =z l -z l-1is as followslThe thickness of the layer of integrated circuit board,t n-1=z n-1-z n-2is as followsn-1 layer integrated circuit board thickness.
In the above expression, if R is calculated1~R6Then nine components of the whole dyadic Green function can be calculated
Etc. to calculate R1~R6The key to (1) calculating the infinite integral of a Bezier function of an integrand containing integrated circuit information, wherein the integrated circuit information comprises: electromagnetic parameters of materials of each layer of the integrated circuit, thickness of each layer, working frequency of the integrated circuit, and distance from a point current source on a copper-clad area of the integrated circuit to a field point of other layers. For example, R1The method comprises the following steps:
the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a point current source on a copper-coated region of the integrated circuit to a field point of other layers acting on the integrated circuit, and the integrated function has different convergence speeds corresponding to source points and field points of different integrated circuits, so that zero point ranges or the number of the Bessel functions are different when Bessel zero points corresponding to the actions of different integrated circuit source points on the field points are calculated.
Infinite integral R by Bessel function1~R6It can be seen that the order of the Bessel function is clearly expressed in the expression, wherein R1, R4, R5Requires the calculation of a Bessel integral, R, of order 12, R3, R6A bezier integral of order 0 needs to be calculated.
Step S2: according to the order of one Bezier function, calculating the zero point of the Bezier function by adopting an iteration method; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
the zero point of the Bezier function is calculated, namely the zero point a of the Bezier function is determined to be the solution of the Bezier function value being 0, namely
The solution of (c) is calculated by the following Halley algorithm, as shown in FIG. 2:
step S1.1, setting p = 1;
step S1.2, setting the initial guess value of the p-th zero point
；
Step S1.3, calculating by the following iterative formula
Nearby Bessel function
P th zero point of (c):
and (3) iteration termination conditions:
wherein
Is a predefined threshold. Wherein the content of the first and second substances,
representing Bessel functions
The first derivative of (a) is,
representing Bessel functions
The second derivative of (a); q denotes the qth iteration, and the value at q =0 is the initial guess value.
Step S1.4, if the calculated zero reaches the specified interval range, the zero calculation is completed; otherwise, let p = p +1 and go to step S1.2.
According to the zero point of the Bessel function, the sectional point forming the integral interval obtained based on the zero point of the Bessel function can be determined as
Wherein r is the space distance acted by the green function, and the integral interval is equivalent to the value range of the integral variable.
Step S3: for the integral kernelThe product of the function and the Bessel function forms an integrand, and the integrand subinterval is set asm=1；
Step S4: according to the accumulated function formed by the product of the integral kernel function and the Bessel function and the zero point, the first self-adaptive Bessel integralmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
the application is for the second threshold not being satisfiedmThe sub-intervals are divided, not all the intervals are divided, but only the sub-intervals which do not meet the first threshold value are further divided, and the original integral value is retained in other sub-intervals which meet the first threshold value. This reduces unnecessary redundant calculations.
In addition, the present application is directed to the second threshold not being satisfiedmA sub-section divided in a manner not in the second divisionmOne more point is added in the middle of the sub-interval as a subdivision point, but in the second placemAdding a plurality of points in the subinterval, i.e. for the second not satisfying the first thresholdmThe subinterval is mainly divided so that the subinterval meets the first threshold as soon as possible, and the dividing process and the formula refer to the steps S4.1-S4.6.
Step S5.1: judging the second of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not;
step S5.2: if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold, at this time, the Bezier integral of the whole integral subinterval is the final integral result of one Bezier integral of the parallel vector Green' S function of the field generated by the source point of the current integrated circuit at the field point calculated by the self-adaptive segmentation method, and the step is turned to step S6;
step S5.3: otherwise (i.e. if the Bessel integral is the secondmThe accumulated integral result after the sub-interval division is more than or equal to a first threshold value)m=m+1, the flow proceeds to step S4；
The step ismThe sub-interval integrals are compared to a first threshold value, and not the accumulated values of all sub-intervals.
Adaptive to the firstmThe subintervals are divided, and the integral of the Bessel function after the mth subinterval is divided is calculated to obtain the integral of the Mth subintervalmThe accumulated integration result after the sub-interval division, as shown in fig. 3, includes the following steps:
step S4.1: setting the number j of segmentation, wherein j is equal to zero before segmentation;
step S4.2: setting j = j +1, and calculating a division point when the number of division is j
The method comprises the following steps of:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
Step S4.3: using said division point
Will be firstmThe integration of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe current source at any point on the copper-clad area of any layer of the order calculation integrated circuit is produced on other layersThe bessel function used in the dyadic green function of the field,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
Step S4.4: integration for each next level subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
：
Wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The time value g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
Step S4.5: when the number of times of division is jmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
；
Step S4.6: if it is satisfied with
If the value is less than the second threshold value, the segmentation is finished to obtain the secondmIntegration results after sub-interval division, i.e.
Otherwise, go to step S4.2.
Taking fig. 4 as an example, the implementation steps of the present application are described in detail, as described in fig. 4,
the superscript of (d) indicates the number j of divisions of the integration interval;
determining the distribution of zeros of the Bessel function, assuming 5 zeros, i.e.
J =0 at this time, the whole integral interval of the Bezier function is directly divided into a plurality of subintervals according to the zero point distribution of the Bezier function;
setting the integral subinterval asm=1；
Adaptive to the firstmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
a pair of subintervals
Set j =1, i.e. divide for the first time, the subintervals become
And
decomposing the Bessel integral of the 1 st sub-interval into integral accumulation of 2 sub-intervals
Bessel integral and subinterval of
And
when the difference between the Bessel integral accumulation values is not less than the second threshold value, the second threshold value is set to be
And
is divided again, i.e. j =2 is set, in which case the sub-interval is divided again
Decomposed into 3 sub-intervals
And
at this time, the subinterval
Bessel integral and subinterval of
And
the difference between the Bezier integrals is less than a second threshold value, and the subinterval
No decomposition occurs.
At this time, the accumulated integration result is accumulated to Bessel integration over the entire interval except for the firstmSub-interval (at this time)m= 1), the other subintervals are not divided, and the second subinterval is determinedmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if said first ismThe accumulated integral result after the sub-interval division is smaller than a first threshold value, the calculation is finished, the Bessel integral of the whole interval is the final integral result of the self-adaptive segmentation method, and otherwise, the Bessel integral result is not smaller than the first threshold valuem=m+1, continue dividingm+1 subintervals, i.e.mSubintervals of = 2. Partition in the same waym+2、m+3 subintervals:
in the same way, sub-intervals
Decomposed into 3 sub-intervals
Sub-interval
Decomposed into 3 sub-intervals
No decomposition is required after that, and for the sub-intervals
Set j =1, split it into 2 subintervals and
then, finding the subintervals
Bessel integral and subinterval of
And
the difference between the Bessel integrals has been smaller than the second threshold, so the subinterval no longer needs to continue the decomposition.
At this time, the accumulated integration result is accumulated to the Bessel integration of the whole interval, at this timemThe subinterval of =1 has been divided 2 times,mthe subinterval of =2 is divided 2 times, the m =3 subinterval is also divided 2 times,mthe number of times of division is 1 for the sub-interval of =4, and the second time is judgedmIf the accumulated integral result after dividing the 4 subintervals is less than a first threshold value; if said first ismAnd if the accumulated integral result obtained after the division of the =4 subintervals is smaller than a first threshold value, the calculation is finished, and at the moment, the Bessel integral of the whole integral subinterval is the final integral result of the adaptive segmentation method.
Step S6: calculating other Bessel integrals in the dyadic Green function in the same way as steps S2-S5 (for example, the Bessel integral R is calculated for the first time1Then all other R's need to be calculated2、R3、R4、R5、R6Bessel integral) based on the bessel integral computation setThe parallel vector Green function of the field generated by any continuous current discrete point current source on the copper-clad region of any layer of the circuit on other layers is calculated, namely the element of the parallel vector Green function is calculated
And further calculating the field generated by any continuous current on the copper-clad area of any layer of the integrated circuit on other layers. On the premise of knowing the dyadic Green function of fields generated by any continuous current on the copper-clad region of any layer of the integrated circuit on other layers, the method calculates the fields generated by any continuous current on the copper-clad region of any layer of the integrated circuit on other layers, belongs to the prior art, refers to the patent CN112989750B in the detailed process, and is not repeated in the application.
In a second aspect, the present application provides an adaptive segmentation method for a bezier integral in an integrated circuit fast computation, as shown in fig. 5, including: the device comprises an order determining module, a zero point determining module and an integral subinterval setting module; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module;
the order determining module, the zero point determining module and the integral subinterval setting module are arranged on the base; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module, wherein the subinterval segmentation module, the subinterval judgment module and the field calculation module are sequentially connected, and the subinterval judgment module is connected with the subinterval segmentation module;
the order determining module is used for calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;
the zero point determining module is used for calculating the zero point of the Bezier function by adopting an iteration method according to the order of one Bezier function; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
the integral subinterval setting module is used for carrying out self-adaptive integration on an integrand formed by the product of the integral kernel function and the Bessel function, and setting the integral subinterval asm=1；
The subinterval division module is used for adaptively carrying out the first Bessel integral according to an integrand function formed by the product of the integral kernel function and the Bessel function and the zero pointmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
the subinterval judging module is used for judging the second part of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold value, and at the moment, the Bessel integral of the whole integral subinterval is the final integral result of one Bessel integral of a field dyadic Green function of a source point of the current integrated circuit generated at a field point and calculated by a self-adaptive segmentation method, and the final integral result is transferred into the field calculation module; otherwisem=m+1, switching to the subinterval division module;
the field calculation module is used for calculating other Bessel integrals in the dyadic Green function by adopting the same method, calculating the dyadic Green function of fields generated by any point current source with discrete continuous current on the copper-clad region of any layer of the integrated circuit on other layers based on the Bessel integrals, and further calculating the fields generated by any continuous current on the copper-clad region of any layer of the integrated circuit on other layers.
The subinterval segmentation module includes: the device comprises a setting unit, a division point calculating unit, a next-stage sub-integration unit, an integration accumulating unit and a threshold value comparing unit;
the setting unit, the division point calculating unit, the next-stage sub-integration unit, the integration accumulation unit and the threshold value comparing unit are sequentially connected, and the threshold value comparing unit is connected with the division point calculating unit;
the setting unit is used for setting the dividing times j, and j is equal to zero before division;
the division point calculation unit is used for setting j = j +1 and calculating the division point when the division frequency is j
Including a start point and an end point;
the next level sub-integration unit is used for utilizing the division point
Will be firstmThe integral of the subinterval is divided into the accumulation of j +1 subinterval integrals of the next level;
the integral accumulation unit is used for integrating for each next-stage subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
；
The threshold comparison unit is used for comparing j times of divisionmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
If it is satisfied
If the value is less than the second threshold value, the segmentation is finished to obtain the secondmIntegration results after sub-interval division, i.e.
And otherwise, switching to the division point calculation unit.
The division point calculation unit calculates the division point when the division frequency is j
The formula is as follows:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
The next level sub-integration unit is used for utilizing the division point
Will be firstmThe integral of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level, and the formula is as follows:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
The integral accumulation unit is used for integrating for each next-stage subinterval
And calculating the integral value of each next-stage subinterval by adopting a Gaussian integration method, wherein the formula is as follows:
wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The value of time g (lambda) is the electromagnetic parameters of the source and field points of the integrated circuit and the materials of the layers of the integrated circuit, the thickness of each layer, and the integrated circuit according to the current calculationThe working frequency of the integrated circuit and the distance from a source point of the integrated circuit to a field point determine an integral kernel function, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
The present applicant has described and illustrated embodiments of the present invention in detail with reference to the accompanying drawings, but it should be understood by those skilled in the art that the above embodiments are merely preferred embodiments of the present invention, and the detailed description is only for the purpose of helping the reader to better understand the spirit of the present invention, and not for limiting the scope of the present invention, and on the contrary, any improvement or modification made based on the spirit of the present invention should fall within the scope of the present invention.
Claims (10)
1. A Bessel integral self-adaptive segmentation method in integrated circuit fast calculation is characterized by comprising the following steps:
step S1: calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;
step S2: according to the order of one Bezier function, calculating the zero point of the Bezier function by adopting an iteration method; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
step S3: performing self-adaptive integration on an integrand formed by the product of the integral kernel function and the Bessel function, and setting an integral subinterval asm=1；
Step S4: according to the accumulated function formed by the product of the integral kernel function and the Bessel function and the zero point, the first self-adaptive Bessel integralmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
step S5: judging the second of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold, at this time, the Bezier integral of the whole integral subinterval is the final integral result of one Bezier integral of the field dyadic Green' S function of the field generated by the source point of the current integrated circuit at the field point calculated by the self-adaptive segmentation method, and the step is turned to S6; otherwisem=m+1, go to step S4;
step S6: and calculating other Bessel integrals in the dyadic Green function according to the same method of the steps S2-S5, calculating the dyadic Green function of fields generated by any point current source with discrete continuous current on the copper-clad area of any layer of the integrated circuit on other layers based on the Bessel integrals, and further calculating the fields generated by any continuous current on the copper-clad area of any layer of the integrated circuit on other layers.
2. The method of adaptive segmentation of Bessel's integrals in integrated circuit fast computation of claim 1, characterized in that the adaptation is to the secondmThe sub-interval is divided intomCarrying out Bessel integration and accumulation on the next-stage subintervals after subinterval division to obtain the first-stage subintervalsmAccumulating the integration result after the sub-interval division comprises the following steps:
step S4.1: setting the number j of segmentation, wherein j is equal to zero before segmentation;
step S4.2: setting j = j +1, and calculating a division point when the division frequency is j
Including a start point and an end point;
step S4.3: using said division point
Will be firstmThe integral of the subinterval is divided into the accumulation of j +1 subinterval integrals of the next level;
step S4.4: integration for each next level subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
；
Step S4.5: when the number of times of division is jmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
；
Step S4.6: if it is satisfied with
If the value is less than the second threshold value, the segmentation is finished to obtain the secondmIntegration results after sub-interval division, i.e.
Otherwise, go to step S4.2.
3. The adaptive segmentation method for Bessel's integral in integrated circuit fast calculation as claimed in claim 2, wherein the segmentation point when the number of segmentation times is j is calculated
The formula is as follows:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
4. The method of adaptive segmentation of Bessel's integrals in Integrated Circuit Rapid computations of claim 2, characterized in that said utilizing said segmentation points
Will be firstmThe integral of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level, and the formula is as follows:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
5. The adaptive segmentation method for Bessel integration in rapid calculation of integrated circuits as claimed in claim 4, wherein the Gaussian integration method is used to calculate the integral value of each subinterval of the next stage, and the formula is as follows:
wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The value of time, g (lambda), is based on the currently calculated electromagnetic parameters of the source and field points of the integrated circuit and the material of the layers of the integrated circuitThe thickness of each layer, the working frequency of the integrated circuit and the integral kernel function determined by the distance from the source point to the field point of the integrated circuit, wherein lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
6. An adaptive segmentation system for Bessel integration in rapid calculation of an integrated circuit, comprising: the device comprises an order determining module, a zero point determining module and an integral subinterval setting module; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module;
the order determining module, the zero point determining module and the integral subinterval setting module are arranged on the base; the device comprises a subinterval segmentation module, a subinterval judgment module and a field calculation module, wherein the subinterval segmentation module, the subinterval judgment module and the field calculation module are sequentially connected, and the subinterval judgment module is connected with the subinterval segmentation module;
the order determining module is used for calculating a dyadic Green function of fields generated by any continuous current discrete point current source on a copper-clad region of any layer of the integrated circuit on other layers, acquiring all Bezier integrals contained in the dyadic Green function, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;
the zero point determining module is used for calculating the zero point of the Bezier function by adopting an iteration method according to the order of one Bezier function; determining the range or the number of the zero points according to the convergence speed of an integrant kernel function and an integrant function formed by the product of a Bessel function, wherein the integrant kernel function is determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from a source point to a field point, and the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad region of the integrated circuit to the field point acting on other layers;
the integral subinterval setting module is used for carrying out self-adaptive integration on an integrand formed by the product of the integral kernel function and the Bessel function, and setting the integral subinterval asm=1；
The subinterval division module is used for adaptively carrying out the first Bessel integral according to an integrand function formed by the product of the integral kernel function and the Bessel function and the zero pointmThe subintervals are divided, Bessel integration is carried out on the subintervals of the next level after the mth subinterval is divided, and the subintervals are accumulated to obtain the first levelmAccumulating the integral results after the sub-intervals are divided, and accumulating the accumulated integral results to Bessel integrals of the whole intervals;
the subinterval judging module is used for judging the second part of the Bessel integralmWhether the accumulated integral result after the sub-interval division is smaller than a first threshold value or not; if the Bessel integral is the firstmThe accumulated integral result after the subinterval division is smaller than a first threshold value, and at the moment, the Bessel integral of the whole integral subinterval is the final integral result of one Bessel integral of a field dyadic Green function generated by a source point of the current integrated circuit at a field point and calculated by a self-adaptive segmentation method, and the final integral result is transferred into the field calculation module; otherwisem=m+1, switching to the subinterval division module;
the field calculation module is used for calculating other Bessel integrals in the dyadic Green function by adopting the same method, calculating the dyadic Green function of fields generated by any point current source with discrete continuous current on the copper-clad region of any layer of the integrated circuit on other layers based on the Bessel integrals, and further calculating the fields generated by any continuous current on the copper-clad region of any layer of the integrated circuit on other layers.
7. The system for adaptive segmentation of Bessel's integrals in Integrated Circuit fast computation of claim 6, characterized in that the subinterval segmentation module includes: the device comprises a setting unit, a division point calculating unit, a next-stage sub-integration unit, an integration accumulating unit and a threshold value comparing unit;
the setting unit, the division point calculating unit, the next-stage sub-integration unit, the integration accumulation unit and the threshold value comparing unit are sequentially connected, and the threshold value comparing unit is connected with the division point calculating unit;
the setting unit is used for setting the dividing times j, and j is equal to zero before division;
the division point calculation unit is used for setting j = j +1 and calculating the division point when the division frequency is j
Including a start point and an end point;
the next level sub-integration unit is used for utilizing the division point
Will be firstmThe integral of the subinterval is divided into the accumulation of j +1 subinterval integrals of the next level;
the integral accumulation unit is used for integrating for each next-stage subinterval
Calculating the integral value of each next stage subinterval by adopting a Gaussian integration method and accumulating the integral value as the first stagemIntegration of sub-intervals
；
The threshold comparison unit is used for comparing j times of divisionmIntegration of sub-intervals
The previous division timemIntegration of sub-intervals
If it is satisfied
Is less than that ofTwo thresholds, then the division is finished to obtain the secondmIntegration results after sub-interval division, i.e.
And otherwise, switching to the division point calculation unit.
8. The Bessel integral adaptive segmentation system in integrated circuit fast calculation as claimed in claim 7, wherein the segmentation point calculation unit calculates the segmentation point when the number of segmentation is j
The formula is as follows:
wherein the content of the first and second substances,dis as followsdThe next level of subintervals.
9. The system for adaptive segmentation of Bessel's integrals in Integrated Circuit Rapid computations of claim 7, wherein the next stage sub-integration unit is used to utilize the segmentation point
Will be firstmThe integral of the subintervals is divided into the accumulation of j +1 subinterval integrals of the next level, and the formula is as follows:
wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vfor the order of the Bessel function,
is the number of divisions jmSub-interval ofdAnd r is the space distance between the source point and the field point of the currently calculated integrated circuit.
10. The integrated circuit fast compute Bessel integration adaptive segmentation system of claim 7, wherein the integration accumulation unit is configured to integrate for each next stage subinterval
And calculating the integral value of each next-stage subinterval by adopting a Gaussian integration method, wherein the formula is as follows:
wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval
Conversion to standard Gaussian integration interval [ -1,1 [ -1]The conversion of the jacobian of (a),
an inverse transform of D;
is the k-th gaussian point and,
is the weight corresponding to the kth gaussian point;
for the function g (λ) taking the value at λ
The time value g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable;
is composed ofvThe order calculates Bessel function used by current source at any point on copper-clad region of any layer of integrated circuit in dyadic Green function of field generated by other layer,vis the order of the Bessel function.
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| CN114781307B (en) * | 2022-06-17 | 2022-08-23 | 北京智芯仿真科技有限公司 | Non-uniform sampling method and device for integrated circuit Hankel transform filter |
| CN115081390A (en) * | 2022-06-28 | 2022-09-20 | 北京智芯仿真科技有限公司 | Non-uniform sampling optimization method and device for integrated circuit Hankel transform filter |
| CN115081390B (en) * | 2022-06-28 | 2022-12-13 | 北京智芯仿真科技有限公司 | Non-uniform sampling optimization method and device for integrated circuit Hankel transform filter |
| CN115442188A (en) * | 2022-08-19 | 2022-12-06 | 哲库科技(北京)有限公司 | Channel estimation method, device, equipment and storage medium |
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