CN108763777A - VLSI global wiring method for establishing model based on Poisson's equation explicit solution - Google Patents
VLSI global wiring method for establishing model based on Poisson's equation explicit solution Download PDFInfo
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Abstract
The present invention relates to the VLSI global wiring method for establishing model based on Poisson's equation explicit solution, and circuit is expressed as hypergraph model；It is two-dimensional electrostatic system by VLSI circuit layout modelings, converts constraint to the constraint of total potential energy N (v)=0 of electrostatic system；Partial differential equations are established based on Poisson's equation, boundary condition and compatibility condition；The analytic expression of density function is established, and substitutes into partial differential equations；The expression formula of potential and electric field is determined according to density function；Determine the convergence of potential and electric field expression formula；According to part and obtain the solution expression formula of potential and electric field；The potential and electric field value of each grid are obtained by quick calculation method, weighting obtains the potential and electric field of module, and VLSI circuit layout model foundations are completed under electric field force effect.The present invention can provide the global wiring of highly effective as a result, especially to large-scale example in fact, can meet the needs of current VLSI global wirings stage.
Description
Technical field
The present invention relates to super large-scale integration (VLSI) physical design automation technical fields, are based particularly on Poisson
The VLSI global wiring method for establishing model of equation explicit solution.
Background technology
As technology development enters the deep nanometer era that 1,000,000,000 transistors integrate, the performance of layout tool is in eda tool
It occupies an leading position in overall quality.Therefore, in the proximal segment time, many layout devices are developed.Main placement algorithm
There are three types of：Method based on simulated annealing, the method based on division and the method based on analysis.It has recently been demonstrated that analysis
The layout device of type can be normally reached more preferably placement quality and be with good expansibility.
In the layout based on analysis, a crucial technology is to reduce the overlapping of intermodule, obtains evenly dispersed cloth
Office.For the layout based on analysis, many documents propose the method for reducing overlapping, such as division, unit movement, frequency control
System, distribution, bell density domination, Helmholtz's density domination and Poisson density control.In these methods, Poisson density controls
By the layout device of some mainstreams, as FDP, Kraftwerk, mFAR and ePlace are used.
In global wiring, ePlace uses Poisson's equation, and based on ISPD2005 and ISPD2006 test cases
In all theoretical type layout devices of son, optimal line length is realized.Each module converter is first a positive charge by ePlace,
Block density is converted to energy of position constraint to handle.The potential and electric field generated followed by given charge density is built
Mould establishes Poisson's equation, and according to setting Nuo Yiman boundary conditions the characteristics of layout and compatibility condition.Poisson's equation is one
Partial differential equation (PDE) solve this partial differential equation by spectral method, and ePlace can quickly calculate a potential and electricity
Field distribution, therefore it can realize that line length minimizes while by module fast spread.
Poisson's equation is usually used in many fields, as electrostatics, computer science, mechanical engineering, Theoretical Physics, astronomy,
Chemistry etc..Have for example, Poisson's equation can utilize the Hermite basic function of bicubic element to establish one in rectangular domain
The system of finite element.Particularly, Poisson's equation is often used in description Distribution of Potential Field caused by given charge or mass density.
The solution of Poisson's equation is divided into two classes：Analytic solution and numerical solution.Analytic solutions are a kind of accurately solutions, while
It is the explicit solution of partial differential equation.And numerical solution is obtained by some numerical methods, such as FInite Element, numerical radius
Method, interpolation method etc..Because numerical solution can only approximation PDE solution, numerical solution inevitably result from some numerical value mistake
Difference.By taking the half-space problem of characteristic strain as an example, people have done many work and have reduced numerical error to the greatest extent.In general, such as
Fruit can find an explicit solution of PDE, then it will inherently be better than numerical solution.
All it is to solve Poisson's equation using numerical solution in the layout device controlled based on Poisson density existing at present.
In Kraftwerk, the more grid solver DiMEPACK of geometry is used to solve Poisson's equation；In ePlace, it is utilized
Fast Fourier Transform (FFT) (FFT) calculates potential and field distribution.But the Poisson's equation of various boundary has different spies
Point, therefore its solution is also different.Generally, it is considered that the explicit solution for obtaining PDE is very challenging, even can not
Can.For example Lorentz force expression formula is suitable and accurate in finite element modelling, but be not particularly suited for calculating.
Invention content
The purpose of the present invention is to provide a kind of VLSI global wiring method for establishing model based on Poisson's equation explicit solution,
To overcome defect existing in the prior art.
To achieve the above object, the technical scheme is that：A kind of VLSI overall situation cloth based on Poisson's equation explicit solution
Office's method for establishing model, includes the following steps：
Step S1：Circuit is expressed as hypergraph model H={ V, E }；
Step S2：It is two-dimensional electrostatic system by VLSI circuit layout modelings, converts constraint to electrostatic system
Total potential energy N (v)=0 constraint；
Step S3：Partial differential equations are established based on Poisson's equation, boundary condition and compatibility condition；
Step S4：The analytic expression of density function is established, and substitutes into partial differential equations；
Step S5：The expression formula of potential and electric field is determined according to density function；
Step S6：Determine the convergence of potential and electric field expression formula；
Step S7：According to part and obtain the solution expression formula of potential and electric field；
Step S8：The potential and electric field value of each grid are obtained by quick calculation method, weighting obtain module potential and
Electric field completes VLSI circuit layout model foundations under electric field force effect.
Compared to the prior art, the invention has the advantages that：
(1) distribution for carrying out analog module using accurate density function, not will produce numerical error；
(2) direct solution Poisson's equation obtains an explicit solution on the basis of accurate density function, and proves understanding
Convergence；Due to using analytic solutions, not will produce numerical error when solving electrostatic system；
(3) this method not only realizes the optimization of line length, also ensures the speed of solution.In the layout device with two big mainstreams
In the comparison of ePlace and NTUplace3, using ISPD2005 and ISPD2006 test sets the result shows that, algorithm of the invention
It can allow line length smaller.Density domination is replaced with the quick solution for calculating Poisson's equation of the software, is embedded into NTUplace3
The experimental results showed that, algorithm can reduce by 11% line length, greatly improve result.
Description of the drawings
Fig. 1 is the flow chart of the VLSI global wiring method for establishing model based on Poisson's equation explicit solution in the present invention.
Specific implementation mode
Below in conjunction with the accompanying drawings, technical scheme of the present invention is specifically described.
A kind of VLSI global wiring method for establishing model based on Poisson's equation explicit solution of the present invention, as shown in Figure 1, including
Following steps：
(1) circuit is expressed as hypergraph H={ V, E }；
(2) location problem is modeled to two-dimensional electrostatic system, converts constraint to total potential energy N (v) of electrostatic system
=0 constraint；
(3) Poisson's equation is utilized, boundary condition and compatibility condition establish partial differential equations；
(4) analytic expression of density function is provided, and substitutes into partial differential equations；
(5) exact expression of potential and electric field is determined using density function；
(6) convergence of potential and electric field expression formula is proved；
(7) using part and obtain the solution expression formula of potential and electric field.
(8) potential and electric field value of each grid being obtained by quick calculation method, weighting obtains the potential and electric field of module,
Layout is completed under electric field force effect.
Further, in step (1), it is [0, W] × [0, H] to give layout areas, and the circuit layout problem of VLSI can
It is modeled to a hypergraph G (V, E), module is expressed as vertex set V={ v1,v2,...,vn, gauze is expressed as super side collection E={ e1,
e2,...,er, module viWidth and it is high be respectively wiAnd hi, center point coordinate is (xi,yi).Wherein i=1,2, n.
VLSI location problems are sought between the modules without the optimum position for determining each module on the basis of overlapping, and total line length
It is optimal：
Without overlapping (1) between min W (v) s.t. modules
Wherein, W (v) is total line length, is calculated by semi-perimeter line length (HPWL).
Further, in step s 2, location problem is modeled to two-dimensional electrostatic system.According to module in layout areas
Position, it may be determined that potential φ (x, y) and electric field ξ (x, y), wherein ξ (x, y)=(ξx,ξy)=- φ (x, y).Module i is seen
It is a positive charge i, area AiIt is expressed as quantity of electric charge qi.Use φi=φ (xi,yi) and ξi=ξ (xi,yi) indicate at charge i
Potential and electric field.Later, charge i is according to electric field force Fi=qiξiIt is moved.Therefore, potential energy of system may be defined asWherein Ni=qiφiIndicate the energy of position of charge i.Constraint is finally converted into the total of electrostatic system
The constraint of potential energy N (v)=0.
Further, in step (3), using Poisson's equation, boundary condition and compatibility condition establish partial differential equation
Group.Based on Gauss law, build the Poisson's equation of electrostatic system, meet meet layout boundary condition and compatibility condition it is same
When obtain partial differential equation：
Wherein, equation (2a) gives Poisson's equation, and wherein ρ (x, y) is density function；Equation (2b) is the boundaries Nuo Aiman
Condition, for avoiding module from running out of the boundary of layout, wherein R andBoundary and the outer-normal direction of layout areas R are indicated respectively；Side
Journey (2c) is compatibility condition, and equation group is made to have unique solution.
Further, in step (4), the density functions of module i in the x direction are defined as：
In order to vector (x, y)=(x1,y1,...xi,yi) distinguish, it usesIndicate continuous variable.It is also possible to
The density functions of module i in y-direction are defined asThen the density function of module i isAll modules exist
Gross density on layout areas is：
Wherein, n is number of modules.
Further, " analytic solutions " part is solved in Fig. 1, concrete mode is as follows：
In order to meet equation (2c), redefine
Using accurate density function, Poisson's equation, relevant boundary condition and compatibility condition (2a)-(2c) become：
Further, in step (5), by equation (6a) and (6b), can obtain their solution shaped like：
Wherein, au,pIndicate that the coefficient of each wave function, u and p indicate integer index.For design factor au,p, by formula (7)
Poisson's equation (6a) is substituted into, calculating is passed throughIt can obtain density functionAnother expression-form：
It is multiplied by simultaneously in the right and left of equation (8)And it integrates and obtains
The integral domain of equation (9) is R=(0, W) × (0, H).Therefore, on the right of equation (9), according to trigonometric function
Orthogonality, first item is only in μ=0, p=η, and Section 2 is only in η=0 and u=μ, and Section 3 is only in p=η and u=μ negated zero
Value.Therefore in μ >=1 and η=0, equation (9) is reduced to：
Therefore it can obtain：
It is also possible to obtain coefficient a0,ηAnd aμ,η, in order to meet equation (6c), a is enabled in equation (7)0,0=0, then may be used
To obtain：
In μ, η >=1, brings formula (5) into formula (12d) and obtain：
It can equally obtain：
With
It should be pointed out that in VLSI location problems, aU, pIt is to be come out by the integral and calculating of accurate density function (5),
Than in ePlace discrete calculation it is more acurrate.
Further, " potential gradient " part is solved in Fig. 1, concrete mode is as follows：
Known by Gauss law, electric fieldEqual to the negative gradient of potentialBy formula (7)Have
Further, in step (6), because in equation (9)It is infinite series, therefore proves
It is convergent.
Lemma 1.
Infinite seriesWithIt is convergent.
Theorem 1.
Infinite seriesIt is absolutely convergent.
It proves：
It notices：
For au,p, u, p >=1 is had by equation (13)：
For other two kinds of situation u=0, p >=1 and u >=1, p=0 have：
With
Therefore
By lemma 1 know three above infinite series be it is convergent, thereforeThere are a convergence upper bounds, soIt is absolutely convergent.
Further, it in step (7), according to theorem 1, only needs to calculate in actually calculatingPart and.This
Outside,It isNegative gradient, can equally usePart and come it is approximate.Because containing in equation (17)
u3Or p3, thereforeRestrain quickly, it is only necessary to K calculating section of iteration and can be obtained by one it is more accurate
Solution.
Therefore, layout areas can be divided into the grid of m × m same sizes, and each grid is denoted as：blj, wherein l
=0,1, m-1, j=0,1, m-1 indicate grid label, then grid bljDensity be just：
Module i is in grid bljIn area determined by the size of module, and with the central point distance and grid of module i
bljDistance be inversely proportional, this method be similar to ePlace in local smoothing and density zoom technology, so as to obtain
Following density function
WhereinIndicate grid bljDensity function.Enable (xl, yj) indicate grid bljCenter point coordinate, that
The density function that can be obtained by all grids for being similar to formula (5) is：
In a acquiredU, pOn the basis of by formula (18) substitute into formula (12b), (12c) and (12d), in u=0 and p>=1 feelings
It is obtained under condition：
Wherein, xlAnd yjIt is determined by the size and layout areas of grid, it will
Substitution formula (19) obtains：
Remaining coefficient can also acquire in the same way.All coefficient a acquiredu,pIt is as follows：
For each grid blj, the potential in (7) formulaIt can recalculate to obtain with following formula：
Known by theorem 1, infinite seriesIt is absolutely convergent, therefore can does such as lower aprons：
Electric field in formula (16)It can be approximately：
Further, in step (8), the density of each grid is calculated according to formula (18)In formula
In (21a)-(21b), ignore the summation of coefficient, the coefficient matrix a ' of m × m can be calculatedu,p：
Coefficient matrix can be calculated by library FFT of calling, then spends m2Time is coefficient matrix a 'u,pIt is updated to
au,p, calculate all coefficient au,p.After calculating all coefficients, it can be counted by the inverse transformation of Fast Fourier Transform (FFT)
Calculate φ (l, j) and ξ (l, j), electric field force Fi=qiξiModule i can be made to move, to complete to be laid out.
Further, " line length gradient " part is solved in Fig. 1, concrete mode is as follows：
In problem (1), W (v) is non-differentiability, and directly optimization is difficult, therefore carrys out approximate HPWL using LSE line lengths.The side x
Upward LSE line length functions are:
Wherein, γ is smoothing parameter, and the functional gradient is asked to can be obtained line length gradient.
Further, " module position optimizes and parameter update " part, concrete mode are as follows in Fig. 1：
In each iteration, unconstrained minimization problem is solved using Nesterov methods, by an iteration, obtained
One new explanation (xk+1, yk+1), which is the module position after optimizing, and then updates penalty parameter λ.
The above are preferred embodiments of the present invention, all any changes made according to the technical solution of the present invention, and generated function is made
When with range without departing from technical solution of the present invention, all belong to the scope of protection of the present invention.
Claims (8)
1. a kind of VLSI global wiring method for establishing model based on Poisson's equation explicit solution, is characterized in that, includes the following steps：
Step S1：Circuit is expressed as hypergraph model H={ V, E }；
Step S2：It is two-dimensional electrostatic system by VLSI circuit layout modelings, converts constraint to the total of electrostatic system
The constraint of potential energy N (v)=0；
Step S3：Partial differential equations are established based on Poisson's equation, boundary condition and compatibility condition；
Step S4：The analytic expression of density function is established, and substitutes into partial differential equations；
Step S5：The expression formula of potential and electric field is determined according to density function；
Step S6：Determine the convergence of potential and electric field expression formula；
Step S7：According to part and obtain the solution expression formula of potential and electric field；
Step S8：The potential and electric field value of each grid are obtained by quick calculation method, weighting obtains the potential and electric field of module,
VLSI circuit layout model foundations are completed under electric field force effect.
2. the VLSI global wiring method for establishing model according to claim 1 based on Poisson's equation explicit solution, feature
It is, in step sl, layout areas is [0, W] × [0, H], n module and r gauze is given, by VLSI circuit layout moulds
Type is expressed as vertex set V={ v as a hypergraph G (V, E), by module1,v2,...,vn, gauze is expressed as super side collection E=
{e1,e2,...,er, module viWidth and it is high be respectively wiAnd hi, center point coordinate is (xi,yi), i=1,2, n；
VLSI circuit layouts model is between the modules without the optimum position of determining each module on the basis of overlapping, and total line length is
Optimal：
min W(v)
S.t. it is not overlapped between module
Wherein, W (v) is total line length, is calculated and is obtained by semi-perimeter line length.
3. the VLSI global wiring method for establishing model according to claim 2 based on Poisson's equation explicit solution, feature
It is, in step s 2, according to position of the module in layout areas, determines potential φ (x, y) and electric field ξ (x, y), wherein ξ
(x, y)=(ξx,ξy)=- φ (x, y)；Using module i as a positive charge i, by area AiAs quantity of electric charge qi, use φi=φ
(xi,yi) and ξi=ξ (xi,yi) it is illustrated respectively in potential and electric field at charge i, charge i is according to electric field force Fi=qiξiIt is moved
Dynamic, potential energy of system isWherein, Ni=qiφiIndicate the energy of position of charge i；It converts constraint to quiet
The constraint of total potential energy N (v)=0 of electric system.
4. the VLSI global wiring method for establishing model according to claim 1 based on Poisson's equation explicit solution, feature
Be, in the step S3, be based on Gauss law, establish the Poisson's equation of electrostatic system, meet layout boundary condition and
Partial differential equation are obtained while compatibility condition, respectively：
Poisson's equation, wherein ρ (x, y) is density function：
▽ ▽ φ (x, y)=- ρ (x, y)
Neumann boundary condition, for avoiding module from running out of the boundary of layout, wherein R andThe boundary of layout areas R is indicated respectively
And outer-normal direction：
Compatibility condition makes equation group have unique solution：
5. the VLSI global wiring method for establishing model according to claim 4 based on Poisson's equation explicit solution, feature
It is, in the step S4, the density functions of module i in the x direction is denoted as：
In order to vector (x, y)=(x1,y1,...xi,yi) distinguish, it usesIndicate continuous variable；By module i in the side y
Upward density function is defined asThen the density function of module i isAll modules are on layout areas
Gross density is：
Wherein n is number of modules；
Redefine ρ (x, y)：
According to density function, then Poisson's equation, boundary condition and compatibility condition become：
6. the VLSI global wiring method for establishing model according to claim 5 based on Poisson's equation explicit solution, feature
It is, in the step S5, according to Poisson's equation and boundary condition that the step S4 is obtained, obtains following solution：
Wherein, au,pIndicate that the coefficient of each wave function, u and p indicate integer index；For design factor au,p, by above-mentioned solution generation
Enter the Poisson's equation that step S4 is obtained, passes through calculatingObtain density functionAnother expression
Formula：
Wherein,
According to Gauss law, electric fieldEqual to potentialNegative gradient；By above-mentioned solutionHave
7. the VLSI global wiring method for establishing model according to claim 6 based on Poisson's equation explicit solution, feature exist
In in the step S7, layout areas being divided into the grid of m × m same sizes, and each grid is denoted as blj, wherein l
=0,1, m-1, j=0,1, m-1 indicate grid label, enable
Grid bljDensity be
Then potential is：
Electric field is：
Wherein：
8. the VLSI global wiring method for establishing model according to claim 7 based on Poisson's equation explicit solution, feature
It is, in the step S8, calculates the coefficient matrix a ' of m × mu,p：
Coefficient matrix is calculated by library FFT of calling, then spends m2Time is by coefficient matrix a 'u,pIt is updated to au,p, calculate
All coefficient au,p；After calculating all coefficients, φ (l, j) and ξ are calculated by the inverse transformation of Fast Fourier Transform (FFT)
(l, j), electric field force Fi=qiξiModule i can be made to move, complete VLSI circuit layout model foundations.
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