單晶片系統驗證之核心技術開發---子計畫六：針對先進晶片設計的熱點驗證之完整熱模型與高效能熱分析(III)

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(1)單晶片系統驗證之核心技術開發子計畫六：針對先進晶片設計的熱點驗證之完整熱模型與高效能熱分析(3/3) Compact Thermal Modeling and Efficient Thermal Simulation for Hot Spots Verifications of Modern IC Designs 計畫編號：NSC. 96－2220－E009－012. 執行期間：96 年. 8 月 1 日至 97 年計畫主持人：李育民. 7月. 31 日. 一、中文摘要在現今的積體電路設計，能夠精確預測電路溫度分佈對於電路時序分析(timing analysis)、減少漏電流、消耗功率評估、熱點的避免和可靠度分析是相當重要的。本計劃主要目的為發展晶片上預測電路溫度分佈的快速分析工具。在計劃的前兩年中，我們已利用一般化的積分轉換(generalized integral transforms)技術針對設計自動化流程前端發展出一套有效率的熱分析工具，並應用此一技術分析 3D 積體電路的熱分佈[R1~R3]；同時亦針對現今在漏電流主導的製程技術下，規劃了一個統計型的晶片溫度分佈分析的初步流程。在計劃的最後一年，我們將完整的規劃此一流程、實現分析的方法並驗證其準確度與效率。關鍵詞：一般化的積分轉換；製程變異；熱分析；溫度剖面；熱點；漏電流二、英文摘要 The capability of predicting the temperature profile is critically important for circuit timing analysis, leakage reduction, power estimation, hotspots avoidance, and reliability concerns during modern IC designs. This work presents an accurate and fast analytical full-chip thermal simulator for the temperature-aware chip design. In the previous two years, we have developed a generalized integral transforms method to solve the transient and steady temperature distribution for the thermal placement stage, and extended the proposed GIT based method to deal with 3D ICs thermal simulation[R1~R3]. In this year, we have developed a stochastic thermal simulation procedure with considering the leakage power variation because of the effects of process variations, implemented the proposed method, and demonstrated its accuracy and efficiency. Keywords ： Generalized Integral Transforms, Process Variations, Spatial Correlation, Thermal Analysis, Temperature Profile, Hotspot, Leakage Current, Leakage Power 三、研究計畫之背景及目的 Because of the drastically increasing power consumption of integrated circuits, the thermal issue has become one of the most important concerns in VLSI design. The high temperature 1.

(2) distribution and thermal gradient variation have serious impacts on the timing, power and reliability of designs [2]–[8]. Conventional thermal simulators [2]–[8] are only conducted by solving the heat transfer equations with the nominal power consumption of the die. However, as the technology scales down, the decreased controllability of processes have caused considerable variations of leakage power [1]. The variations of leakage power are expected as high as 20 times caused by 30% within-die process variations [1], and the related fluctuations of temperature distribution are significant. Those unreliably optimistic estimations [2]–[8] might guide designers to the wrong design direction and lead to low yields. On the contrary, the deterministic simulation with the worst-case parameters can result in the immoderate guard-banding and can cause low performance [11]. These undesirable phenomena lead the statistical thermal simulation to be essential, especially for the leakage power dominated technology. Thermal simulations can be generally divided into two categories as transient-analysis and steady-state analysis. Transient-analysis is concerned with the evolution of temperature distribution within a chip given a time-varying power density distribution. As indicated in [2], [6], [13], the thermal time constant of heat conduction is much larger than the clock period of circuit. This fact leads to steady-state thermal analysis is more interested to study the stability of temperature distribution with a given power density distribution averaged over time. In this work, we will focus on the steady-state thermal analysis with considering withindie process variations with spatial correlation. Although we do not consider electro-thermal coupling due to the scope of this paper, our simulator can be readily combined with the temperature-dependent electrical modeling to perform an iteratively update scheme, for example, consider the temperature-dependent subthreshold leakage power in the thermal simulation. By using KL expansion [12], we transform the physical parameters with variations to a set of uncorrelated random variables and employ the PCs scheme [12] and stochastic Galerkin procedure to convert the stochastic thermal problem to a set of deterministic problems. After that, any existing deterministic thermal simulator such as [2]–[8] can be used to solve those deterministic heat transfer equations. Finally, the mean and variance profiles of the steady-state temperature for the full chip can be evaluated. Our major contributions are 1) To the authors’ best knowledge, this is the first stochastic thermal simulator considering within-die process variations with spatial correlation. We also demonstrate that the deterministic simulators with nominal physical parameters underestimate the temperature distribution, and are unreliable in the nanometer technology. 2) Our simulator can accurately and efficiently provide the mean and standard deviation profiles of the temperature to guide designers avoiding thermal failures due to process variations. 3) Experimental results indicate that ignoring process variations with spatial correlation during the thermal simulation is not allowable, and can induce several issues of design and reliability. The rest of this report is organized as follows. The problem formulation is introduced in section 四、A. The flowchart of proposed stochastic thermal simulation is 2.

(3) presented in section 四、B. Then, the modeling of physical parameters are described in section 四、C, the leakage power modeling is illustrated in section 四、D, and the stochastic Galerkin procedure is addressed in section 四、E. The experimental results are given in section 四、F. Finally, the conclusion and achievements are given in sections 五 and 六, respectively. 四、研究方法 A. Problem Formulation. Fig. 1. Compact thermal model of physical design. The typical compact thermal model for physical design stages [2]–[8] is shown in Fig. 1. It consists of three portions. The primary heat flow path is composed of thermal interface material, heat spreader and heat sink. The secondary heat flow path contains interconnect layers, I/O pads and the print circuit board. The functional blocks on the die are modeled as many power generating sources attached to the thin layer close to the top surface of the die with the thickness being equal to the junction depth of device [9]. The devices consuming the dynamic and leakage power are the mainly heat sources. Generally, the dynamic power is insensitive to process variations and can be assumed to be deterministic [10]. The leakage power is greatly affected by physical parameters such as the channel length and oxide thickness, and needs to be treated as spatial random processes [11]. By combining the compact thermal model and statistical power dissipation, the steady state temperature bT(r; _;$) of die is governed by the following stochastic steady-state heat transfer equation..  . .    r,Tˆ Tˆ  r, ,    p  r, Lch  x, y,  , Tox  x, y,   , Subject. to the following boundary condition. 3. (1).

(4) .  rbs , Tˆ. . . Tˆ rbs , , nbs.   h Tˆ bs. r. bs. .  . , ,  fbs rbs ,. (2). where r=  x, y, z   D , D   0, Lx    0, Ly     Lz , 0  is the domain of die, Lx and Ly are.  . lateral sizes of die, Lz is the thickness of die,  r, Tˆ. is the thermal conductivity W m  C . of die,  is the diverge operator, bs is any specific boundary surface of the die, rbs is the position located on bs, hbs is the heat-transfer coefficient on bs , hbs is the heat flux function on bs ,  nbs is the differentiation along the outward direction normal to bs ,  and  are sampling values of manufacturing outcomes Lch and Tox for the channel length and oxide thickness, respectively, Lch  x, y,  and Tox  x, y,  are the random processes of the device channel length and the oxide thickness, respectively, p  r, Lch  x, y,  , Tx  x, y,   is the random process of power density profile which consists of dynamic power density profile pd  r  , subthreshold leakage power density profile ps  r, Lch  x, y,   , and gate leakage power density profile pg  r, Tx  x, y,   . Since the major part of device current passes through the channel, the power density distribution has its value only when r   0, Lx    0, Ly     jd , 0  . Here, jd is the junction depth of device [9]..  . Generally, the values of  r, Tˆ. are temperature dependent. For the deterministic thermal. simulation, the difference of peak temperature is about 5 C between the result with temperature-dependent thermal parameters and the result with constant thermal parameters at 25 C [5]. In current VLSI design, the on-die temperature can be in the degree of 100  C . Under this situation, this difference may lead to about 5% error for the peak temperature of die. Since. the effort to amend this error is relatively high1, for practical purposes, these thermal parameters are usually treated as appropriate constants while performing temperature aware floor-planning and placement [13]. In this work, the reasonable value of each thermal parameter is set at the roughly steady-state average mean temperature of die which is got by using an iteratively computational scheme to the simplified 1-D thermal model shown in Fig. 2. Please see section 四、F for the detail of using 1-D thermal model. By using these estimated thermal parameters, the error of peak steady-state temperature can be reduced. 4.

(5) Fig. 2. Simplified 1-D thermal model for obtaining the roughly average rising mean temperature of die. The modeled thermal resistance network is shown in the right hand side. The values of thermal resistors are Rs  1 Adz hs , R p  1 Adz hp and Rdie  DT  Adz . Tavg  z  is the average rising mean temperature with respect to the room temperature Ta of lateral planes at arbitrary z position of the die. Here, Rdie can be viewed as a variable resistor when obtaining Tavg  z  at certain z position. PT is the total mean power consummation of the die. Adz is the cross area of die normal to the z-direction and DT is the thickness of the die. With the above description, the stochastic heat transfer equations for the steady state rising temperature T  r, ,   Tˆ  r, ,   Ta of die can be written as. 2T  r, ,    p  r, Lch  x, y ,  , Tox  x, y ,   ,. (3). subject to the boundary condition. . . T rbs , , nbs. h T bs. r. bs. , ,  fˆbs rbs ,. .  . (4). where  is the thermal conductivity of die got by using the roughly steady-state average mean.  . temperature, fˆbs rbS. is a modified heat flux function on bs , and Ta is the ambient temperature.. With the above stochastic steady-state heat transfer equations, we are going to evaluate the mean and variance profiles of the steady state full-chip temperature. B. Stochastic thermal simulation flowchart The executing flow of the proposed stochastic thermal simulator is summarized in Fig. 3. Given the spatial covariance functions of physical parameters, we transfer spatially the correlated physical parameters such as the channel length and the oxide thickness into a set of uncorrelated 5.

(6) random variables by using the KL expansion. With these uncorrelated random variables, the PCs are built to serve as polynomial bases for approximating the die temperature. Then, the leakage current models for different types of gates are built for modeling the power of gates by applying the minimal least square fitting to the simulation results of HSPICE under the TSMC 65 nm technology.. Fig. 3. Stochastic thermal simulation flowchart. After the chip geometry, the gate level placement, the package configuration, the dynamic and the stochastically leakage power density profiles are obtained, the compact thermal model shown in Fig. 1 is constructed. Then, the stochastic Galerkin projection method [12] is employed to convert the stochastic heat transfer equations to a set of deterministic heat transfer equations. The number of these deterministic heat transfer equations is equal to the number of total PCs. Finally, an efficiently deterministic thermal simulator [8] is utilized to solve these deterministic heat transfer equations, and the mean and variance profiles for the steady-state full-chip temperature are obtained. C. Parameter modeling The number of random variables and the computational complexity of simulation severely increase when considering the spatial correlation of within-die process variations. A well known technique to reduce the above difficulties is the Principal Component Analysis (PCA) [14], which is a grid-based method. However, the nature of PCA has the limitation of high-dimensional parameter modeling. An alternative formulation without the drawback of grid-based methods for tackling with the correlated parameters is the KL expansion. With the same level of accuracy, the number of random variables used by the KL expansion is significantly smaller than the PCA’s [15]. 6.

(7) (1) Karhunen-Loeve expansion The KL expansion [19] of a second-order random process  ( x, y,  ) with a continuous spatial covariance function is .   x, y,      x, y     k k  x, y k   ,. (5). k 1. where  ( x, y ) is the mean of  ( x, y,  ) , and each  k and each k ( x, y ) are the eigenvalue and the eigenfunction derived from the following Fredholm integral equation..  C  x , x    x  dx 1. 2. k. 2. 2.   kk  x1  .. (6). Do. Here,. C (x1 , x 2 ). is the covariance function of the random process  ( x, y,  ) ,. ( x, y )  D0  (0, Lx )  (0, Ly ) is the plane at the top surface of die,. x1  ( x 1 , y1 ) ,. x 2  ( x 2 , y2 ) ,  is the sampling event of sample space  , and { k ( )} is a set of uncorrelated random variables with each  k ( ) being zero mean and unit variance. The KL expansion satisfies the following properties [12] : 1) The minimized mean-square error property for a finite-term representation of a random process. 2) It is unique for a random process with a given covariance function. 3) { k ( )} is a set of independent standard normal random variables if the target random process is Gaussian. Since values of physical parameters such as the oxide thickness and channel length are bounded, they are second-order random processes [19]. Moreover, as indicated in [15], [20]–[22], continuous spatial correlation functions for physical parameters such as exponential, Gaussian, linear, or a fitting form from the experimental data are suggested to be used. Combining with the second-order property and the continuity of above covariance functions, practically, the KL expansions of physical parameters are valid. (2) Spatial correlation modeling As indicated in [20], the spatial covariance function is not monotonically decreasing as the distance increases because the decreasing rates of the spatial covariance in the x- and y-directions are different. In order to model the spatial covariance function with the above characteristic, we adopt the following spatial covariance function which was proposed by [15] instead of adopting the purely distance dependent spatial covariance function [21], [22].  x x C  x1 , x 2    2 exp   1 2 x .  y1  y2   exp   y  .   , . (7). where  x and  y are correlation lengths of the target random process in the x- and y-directions, respectively,  is the standard deviation of the target random process. Closed-form expressions of eigen-functions and eigen-values which satisfy equation (6) with C (x1 , x 2 ) being stated in equation (7) can be found in [23]. Due to the limitation of 7.

(8) space, expressions of eigenfunctions and eigen-values are not presented in this paper. With the closed-form expressions of the eigenvalues and eigenfunctions for the spatial covariance functions in equation (7), KL expansions of Lch ( x, y,  ) and Tox ( x, y , ) can be obtained as N Tox. Lch  x , y ,    Lch  x , y     m qm  x , y   m   ,. (8). n 1. NTox. Tox  x, y ,   Tox  x, y     n f n  x, y   n   .. (9). n 1. Here, Lch ( x, y )  E{Lch ( x, y,  )} , Tox ( x, y )  E{Tox ( x, y, )} , f n ( x, y ) and qm ( x, y ) are eigenfunctions of Tox ( x, y , ) and Lch ( x, y,  ) , respectively,  n and  m are eigenvalues of Tox ( x, y , ) and Lch ( x, y,  ) , respectively, { n ( )} and { n ( )} are independently standard normal random variables because Tox ( x, y , ) and Lch ( x, y,  ) are physically similar to Gaussian process [20], and indeed, the random processes of oxide thickness and channel length are assumed to be independent. In this work, we employ the criterion N 1  N 1 /  k 1  k   with.   0.001 to obtain the specified truncation numbers NT and N L ox. ch. for Tox ( x, y , ) and Lch ( x, y,  ) , respectively. For the sake of notation simplicity, { n ( ,  )}nNKL1 is set as the union of { n ( )}nN1 and { n ( )}nN1 , N KL  NTox  N Lch , and  n , Tox. Lch.  n and  n are used to represent  n ( , ) ,  n ( ) and  n ( ) , respectively, for the rest of this report. D. Leakage power modeling The analytical models of two major leakage currents, gate tunneling and subthreshold leakage currents, will be introduced in this section. The leakage current of each functional gate is input pattern dependent [11]. By applying different input patterns to different types of gates, their average leakage currents are measured to construct a set of cell leakage powers by using HSPICE. Then, the fitting model of empirical current for each type of gates is built by using the minimal least square fitting. The maximum error of fitting models compared with the results of HSPICE is less than 2%. (1) Gate tunneling leakage current Since the variations of gate leakage current are excessively sensitive to the variations of oxide thickness, the influence of channel length variations can be securely ignored [10]. Thus, the gate tunneling leakage current for a specific type of gate can be model as [11]. I g  a0 exp  a1Tox  ,. (10). where a0 and a1 are fitting constants. By substituting equations (9) into equation (10) and multiplying it by the supply 8.

(9) voltage Vdd , the stochastic gate tunneling leakage power for a specific type of gate located at.  x , y  can be expressed as *. *. Pg ( x * , y * ,  )  Pg exp(a1 where. a0. P g  a 0Vdd ,. and. a1. T* f. are. T. ),. (11). known. . values,.    1 ,  2 , ,  NT   ox . T. ,. . * * * f *   f1* ,  , f n*  , f N*T  , and each f n   n f n x , y .  ox . (2) Sub-threshold leakage current The sub-threshold leakage current is temperature dependent. For simplicity, we apply the following empirical form introduced in [26] at a suitable reference temperature. I s  b0 exp(b1Lch  b2 L2ch ),. (12). where b0 , b1 and b2 are fitting constants. Substituting equations (8) into equation (12) and multiplying it by Vdd , the sub-threshold leakage power for a specified type of gate located at Ps ( x* , y * ,  )  Ps exp(b1 T q*  b2 T A * ),.  x , y  can be given as *. *. (13) T. where Ps  b0Vdd , b0 , b1 and b2 are known values,    1 ,  2 ,  ,  N L  ,  ch  T. q*   q1* ,  , qm* ,  , q*N L  , A* is a N Lch  N Lch symmetric matrix with each entry  ch . Anl*  2 nl 1 qn* ql* , and each qm*   m qm  x* , y*  .. E. Stochastic Galerkin procedure via Hermite polynomial chaos The Taylor expansion method has been widely used in the statistical timing and circuit performance analysis [16], [17]. However, the assumption of small variation of the desired solution with respect to the random variables is not appropriate for approximating the temperature distribution, because the leakage power exponentially depends on the physical parameters and the temperature is directly affected by the leakage power. With a similar situation, the inaccuracy of the second order Taylor expansion method for solving a power grid system with variations of physical parameters for log-normal leakage currents was indicated by [18]. On the contrary, the PC based method [12] is adopted because it can handle the desired solution with large variation with respect to the random variables and can achieve a minimal mean square error approximation. Moreover, the projected deterministic heat transfer equations in this work are un-coupled for PCs. Hence, the efficiency is equal to applying the Taylor expansion method to the stochastic heat transfer equations (3)–(4).. 9.

(10) (1) Stochastic Galerkin procedure With { n }nNKL1 , a set of N KL -dimensional Hermite Polynomial Chaoses (H-PCs) [12] can be constructed to serve as bases to expand a general second-order random process u ( ,  ) . According to the theorem of Cameron and Martin [25], the random process of rising temperature distribution T (r, ,  ) can be approximated as N PC. T (r, ,  )   Tk (r ) k ( ),. (14). k 0. where each Tk (r ) is the projected temperature coefficient function of the k -th H-PC,  k ( ) is the k-th H-PC4, and N PC is the truncation number. The relation between N PC and N Lch and NTox will be described in section 四、E.(4). The stochastic Galerkin projection is executed as follows. Due to the limitation of space, the detail derivation is ignored. 1) Obtain the residual functions by substituting equation (14) into equations (3) and (4). 2) Enforce residual functions to be orthogonal to each H-PC. Then, we obtain the following decoupled deterministic heat transfer equations for solving each Tk (r ) for each different k ..  2Tk (r )  . pk ( r ) , R{ 2k ( )}. subject to the boundary condition T (r )  k bs  hbsTk (rbs )  fˆbs (rbs ) 0 k for each bs , nbs. (15). (16). where pk (r )  E{ p(r,  ) k ( )} is equal to pk (r )  pd (r, t ) 0k  pgk (r )  psk (r ).. Here,. pgk (r )  E{ pg (r,  ) k ( )}. and. (17). psk (r )  E{ ps (r,  ) k ( )} are. the. projected. gate-leakage and subthreshold-leakage power density profiles of the k -th H-PC, respectively. The term  0k in both equations (16) and (17) is because E{ k ( )}   0 k [12]. After pk (r ) is calculated, any existing deterministic thermal simulator [2]–[8] can be utilized to obtain each Tk (r ) . The above un-coupled deterministic heat transfer equations have an advantage for both numerical and analytical thermal simulators. For example, if the numerical simulators [2]–[6] are employed to solve the deterministic heat transfer equations, the system matrices of the above deterministic heat transfer equations are the same. Hence, the system matrices handling, such as the LU decomposition [6], building the multi-grid cycle [4], and setting up the tri-diagonal matrix in each direction [2], can be performed only once. After that, all deterministic heat transfer equations can share the same post-process matrix to obtain the solution. In this work, we utilize an efficient early-stage thermal simulator [8] to serve as the 10.

(11) deterministic thermal simulator. The mean and variance profiles of the steady-state temperature can be obtained as E{T (r, ,  )  Ta }  T0 (r)  Ta , (18) N PC. Var{T (r , ,  )  Ta }   Tk2 (r )E  k2 ( ).. (19). k 1. Note that only one deterministic heat transfer equation is needed to solve for obtaining the spatial mean temperature distribution. Two algorithms are proposed in the following two subsections to calculate the projected leakage powers for a specific type of gate located at arbitrary position of the die up to the second order of H-PCs. By those two algorithms, pk (r ) can be obtained for solving Tk (r ) . (2) Gate leakage power projection By using equation (11), the projected gate tunneling leakage power of the k -th H-PC for a specific type of gate located at a reference position. x , y  *. *. is. E Pg ( x* , y * ,  ) k ( )  Pg E exp( a1 T f * ) k ( ).. (20). Fig. 4 shows an algorithm for calculating equation (20) up to second order of H-PCs . Steps. 4 ~ 5 are owing to the independence of  i  and  i  . The rest steps of Fig. 4 can be derived by utilizing 0-th, 1-th and 2-th derivatives of the moment generating function of independent standard normal random variables.. Fig. 4. Gate tunneling leakage power projection algorithm (3) Sub-threshold leakage power projection By using equation (13), the projected subthreshold leakage power of the k-th H-PC for a specific type of gate located at a reference position (x_; y_) is. . .  . . . E Ps  x* , y * ,    k    Ps E exp b1 T q*  b2 T A*  k  . (21). Fig. 5 shows an algorithm for calculating equation (21) up to the second order of H-PCs. Due to the limitation of space, the derivation is ignored.. 11.

(12) Fig. 5. Sub-threshold leakage power projection algorithm. As indicated in [11], the number of reference points for modeling physical parameters can be much less than the number of gates while maintaining an acceptable accuracy. The simulated chip is divided into Ng grids for modeling physical parameters, and the central point of each grid is set to be a reference point.  x , y  . Gates located in the same grid *. *. share the same modeled physical parameters; hence, they have the same A* . Therefore, the number of eigen-decompositions for all A* is N g instead of the number of gates. Moreover, the eigenfunctions and eigenvalues of the channel length only depend on the spatial covariance function of the channel length, and each A* can be known after the information of spatial covariance function is given. Therefore, the eigen-decomposition of each A* and the projected power for each type of gate can be calculated before the thermal simulation. After the projected power for each type of gate at each grid is obtained, the rest computational cost for obtaining the steady-state sub-threshold-leakage power density profile of the k -th H-PC is O(#Gates). (4) Truncated number of H-PCs The original truncated number of H-PCs is [12] p. 1 n 1   N KL  r , n 1 n ! r  0. N pc  1  . where p is the order of H-PCs, and N KL  NTox  N Lch . 12. (22).

(13) The projection values of Pg  x, y,   and Ps  x, y,   upon the H-PC which simultaneously contains random variables in  i  and  i  are equal to zeros. Therefore, the number of H-PCs is reduced to p. p 1 n 1 1 n 1 N Tox  r    N Lch  r .  n 1 n ! r  0 n 1 n ! r  0. N pc  1  . . . . . (23). Since reduced truncated number is much less than the original truncated number, the computational effort is significantly reduced. F. Experimental Results Our stochastic thermal simulator is implemented in C++ language and tested on a HPxw9300 workstation with 16GB memory. The die size is 5mm  5mm  0.5 mm. The device junction depth is set to be 20nm which is the nominal value of the device junction depth for the 65nm technology [9]. The test chip floorplan is shown as rectangular blocks in Fig. 7(b). Numerous functional gates with the 65 nm technology are inserted into each rectangular block of Fig. 7(b), and the number of functional gates on the test chip is around 4.7 millions. The internal gates of each rectangular block in Fig. 7(b) are not shown for the sake of clarity. The nominal value of oxide thickness is 1.4 nm and the 3  values of parameter variations for the channel length and the oxide thickness are 20% of their nominal values. Both  x / Lx and.  y / Ly are set to 0.31 which means the correlation between two devices located half of the chip dimension away in either the x-direction or the y-direction is 0.2 [20]. The number of reference points is set to be 16 for the parameter modeling of the channel length and the oxide thickness. Based on the criterion stated in section IV-B, the truncation points of KL expansions for the channel length and the oxide thickness are chosen as N Lch = 82 and N Tox = 82, respectively. The values of hp and hs for executing [8] are obtained as follows. Based on the same setting. . . of chip geometry as [7], hp is obtained as 8700W / m2  C . The value of hs is got by using the modeling techniques of the equivalent thermal resistance for the C4/CBGA package [27] and effective thermal conductivity for interconnect layers [7]. To set the thermal conductivity of die at the roughly steady-state average mean temperature of die, we apply the 1-D thermal model shown in Fig. 2 and the following iteratively computation scheme. Initially, Tavg is set to be Ta , then the initial value of Rdie can be obtained. With this Rdie and calculated PT which can be got by using 0-th order projected powers of H-PCs sated in section VI-B and VI-C, we update Tavg by the 1-D thermal model. The above procedure is repeated until Tavg converges. Here, the room temperature Ta is set to be 27 C . With the above 13.

(14) calculation, the related thermal parameters and boundary conditions for executing the deterministic simulator [8] are summarized in Fig. 6. The boundary condition of each vertical surface. Fig. 6. Thermal conductivity of the die calculated at Tavg + Ta, and the equivalent heat transfer coefficients hp and hs for the primary and the secondary heat flow paths, respectively. is set to be isothermal [7], [8]. The top surface of the simulated die is divided into 1024  1024 grids for solving the deterministic heat transfer equations with respect to each H-PC. (1) Accuracy and efficiency. (a) (b) Fig. 7. The temperature distribution at the top surface of the test die. (a) The nominal temperature distribution and (b) the spatial mean temperature distribution. The Monte Carlo (MC) method with 105 samples and 256 reference points for modeling parameters are used as the reference solution to demonstrate the accuracy. As shown in Table I, the maximum errors of our simulator are less than 2% in both the spatial mean and the spatial standard deviation distributions with NKL being equal to 164 and the order of H-PCs being equal to 1. The runtime is only 113 seconds. Since Ng is set to be 16 in our simulator, the result demonstrates that Ng can be quite small without sacrificing the accuracy. The average mean temperature at the top surface of the die calculated by our simulator is 114.73 C . Note that, the average mean temperature at the top surface of the die calculated by the 1-D thermal model is 114.75 C , which is consistent with the value got by our simulator. This verifies the ability of the 1-D thermal model for predicting the average steady-state mean temperature of the die. Thus, the thermal parameters are set at an accurate average steady-state mean temperature of the die. To demonstrate the efficiency of the proposed method, the runtime comparison 14.

(15) between the proposed method and the Monte Carlo method is shown in Table I. Here, the number of sampling times for the Monte Carlo method is set for achieving the same standard deviation error level as our simulator. The results show that our simulator can be orders of magnitude faster than the Monte Carlo method. It can be observed that the maximum error of the mean profile of the temperature only depends on N KL rather than the order of H-PCs. However, the maximum error of the spatial standard profile of the temperature relies not only on N KL but also on the order of H-PCs. As shown in Table I, using the first order of H-PCs with large N KL can provide an accurate solution and the complexity is linear to N KL . Using the second order of H-PCs with small N KL can also provide an accurate solution but the number of the N PC increases quadratically. Based on the above observation, the following strategy can be used to further improve the accuracy without sacrificing the efficiency. After the initial N KL is decided by the criterion stated in section IV-B, the temperature coefficient function for each first order polynomial chaos can be obtained. Then, the temperature coefficient function of the second order polynomial chaos with respect to the decreasing order of eigenvalues one by one is obtained until there is no significant change of the standard deviation profile of the temperature. (2) Deterministic v.s. stochastic thermal simulators The nominal and mean temperature profiles on the top surface of the die are shown in Fig. 7(a) and (b), respectively. The difference between them is over 16%. This indicates that the deterministic thermal simulator with the nominal power underestimates the hottest value and profile of the die temperature. (3) Without or with including the effect of spatial correlation Fig. 8 reveals the dramatic difference of the standard deviation profiles for the temperature between the result considering the spatial correlation of physical parameters and the result ignoring the spatial correlation of physical parameters. Although their spatial mean temperature distributions are equal because of equation (18), their spatial standard deviations profile of temperature are drastically different. The values presented in Fig. 8(b) are 3~4 times of Fig. 8(a). Hence, the spatial correlation of the physical parameter should be taken into account in the stochastic thermal analysis. TABLE I ACCURACY AND EFFICIENCY COMPARED WITH THE MONTE CARLO METHOD.. 15.

(16) (a) (b) Fig. 8. The spatial standard deviations for the temperature distribution at the top surface of the test die. (a) The spatial standard deviations without including the effect of spatial correlation and (b) the spatial standard deviations with including the effect of spatial correlation. (4) Temperature variation trend with respect to variation of physical parameters To further study the trend of temperature variation with respect to the variation of physical parameters, we sweep the 3 ranges of channel length and oxide thickness and show the corresponding maximum mean and maximum standard deviation of die temperature in Fig. 9. As we can see, both of the maximum mean and the maximum standard deviation are exponentially dependent on the 3 ranges of channel length and oxide thickness.. Fig. 9. (a) The maximum mean of die temperature. (b) The maximum standard deviation of die temperature. 五、結論與討論 In this report, we have developed a stochastic thermal simulator with considering spatial correlated within-die process variations. The experimental results have demonstrated that our 16.

(17) simulator can efficiently provide very accurate estimates. Our simulator can readily provide a simulating kernel of the elector-thermal simulating loop. Our future work is to combine our simulator with the elector-thermal simulating loop for providing a more accurate thermal estimation under the spatial correlated within die process variations. 六、成果 In the last year of this project, we have published three international conference papers [R1, R2, R6] and two domestic conference papers [R4, R5], one regular paper [R3] has been accepted by TVLSI, and two international conference papers [R7, R8] has been accepted by ASPDAC 2009.. [R1] Pei-Yu Huang, Chih-Kang Lin, and Yu-Min Lee, “Full-Chip Thermal Analysis via Generalized Integral Transforms”, the 14th Workshop on Synthesis and System Integration of Mixed Information Technologies (SASIMI), 2007. [R2] Pei-Yu Huang, Chih-Kang Lin, and Yu-Min Lee, “Full-Chip Thermal Analysis for the Early Design Stage via Generalized Integral Transforms”, the 13th Asia and South Pacific Design Automation Conference (ASPDAC) 2008, pp. 462-7, 2008. [R3] Pei-Yu Huang and Yu-Min Lee, “Full-Chip Thermal Analysis for the Early Design Stage via Generalized Integral Transforms”, accepted by IEEE Transactions on Very Large Scale Integration Systems (TVLSI). [R4] Pei-Yu Huang, Jia-Hong Wu, Yu-min Lee, and Huai-Chung Chang, “Stochastic Thermal Simulation Considering With-in Die Process Variation”, the 19th VLSI Design/ CAD Symposium (VLSI/CAD 2008). [R5] Shih-An Yu, Pei-Yu Huang and Yu-Min Lee, “Power Optimization in 3D ICs Considering Process Variations and Thermal Effect”, the 19th VLSI Design/ CAD Symposium (VLSI/CAD 2008). [R6] Pei-Yu Huang, Chih-Kang Lin, and Yu-Min Lee, “Hierarchical Power Delivery Network Analysis using Markov Chains”, IEEE International SOC Conference (SOCC) 2007. [R7] Pei-Yu Huang, Jia-Hong Wu and Yu-Min Lee, “Stochastic Thermal Simulation Considering Spatial Correlated Within-Die Process Variations”, to appear in Asia South Pacific Design Automation Conference (ASPDAC) 2009. [R8] Shih-An Yu, Pei-Yu Huang and Yu-Min Lee, “A Multiple Supply Voltage Based Power Reduction Method In 3-D ICs Considering Process Variations And Thermal Effects”, to appear in Asia South Pacific Design Automation Conference (ASPDAC) 2009.. 17.

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VJG XQNWOG OGUJKPI C JWIG UGV QH NKPGCT GSWCVKQPU HQT VJG WPKPVGTGUVGF VGORGTCVWTG KP UWDUVTCVG UVKNN PGGF VQ DG JCPFNGF GXGP KH QPN[ VJG VGORGTCVWTG FKUVTKDWVKQP ENQUG VQ VJG FGXKEG NC[GT KU QH KPVGTGUV 1P VJG EQPVTCT[ CPCN[VKECN OGVJQFU CTG IQQF ECPFKFCVGU HQT VJG GCTN[ FGUKIP UVCIG DGECWUG VJG[ CXQKF FKTGEVN[ RGTHQTOKPI VJG XQN WOG OGUJKPI QH GPVKTG UWDUVTCVG CPF JCXG ENQUGFHQTO TGRTGUGPVC VKQPU HQT VJG VGORGTCVWTG FKUVTKDWVKQP 1PG CPCN[VKECN ECVGIQT[ QH VJGTOCN UQNXGTU KU VJG )TGGPŏU HWPEVKQP DCUGF OGVJQF =? +P =? VJG CWVJQTU CRRNKGF VJG )TGGPŏU HWPEVKQP VQ VJG VKOGKPFGRGPFGPV 2QUUKQPŏU GSWCVKQP CPF WUGF HCUV (QWTKGT VTCPUHQTO ((6 VQ GXCNWCVG VJG UVGCF[UVCVG VGORGTCVWTG FKUVTKDWVKQP *GPEG VJG EQORWVCVKQPCN EQUV ECP DG QPN[ O(M N log2 M N ) YJGTG M CPF N CTG PWODGTU QH FKXKUKQPU KP VJG RQYGT FGPUKV[ OCR CNQPI x CPF yFKTGEVKQPU TGURGEVKXGN[ *QYGXGT VJG EQPXGTIGPV TCVG QH VJGKT HQTOWNCVKQP KU PQV HCUV GPQWIJ DGECWUG VJG IGPGTCVGF EQUKPG UGTKGU DCUGF QP VKOGKPFGRGPFGPV 2QUUKQPŏU GSWCVKQP =? ECP PQV HWNN[ ECRVWTG VJG VTCPUKGPV EJCTCEVGTKUVKEU QH QTKIKPCN JGCV GSWCVKQPU #U UJQYP KP =? VJG VTWPECVKQP RQKPVU PGGF VQ DG NCTIG GPQWIJ VQ CEJKGXG UOCNN TGNCVKXG GTTQT (WTVJGTOQTG KV KU QPN[ HQT UVGCF[ UVCVG VGORGTCVWTG ECNEWNCVKQP DWV VJG VTCPUKGPV CPCN[UKU KU CNUQ PGEGUUCT[ YJKNG RGTHQTOKPI VJG F[PCOKE VJGTOCN OCPCIGOGPV CPF TWPVKOG VJGTOCN CPCN[UKU =?Ō=? 6Q QXGTEQOG VJGUG UJQTVEQOKPIU QWT OCLQT EQPVTKDWVKQPU CTG • 9G KORTQXG VJG EQPXGTIGPEG TCVG QH CPCN[VKECN UQNWVKQP HQT UVGCF[ UVCVG VGORGTCVWTG FKUVTKDWVKQP CPF RTQXKFG C VTCP UKGPV VGORGTCVWTG UKOWNCVKQP D[ WVKNK\KPI IGPGTCNK\GF KPVGITCN VTCPUHQTOU )+6 =? VQ EQPUVTWEV C UGV QH URCVKCN DCUGU CPF VJGKT EQTTGURQPFKPI VKOGXCT[KPI EQGHſEKGPVU 6JG RTQRQUGF OGVJQF ECP CEEWTCVGN[ GUVKOCVG VJG VGORGTCVWTG FKUVTKDWVKQP QH HWNNEJKR YKVJ XGT[ UOCNN VTWPECVKQP RQKPVU Nx CPF Ny QH DCUGU KP VJG URCVKCN FQOCKP 6JG GZRGTKOGPVCN TGUWNVU RTGUGPVGF KP UGEVKQP +8 UJQY VJCV Nx Ny ECP DG HCT NGUU VJCP M N YKVJQWV NQUKPI CP[ CEEWTCE[ EQORCTGF YKVJ =? • 9G FGXGNQR C ((6 NKMG GXCNWCVKPI CNIQTKVJO VQ GHſEKGPVN[ GXCNWCVG VJG VGORGTCVWTG OCR QH CNN ITKF EGNNU CPF KVU EQORWVCVKQPCN EQUV KU KP QTFGT QH O(M N log2 Nx Ny ) YJGTG Nx CPF Ny CTG VTWPECVKQP RQKPVU QH DCUGU CNQPI x CPF y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ƀQY RCVJ VJG UGEQPFCT[ JGCV ƀQY RCVJ CPF VJG JGCV VTCPUHGT EJCTCEVGTKUVKE QH GCEJ OCETQDNQEM QP UKNKEQP FKG =? CU UJQYP KP (KI 6JG RTKOCT[ JGCV ƀQY RCVJ KU EQORQUGF QH VJGTOCN KPVGTHCEG OCVGTKCN JGCV URTGCFGT CPF JGCV UKPM 6JG UGEQPFCT[ JGCV ƀQY RCVJ EQPVCKPU KPVGTEQPPGEV NC[GTU +1 RCFU CPF RTKPV EKTEWKV DQCTF. 462.

(21) 5C-2 Ambient Air. HQT VJG GCTN[ FGUKIP UVCIG ECP DG TGYTKVVGP CU κ∇2 T (x, y, z, t) = σ. hs. I/O Pads & PCB. Secondary Heat Flow Path. Interconnect Layers. pmn H mn t

(22). x=Lx. N Divisions. y=Ly. M Divisions. x y. z Die Thermal Interface Material Heat Spreader. 'x 'y. z=0. Die Substrate. z=-Lz. Primary Heat Flow Path. hp. Ambient Air %QORCEV VJGTOCN OQFGN QH VJG GCTN[ FGUKIP UVCIG. 2%$ 6JG HWPEVKQPCN DNQEMU QP VJG FKG CTG OQFGNGF CU OCP[ RQYGT UQWTEGU CVVCEJGF QP VJG VQR UWTHCEG QH FKG 6JG TKUKPI VGORGTCVWTG T (r, t) QH FKG EQTTGURQPFKPI VQ VJG CODKGPV VGORGTCVWTG ECP DG IQXGTPGF D[ VJG HQNNQYKPI JGCV VTCPUHGT GSWCVKQPU =? ∇ · (κ(r)∇T (r, t)) = σ(r). ∂T (r, t) ;r ∈ D ∂t. ∂T (r, t) + hbs T (r, t) = fbs (r) κ(r) ∂nbs. ; (x, y, z) ∈ D. . . . . z=0. Heat Sink. (KI . ∂T (x, y, z, t). ∂t ∂T (x, y, z, t) ∂T (x, y, z, t) = =0 ∂x ∂y y=0,Ly x=0,Lx ∂T (x, y, z, t) κ = hp T (x, y, −Lz , t) ∂z z=−Lz ∂T (x, y, z, t) κ = hs T (x, y, 0, t) + p(x, y, t) ∂z. . . YJGTG r = (x, y, z) κ(r) KU VJG VJGTOCN EQPFWEVKXKV[ W/m·◦ C QH FKG σ(r) KU VJG RTQFWEV QH VJG FGPUKV[ QH OCVTKCN CPF URGEKſE JGCV J/m3 ·◦ C QH FKG ∇ KU VJG FKXGTIG QRGTCVQT D Lx × Ly × −Lz KU VJG FKOGPUKQP QH FKG Lx CPF Ly CTG VJG NCVGTCN UK\GU QH FKG Lz KU VJG VJKEMPGUU QH FKG hbs KU VJG JGCVVTCPUHGT EQGHſEKGPV QP VJG DQWPFCT[ UWTHCEG bs QH FKG fbs (r) KU VJG JGCV ƀWZ HWPEVKQP QP VJG DQWPFCT[ UWTHCEG CPF ∂/∂n bs KU VJG FKHHGTGPVKCVKQP CNQPI VJG QWVYCTF FKTGEVKQP PQTOCN VQ VJG DQWPFCT[ UWTHCEG 6Q RTQXKFG TGCUQPCDNG CEEWTCE[ YKVJ NKVVNG EQORWVCVKQPCN GHHQTV FWTKPI VJG GCTN[UVCIG VGORGTCVWTGCYCTG QRVKOK\CVKQP RTQEGFWTG JGCVVTCPUHGT EQGHſEKGPVU QP VJG DQWPFCT[ UWTHCEGU QH FKG UJQWNF DG CRRTQRTKCVGN[ OQFGNGF $CUGF QP VJG OQFGN RTQRQUGF KP =? VJG JGCV VTCPUHGT EQGHſEKGPVU QH RTKOCT[ RCVJ ECP DG GSWCNK\GF VQ CP GHHGEVKXG JGCV VTCPUHGT EQGHſEKGPV hp D[ EQODKPKPI VJG GHHGEV QH GCEJ EQORQPGPV QP VJG RTKOCT[ RCVJ 5KPEG VJG FGVCKN NC[QWV QH KPVGTEQPPGEVU KU PQV CXCKNCDNG KP VJG GCTN[ FGUKIP UVCIG =? OQFGNGF VJG KPVGTEQPPGEV NC[GT CU CP GSWKXCNGPV VJGTOCN TGUKUVCPEG D[ GUVKOCVKPI VJG FGPUKV[ DCUGF QP VJG TGIWNCTKV[ UVTWEVWTG CUUWORVKQP QH OGVCN CPF FKGNGEVTKE OCVGTKCN 9KVJ VJG OQFGN QH GCEJ KPVGTEQPPGEV NC[GT VJG JGCV VTCPUHGT EQGHſEKGPVU QH UGEQPFCT[ RCVJ ECP DG UKORNKſGF VQ DG CP GSWKXCNGPV JGCV VTCPUHGT EQGHſEKGPV hs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κ(r) CPF σ(r) QH FKG CTG RQUKVKQPFGRGPFGPV VJG XCTKCVKQPU QH VJGUG VJGTOCN RCTCOGVGTU CTG WUWCNN[ PQV UKIPKſECPV CPF CU UWIIGUVGF KP =?Ō=? VJGUG RCTCOGVGTU ECP DG VTGCVGF CU EQPUVCPVU YJKNG RGTHQTOKPI VJG VGORGTCVWTGCYCTG ƀQQTRNCPPKPI CPF RNCEGOGPV $CUGF QP VJG CDQXG OQFGN VJG JGCV FKHHWUKQP GSWCVKQPU QH FKG. *GTG p(x, y, t) KU VJG RQYGT FGPUKV[ W/m2 QP VJG VQR UWTHCEG QH FKG CPF T (x, y, z, 0) $[ FKUETGVK\KPI VJG RQYGT UQWTEG RNCPG QP VJG VQR QH FKG KPVQ M N ITKF EGNNU YJGTG M CPF N CTG PWODGTU QH FKXKUQPU CNQPI x CPF yFKTGEVKQPU TGURGEVKXGN[ VJG RQYGT FGPUKV[ p(x, y, t) ECP DG TGYTKVVGP CU N −1 M −1 p(x, y, t) =. pmn Πmn (x, y)Hmn (t),. . n=0 m=0. YJGTG Πmn (x, y) KU VJG KPFKECVG HWPEVKQP YKVJ PQP\GTQ XCNWG GSWCNKPI VQ QPN[ YJGP (x, y) KU KP [mΔx, (m + 1)Δx] × [nΔy, (n + 1)Δy] ΔxLx M ΔyLy N m CPF n CTG KPFKEGU QH FKXKUKQPU CPF pmn CPF Hmn (t) CTG VJG CXGTCIG RQYGT FGPUKV[ CPF VJG VWTPKPI QPQHH HWPEVKQP QH ITKF EGNN m n TGURGEVKXGN[ #U ECNEWNCVKPI VJG UVGCF[ UVCVG VGORGTCVWTG Hmn (t) KU C WPKV UVGR HWPEVKQP 1VJGTYKUG KV KU CP KPUVTWEVKQP URGEKſ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ſEKGPE[ EQPUKFGTCVKQP #HVGT DCUGU DGKPI EQP UVTWEVGF VJG VGORGTCVWTG FKUVTKDWVKQP ECP DG GZRTGUUGF D[ VJQUG DCUGU YKVJ UWKVCDNG VKOGXCT[KPI EQGHſEKGPVU +P PGZV VYQ UWDUGEVKQPU JQY VQ CRRN[ VJG CDQXG RTQEGFWTG VQ VJG HWNNEJKR VJGTOCN CPCN[UKU YKNN DG FGUETKDGF KP FGVCKN #HVGT VJCV VJG EQORCEV HQTOWNC YKNN DG IKXGP VQ ECNEWNCVG VJG CXGTCIG UVGCF[UVCVG VGORGTCVWTG FKUVTKDWVKQP CPF KVU EQPXGTIGPEG TCVG KORTQXGOGPV QXGT VJG )TGGPŏU HWPEVKQP DCUGF OGVJQF =? YKNN DG RQKPVGF QWV (KPCNN[ YG YKNN FGXGNQR HCUV GXCNWCVKPI CNIQTKVJOU HQT QWT )+6 DCUGF HQTOWNCVKQP CPF WVKNK\G QWT OGVJQF VQ RGTHQTO VJG VTCPUKGPV VJGTOCN UKOWNCVKQP # #WZKNKCT[ 2TQDNGO HQT )GPGTCVKPI #RRTQRTKCVG 5RCVKCN $CUGU 6JG CWZKNKCT[ RTQDNGO ECP DG KPVTQFWEGF D[ EQPUKFGTKPI VJG JQOQIGPGQWU RTQDNGO YJKEJ VJG UQNWVKQP QH VGORGTCVWTG FKUVTK DWVKQP UCVKUſGU JQOQIGPGQWU IQXGTPOGPV GSWCVKQPU YKVJ p(x, y, t) = 0 #U UVCVGF KP =? VJG CWZKNKCT[ RTQDNGO ECP DG VJG HQNNQYKPI 5VWTO.KQWXKNNG RTQDNGO YKVJ URGEKſE DQWPFCT[ EQPFKVKQPU. 463. ∇2 φilq (x, y, z)+λ2ilq φilq (x, y, z)=0; (x, y, z) ∈ D. ∂φilq (x, y, z) ∂φilq (x, y, z) = =0 ∂x ∂y x=0,Lx y=0,Ly ∂φilq (x, y, z) κ = hp φilq (x, y, −Lz ) ∂z z=−Lz ∂φilq (x, y, z) κ = hs φilq (x, y, 0) ∂z z=0. . . . .

(23) 5C-2 6JG UQNWVKQPU QH 5VWTO.KQWXKNNG RTQDNGO HQTO C UGV QH EQO RNGVGN[ QTVJQPQTOCN DCUGU KP VJG URCVKCN FQOCKP QH FKG CPF VJGKT IGPGTCN HQTO ECP DG HQWPF KP =? $[ CRRN[KPI VJG IGPGTCN HQTO KPVQ QWT RTQDNGO UGVVKPI hs VQ DG \GTQ 6JKU CUUWORVKQP KU QPN[ HQT EQORCTKPI QWT OGVJQF YKVJ =? WPFGT VJG UCOG GZRGTKOGPVCN UGVVKPI 1WT UQNXGT ECP GCUKN[ VCMG KPVQ CEEQWPV VJG GHHGEV QH UGEQPF JGCV ƀQY RCVJ CPF YKVJ UGXGTCN OCPKRWNCVKQPU φilq (x, y, z) ECP DG IQV CU φilq (x, y, z) =. ) cos( lπy ) cos(λzq z) cos( iπx Lx Ly. . Nilq. =. λ2xi. +. λ2yl. +. 2. ,. λ2zq ,. . 2. $ 5[UVGO 6TCPUHQTOCVKQP HQT 6KOG8CT[KPI %QGHſEKGPVU 5KPEG VJG IGPGTCVGF DCUGU CTG EQORNGVGN[ QTVJQPQTOCN KP VJG URCVKCN FQOCKP QH FKG T (x, y, z, t) ECP DG CRRTQZKOCVGF D[ T(x, y, z, t) CU. YJGTG pilq (t) = 0 0 p(x, y, t)φilq (x, y, 0)dxdy ≤ i ≤ Nx ≤ l ≤ Ny CPF ≤ q ≤ Nz 1DVCKP VJG IGPGTCN UQNWVKQP QH GCEJ ψilq (t)ŏU CU HQNNQYKPI t k 2 1 ψilq (t) = pilq (τ )e− σ λilq (t−τ ) dτ.. σ. q=0. l=0. ψilq (t)φilq (x, y, z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ψilq (∞) = pilq (∞)/(kλ2ilq ) ECP DG CPCN[VKECNN[ QDVCKPGF HTQO GSWCVKQP YKVJQWV CP[ VKOG UVGR GXCNWCVKQP #HVGT VJCV RNWIIKPI φilq (x, y, z)ŏU CPF ψilq (∞)ŏU KPVQ GSWCVKQP VJG CXGTCIG UVGCF[ UVCVG TKUKPI VGORGTCVWTG T mn QH ITKF EGNN (m, n) QP VJG VQR UWTHCEG KU (n+1)Δy (m+1)Δx 1 T mn = T(x, y, 0, ∞)dxdy ΔxΔy. =. ∂(x, y, z, t) r(x, y, z, t) = κ∇2 (x, y, z, t) − σ , ∂t. •. Kil cos. i=0. mΔx. iπ(2m + 1) 2M. cos. lπ(2n + 1). , 2N. YJGTG Kil =. Nz −1 Pil Cilq , κ Nilq. . q=0. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Cilq =. . YJGTG (x, y, z, t) = T (x, y, z, t) − T(x, y, z, t) 6Q CEEWTCVGN[ CRRTQZKOCVG T (x, y, z, t) D[ GSWCVKQP VJG PQTO QH r(x, y, z, t) UJQWNF DG CU UOCNN CU RQUUKDNG +P QTFGT VQ CEJKGXG VJKU IQCN VJG HQNNQYKPI UVGRU CTG RGTHQTOGF VQ ſPF VJG FGUKTGF GZRTGUUKQP QH ψilq (t) &WG VQ VJG NKOKVCVKQP QH URCEG YG QPN[ NKUV VJG FGTKXGF UVGRU QH ψilq (t) KP DTKGH CPF VJG FGVCKN FGTKXCVKQP KU KIPQTGF • 'ZRCPF r(x, y, z, t) D[ WUKPI φilq (x, y, z) CU. . i=0. YJGTG ψilq (t) KU VJG VKOGXCT[KPI EQGHſEKGPV Nx Ny CPF Nz CTG VTWPECVKQP RQKPVU KP x y CPF zFKTGEVKQPU TGURGEVKXGN[ *GTG QWT OCLQT IQCN KU VQ ſPF CP CPCN[VKECN GZRTGUUKQP QH ψilq (t) HQT CEJKGXKPI CP CEEWTCVG VGORGTCVWTG CRRTQZKOC VKQP 5WDUVKVWVKPI GSWCVKQP KPVQ 3 VJG TGUKFWCN HWPEVKQP r(x, y, z, t) KU GSWCN VQ. •. nΔy. Ny −1 Nx −1. l=0. . 0. 'SWCVKQP ECP DG CRRNKGF VQ QDVCKP VJG UVGCF[UVCVG VGORGTC VWTG YKVJQWV CP[ VKOG UVGR GXCNWCVKQP CPF GSWCVKQP KU CRRNKGF VQ QDVCKP VJG VTCPUKGPV VGORGTCVWTG FKUVTKDWVKQP. Nz −1 Ny −1 Nx −1. . . Ly Lx. . YJGTG λ2xi (iπ/L x ) CPF λ2yl (lπ/L y ) 6JG λ2xi λ2yl λ2zq CTG GKIGPXCNWGU KP x y CPF zFKTGEVKQPU CPF λzq ECP DG UQNXGF D[ CRRN[KPI VJG 0GYVQP4CRJUQP OGVJQF =? 'UUGPVKCNN[ VJG GSWKXCNGPV VJGTOCN EQPFWEVKXKV[ QH UGEQPF JGCV ƀQY RCVJ UJQWNF PQV DG \GTQ 5GVVKPI hs VQ DG \GTQ YJKEJ KU VJG UCOG CU =? KU QPN[ HQT EQORCTKPI QWT OGVJQF YKVJ =? WPFGT VJG UCOG GZRGTKOGPVCN UGVVKPI 6JG GHHGEV QH UGEQPF JGCV ƀQY RCVJ ECP DG GCUKN[ VCMGP KPVQ CEEQWPV KP QWT UQNXGT DGECWUG VJG IGPGTCN UQNWVKQP QH 5VWTO.KQWXKNNG RTQDNGO CNTGCF[ VCMGU hs KPVQ CEEQWPV 6JWU QPN[ VJG EQORWVCVKQPCN HQTOWNCU QH Nzq CPF λzq CTG PGGFGF VQ DG OQFKſGF CPF VJKU YKNN PQV KPƀWGPEG VJG FGTKXCVKQP QH EQORWVCVKQPCN HQTOWNC CPF GXCNWCVKPI CNIQTKVJO QH VJG VGORGTCVWTG KP VJG TGOCKPKPI RQTVKQP QH VJKU YQTM. T(x, y, z, t) =. σψilq (t) = −κλ2ilq ψilq (t) + pilq (t), and ψilq (0) = 0;. •. YJGTG Nilq θil Lx Ly Nzq θ00 1/2 θi0 θ0l θil 2 YKVJ i= CPF l= Nzq Lz κhp ×sin (λzq Lz ) CPF κhp ×λzq cot λzq Lz 6JQUG UQNWVKQPU φilq (x, y, z)ŏU CTG ECNNGF GKIGPHWPEVKQPU 'CEJ QH VJGO JCU C EQTTGURQPFKPI PQP\GTQ GKIGPXCNWG λ2ilq CPF Nilq KU VJG PQTOCNK\GF XCNWG 'CEJ GKIGPXCNWG KU GSWCN VQ λ2ilq. #RRN[ GSWCVKQP CPF VJG QTVJQPQTOCNKV[ QH φilq (x, y, z)ŏU VQ VJG TGUWNVGF GSWCVKQP HQT IGVVKPI VJG HQNNQYKPI WPEQWRNGF U[UVGO. •. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩. ΔxΔy ; λ2. i = 0, l = 0. ilq. lπ 4N Ly Δx sin2 ( 2N ). l2 π 2 λ2. ilq. i = 0, l = 0. ;. iπ 4M Lx Δy sin2 ( 2M ). i2 π 2 λ2. ilq. iπ lπ 16M N Lx Ly sin2 ( 2M ) sin2 ( 2N ). i2 l2 π 4 λ2. ilq. CPF Pil =. . M −1 N −1. m=0 n=0. pmn cos. iπ(2m + 1) 2M. . i = 0, l = 0. ;. ; i = 0, l = 0. cos. lπ(2n + 1) 2N. . . $GHQTG KPVTQFWEKPI QWT GXCNWCVKPI CNIQTKVJOU HQT ECNEWNCVKPI VJG CXGTCIG UVGCF[ UVCVG TKUKPI VGORGTCVWTG YG ſTUV RTQEGGF VJG EQP XGTIGPV CPCN[UKU VQ UJQY VJG DGPGſV QH QWT VGORGTCVWTG ECNEWNCVKPI ∞ ∞ ∞ r(x, y, z, t) = rilq (t)φilq (x, y, z),. HQTOWNC 6JG GTTQT QH QWT VGORGTCVWTG ECNEWNCVKPI HQTOWNC ECP DG DQWPFGF D[ VJG HQNNQYKPI VJGQTGO q=0 l=0 i=0 6JGQTGO 6JG CDUQNWVG VTWPECVKQP GTTQT QH CXGTCIG UVGCF[ 0 L L YJGTG rilq (t) −Lz 0 y 0 x r(x, y, z, t)φilq (x, y, z)dxdydz UVCVG VGORGTCVWTG HQT GCEJ ITKF EGNN (m, n) D[ WUKPI VJG )+6 DCUGF HQTOWNCVKQP YKVJ VTWPECVKQP RQKPVU Nx Ny CPF Nz KP x 2GTHQTO VJG )CNGTMKPŏU UEJGOG =? YJKEJ UGVU VJG rilq (t)ŏU y CPF zFKTGEVKQPU KU DQWPFGF D[ VQ DG \GTQU WR VQ VTWPECVKQP RQKPVU Nx Ny CPF Nz α2 α3 α4 α1 6TCPUHGT VJG TGUWNVGF GSWCVKQP VQ VJG HQTO YJKEJ RTGUGTXGU + + + ,. 2 2 2 2 2 i l λilq i λi0q l2 λ20lq λ200q VJG NCY QH EQPUGTXCVKQP D[ WUKPI &KXGTIGPEG 6JGQTGO =? q∈S4 (i,l,q)∈S1 (i,q)∈S2 (l,q)∈S3. 464.

(24) 5C-2 #NIQTKVJO 4CFKZVYQ &56.((6 +PRWV Complex vector f with length Nx 1WVRWV Complex vector F with length 2M $GIKP fR 4GXGTUGDKV f L 4M/Nx NSubDF T s Nx /2 (QT SubIndex 0 VQ NSubDF T s − 1 k L × SubIndex i × SubIndex (QT SubK 0 VQ L − 1 F [k] F [k] fR [i] fR [i + 1] ×ej2π×SubK/L kk 'PF(QT 'PF(QT #RRN[ VJG DQVVQO WR RTQEGFWTG QH UVCPFCTF ((6 VQ GZGEWVG log2 Nx − 1 VKOGU &CPKGNUQP.CPE\QU NGOOC QH HQT GXCNWCVKPI VJG ſPCN F 'PF. YJGTG S1 [1, Nx ] × (Ny , ∞) × [0, ∞) ∪ (Nx , ∞) × [1, Ny ] × [0, ∞) ∪ [1, Nx ] × [1, Ny ] × (Nz , ∞) S2 [1, Ny ] × (Nz , ∞) ∪ (Nx , ∞) × [0, ∞) S3 [1, Nx ] × (Nz , ∞) ∪ (Ny , ∞) × α1 128M 2 N 2 PT /(Lx Ly Lz κπ 4 ) [0, ∞) S4 (Nz , ∞) 2 2 α2 16M PT /(Lx Ly Lz κπ ) α3 16N 2 PT /(Lx Ly Lz κπ 2 ) CPF α4 2PT /(Lx Ly Lz κ) &WG VQ VJG NKOKVCVKQP QH URCEG VJG RTQQH KU KIPQTGF 6JG CDQXG TGUWNV UJQYU VJCV VJG FGEC[KPI TCVG QH VJG VTWPECVKQP GTTQT QH QWT )+6 DCUGF OGVJQF ECP DG KP VJG QTFGT QH i2 l2 ((iπ/L x )2 + (lπ/L y )2 + λ2zq ) 6Q EQORCTG VJG EQPXGTIGPV TCVG YG CNUQ QDVCKP VJG VTWPECVKQP GTTQT DQWPF QH HQTOWNCVKQP KP =? CU HQNNQYKPI β1 β2 β3 (i,l)∈B1. i2 l2 γil. +. i∈B2 ,l=0. i2 γil. +. i=0,l∈B3. l2 γil. ,. . 3. f7. F31. (a). First Phase. Fî ee Fî eo. Level 1 Second Phase Level 2. Fî e. 1. f1 3 fˆ 3. fˆ1 5. Fˆ0 …. √ YJGTG Nx <M CPF GCEJ KU RQYGT QH 2 j −1 CPF fi ŏU CPF F k ŏU CTG EQORNGZ KPRWV CPF QWVRWV FCVC TGURGEVKXGN[ 5KPEG VJG NGPIVJ QH F k ŏU KU NCTIGT VJCP VJG NGPIVJ QH fi ŏU VJG \GTQURCFFKPI UVGR QH fi ŏU WUGF KP UVCPFCTF ((6 CNIQTKVJO HQT GXCNWCVKPI F k ŏU. F23 F24. fˆ1 4 fˆ. …. . f1 f5. fˆ1 2 fˆ 2. …. k = 0, · · · , 2M − 1,. F1 5 F1 6. fˆ 0. …. ;. F0 F7 F8. f2 f6. f. Level 3. …. fi e. i=0. Second Phase. …. Fk =. Level 2. …. C &56.((6 6JG RTQVQV[RG QH &56.((6 KU Nx −1 j2πik/2M. Level 1. …. % (CUV 'XCNWCVKPI #NIQTKVJOU HQT )+6 (QTOWNCVKQP 6Q GHſEKGPVN[ TGCNK\G QWT HQTOWNCVKQP QH UVGCF[ UVCVG VGORGTCVWTG FKUVTKDWVKQP YG ſTUV FGTKXG C QPGFKOGPUKQPCN TCFKZVYQ DCUGF ((6 NKMG GXCNWCVKPI CNIQTKVJO HQT VJG NGPIVJ QH QWVRWV FCVC DGKPI NCTIGT VJCP VJG NGPIVJ QH KPRWV FCVC &56.((6 6JGP DCUGF QP &56.((6 YG FGXGNQR CPQVJGT QPGFKOGPUKQPCN ((6 NKMG GXCNWCVKPI CNIQTKVJO HQT VJG NGPIVJ QH QWVRWV FCVC DGKPI UOCNNGT VJCP VJG NGPIVJ QH KPRWV FCVC &.65((6 (KPCNN[ VJGUG VYQ CNIQTKVJOU CTG KPVGITCVGF VQ ECNEWNCVG GSWCVKQPU CPF D[ VJG TQYEQNWOP RTQEGFWTG CPF VJG EQORWVCVKQPCN EQORNGZKV[ QH QWT )+6 DCUGF VJGTOCN UKOWNCVQT ECP DG CPCN[\GF VQ DG QPN[ O(M N log2 Nx Ny ). First Phase. f0 f4. …. YJGTG γil (iπ/L x )2 + (lπ/L y )2 B1 (Nx , ∞) × (Ny , ∞) B2 = (Nx , ∞) B3 (Ny , ∞) β1 64M 2 N 2 PT /(Lx Ly κπ 4 ) β2 = 8M 2 PT /(Lx Ly κπ 2 ) CPF β3 = 8N 2 PT /(Lx Ly κπ 2 ) 6JKU DQWPF UJQYU VJCV VJG FGEC[KPI TCVG QH VJG VTWPECVKQP GTTQTQH )TGGPŏU HWPEVKQP DCUGF HQTOWNCVKQP KU KP VJG QTFGT QH i2 l2 (iπ/L x )2 + (lπ/L y )2 6JGTGHQTG VJG EQPXGTIGPEG TCVG QH )+6 DCUGF OGVJQF KU OWEJ HCUVGT VJCP )TGGPŏU HWPEVKQP DCUGF OGVJQF =? 6JG TGCUQP KU VJCV VJG )+6 DCUGF OGVJQF IGPGTCVGU VJG QTVJQPQTOCN URCVKCN DCUGU HQT VJG VTCPUKGPV JGCV FKHHWUKQP GSWCVKQP CPF QDVCKPU VJG ENQUGHQTO QH UVGCF[ UVCVG UQNWVKQP D[ WUKPI VJGUG URCVKCN DCUGU YJKEJ ECP HWNN[ ſNN VJG GKIGPURCEG QH JGCV FKHHWUKQP GSWCVKQP 1P VJG QVJGT JCPF =? EQPUVTWEVU VJG URCVKCN CRRTQZKOCVGF HWPEVKQP D[ CRRN[KPI )TGGPŏU HWPEVKQP VQ 2QUUKQPŏU GSWCVKQP YJKEJ FQGU PQV EQPVCKP VJG VGORQTCN KPHQTOCVKQP #U C TGUWNV VJG IGPGTCVGF )TGGPŏU HWPEVKQP EQWNF PQV HWNN[ ſNN VJG GKIGPURCEG QH VTCPUKGPV JGCV FKHHWUKQP GSWCVKQP HQT CRRTQZKOCVKPI VJG VGORGTCVWTG 6JG EQPXGTIGPEG TCVG QH QWT )+6 DCUGF HQTOWNCVKQP KU PQV QPN[ HCUVGT VJCP =? VJG GZRGTKOGPVCN TGUWNVU CNUQ FGOQPUVTCVG VJCV KV ECP OCKPVCKP VJG UCOG CEEWTCE[ CU =? GXGP KH VJG VTWPECVKQP RQKPV Nx QT Ny KU HCT NGUU VJCP VJG PWODGT QH FKXKUKQPU M QT N #NVJQWIJ VJG VTWPECVKQP RQKPVU Nx Ny ECP DG HCT NGUU VJCP VJG PWODGT QH ITKF EGNNU M N VJGTG KU PQ CEVWCN GHſEKGPE[ KORTQXGOGPV QXGT =? KH YG FKTGEVN[ CRRN[ VJG UVCPFCTF ((6 VQ GXCNWCVG GCEJ T mn DGECWUG VJG UVCPFCTF ((6 PGGF RCF \GTQU VQ VJG KPRWV FCVC YJGP VJG FKOGPUKQPU QH KPRWV CPF QWVRWV FCVC CTG PQV GSWCN 6Q QXGTEQOG VJKU NKOKVCVKQP YG RTQXKFG HCUV GXCNWCVKPI CNIQTKVJOU HQT QWT )+6 HQTOWNCVKQP YKVJQWV VJG \GTQ RCFFKPI. (KI 2TQEGFWTG QH &56.((6 6JG ő4GXGTUGDKVŒ OGCPU VJG TGXGTUGDKV CNIQTKVJO =?. Fî oe Fîoo. Fî o. Fˆ7. (b). (KI %QORWVCVKQPCN ƀQY ITCRJU QH &56.((6 CPF &.65((6 YKVJ Nx = 8 CPF M = 16 C 6JG &56.((6 D 6JG &.65((6. UJQWNF DG CXQKFGF VQ UCXG TWPVKOG 6JGTGHQTG QWT &56.((6 CNIQTKVJO C OQFKſGF ((6 CNIQTKVJO KU FGXGNQRGF CU UVCVGF KP (KI VQ ECNEWNCVG 'SWCVKQP YKVJQWV \GTQURCFFKPI (KTUVN[ VJG ő4GXGTUGDKV f Œ RGTHQTOU log2 Nx VKOGU QH VJG &CPKGNUQP .CPE\QU NGOOC =? VQ 'SWCVKQP CPF TGQTFGTU VJG KPRWV FCVC 6JGP VJG &56.((6 CNIQTKVJO GXCNWCVGU VJG QWVRWV QH VJQUG L UWD &KUETGVG (QWTKGT 6TCPUHQTOU &(6U KP VJG DQVVQO NGXGN D[ WUKPI .KPG ∼ CPF RGTHQTOU .KPG VQ IGV VJG QWVRWV QH TGUV NGXGNU #P GZCORNG YKVJ M = 16 CPF Nx = 8 KU IKXGP KP (KI C VJGTG CTG DKUGEVKPI NGXGNU CPF UWD &(6U KP VJG DQVVQO NGXGN #HVGT RGTHQTOKPI VJG TGXGTUGDKV CNIQTKVJO VQ KPRWV FCVC VYQ RJCUGU CTG GZGEWVGF 6JG ſTUV RJCUG KU FQPG D[ WUKPI .KPG ∼ QH (KI 6JGP VJG UGEQPF RJCUG KU VQ IGV QWVRWV QH TGUV NGXGNU D[ GZGEWVKPI VJG DQVVQO WR RTQEGFWTG QH UVCPFCTF ((6 CU UVCVGF KP .KPG QH (KI 6JG EQORNGZKV[ QH &56.((6 KU O(M log2 Nx ) DGECWUG VJGTG CTG log2 Nx DKUGEVKPI NGXGNU CPF VJG EQORNGZKV[ QH GCEJ NGXGN KU O(M ) D &.65((6 6JG RTQVQV[RG QH &.65((6 KU M −1 Fi = fm ej2πim/2M ; i = 0, · · · , Nx − 1,. . m=0. YJGTG M > Nx CPF Fi CPF fm CTG EQORNGZ QWVRWV CPF TGCN KPRWV FCVC TGURGEVKXGN[ 4GRGCVKPI VJG &CPKGNUQP.CPE\QU NGOOC YKVJ log2 (M/N x ) + 1 VKOGU Fi ECP DG YTKVVGP CU VJG UWO QH 2M/N x UWD &(6U CPF GCEJ UWD &(6 JCU VJG UCOG HQTO CU VJG &56. ((6 YKVJ KPRWV CPF QWVRWV NGPIVJ CTG Nx /2 CPF Nx TGURGEVKXGN[ 6YQ RJCUGU CTG WVKNK\GF VQ GXCNWCVG Fi CPF VJG &.65((6 CNIQTKVJO KU UWOOCTK\GF KP (KI (KTUV .KPG RGTHQTOU VJG TGXGTUGDKV CNIQTKVJO VQ VJG KPRWV FCVC CPF .KPG ∼ WUG VJG & 56.((6 CNIQTKVJO VQ QDVCKP GCEJ DKUGEVGF UWD &(6 #HVGT GCEJ UWD &(6 DGKPI FQPG C DQVVQO WR RTQEGFWTG KU CRRNKGF VQ VJG TGUV log2 (M/N x )+1 DKUGEVKPI NGXGNU HQT ſPFKPI Fi CPF VJG GZGEWVKPI UVGRU CTG HTQO .KPG ∼. 465.

(25) 5C-2 #NIQTKVJO 4CFKZVYQ &.65((6 +PRWV Real vector fwith length M with length Nx 1WVRWV Complex vector F $GIKP fR 4GXGTUGDKV f NSubDF T s = 2M/Nx (QT Subi 0 VQ NSubDF T s − 1 Start Subi × Nx End Start + Nx : Ft (Start : End − 1) &.65((6 fR ( Start 2 'PF(QT L Nx (QT level 0 VQ log2 (M/Nx ) N ext∗ Subi NSubDF T s NSubDF T s 9JKNG Subi < NSubDF T s (QT i VQ Nx − 1 b1 i Subi × Nx b2 b2 Nx n i N ext∗ Ft [n] Ft [b1] Ft [b2]× ej2πi/L 'PF(QT Subi Subi N ext∗ N ext∗ Nx 'PF9JKNG L2×L 'PF(QT Ft (0 : Nx − 1) F 'PF. (KI . End 2. − 1). . #NIQTKVJO 4CFKZVYQ &56.((6 +PRWV Complex matrix K with length Nx × Ny 1WVRWV Complex matrix F with length 2M ×2N $GIKP (QT K VQ Nx − 1 TRow (i, 0 : 2N − 1) &56.((6 K(i, 0 : Ny − 1) 'PF(QT (QT L VQ 2N − 1 F (0 : 2M − 1, j) &56.((6(TRow (0 : Nx − 1, j)) 'PF(QT 'PF. . (KI . 2TQEGFWTG QH &.65((6. #P GZCORNG YKVJ M = 16 CPF Nx = 8 KU UJQYP KP (KI D +P VJG ſTUV RJCUG VJG KPRWV FCVC CTG TGQTFGTGF D[ WUKPI VJG TGXGTUGDKV CNIQTKVJO CPF VJGUG TGQTFGTGF FCVC CTG HGF KPVQ VJG EQTTGURQPFKPI &56.((6 DNQEMU 6JKU ECP DG FQPG D[ WUKPI .KPG ∼ KP (KI 6JGP VJG QWVRWV QH VQR DNQEM KP VJG NGXGN QH UGEQPF RJCUG KU ECNEWNCVGF D[ Fie = Fiee + ej2πi/16 Fieo ,. . CPF Fio ECP DG ECNEWNCVGF D[ WUKPI C UKOKNCT YC[ (KPCNN[ Fi KU GSWCN VQ Fi = Fie + ej2πi/32 Fio .. . 6JG CDQXG EQORWVCVKQPCN ƀQY QH VJG UGEQPF RJCUG KU UWOOCTK\GF HTQO .KPG ∼ KP (KI (QT VJG IGPGTCN ECUG VJG UWD &(6U KP GCEJ NGXGN QH VJG UGEQPF RJCUG ECP DG QDVCKPGF D[ EQODKPKPI VJQUG UWD &(6U QH VJGKT RTGXKQWU NGXGN YKVJ VJG UKOKNCT HQTOWNC QH GSWCVKQP D[ TGRNCEKPI VQ DG 2Nx 4Nx · · · 2M KP GCEJ NGXGN 6JG EQORWVCVKQPCN EQORNGZKV[ QH ſTUV RJCUG KU O(M log2 Nx ) DGECWUG VJG &56.((6 PGGF VQ DG GZGEWVGF 2M/N x VKOGU CPF GCEJ EQORNGZKV[ KU O(Nx log2 Nx ) 6JG EQORNGZKV[ KU O(M ) HQT VJG UGEQPF RJCUG *GPEG VJG EQORWVCVKQPCN EQORNGZKV[ QH &.65((6 KU O(M log2 Nx ) E 6GORGTCVWTG 'XCNWCVKQP 6JG CXGTCIG UVGCF[ UVCVG TKUKPI VGORGTCVWTG T mn UJQYP KP GSWCVKQP ECP DG GXCNWCVGF CU 1 T mn = Re F m,n + F 2M −(m+1),n , 2. . YJGTG Re {·} KU VJG TGCN RCTV QRGTCVQT CPF Nx −1 Ny −1. F k1 ,k2 =. i=0. K il e. j2πik1 2M. e. j2πlk2 2N. .. . l=0. *GTG 0≤ k1 ≤2M −1 0≤k2 ≤2N −1 K il Kil ej2πi/4M ej2πl/4N CPF GCEJ Kil KU GSWCN VQ GSWCVKQP +P VJG HQNNQYKPI YG CTG IQKPI VQ WVKNK\G VJG &56.((6 CNIQ TKVJO VQ FGXGNQR C TQYEQNWOP RTQEGFWTG VQ ECNEWNCVG F k1 ,k2 ŏU 6JG Kil ŏU ECP CNUQ DG QDVCKPGF D[ WUKPI C UKOKNCT RTQEGFWTG YKVJ VJG &.65((6 CNIQTKVJO 6JG TQYEQNWOP DCUGF &56.((6 OGVJQF HQT ECNEWNCVKPI F k1 ,k2 ŏU KU UWOOCTK\GF KP (KI .KPG ∼ RGTHQTOU &56.((6 VQ GCEJ TQY QH VJG KPRWV OCVTKZ K YJKEJ GCEJ il GPVT[ KU GSWCN VQ K il CPF VJGP .KPG ∼ CRRNKGU. 2TQEGFWTG QH &56.((6. &56.((6 VQ GCEJ EQNWOP QH QWVRWV OCVTKZ IQV HTQO VJG TQY RTQEGFWTG VQ QDVCKP VJG FGUKTG OCVTKZ F 5KPEG VJG EQORNGZKV[ QH &56.((6 KU O(M log2 Nx ) VJG VQVCN EQORNGZKV[ QH GXCNWCVKPI F k1 ,k2 ŏU D[ VJKU TQYEQNWOP RTQEGFWTG KU O(M N log2 Nx ) 6Q QDVCKP GCEJ Kil HTQO GSWCVKQP Pil ŏU PGGF VQ DG MPQYP HTQO GSWCVKQP 6JGTGHQTG VJG VYQ FKOGPUKQPCN V[RG QH GSWCVKQP KU PGGFGF VQ IGV TGNCVGF Fi,l ŏU HQT KPRWV FCVC DGKPI pmn ŏU 5KOKNCTN[ C &56.((6 DCUGF TQYEQNWOP RTQEGFWTG ECP DG WUGF VQ IGV VJQUG TGNCVGF Fi,l ŏU *QYGXGT VJG HQTO QH GSWCVKQP ECP PQV DG WVKNK\GF VQ ECNEWNCVG Pil ŏU DGECWUG VJG NGPIVJU QH VJQUG TGNCVGF Fi,l ŏU KP TQY CPF EQNWOP FKTGEVKQPU CTG NGUU VJCP 2M CPF 2N TGURGEVKXGN[ 6JGTGHQTG VJG EQORNGZ EQPLWICVGU QH Fi,l ŏU CTG TGSWKTGF VQ EQORNGVG VJG ECNEWNCVKQP QH Pil ŏU (QTVWPCVGN[ VJG EQORNGZ EQPLWICVG QH VJG QWVRWV HTQO GCEJ UWD &56.((6 KP ECNEWNCVKPI Fi,l ŏU ECP DG FKTGEVN[ QDVCKPGF D[ TGXGTUKPI VJGUG UWD &(6U HQT GZCORNG (Fiee )∗ FNeex −i KP (KI 6JGTGHQTG VJG EQORNGZ EQPLWICVG QH Fi,l ŏU ECP DG IQV D[ ſTUVN[ TGXGTUKPI VJG FCVC QH Ft HTQO .KPG KP (KI CPF RGTHQTOKPI .KPG ∼ KP (KI FWTKPI VJG TQYEQNWOP RTQEGFWTG QH Fi,l ŏU 5KOKNCT VQ VJG CPCN[UKU QH F k1 ,k2 ŏU VJG EQORNGZKV[ HQT GXCNWCV KPI Fi,l ŏU KU O(M N log2 Ny ) 6JG EQORNGZKV[ QH ECNEWNCVKPI VJG PGICVKXG HTGSWGPE[ EQORQPGPVU QH Fi,l ŏU KU O(M N ) + O(Ny N ) UKPEG QPN[ VJG UGEQPF RJCUG PGGF VQ DG TGEQORWVGF 6JGTGHQTG VJG EQORNGZKV[ HQT EQORWVKPI GSWCVKQP KU O(M N log2 Ny ) (TQO VJG CDQXG FKUEWUUKQP YG EQPENWFG VJCV VJG EQORNGZKV[ QH QWT )+6 DCUGF VJGTOCN UKOWNCVQT KU O(M N log2 Nx Ny) % 6TCPUKGPV 5KOWNCVKQP 6Q RGTHQTO VJG VTCPUKGPV UKOWNCVKQP VJG VWTPKPI QPQHH HWPEVKQP QH GCEJ ITKF Hmn (t) KU C VKOG KPVGTXCN HWPEVKQP URGEKſGF D[ KPUVTWEVKQP #HVGT CRRN[KPI ſPKVG FKHHGTGPEG UEJGOGU (QT UKORNKEKV[ YG WUG VJG DCEMYCTF'WNGT OGVJQF VQ GSWCVKQP t CV VJG UCORNKPI VKOG t KU GCEJ VKOGXCT[KPI EQGHſEKGPV ψilq t ψilq =. σ t−Δt Δt ψ + Pilt , Rilq ilq Rilq Nilq. . YJGTG Δt KU VJG VKOG UVGR Rilq = σ + κλ2ilq Δt Pilt KU GSWCN VQ t t CPF Hmn KU VJG GSWCVKQP YKVJ pmn TGRNCEGF D[ pmn Hmn XCNWG QH VWTPKPI QPQHH HWPEVKQP CV VKOG UVGR t #HVGT VKOGXCT[KPI EQGHſEKGPVU CV VKOG t DGKPI ECNEWNCVGF VJG CXGTCIG VGORGTCVWTG KP GCEJ ITKF EGNN CV VKOG UVGR t ECP DG QDVCKPGF D[ GSWCVKQP YKVJ VJG UCOG GXCNWCVKPI OGVJQF RTGUGPVGF KP RTGXKQWU UWDUGEVKQP +8 ' :2'4+/'06#. 4 '57.65 9G KORNGOGPV QWT )+6 DCUGF VJGTOCN UKOWNCVQT CPF VJG #NIQ TKVJO ++ QH C JKIJN[ GHſEKGPV )TGGPŏU HWPEVKQP DCUGF OGVJQF =? KP % NCPIWCIG 6JG UVCVGQHVJGCTV ((6 RCEMCIG ((69 =? KU WUGF VQ TGCNK\G VJG &%6 CPF +&%6 HQT =? #NN OGVJQFU CTG VGUVGF QP C *2 ZY YQTMUVCVKQP YKVJ )$ OGOQT[ 6JG TGUWNVU CTG EQORCTGF YKVJ C EQOOGTEKCN EQORWVCVKQPCN ƀWKF F[PCOKE 4 UQHVYCTG ANSYS. 466.