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Techniques for VLSI Circuit Optimization Considering Process Variations

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Techniques for VLSI Circuit Optimization Considering Process Variations

Mahalingam Venkataraman, PhD Defense

Date: 3/23/2009

Mahalingam Venkataraman

Department of Computer Science and Engineering

University of South Florida, Tampa, FL, 33620

Chair: Prof. Babu Joseph

Major Professor: Prof. Nagarajan Ranganathan

Committee Members: Prof. Srinivas Katkoori

Prof. Hao Zheng

Prof. Justin E. Harlow

Prof. Kandethody Ramachandran

Prof. Sanjuktha Bhanja

VLSI Circuit Complexity

- Transistor
- Count

Source: Intel

Wolfdale

410 Mill.

Yonah

151 Mill.

Prescott

125 Mill.

Northwood

55 Mill.

Yonah,

151 Mill.

Nanometer Dimensions

1 m

10 cm

100 nm

1 mm

10 µm

100 µm

1 cm

Source: Spektrum der Wissenschaften

65 nm Transistor

Source: Intel

Courtesy: Sill, PGPEE 2008

Process Variations

Process variations, in general, refer to the difference between the intended and obtained values in voltage and process parameters prior and post fabrication of the circuit.

The variations are more pronounced in nanometer era due to the limitations in fabrication equipment and lithography process

Process variations in nanometer era has a impact on the failure probability and hence the timing yield of integrated circuits

VLSI Circuit Optimization

- Circuit optimization in the nanometer era, is formally defined as the process of designing circuits with best possible power, delay and noise parameters
- Common methods
- Transistor/Gate Sizing, Wire sizing, Incremental placement
- Multiple supply, threshold voltages, Buffer insertion
- The relationship among the parameters are conflicting
- Circuits with optimal power can have a poor performance and/or noise value
- Process variations have made the relationships among the conflicting optimization objectives complex and hence more difficult to optimize

Motivation: Dissertation Research

- Corner based circuit optimization ignoring variation effects can negatively impact timing yield
- Worst case consideration of variations, guarantees good yield, but can lead to severe over design.
- In this context, there is a strong need for re-invention of circuit optimization techniques in a statistical perspective.
- The methodology
- has to consider multiple conflicting objectives
- model variation effects without assumptions regarding distributions
- has to be efficient enough to handle large circuits.
- Hence, in this dissertation, we model and develop novel statistical and runtime variation aware solutions for circuit optimization considering process variations.

Statistical Timing Analysis

Static Timing Analysis

Min/Max

Min/Max

Min/Max

Min/Max

Min/Max

Element delay

Circuit delay

Statistical Timing Analysis

- Variation awareness in VLSI started with PDF/CDF propagation
- in timing analysis.
- Circuit optimization frameworks were then built on top of the
- SSTA engine to optimize performance considering variations.

Element delay

as PDF/CDF

Circuit delay as PDF/CDF

Mathematical Programming based Circuit Optimization

- SSTA based iterative circuit optimization require a number of complicated operations at each node and hence incur a prohibitive runtime [Schmidt, EJOR 2000, Karkowski, ICSS 1995].
- Hence, the authors in [Mani, DAC 2005, Mani ICCD 2004], proposed stochastic mathematical programming based circuit optimization.
- Mathematical programs are fast and has the capability to handle large circuits
- Several circuit optimization problems like gate sizing, buffer insertion and placement have well defined mathematical programming formulations
- The stochastic programming technique is reasonably fast, but can be conservative in terms of yield and hence lesser savings in area or power [Buckley, IJFSS 1990]

Fuzzy Mathematical Programming (FMP)

- FMP is a special case of Mathematical programming
- with fuzzy variables in constraints or objective functions.
- variations are modelled as fuzzy numbers.
- Similar to stochastic programming, fuzzy programming involves a relaxation step
- FMP has been used to model uncertainty in
- scheduling, binding, testing, robotics,

pattern matching and artificial intelligence.

- A fuzzy number (linear, trapezoidal or non-linear) is defined as a number whose precise value is somewhat uncertain.

Motivation: Fuzzy Programming

- The author, in [Buckley, IJFSS 1990], highlighted that fuzzy programming guarantees solutions better or at least as good as stochastic programming and proved the same using Monte-Carlo simulations.
- The bound constraints in fuzzy programming allows the FMP to search for the optimal value instead of averaging a list of close to optimal values as in stochastic programming.
- Fuzzy programming also handles variation parameter in the objective function as opposed to constraints in stochastic
- Hence, we planned to use fuzzy programming based modeling and solution for uncertainty aware VLSI circuit optimization.

Motivation: Dynamic Clock Stretching

- The proposed statistical design methods (fuzzy or stochastic) are quite effective in the presence of variations incurring reasonable overheads.
- However, when there are no variations occuring in critical paths, the overheads still remain.
- To avoid this, we investigate a completely different approach to handle process variations.
- A dynamic delay detection and clock stretching technique is proposed to combat the effects of process variations

Variation Aware Gate Sizing (VA-GS)

Gate sizing is one of the simplest, yet effective technique for improving power/performance trade-off in VLSI circuits

Increasing size of a gate increases performance and power consumption.

The problem of gate sizing is well suited to be formulated as a mathematical programming problem

In this work, we formulate variation aware gate sizing as a fuzzy linear programming problem, maximizing timing yield with power and delay as constraints.

Variation Aware Gate Sizing – Outline

Step 1: Formulation of linear models for gate delay and dynamic power as functions of gate sizes.

Step 2: Modeling process variation in gate delay coefficients by treating them as triangular fuzzy numbers.

Step 3: Formulating and solving the LP for Deterministic Gate Sizing by setting the variation parameters to worst and typical case -> we get bounds for fuzzy formulation.

Step 4: The bound values generated above are used to convert fuzzy formulation into a corresponding crisp formulation using symmetric relaxation.

Step 5: The crisp optimization problem is then solved through a commercial nonlinear optimization solver.

Step 1: Power and Timing Models

The power consumption of a gate is fitted as a linear function of the gate size (si) only.

Linear approximation for gate delayis adopted from [Berkelaar, EDAC 90]

where a, b, c : constant coefficients from spice simulations

fo(i): fan-out of gate i;

si: size of gate i;

The above equation describes, gate delay (di) as a function of gate size (si) and sizes of its fan-out gates

Step 2: Modeling Variations

The variations in gate length and oxide thickness are translated to coefficients b and c in the delay equation

The actual physical variability of these coefficients are unknown, but they closely approximate gate length and oxide thickness [Mani, ICCD 04]

The fuzzy coefficients are modeledas triangular fuzzy numbers of the form (bi,bi–gi, bi+gi) and (ci,ci–hi,ci+hi)and the coefficients gi and hi represent the maximum variations

Step 3: Deterministic Gate Sizing: LP Formulation

In this work, we use a delay constrained power minimization formulation for gate sizing

The deterministic version of the gate sizing optimization problem can be shown as

where Pi is the power consumption of gate i, Dp is the delay of path p and Tspec is the required timing specification of the circuit

The variations in delay are transferred to the coefficients b and c in the delay equation

Step 3: Pre-Processing for Creating Crisp Problem

The deterministic LP problem is solved with gate delay set to worst case (wc_sizing)

Next, the deterministic LP problem is also solved with delay of a gate set to nominal case (nc_sizing)

The solution to these optimizations represent the lower and upper bound values for variation aware fuzzy gate sizing problem

Using these bound values from the pre-processing step and a variationparameter lambda ) the fuzzy linear programming problem shown below is converted to crisp programming problem.

The solution to the crisp problem is in between the bound values and represents an overall degree of satisfaction of the variation parameters and the objectives of the optimization problem.

Step 4: Variation Aware Fuzzy Gate SizingStep 4: Crisp Nonlinear VA-GS Problem

The crisp problem for VA-GS is given by,

Where is the variation parameter, ncsizingand wcsizingrepresent the values of the objective functions from the deterministic pre-processing optimizations and varies from 0 to 1.

The crisp problem maximizes the variation resistance (robustness), bounds the power value and satisfies the delay constraints in an optimal fashion

Step 5: VA-GS Simulation

VA-GS was tested on ITC’99 circuits

AMPL – mathematical programming language format.

KNITRO a commercial non-linear optimization solver.

A variation of 25% in gate delay was assumed in accordance with [Nassif, ISSCC 2000].

Experimental Results

The variation aware fuzzy gate sizing approach provides an average improvement of 18% compared to DWC and 9% compared to stochastic gate sizing without compromising on timing yield.

Monte-Carlo simulation

The solution of the fuzzy technique is verified for timing yield values using Monte-Carlo simulation

We generated 10000 copy of all benchmark circuits with random gate delay coefficients and fixed gate sizes from the solution of the fuzzy approach

The delay coefficients corresponding to gate length and oxide thickness were treated as random numbers within the nominal case and worst case range.

The timing yield defined as the number of times delay of the random circuit is less than Tspec value.

The proposed fuzzy approach indicates a timing yield of 99% for the ITC benchmark circuits.

Timing Based Placement (TBP)

Incremental placement for delay improvement is a crucial step in the post layout timing convergence flow

The TBP performs small changes to the cell locations, after wire length driven standard cell placement, with the objective of improving worst negative slack

Previous works on timing driven placement [Choi, ICCAD 03] has shown significant improvements of (upto 20%) in worst negative slack

Variation Aware TBP

The objective of timing based placement is to find optimal locations of cells in a critical sub-circuit such that the critical delay of the circuit is minimized.

The timing based placement technique requires a nonlinear programming approach, as net delay has a quadratic dependence on net length

We proposed two new solutions:

(i) A fuzzy nonlinear program based solution

(ii) A stochastic chance constrained programming based solution

for variation aware timing based placement.

uppery

Net

lowery

leftx

rightx

Location Constraints and HPWL- The variables leftx, rightx, lowery and uppery are defined for every net.
- For every cell at location (x,y)connected to net, following constraints are required,
- Half perimeter wire length (HPWL) of this net is then given by,

Variation Aware Fuzzy TBP – Outline

Step 1: Formulationof linear model for gate delay and nonlinear model for interconnect delay.

Step 2: Modeling process variation in delay coefficients by treating them as triangular fuzzy numbers.

Step 3: Estimate critical cells and calculate move distance.

Step 4: Formulating and solving the NLP for TBP by setting the variation parameters to worst and typical case -> we get bounds for fuzzy formulation.

Step 5: The bound values generated above are used to convert fuzzy formulation into a corresponding crisp formulation using symmetric relaxation.

Step 6: The crisp optimization problem is then solved through a commercial nonlinear optimization solver.

Step 1: Gate and Interconnect Delay models

- We model gate delay as linear function of gate size (si) and capacitance (Cpi). In timing based placement, the gate size (si) does not change and only load seen by the gate changes, due to change in interconnect length.
- The interconnect delay is modeled as a quadratic function of the net length and can be shown as,
- Hence, in this work, we model timing based placement as a nonlinear programming problem to maximize timing yield with delay and location constraints

Step 4: Deterministic TBP Formulation

The deterministic version of the incremental timing based placement problem can be shown as,

The HPWL and location constraints are not shown here as they are not affected by process variations. Here, arris the arrival time variable of gate and nets and Tspec is the required timing specification of the circuit

The problem is formulated to maximize the timing specification (a pseudo for worst negative slack) with node based required arrival time constraints.

Using these bound values from the pre-processing step and a variation parameter lambda ) the uncertain nonlinear programming problem is converted to a crisp nonlinear problem.

The problem aims to maximize variation resistance (l) and maintains the timing specification in between the bound values ( wc_tbp and nc_tbp)

Step 5: Crisp TBP FormulationStochastic Timing Based Placement

The stochastic formulation is cast as a robust mathematical program, which captures variation effects on the constraints using the mean and variance of the uncertain parameters.

The stochastic chance constrained programming technique models uncertainty in delay using probabilistic constraints.

Probabilistic Constraints

The uncertain arrival time constraints modeled as probabilistic constraints:

Where, (h) the probability at which the constraint has to be met corresponds to the timing yield of the circuit

The probabilistic constraints are relaxed to the equivalent formulation with mean, cumulative distribution and standard deviation

Stochastic TBP

The resultant stochastic TBP problem can be shown as,

Here, (s) is the standard deviation and is the inverse cdf value of the distribution.

In accordance with previous works [Prekopa, Kluwer 95], a inverse cdf value of 3 is used for timing yield of 99.7%

Step 6: VA-TBP Simulation

VA-TBP was tested on ITC’99 benchmark circuits

KNITRO solver available through NEOS is used for both formulations described in AMPL format

Experimental Results

The variation aware fuzzy placement approach provides an average improvement of 12% compared to DWC and the stochastic placement methodology provided a 10% compared to DWC

Buffer Insertion and Driver Sizing (BIDS)

Impact of interconnect driven performance optimization is increasing in the nanometer era.

In prior buffer insertion techniques, wires have been divided into smaller segments and bring the wire delay to almost linear in terms of its length.

It has also been pointed out in [Saxena, TCAD 04], that 35% of the total standard logic cells in a circuit will be buffers at the 65nm technology level.

Further, several works have pointed out that buffer insertion coupled with driver sizing, in the optimization phase, can reduce the number of buffers inserted.

Logic Level Variation Aware BIDS

- We formulate the buffer insertion and driver sizing problem at the logic level as a piece-wise linear program with variations modeled as fuzzy numbers.
- Piece-wise linear constraints are used for modeling buffer insertion, when multiple buffers are to be inserted in a net segment
- A look-up table based approximation is used for net length modeling at the logic level
- Number of buffers and gate sizes used as pseudonym for dynamic power consumption during BIDS

Logic Level Net Length Estimation

Accurate modeling of the interconnect length at the logic level is crucial to optimization at this level

In this work, we estimate wire length using a fast and accurate lookup table based estimation.

Previous works, have used the Rent’s rule to derive the upper bounds for interconnection lengths

The rent’s rule however, does not hold true at all levels of partition hierarchy in the nanometer era

Hence, we use a table based methodology with number of cells/interconnects and fan-out count of each cell as the address for look-up

Logic Level Net Length Estimation

The look-up table is created with layout-level wire length results of sample benchmark circuits

MCNC benchmark suite with gate complexity ranging from 500 to 10000 gates were used for estimation

Interconnects with same fan-out count is grouped and the average net length for each fan-out count is calculated

For each fan-out count, nets are averaged again based on gate count in the second dimension

A maximum fan-out size of 20 is assumed and all nets with more than 20 fan-out count are rounded to 20

Deterministic-BIDS

- The equation below shows the BIDS problem formulated to minimize buffer and gate cost with piece-wise required time constraints

Conversion to Crisp Formulation

- The Objwcand Objnc from the deterministic-BIDS are the worst case and nominal case objective values
- Now with these pre-processed objective (Obj) values and a variation resistance parameter (lambda), the fuzzy problem is converted to the following crisp problem,

Experimental Setup

The simulation flow for the fuzzy-BIDS is shown in Figure.

Fuzzy-BIDS was tested on ITC 99 benchmark circuits mapped to user defined technology library

AMPL – mathematical programming language format

KNITRO –interior point non-linear optimization solver

Experimental Results

The variation aware logic level fuzzy-BIDS approach provides an average improvement of 35% on the number of buffers and gate cost required to meet performance and yield targets

Layout Level Variation Aware BIDS

The variation aware buffer insertion at the layout level is formulated to optimize variation resistance with delay and cost (number of buffers and gate sizes) as constraints.

The layout level buffer insertion, however, has restriction on the candidate buffer location to avoid repeating the place and route step.

The generation of candidate buffer locations is performed by dividing the routed wires into channels.

Sparse channels were preferred as candidates compared to denser ones.

A incremental legalization step is performed after the layout level buffer insertion to remove overlaps

Layout Level BIDS Simulation

The benchmark circuits for layout level BIDS were placed and routed using cadence design encounter tool to estimate actual wire lengths

Similar to the logic level simulations, the layout level AMPL models were solved with KNITRO nonlinear programming (NLP) solver

The AMPL models were rebuilt for layout level with worst-case, nominal-case and fuzzy modeling

Logic Level versus Layout Level BIDS

The cost function (number of buffers plus gate size increments) comparing logic and layout level BIDS for various benchmarks is shown in Figure.

The average difference (among all benchmarks) in buffer plus gate cost between logic and layout level simulations is within 10%

Dynamic Clock Stretching: Motivation

Statistical optimization methods (fuzzy, stochastic) have been effective in improving the yield/cost tradeoffs for circuits in the nanometer era

However, statistical design methods over consume power/delay even in the absence of variations

Hence, solutions which can dynamically detect delay due to variations and perform corrective/preventive action is becoming necessary

Here, we propose a dynamic delay detection and clock stretching technique to prevent timing violations

Related Work: Dynamic Error Correction using Adaptive Voltage Scaling

- The methodology uses a shadow latch to capture delayed transitions and generates error signal, which is sent to the voltage controller
- The technique, based on current timing failures, corrects them from happening in later cycles

RAZOR: Dan Ernst et. al., MICRO 2003.

Related Work: Critical Path Isolation Based Variation Tolerance [Ghosh, ICCAD 2003]

The methodology isolates critical paths.

Evaluates the data in two cycles whenever critical paths are activated.

Works well on special designs with few critical paths, but incurs delay overhead on random designs.

Related Work: Other Methods

Adaptive voltage scaling based on Critical path duplication[Burd, ISSCC 2000]

Clock phase adjustment based on dynamic delay buffer cell [Semiao, DDECS 2008]

The dynamic delay buffer and critical path duplication do not consider spatial correlation

Dynamic delay buffer design considers variations in process parameters and ignores temperature and voltage variations

Basic Idea

Irrespective of the variations occurring (P, V or T), we would like to investigate solutions at circuit level to combat variations with significantly less overheads.

Identify and capture the delay due to process variations early in the clock period

Employ a delay detection circuit to identify if a transition is delayed in the critical paths

Delay the clock (or select a delayed clock) in the event that the arrival of a signal is delayed due to process variations.

Proposed Work: Discussion

An important pre-processing step would be the identification of critical locations (interconnects), halfway in the critical path

In the presence (absence) of variations, the transitions have to be after (before) the negative edge of the clock

The positive level triggered latch, shown in Figure captures the value floating on critical interconnect at the positive level of the clock.

Proposed Work: Discussion

If the transition is delayed due to process variations, then the inputs to the XOR gate will be different.

The multiplexor selects the normal (undelayed) or delayed clock for the destination flip-flop based on the value of the XOR gate output

In the proposed approach, the delayed clock can be dynamically selected, in case the signal propagation is delayed in the data path due to process variations.

Proposed Work: Discussion

The delay detection and clock stretching logic (CSL) is added to the critical and near critical paths that can potentially have timing failure due to process variations

Unlike voltage or frequency scaling, the proposed methodology can provide immediate activation and enable prevention of timing failures

Since the detection circuit monitors data transitions on critical interconnects, the methodology is independent of the type of process variation (PVT).

Example Circuit Evaluation

- Next, we show the simulation snapshots for the example circuit simulation

A chain of inverters in between two flip-flops stages is chosen as the example circuit.

In this circuit, all interconnects in the path switch making the net halfway in the path, the necessary critical interconnect.

Short paths and Pipelined critical paths

In the context of clock stretching, the issue of short paths and consecutively pipelined critical paths has to be addressed.

In nanometer designs, short paths are usually rare due to the multiple objectives of power, performance and yield

Plus, in this work, we only use a small margin for clock stretching (approximately 10%), hence minimizing the possibility of short path failures

Secondly in pipeline circuits if a critical path is followed by another critical path in the following pipeline stage, the CSL methodology can cause timing failures.

This is because the delayed clock circuitry reduces the data capture time available in subsequent pipeline stage.

Monte-Carlo Flow for Timing Yield Estimation

The simulation flow for timing yield estimation is shown in Figure.

A simple C program was developed to estimate timing yield, with place-route and timing analysis reports

Timing Yield Results on Benchmarks

Number of critical paths can be reduced by incremental sizing/placement to improve CSL overhead

Clock Stretch Range Versus Timing Yield

Graph showing impact of clock stretching on timing yield.

- It can be clearly seen that 10% is a good choice considering the objective of high timing yield and short path failures

Conclusions

In this research, we have proposed solutions for improving timing yield considering variations without significant over design.

The fuzzy modeling is shown to effectively model variations in linear, nonlinear and piece-wise linear circuit optimization problems.

Hence, the various algorithms and circuit optimization methods proposed in this dissertation research represent significant additions to the VLSI CAD tools in the context of variation aware design.

Conclusions

The proposed circuit level technique can be used to dynamically detect delay in signals that occur due to variations and stretch the clock to add the required extra slack.

This method is expected to make a significant impact in the industry and a totally different approach from the previous works.

Acknowledgements

Semiconductor Research Corporation contract 2007-HJ-1596

NSF Computing Research Infrastructure grant CNS-0551621

Publications

V. Mahalingam, N. Ranganathan and J.E. Harlow, ”Fuzzy Optimization Approach for Gate Sizing in the presence of Process Variations”, IEEE Transactions on VLSI Systems, 16(8), Pages 975-984, Aug 2008

V. Mahalingam and N. Ranganathan, ”Timing Based Placement Considering Uncertainty due to Process Variations”, Accepted for Publication (Feb 2009) in IEEE Transactions on VLSI Systems

V. Mahalingam and N. Ranganathan, ”Improving Accuracy in Mitchells Logarithmic Multiplication using Operand Decomposition”, IEEE Transactions on Computers, 55(12), Pages 1523-1535, Dec 2006

V. Mahalingam, K. Bhattacharya, N. Ranganathan, H. Chakravarthula, R. Murphy and K. Pratt,”An Efficient VLSI Architecture for Accurate Computation of Lucas-Kanade based Optical Flow”, Accepted for Publication (Sep 2008) in IEEE Transactions on VLSI Systems

N. Ranganathan, U. Gupta and V. Mahalingam, ”Simultaneous Optimization of Total Power, Crosstalk Noise, and Delay Under Uncertainty”, Great lakes symposium in VLSI (GLSVLSI), Pages 171-176, May 2008

V. Mahalingam and N. Ranganathan, ”A Fuzzy Optimization Approach for Process Variation Aware Buffer Insertion and Driver Sizing”, IEEE Computer Society Annual Symposium on VLSI (ISVLSI), Pages 329-334, Apr 2008

V. Mahalingam and N. Ranganathan, ”Variation Aware Timing based Placement using Fuzzy Programming”, IEEE International Symposium on Quality Electronic Design (ISQED), Pages 327-332, Mar 2007

V. Mahalingam, N. Ranganathan and Justin E. Harlow, ”A Novel Approach for Variation Aware Power Minimization during Gate Sizing”, IEEE International Symposium on Low Power Electronic Design (ISLPED), Pages 174-179, Oct 2006

V. Mahalingam and N. Ranganathan, ”Variation Aware Circuit-Wise Buffer Insertion and Driver Sizing at the Logic Level”, Submitted to Design Automation Conference (DAC), 2009

V. Mahalingam,N. Ranganathan, N. Ahmed and H. Towfique, “A Variation Aware Circuit Design using Dynamic Clock Stretching”, Submitted to IEEE International Symposium on Low Power Electronic Design (ISLPED), 2009

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