# Presentation: 906541

## Large scale electromagnetic and electrostatic simulations

_*Large scale electromagnetic and electrostatic simulations*_

- Sharad Kapur         David E. Long
- Agere Systems

## Simulation of devices and interconnect

- Modeling of passive structures
- Interconnect  (wires on a chip)
- High frequencies cause severe coupling, glitches, crosstalk, delay, etc.
- Components (for RF/Optical circuits)
- Inductors, filters need accurate modeling
- Models used in higher level simulators
- Spice, HB, delay calculators, Reduced order modeling tools

## The physics

- The problems are well described by Maxwell’s equations
- Low-frequency Helmholtz or Laplace’s equation in layered dielectric media
- Traditionally two approaches to solving these problems
- Finite element/Finite Difference methods
- Integral-equation or boundary element methods

## Integral equation solutions

- The fundamental advantage of integral approaches over finite-element methods is that they exploit the known analytic solutions of Maxwell’s equations
- Instead of discretizing the operator as in FE methods, the solution is composed of a linear combination of solutions that satisfy the underlying PDE.
- It is sufficient to discretize boundaries between materials as opposed to all of space
- Very well conditioned linear systems amenable to iterative techniques

## Capacitance formulation

- The potential is computed by adding the influence of each surface charge
- In discretized form, we get a matrix equation

## Why integral equations? cont.

- Integral methods lead to a dense system of linear equations, as compared to sparse systems that arise from finite element approaches
- Because of the O(n3) cost of computing and solving the system, integral equations were largely abandoned
- Modern numerical methods reduce the cost to O(n)
- Iterative techniques for solving linear systems
- Fast matrix-vector products for the sorts of matrices that arise from integral equations

## Fast Matrix-vector products

- Black box approaches
- Methods based on the FFT
- Methods base on low-rank decompositions (SVDs)
- Kernel based approaches
- Fast-multipole and fast-multipole like methods
- Both the Fast Multipole methods and the SVD based methods are based on efficient approximation of potential kernels of the form 1/r

## Low-rank nature of matrices 

- Key observation: With well-separated points interaction matrix is numerically _low rank_.

## SVD compression

- For an *N x N* matrix A* *of rank *r* the SVD is used to factor
- where U and V are *N by** **r* matrices
- Matrix vector product
- Directly: requires *O(N**2**)* operations
- Using the UV representation requires *2 r N *operations
*When r << N this is far more efficient*
- FMM based on similar factorization with efficient multipole representation

## IES3

- IES3 is a method for matrix compression based on the singular value decomposition
- Order points, and recursively subdivide space into well-separated regions
- Primarily used to solve time-harmonic Maxwell
- Has been successfully used for a few years both internally and commercially for component level simulation

## Excellent predictive capabilities

- Inductor design

## Entire VCOs

## Baluns and Hybrids (with R.Frye and R.Melville)

**2 GHz Hybrid Coupler**

**1-6 GHz Balun**

- Use inductive coupling to change phase
- Replace off-chip components or non-linear elements
- for wireless circuits

## Simulation vs coupler measurements

- Hybrid

- Balun

## Not good enough

- IES3 can tackle relatively tiny problems.
- Needed some significant improvement
- Could handle problems from 105 to 106 unknowns with standard discretizations
- New approach:
- Change the discretization strategy
- Change to a version of the Fast Multipole method specialized to IC geometries
- Approximate geometry

## Nebula

- IES3 is typically used for single a small ensemble of components. Inadequate for large structures
- Chip level capacitance calculation
- The scale of the geometric description is overwhelming
*Billions* of geometric features

## Use a variant of the fast Multipole method

- Subdivide space in an octtree
- Interactions between all leaves
- Close interactions done directly
- Far interactions are done via a legendre expansions (multipole expansion) of the Green’s function
- Precompute all interaction matrices with a given Green’s function
- 10x-50x faster than IES3

## Coarse representation of geometry

- Only a few numbers are needed to capture the far field interactions

- Approximate characteristic function of
- geometry with moments

## RF Chips

- 1.3mm on a side
- 92,000 rectangles
- Boxes show typical discretization for an individual net using Nebula
- Far away boxes have hundreds of conductors

## 258,000 rectangles, 838 nets0.5mm on a side

- Section of digital chip

## Efficiency issues

- Even with all advances field solving approach is very slow compared to pattern matching approaches
- Always trying to come up with better discretizations
- Adaptive refinement is too conservative and slow
- Many heuristics, basically guessing form of the solution put into mesh generation

## What constitutes a good answer?

- 1% accuracy compared to measurement is considered excellent
- Simulation accuracies are usually set to 1%
- How does this make sense if process variation can be up to 20%?
- Often in circuit design the absolute number does not matter but a relative number is more important
- Differential design and symmetry can further isolate errors due to process variations

## New directions

- Modeling for optical circuits
- In the future there will be a need for optical circuit simulators
- Lasers take the role of transistors
- Waveguides/Filters take the role of passives (RLC)
- Accelerating Nebula using FPGAs

## Optical structure modeling

- Integrated optics will require accurate modeling of optical structures (e.g., waveguides, filters, etc.)
- In the future when dielectric differences become large it will be possible to construct sophisticated passive optical components on a chip
- Methods such as beam propagation and FDTD will not work in such an environment
- Preliminary research into making such a tool

## Integral formulation

- Representation in terms of Electric and Magnetic currents at interfaces
- Construct an integral-equation operator describing interactions between currents

## Currently...

- Setting up the infrastructure...
- Formulation, numerical discretization, eigensolution method
- Works surprisingly well for solving for eigenmodes of a metallic and dielectric waveguides
- Integrated with both IES3 and a high frequency FMM

## Accelerating Nebula with FPGAs

- Oskar Mencer (Bell Labs)
- Has a methodology for accelerating floating point computations using FPGAs
- A bottleneck in Nebula is the computation of certain double integrals (50% of the time is currently spent doing this)
- The double integral is mapped to an FPGA and run on a PCI board
- Potential 100x speedup over software

## Conclusion

- Integral equation methods coupled with iterative methods and Fast Matrix vector products have been successful in modeling interconnect and devices
- Orders of magnitude faster than traditional BEM methods and FE/FD methods
- Acceleration schemes for chip level calculations
- Specialized FMM methods
- Complex conductor geometries hierarchically summarized by few numbers

## People we work(ed) with

- Designers: P. Kinget, H. Wang, R. Frye, R.Melville
- Measurement:  P. Smith, M. Frie, S. Moinian
- ALC:  K. Singhal, J. Finnerty R.Gupta
- Cadence: C-Lo, S. Nahar
- Ansoft: R. Hall, D. Zheng
- Summer students: J. Zhao, F. Ling
- External: L. Greengard, V. Rokhlin (Yale)
- Friendly competition: (MIT) J. White, J. Phillips, K. Nabors, etc.