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# 3D FEM Analysis Software for Solving Complex Problems

#### Spectral & Domain Decomposition

#### MMIC Example

#### Conclusion

July 1, 2009

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Electromagnetic (EM) analysis software has evolved from its traditional place as a back-end verification tool to a necessity throughout the design flow. However, this role can impede the design flow if it requires designers to spend precious time performing file conversions, transfers, and other minutiae, or slow it to a crawl when tackling large planar or arbitrary three-dimensional (3D) problems. AWR®, a company founded by microwave engineers who have experienced first-hand the complex issues that are part and parcel of high frequency design, has addressed this issue with its AXIEM™ 3D planar EM simulator, which can process upwards of a million unknowns at very high speed. The company has now expanded these capabilities to include a full 3D finite element method (FEM) EM analysis with its Analyst™ 3D FEM EM acquisition. Analyst technology is unique among 3D EM tools because it can efficiently use clusters of computers in parallel employing both spectral and domain decomposition techniques. Together these proprietary algorithms produce an orders-of-magnitude reduction in computational time while improving results. The complexity of the problems Analyst can solve are limited only by the computational resources available to the designer and not the abilities of the software itself.

Even relatively straight-forward geometries presented to a 3D FEM EM solver can contain many thousands of coupled equations (see ** Figure 1**). While the solutions to these are necessary, a single computer’s processing time and memory requirements for such calculations rapidly becomes a bottleneck in the design process; it is sometimes not even possible to solve problems above a certain size and complexity. To mitigate these limitations, multiple processors must be employed in parallel and problems must be solved using spectral and domain decomposition. To understand the latter two techniques, it is necessary to describe the unique characteristics of each one.

**Figure 1** SEM photo reveals the arbitrary 3D features of a FET making the Analyst FEM solver a good fit.

When solving a problem over a range of frequencies using spectral decomposition, a subset of the frequency domain is sent to each processor, which solves the design within a portion of the spectrum assigned by the software. The processors communicate only when they collectively present their results to the user after performing their tasks, which cuts overall computation time by a factor of nX, where n is the number of processors available. For example, if 10 processors are available to the user, a problem can be solved in one tenth the time by sending one tenth of the spectrum to each processor, instead of processing the entire spectrum on one processor and serially solving one frequency after another. Spectral decomposition is neither new nor difficult to implement and is commonly employed by EM tools from many vendors, including AWR’s own AXIEM method of moments (MoM) solver. However, its implementation within Analyst is in conjunction with the much more complex and powerful approach of domain decomposition, providing unique benefits in terms of capacity and memory utilization.

Spectral decomposition works well if each processor can handle its designated portion of the spectrum on its own. However, if one processor alone cannot solve one frequency and spectral decomposition is the only technique available, the problem cannot be solved. The answer lies in domain decomposition, which can overcome the hardware-limitation of spectral decomposition. Domain decomposition grows the possibilities of what can be solved by subdividing the finite element mesh among as many processors as are available to the user. Domain decomposition and its complementary nature to spectral decomposition are unique to the Analyst 3D FEM EM tool and have been developed and refined over 10 years, with funding from the US Department of Energy, by AWR’s Analyst design team.

**Figure 2** Analyst and analytic transmission phase agree to four decimal places of accuracy for this circular waveguide.

Somewhat less efficient than spectral decomposition, domain decomposition provides not quite an nX reduction in computation time given that all processors must communicate with each other when working on the same frequency. Consequently, domain decomposition provides a 60 to 80 percent efficiency in reducing computation time over each additional process, rather than spectral decomposition’s typical near-linear scaling with the number of processors. In the 10 processor example mentioned above, a fraction of the overall mesh is sent to each processor and all 10 work on a single frequency/frequency spectrum together. This makes it possible for Analyst to solve problems of nearly unlimited size, bounded only by the number of processors available. Up until this point, “large” has been described in terms of a problem’s component count and wavelengths; however, a problem can likewise be large as dictated by numerical complexity versus physical complexity (see ** Figure 2**). Here a physically simple waveguide structure is shown with a highly refined mesh as a very precise calculation for phase was desired for comparison to an analytic formula. The result is a “large” number of unknowns and a numerically challenging calculation.

**Figure 3** Analyst's complementary utilization of spectral and domain decomposition makes it possible to solve complex problems accurately.

Whether large is defined in terms of physical design complexity or numerical accuracy required, Analyst can solve large problems and do so efficiently. The Analyst 3D FEM EM engine further boasts resource optimization heuristics that automatically determine the optimum number of processors to use for a given problem at a given frequency. If the problem is too large to fit on one processor but fits on two, communication time is reduced because the software has reduced the number of processors to its optimal number—and two processors do not need to communicate nearly as frequently as 10. This unique combination of spectral and domain decomposition is illustrated in ** Figure 3** and uniquely available within the Analyst tool.

**Figure 4** Results from Analyst for a 3D PHEMT FET at 60 GHz show the distributive nature of the current in the gate, drain and source. The circled area shows a region of high current density.

As the F_{t} of process technology pushes into the low THz range, MMIC design at mm-wave frequencies looks much less like distributed-design-on-a-chip. As such, the metal on the FET now needs to be extracted separately and understood apart from the intrinsic device. Given the complex cross-sectional structure of the gate, mesa-etched epitaxial layers, and metallization effects at mm-wave (see Figure 1), 3D FEM becomes necessary as it gives greater insight into the interplay of metal and device than a 3D planar EM simulation alone. In this example (courtesy of WIN Semiconductor), a four-fingered PHEMT device is analyzed at mm-wave frequencies to reveal the impact of the the 3D nature of not only the trapezoidal vias but also the arched airbridge and stepped metal that shorts the source and drain stripes, as shown in ** Figure 4**. The circled feature is actually a high current area due to a step in the metal in the z-direction.

Analyst transcends the limitations of conventional 3D FEM EM solvers in three ways. First, it was designed from its inception to efficiently perform calculations in parallel using computer clusters rather than a single computer, regardless of how many processors it employs. Second, it optimally and automatically employs the minimum number of processors required to solve a problem because too many processors employed inefficiently can actually increase computational time. Finally, by using spectral and domain decomposition, Analyst can exploit the benefits of both while minimizing their shortcomings. The result is computational times orders of magnitude faster than conventional tools when solving very large problems, the ability to solve large problems with highly accurate results, and the ability to scale linearly to accommodate problems of virtually any size the user demands.

AWR Corp., El Segundo, CA

(310) 726-3000

info@awrcorp.com

www.awrcorp.com.

RS No. 302

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