# Multiphysics Topology Optimization On Manufacturing, Heat, and EMI

December 19, 2019

Multiphysics topology optimization is ideal for improving the design of this mechanical part.

Newer structural, electronic, and optical products are becoming progressively more complex, and product designers continuously resort to simulations to verify their design choices. These simulations are excellent for identifying performance problems prior to producing a prototype, and for generating a set of baseline measurements prior to testing. With the right simulation tools, you can go far beyond simply verifying a given design model and actually optimize the structure and geometry of your new product.

Real products are governed by multiple physical processes, all of which need to be considered during product design and simulation. These various physical processes cannot be considered in isolation during the design phase, and your design’s structure and geometry should be optimized while meeting important performance requirements. This is where you can use multiphysics topology optimization to determine the best design for your system such that it meets the performance requirements of your new product.

## What is Multiphysics Topology Optimization

If you start reading up on this mathematically complex field, it may be difficult to see why this process is important and how it can help you design better products. First, let’s look at an example to see why this is helpful as part of structural and electronic design. Consider the metal bracket in the image shown below. The goal is to design the bracket such that it has the smallest possible mass while still being able to support a specified maximum load. The mounting holes in the bracket can be thought of as boundary conditions, meaning that they delineate between an area with metal and an empty area.

Topology optimization takes this proposed structure and removes unnecessary portions of the volume incrementally while still checking that the bracket can provide the strength required to support the desired load. As the optimization algorithm proceeds, this yields the bracket shown in the right side of the figure. Both brackets will be able to support the load defined by the designer, but the bracket on the right will require less raw material to manufacture.

Bracket design with multiphysics topology optimization

In the above example, the volume occupied by the part was minimized subject to some particular constraints, namely that the strength of the part be larger than the load defined at the top of the bracket. This can easily be extended to consider multiple physical processes, such as heat transfer, fluid flow, or deformation along one or more directions. This process of extending the spatial optimization of a system to more than one physical process is multiphysics topology optimization.

Some of the current research on structural multiphysics topology optimization focuses on manufacturability. This is especially important when we consider traditional subtractive manufacturing processes as machinery used in these processes can place severe constraints on design, just as is the case in PCB fabrication. These manufacturing constraints, such as the smallest distance between gaps and voids in a part or the smallest radius of curvature of bends in a part, can be defined using additional constraints in the optimization procedure. Additive manufacturing systems are alleviating these manufacturing constraints as they allow free-form deposition of mechanical parts and unique PCBs.

There are other benefits to this process beyond reducing the size or weight of a new product. The optimized structure can be used in later simulations to examine any other physical process you like. Using the optimized structure in these later simulations provides two benefits:

1. Because the optimized structure occupies a smaller volume, you can use a finer mesh in an FEM/FDTD technique to create higher resolution simulations without decreasing the convergence rate. Alternatively, the convergence rate will be faster without decreasing your previously-chosen mesh size.

2. You can perform simulations with the optimized and unoptimized designs and compare them. This is particularly useful when upgrading or redesigning an existing product as it informs other design choices.

## Defining Multiphysics Topology Optimization Problems

The volume spanned by simple structures, particularly mechanical structures, can be easily determined by hand when considering a single linear physical interaction. These types of problems can be easily solved using Lagrange multipliers, linear programming, or quadratic programming techniques. When you’re examining multiple physical processes, such as electrical behavior, fluid flow, heat transfer, and mechanical behavior, these problems quickly become intractable and must be solved numerically.

In general, multiphysics topology optimization problems are coupled multiobjective optimization problems. In general, the objective functions and constraints can be integrals or derivatives or nonlinear functions. We can generally define this as a multiobjective problem with M objective functions (i.e., M important physical processes) and N design variables:

Defining multiphysics topology optimization as a multiobjective problem

The above multiobjective (possibly coupled, possibly nonlinear) optimization problem can be solved using gradient descent methods or genetic algorithms. These optimization problems are often defined in terms of an integral in the objective functions. As gradient descent and genetic algorithms are iterative procedures, the integral in the above objective function must be evaluated at each step as the solution algorithm searches for the optimum solution. These integrals may also be integrals of nonlinear functions, and these integrals may not have analytic solutions. Add to this the fact that the geometry is generally not smoothly defined within the solution space, and you must evaluate any derivatives or integrals in the objective functions numerically at each iteration.

If you are familiar with multiobjective problems, then you may know that it is unlikely (in general) that you can locate a single solution that perfectly minimizes all objective functions subject to a single set of constraints. This means you will have a set of different structures that illustrate tradeoffs between different performance parameters.

Take an example from a differential pair interconnect with a nonlinear driver and nonlinear receiver. You may desire to optimize the interconnect geometry or your optimization variables to minimize ringing, harmonic distortion, radiated EMI, and heat dissipation; your objective functions subject to specific impedance matching conditions at each end; and impedance tolerances or your constraints. You may not be able to minimize signal distortion, ringing, and heat dissipation simultaneously for your optimized structure

Working with the tools in a powerful PCB design and simulation package allows you to implement multiobjective topology optimization in advanced electronics systems directly from your design data. Allegro PCB Designer and Cadence’s full suite of analysis tools include the simulation features you need to optimize the structure of your next design to ensure it meets important electrical and thermal demands.

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