The sooner an engineering team understands the physical processes behind a proposed product, the sooner its problems can be identified and solved. These problems can be a combination of heat, stress, flow, electric fields, and diffusion. However, interactions between these processes often catch engineers by surprise, creating problems not easily foreseen despite experience and training.
Fortunately, FEA tools are starting to address these problems through multiphysics analysis. The best programs let analysts mix and match physical processes and either simultaneously or sequentially solve for them. In addition, the software lets users define and modify mathematical models to simulate these processes. The four increasingly complex examples presented here are solved with Femlab from Comsol, Burlington, Mass. (www.Femlab.com).
Finding stress from heat
This first example is a multiphysics problem solved as a sequence of single-physics steps. A U-shaped stainless-steel tube cast in a ceramic block has a surface heater bonded to one face. Find the temperature increase in water passing through the tube, as well as the thermal expansion stresses between tube and ceramic block. This is a classic sequentially solvable problem. Sequential solutions keep the problem size down, because the system matrix needs only the variable to be solved at one point. Two or three small system matrices are collectively smaller than one complete simultaneous-solution matrix. A first step sets up and solves for the flow distribution. Then, using the flow results in the water, solve for the temperature distribution. Finally, given the temperature distribution, solve for the thermal-expansion stresses.
The geometry is modeled as an assembly of three parts: the water, stainless-steel tube, and ceramic block. The parts become subdomains in Femlab. Start with the water. Select Incompressible Navier-Stokes from the multiphysics menu and make the flow equations inactive in the tube and block subdomains, leaving them active in the water subdomain. Define material properties and boundary conditions. Finally, mesh the entire assembly. Only elements in the water subdomain are used at first, but all are used in the heat analysis. Then solve for the velocity distribution.
Now add a new set of physics to the thermal problem by selecting Convection-conduction from the multiphysics menu. There are now two physics defined: flow and convection-conduction. We already know the flow results so it's not necessary to recalculate velocities. Turn off the simultaneous solution of the flow problem to reduce the system matrix to just thermal. This substantially reduces matrix size and memory demands.
Finally, define the coupling between the flow and convection-conduction equations. This step, critical in all multiphysics analyses, defines how the equations are interrelated. In this case there is only one coupling: the pipe-flow velocity distribution, which is used in the velocity terms of the heat equation in the water subdomain. Set the velocity terms to zero for the same equations in the pipe and ceramic-block domains. This is easily done in the subdomain menu. Restarting the solution calculates the temperature distribution throughout the water, pipe, and ceramic block.
Follow the same process to calculate thermal stress: Add structural-analysis physics, turn off both flow and convection-conduction calculations, make the physics active only in the tube and block subdomains, and link the thermal calculation to the structural one by using the already calculated thermal field. Restart to solve for the thermally induced stress.
Alternatively, this problem could have been solved as one simultaneous run of all three physics, provided the computer has enough memory. Solving in steps, however, keeps the numerical size manageable. Not all problems can be separated like this.
An electric bus bar carries high current causing it to heat up. But the temperature increase affects the electrical conductivity, changing the electric current. This is a classic case of bidirectional coupling between two physics problems. The electric current depends on temperature and temperature depends on electric current. Such problems are best solved simultaneously when computer memory allows. Iterating between the two physics is also possible but solutions may converge slowly, or not at all, particularly if they are nonlinear.
This problem is set up much like the previous one with a few exceptions: Both the thermal-diffusion physics and dc-conductive physics are included from the start and both are active in the entire regime. Also, because of bidirectional coupling, both physics are active simultaneously. A simple equation as a function of T (temperature) is typed into the conductivity field to link its temperature dependence. Coupling in the opposite direction appears as a volumetric heat-source term that is a function of V (voltage). Femlab provides a number of ways to define these: as separate expressions, defined constants, or directly in the subdomain menus.
In this case we type the expressions directly in the subdomain menus using constants defined in the options menu. The conductivity is a linear function of temperature, while the volumetric heat source is a function of the square of the derivative of the voltage. Local values for temperature and voltage are used to calculate the local coupling. The entire problem is solved simultaneously.
In some cases, bidirectionally coupled problems need iterative solutions. A scripting language is then used to solve first for one physics, then solve for the second. The second solution is used to set up the first physics to complete the loop. The scripting language directs the iteration from one set of physics to the other until both converge on a stable result.
Fully linked multiphysics
In this example, a MEMs switch is deflected by the electrostatic force between a cantilever beam and base. Structural equations govern the shape and deflection of the switch, given the imposed electrostatic forces. The electrostatic solution depends on the deflected shape of the cantilever, so the smaller the gap the larger the force.
The problem calls for two separate geometries or models. The tactic here is to solve for the electrostatics around the undeformed switch and then calculate the resultant distributed force on the beam. A second geometry models the undeformed beam. The software calculates the beam deflection caused by electrostatic forces. The scripting language is used to create a new space around the deformed switch, and electrostatic fields and forces are recalculated. The new force is applied to the structural problem to calculate a new deflection. The scripting language iterates between the two problems to converge on a balanced set of electrostatic forces and the deflected beam. Femlab scripts are a combination of the Femlab command language and Matlab m-files (from Mathworks Inc.). While Femlab can be run as a stand-alone product, enhanced capability such as scripting is available to users with Matlab. Problems created with the Femlab user interface can be saved as m-file scripts, which can be edited to add loops or other functions.
An iterative approach like this is appropriate when problem size is an issue. For example, two medium-sized system matrices are smaller than one simultaneous system matrix, or when a mesh needs radical reshaping, as in the electrostatic portion of the problem.
Simplify through extended multiphysics
With some problems, the best practice is to simplify one aspect then map its solution into the larger problem. For example, a large chemical mixer has axisymmetric flow, but nonsymmetric mass diffusion.
Rather than solving the entire problem in 3D, we could solve the flow as a 2D axisymmetric problem and map results to a 3D mesh to solve the nonsymmetric convective-diffusion problem. This makes the problem considerably smaller. Femlab coupling variables map results from one geometry to another, or from one part of a problem into another part. They can map averaged results or node-by-node details (such as the flow problem above). This powerful tool can be used to simplify problems and reduce memory requirements or to link different physical processes together.
A second example of this technique involves acoustic pressure waves in a cylinder. They are linked to the deflection of the cylinder's end plate to determine the natural resonance of the endplate and sound-chamber system.
Multiphysics FEA is a powerful analysis tool for complex engineering challenges. Like traditional FEA, problem size is limited by computer memory. While multiphysics analysis inherently makes more demands on computing resources, many problems can be successfully solved with the techniques above and similar ones.
Comsol Inc., (781) 273-3322, www.comsol.com
MathWorks, (508) 647-7000, www.mathworks.com
A word about computer memory
As in single-physics FEA, problems are limited by computer memory and speed. For the same mesh, each added "physics" considerably increases the degrees of freedom (number of variables solved) in the problem -- particularly in 3D. It is easy to inadvertently turn a medium-size problem into a large one, and make a large problem nearly impossible to solve, simply by adding physics. Good multiphysics codes have tools to get around the size problem, such as allowing sequential as well as simultaneous solutions, providing levels of approximation, a variety of linkages between aspects of the problem, a scripting language, and an ability to link simplifications to more complex portions of the problem. Each of the problems here was solved with a relatively standard laptop computer with extra memory (1.5-GHz Pentium-IV processor with 1-Gbyte RAM). Because of the numerical matrices, I recommend at least 500 Mbytes of RAM. Users with large, fully coupled 3D multiphysics problems will appreciate having a gigabyte or more of RAM.