Simulating MicroNanoSystems
Manufacturing MEMS and nano devices requires a great deal of forethought before the actual manual activity occurs. Manufacturers require the highest possible yields and here Integrated Engineering Software discuss the importance of simulation to the manufacturing process.
When simulating MEMS or NANO systems typically thermal, mechanical, electromagnetic and CFD computations may be required. And in reality, the thermal calculations may depend on the electromagnetic calculations, as an example. Or in other words, the different fields need to be coupled for a true solution.
The most common method for handling the thermal, mechanical and CFD problems is the Finite Element Method (FEM). This method can certainly be applied to electromagnetic problems as well. However, the electromagnetic calculations need to include the surrounding space around the object being analysed.
For these problems, the Boundary Element Method (BEM) can have huge advantages over the traditional FEM. Thus the ideal software would include the capability to couple various types of field calculations (i.e. thermal, mechanical, electromagnetic, and CFD) and be able to use the best simulation method (FEM, BEM) or a hybrid of both.
Simulation Today
With the advent of cheap personal computing over the last twenty years, computer simulation of physical processes has become the norm for design engineers. No longer is it a question whether to simulate or not, but to simply select the best tools and methods to do the simulation. Aside from system and circuit simulations, the most common modelling problems encountered are those of calculating stress/strain relationships and thermal fields within a given model. The preferred method of simulation for these classes of physical problems has been almost exclusively the Finite Element Method (FEM). Prior to the development of finite elements another method called Finite Differences (FDM) was explored to solve the partial differential equations associated with these mechanical and thermal problems. For some special applications the Finite Difference method is still employed, such as time domain calculations for high frequency electromagnetic problems. For this class of problems, often huge numbers of unknowns are required. The number of unknowns is determined not only by the complexity of the geometry, but more importantly by the wavelength or time step required to solve the problem. Further details of this method are widely available and as mentioned earlier still has applicability for some specific problems.
It should also be mentioned that the Finite Volume Method (FVM) is commonly used for computational fluid dynamic calculations. This method is related to the Finite Difference Method. More recently the Finite Volume Method has been combined with finite elements to solve certain classes of problems. We will be keeping our attention focused on the Boundary Element and Finite Element Methods.
Comparison of Boundary Elements and Finite Elements
Although boundary elements and finite elements are used to model the same problem, the way they get to the solution is completely different. So without a doubt, both methods should arrive at the same solution (within some numerical accuracy), whether that solution be temperature, stress, flow, or a magnetic field. The difference between the two is that one solves the governing set of equations in so called differential form and the other uses the integral form. The differential form is associated with finite elements and the integral form with boundary elements.
The main point is to discuss the practical implications of using either method.
The practical difference that sets the two methods completely apart is the distribution of unknowns. For finite elements, in real three dimensional space, the geometry needs to be divided up into small pieces called elements. Typical shapes for these small volumes (or elements) are bricks, tetrahedra, prisms, and pyramids. Unknowns are typically placed at the points where elements join other elements. Thus the numbers of unknowns is proportional to n3 for full 3D problems. This is quite different for boundary elements where only the surfaces of the volumes need to be divided up into smaller pieces. Typically shapes for boundary elements in three dimensions are triangles and quadralaterals. Thus for boundary elements the number of unknowns is proportional to n2 for full 3D problems.
So at first glance boundary elements appear too good to be true. An entire three dimensional problem, in which we want to calculate the full 3D temperature distribution as an example, can be done by only putting unknowns on surfaces. How can that be? Without loss in generality this can be demonstrated by a simple two dimensional model.
Figure 1 shows a Simple two-dimensional electrostatic model, above is two conductors at voltages of one and minus one with the system grounded at 0 volts. The electrical conductivities of the two regions are shown. For those more familiar with thermal problems simply replace voltage with temperature and electrical conductivity with thermal conductivity. For 2D the number of unknowns is proportional to n2 for finite elements and n for boundary elements. Figure 2 shows the problem divided into triangular finite elements and Figure 3 shows the same problem divided into boundary elements. For this problem the boundary elements are the smaller line segments between the dots. As shown here the number of finite elements is 222 and the corresponding number of unknowns is 143. For so called linear elements the number of unknowns is simply the number of points all the triangles connect to. The same model using boundary elements results in 34 boundary elements and 33 unknowns. The unknowns are located at the ends (or points as drawn) of each element.
So looking at the two methods so far it would appear that boundary elements would be the logical choice to solve this problem as creating the boundary elements is trivial. We just divide geometry segments into smaller pieces. By contrast, generating the finite element mesh requires dividing the regions into many triangles. So why would anyone use FEM as the mesh generation is far more difficult and you end up with far greater numbers of unknowns?
The answer is due to two reasons, the lack of generality of the boundary element method and the speed of solution. Although the number of unknowns is far less, the intermediate memory and computer time required to solve the problem illustrated is of the same order. This solution time and memory requirements are due to the fact that the finite elements generate a sparse matrix and boundary elements generate a dense matrix.
The lack of generality is due to handling nonlinear and transient problems. Although both can theoretically be done the nonlinear problem requires a mesh similar to finite elements. Inclusion of a mesh to solve the nonlinear problem can ameliorate one major advantage of Boundary Element Method.
Transient problems can involve difficult formulations. Although a lot of detail is being left out in this discussion the practical implications are not.
We should at this point say what each method has actually done. For finite elements the unknowns are simply the voltage or temperature at each node of the finite element mesh. The electric field or temperature gradient is calculated by differentiating the voltage or temperature respectively.
For boundary elements the solution at the nodes is an equivalent source. To calculate the voltage, temperature, electric field, or temperature gradient an integral operation has to be performed over all the boundary elements. Again we will not delve into the math but point out what this practically means.
When performing a numerical differentiation a lot of accuracy is lost. So the calculation of the electric field or temperature gradient is numerically bad for finite elements. For boundary elements we integrate the equivalent source to calculate the field. This produces very accurate and continuous fields. However there is a price to be paid in terms of computation time. Post processing with finite elements is far faster than with boundary elements but at the expense of accuracy.
The example, however, illustrates some general statements about boundary elements and finite elements. For linear problems the mesh generation requirement for boundary elements is trivial compared to finite elements. This is especially true for 3D. For nonlinear problems the meshing advantage is lost. Finite elements can be more easily applied to more complicated problems such as transients. Post processing is a lot faster using finite elements but less accurate than boundary elements. Final solution storage requirements are far less for boundary elements.
So from what we have looked at so far both methods have some advantages and disadvantages. The next section will clearly illustrate why for some problems boundary elements is the only solution.
The Boundary Element Advantage
The previous example illustrates a problem engineers and physicists from almost any background would be familiar with. Technically we would call it a closed region problem. For many problems encountered in electromagnetics and acoustics an open region problem exists. For mechanical and thermal problems (excluding radiation) this type of problem does not exist. So the volume of interest is always finite and in fact, it is inside the boundary of the part being analyzed.
To illustrate this consider a simple MEMS electrostatic actuator with voltage contours as shown in Figure 4. As before the boundary elements are the small segments between the dots. A total of 36 unknowns was used to solve this problem.
Figure 3 illustrates boundary element distribution. For this type of problem, the electric field exists within the dielectric material and all the exterior space around it. To model this problem with finite element an artificial region enclosing the actuator would be required with a corresponding boundary condition. A typical finite element mesh is shown in Figure 5.
The advantage of not having to create an artificial boundary and then meshing the air space around the actuator is huge for this example, fig. 5. 806 unknowns were required. For real 3D open region problems where the geometry can be far more complicated or where there is very large dimensional changes, the boundary element method can enable very efficient solutions. For some problems the number of finite elements required can be so large that the boundary element is the only method of solution.
So the important point is that boundary elements can have a great advantage for open region problems. The main reason it is not so widely used is that most problems are inherently closed region.
Solved Problems using BEM
Figure 6 shows a typical array of nano tubes. Due to the symmetric nature of the problem only one tube actually needs to be modelled. The other tubes are accounted for using symmetry conditions within the boundary element formulation.
As only the tube requires boundary elements the solution is calculated extremely fast, fig. 7.
Conclusion
For modelling most MEMS and NANO systems the finite element approach is adequate. For problems involving electromagnetic calculations the BEM can solve problems which the FEM cannot solve or will take extreme computation time. Having both solution methods allows for the optimal solution of uncoupled and coupled field problems.
