*Boundary Element Method can reduce computational and modeling times*

*The key purpose of design software is quite simply to make existing products work better for less, and to allow designers to test completely new ideas without the often prohibitive cost of building, testing and making mistakes. To build almost anything we start with items such as nuts, bolts, laminations and wire and produce a part such as a motor or a solenoid.*

*By Bruce Klimpke, Technical Director at INTEGRATED Engineering Software and Dr Peter vanDuijsen, Technical Director at Simulation Research*

If we are building circuit boards, we start with a substrate and a set of traces connecting the various chips. Again, this will be a part in the whole system. We can then combine a range of parts or components to create assemblies. Finally, we put all the assemblies together to create the final product, which may be a car, a computer, an NMR system or a treadmill.

### Simulation Overview

Designers at the top level must integrate systems which may involve all disciplines of physics. These could include fluid flow, stress-strain, thermal, electromagnetic, vibration, acoustics and other physical phenomena. To put all these disciplines together, we need a ‘system’ simulator. Although we live in a four dimensional world, three spacial and one in time (x, y, z, and t), the system simulator mainly works with the dimension of time. Individual parts or components, such as motors or circuit boards, are modeled with component simulators that build models which the system simulator can use. Component simulators require the actual geometry of the problem and can include time dependence as well. Thus component simulators undertake detailed analysis of a motor for example, and the model is then used within the system simulator. For some truly dynamic simulations the component solver must be included within the time simulation. In these cases the system simulation needs to be combined with the component simulation. Being able to do this combined simulation enables the designers to model the real world problem with a greater degree of realism.

### Component Simulation

Although it is hard to give a precise description of a component, it is generally thought to be a part of an assembly. So, for example, a motor would be a component of thousands of devices. The wing of an aircraft would be a component of the aircraft body. To perform a simulation, the geometry of the component has to be modeled in addition to any time dependant sources. As well as at the component level, we must solve the governing physical equation. For an electromagnetic analysis we must solve Maxwell’s equations, and for fluid flow, we require the solution of Navier-Stokes equations.

For practical problems the analytic solutions or series approximations to solve these governing equations do not exist. However, an approximate solution can be attained by numerical methods. The most common numerical method is the Finite Element Method (FEM).

The Finite Element Method has been used extensively in all areas of engineering for many years. Basically, the geometry of the problem is divided into smaller pieces called elements. These elements are illustrated in the top left corner of the figure below:

The more elements used, the greater the accuracy of solution. Of course, with greater computer time required to arrive at the solution. Another less common solution of electromagnetic problems is the Boundary Element Method. Unlike the Finite Element Method, only the boundaries of the geometry need to be divided and not the volume. This approach can reduce computational and modeling times radically for certain classes of problems (visit http://www.integratedsoft.com for a comparison of both models).

Once the problem is solved, the designer can review a wide range of parameters such as force, torque, field strength and field density. The problem above illustrates a motor where the magnetic field density is shown. This highlights regions of saturation and also regions where the steel is not being used effectively in the magnetic path. As the laminations and windings generate heat, the electromagnetic solution could be used to calculate the heat generated. This would then be the input for a thermal analysis to find the temperature distribution of the motor.

Thus, the benefit to the designer becomes immediately clear. Now being able to see the magnetic flux path, the dimensions of the motor can be adjusted to obtain greater torques with less steel. The shape of the teeth can be modified to minimise copper usage without degrading performance. Therfore, significantly greater performance can be achieved with less cost.

### System Simulation

The components that were calculated in the field solver are part of a dynamic system. An example of the drive train of a Hybrid Electrical Vehicle (HEV), shown in the figure below, shows the coupled power electronics with mechanical drive train model. From left to right, it shows the models for the battery, DC/DC converter, inverter with vector control, PMSM, gearbox, planetary gearbox, clutch, differential, brakes and finally the wheels. The combustion engine is coupled via a clutch and gearbox to the planetary gear.

Each component is modeled in detail, and in the complete drive train model, these models are combined to form one dynamic simulation model. The model of the PMSM is calculated in the field solver, where the position- and current-depending parameters and/or look-up tables are generated. This data is directly connected to the model of the PMSM in the drive train model in Caspoc. Depending on the position of the rotor and the current in each phase, the PMSM model is evaluated and forms the interface between the power electronics and control on one side and the mechanical shaft on the other side.

The model of the vector controlled PMSM drive is shown more in detail in the figure below. Here the electric circuit of the power electronics together with the system model for the field-oriented control is shown. The field-oriented control can be build up from analog control blocks, or a digital model with fixed-point calculations could be implemented. In either way, the C code for implementing in a microprocessor or DSP can be generated automatically. Libraries for Qmath or IQmath are available for direct implementation of the produced code. The semiconductor models can be either ideal models where on-state and switching losses are predicted based on loss figures from the datasheet, or detailed dynamic models including the non-linear miller capacitance in Mosfets or IGBTs. Coupling the semiconductor model with a thermal model is required to simulate the semiconductor dependence on its own on-state and switching losses.

### Combined Simulation

As is clearly evident from the preceding sections, finite element modeling is required to optimise the geometry or shape of a component as well as other parameters such as materials.

At the system level, the actual shape of the component is no longer of any consequence. What are important are bulk mechanical parameters such as weight, moment of inertia, frictional losses, torques as a function of voltage, and electrical parameters such as capacitance, inductance and input impedance. Thermal models may also be required, such as the thermal capacitance of the component. For most applications, the results of the finite element analysis can be simple scalar values, such as average forces, inductances, and heat capacity. These values are simply entered into the models that the system simulator provides.

More detailed models may be required in which the finite element software generates forces and inductances, for example, as a function of applied voltage. In this case a table of values is generated by the finite element code and is the input data for the system simulator model. The most general case is the situation where the system simulator requires results that are time dependant on the FEM solver. In such a case, the system simulator may need to call the FEM solver at certain time intervals. Thus a true back and forth coupling is required in this instance.

An example of this would be a levitating device in which the Eddy Current calculations are truly transient. The levitating disk problem shown below is such a case. The system simulator (CASPOC): knowing the starting conditions of the disk (location, speed) and the voltage excitation on the coils and the mass of the disk can interact with the magnetic solver (OERSTED) to determine the disk location as a function of frequency of the *driving* voltage. A plot of the combined simulation, using an approximate velocity calculation for the Eddy currents, is shown with the real and simulated results.

From this simple example, it’s clear the benefits derivable by this coupled simulation. Different drive circuits could be applied, along with changing the shape of the structure, to achieve the desired forces or velocity as a function of time.

### Future Development

As processing speeds continue to increase, the demand for more sophisticated simulation will increase. With the advent of 64 bit computing in the PC world, it is now economical to purchase machines with 24 Gb of memory. The revolution is also occurring in the parallel processing arena, where it is now common to buy processors and boards with over eight threads available. These improvements in hardware and the subsequent improvements in software to take advantage of these new machines will again revolutionise the level of sophistication available to the design engineer. Simulations that potentially would have taken days, will be done in hours or eventually minutes. The other major advancement will be continued automatic optimisation of designs. Continuing development in this area will certainly reduce product costs. Still, the biggest advantage remains the ability to explore creative designs that would otherwise be too expensive to prototype.