Spend Your Time Engineering, Not on Differential Equations

Dec. 10, 2008
Simulation software based on a symbolic computational approach bears the brunt of the math.

Digital prototyping has become an essential tool to speed design cycles. It lets designers replace expensive hardware prototypes with virtual models to predict system behavior, providing insight into new designs.

The concept of virtual modeling is not new: Designers create digital models of physical engineering systems, such as spacecraft mechanisms and hybrid-electric vehicle drivelines, and then simulate their behavior before physical prototypes are built. This lets engineers isolate and solve problems early in the design cycle. Here, changes are much easier and faster to implement than they would be after the hardware has already been built. Depending on the project, virtual prototyping can cut days, weeks, and even years off of the time needed to develop a product. In fact, it can sometimes save a company millions of dollars.

Signal-flow modeling

A downside to the signal-flow modeling tools found in conventional control-system design is they force users to spend a lot of time working with the mathematics representing the system rather than helping users develop, simulate, and evaluate designs. Thus physical modeling using this scheme requires users to manually derive a set of equations before implementing them as a block diagram for simulation. Such derivations are time consuming, error prone, and require advanced mathematical skills.

Even when working with something as simple as a spring-mass-damper system, for example, users must draw the free body diagram and then generate the governing equations between the physical components. Next, users must derive the differential equations for the system, convert them to integral form, and, finally, break the equations down to represent blocks. The resulting block diagram looks nothing like the original system representation. Several pages of derivations are required just to translate a free-body diagram to a differential equation. Imagine using this method to model an entire vehicle.

Obviously, it would be better to intuitively model and simulate a system while letting the software bear the brunt of the math. In this scenario, engineering designs would be described using components representing their actual physical counterparts. The software would represent a spring-mass-damper system, for example, with a compact, intuitive component diagram. Similarly, electric circuits would be built using resistors and inductors, and mechanical transmissions would be built with gear sets and driveshafts. It would not be necessary to derive or manually enter equations. Instead, the software stores and manages the necessary relational, physical, and mathematical information. Most importantly, the software generates and simplifies model equations to produce concise models that permit high-speed simulations of sophisticated systems.

Fostering multidomain simulation

An example of this kind of software is MapleSim, a new multidomain simulation tool. The software provides a broad range of components across several physical domains, including thermodynamics, multibody mechanics, rotational and translational mechanics, and analog, digital, and multiphase electric circuits. Each component contains information about which physical laws it must obey. Two connected components exchange information about which physical quantities, such as energy, voltage, torque, and heat and mass flows, must be conserved.

Lingo to know

In conventional control-system design, signal-flow modeling generates input-output models of systems to be controlled. The causal, signalflow representation of a physical system is an idealization.

Numerical computation starts from an initial guess. Numerical computations use iterative methods which form successive approximations that converge to a solution.

Examples: Computer simulations of car crashes solve partial differential equations numerically.

Computing the trajectory of a spacecraft involves the numerical solution of a system of ordinary differential equations.

Symbolic computation uses computers to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols.

MapleSim is built on a Maple math software foundation, so it contains the same numerical and symbolic computation capabilities. Maple uses a powerful computational engine to derive and solve complex sets of equations, simplify large sets of equations, and develop advanced mathematical models.

Unlike purely numeric computations, symbolic computations can directly convert a physical-system representation to mathematical equations. The upshot is MapleSim rapidly formulates the simulation model from the model diagram without the errors associated with manual derivations.

The simulation software generates equations symbolically, so it can simplify complex models using sophisticated techniques, such as differential elimination, before solving them numerically. Unlike purely numeric solvers, these models do not rely on iterative numeric routines. This boosts simulation speed without compromising fidelity in the results. Users notice this when simulating large systems comprising thousands of equations. Simplification reduces equations to a simpler set that can be solved quickly.

Furthermore, the simulation model is fully parameterized. This gives users the flexibility to choose variables for which to solve, and allows operations such as parameter sweeps. Users can thus analyze and optimize a system with minimal effort.

MapleSim has been used to model such complex systems as filters, vehicle dynamics, microrobotics, biomechanical devices, spacecraft mechanisms, hybrid-electric vehicle powertrains and drivelines, and mechatronic multidomain equipment. A recent example comes from a semitrailing arm vehicle suspension, used in many passenger vehicles. The independent suspension has one or more links, or “arms,” connecting the axle and the chassis. The trailing arms are pivoted at inclined angles.

This system is particularly difficult to model because it is moving in three dimensions. Also, two large rubber bushings at the chassis joints exhibit a dynamic response.

The software’s multibody modeling capabilities let engineers quickly and easily define the geometric topology of the suspension mechanism, from which the kinematic and dynamic behavior are computed. Accounting for the dynamics introduced by the bushings is difficult in any modeling software. MapleSim lets users enter the appropriate differential equations in Maple and then include them in MapleSim as a parameterized user-defined block.

Toyota is one of the first industrial companies to use virtual prototyping based on a symbolic computational approach. In fact, the automaker and the Maplesoft have partnered to produce advanced physical-modeling tools. The tools are intended to help the automaker redesign its product development cycle and take model-based development to the next level. This should cut costs, improve time-tomarket, and maintain high-quality standards.

Anyone dealing with the design of “real” systems, such as automotive designers, power engineers, and rocket scientists, will find the symbolic modeling tool useful. It lets engineers sidestep lengthy mathematical work and spend more time on design and analysis. In other words, engineers can get their products out the door faster, perhaps the most important task in today’s marketplace.

About the Author

Laurent Bernardin | Executive Vice President, Research and Development

Laurent Bernardin has worked at Maplesoft since 1999 and led R&D teams to Maple 15 and MapleSim releases. A researcher and an authority in scientific computation, Bernardin is regularly asked to serve at conferences for ISSAC (International Symposium on Symbolic and Algebraic Computing), IAMC (Internet Accessible Mathematical Computations), CASC (Computer Algebra and Scientific Computing), and ACA (Applications of Computer Algebra).

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