While most of our readers are familiar with parametric modeling using 3D CAD tools, I'm going to venture a guess that probably a smaller number have experienced parametric modeling using a mathematical tool for computing physical models and simulations.

In the world of parametric mathematical modeling, the tandem combination of flagship products from Maplesoft – Maple 15 and MapleSim 5 – create a bellwether environment for learning and applying this important engineering technology set. While we'll touch on the Maple 15 math engine, the majority of our discussion will focus on MapleSim 5 for using mathematical methods for exploring physical models and simulation.

**Mathematical Modeling Similarities and Differences**

Before we delve into Maple 15 and MapleSim 5 from Maplesoft, we'll briefly cover what parametric mathematical modeling is and how it differs from parametric CAD modeling. A parametric CAD model is a geometric representation of a physical model, where parameters are a subset of the CAD application's parameters and are used to define the geometry in a CAD system.

**MapleSim 5 is a parametric physical modeling and simulation tool built on a foundation of symbolic computation technology. It efficiently handles all of the complex mathematics involved in the development of engineering models, including multi-domain systems, plant modeling, and control design.**

On the other hand, a parametric mathematical model is a description of a physical system using mathematical concepts, where parameters are expressed as variables in mathematical expressions used to describe conditions affecting a physical system.

In its most rudimentary form, mathematical models represent physical systems using mathematical concepts, such as equations and functions. A mathematical model begins by stating a real world problem, moving to abstraction as the model is built, solving the model using mathematical techniques, and returning to the real world with solutions and results based on the calculations performed in the mathematical modeling tool.

Keep in mind that mathematical models are not just used to mimic real world problems, but actually solve real world problems using mathematical techniques with results that can be applied to physical systems. Generally, the success of a mathematical model is judged on how accurate its predictions are, as well to what degree it can be applied.

**Starting the Math Engine: Maple 15**

Maple 15 is Maplesoft's primary technical computing product and the math engine that supplies the computing horsepower to MapleSim 5 through a direct connection. It can handle math problems from simple algebra to advanced number theory with myriad expressions and everything in between. Maplesoft says it can also compute symbolic solutions to differential equations that no other system can handle, as well as solve 96% of the standard benchmark standard for differential equations (Differentialgleichungen by Kamke). Differential equations are especially important when using MapleSim 5 because the behavior of a mathematical model is typically defined with ordinary differential equations (ODEs).

**Using the smart document environment provided by Maple, you can automatically capture all of your technical knowledge in an electronic form that combines calculations, explanatory text and math, graphics, images, sound, and diagrams.**

Like an increasing number of computationally intensive CAD applications, Maple 15 takes advantage of a computer's CPU by automatically detecting and using all available cores for performing computations in parallel. Memory management and allocation is also improved in Maple 15.

CAD users are familiar with parametric design, and on the mathematical side Maple 15 provides parametric solving. For example, you can get complete solutions to parametric polynomial equations showing all of the different solutions in terms of the properties of the unknown parameters. Sort of a mathematical equivalent of “what-if” scenarios in CAD simulation.

I found one of the best ways for getting up to speed with Maple 15 were the Clickable Math tools that include demonstrations for exploring and learning common mathematical concepts, such as methods of integration and derivatives.

A word of advice – time spent brushing up on some basic math concepts in Maple 15 will help flatten your learning curve with MapleSim 5, so plan on setting aside a least a little time to do so. All in all, Maple 15 is an excellent tool for learning math at many different levels.

**Physical Modeling and Simulation**

MapleSim 5 is a multi-domain (mechanical, electrical, etc.) modeling and simulation tool for physical systems, including complex electromechanical (mechatronic) systems. In MapleSim vernacular, physical modeling, (or better described as physics-based modeling), involves using mathematics and physical laws to describe the behavior of a single engineering component or a system of interconnected components.

Unlike competing mathematical modeling products that use a signal-flow approach and require explicit definition of system inputs and outputs, MapleSim employs a unique topological representation for connecting components with no need for considering how signals flow between them. A big advantage to topological representation is that it translates to its mathematical representation, and this ability allows MapleSim to automatically generate system equations. The system equations are then simplified (redundant equations are removed, expressions are combined and reduced without fidelity loss, etc.), solved, and results are displayed.

Because it’s different than familiar CAD tools, before we get started, let's take a quick tour of the MapleSim 5 user interface.

**The MapleSim 5 User Interface**

- Main Toolbar – Tools for running a simulation.
- Navigation Toolbar – Tools for browsing models and changing model views.
- Model Workspace Toolbar – Tools for laying out and selecting objects, and adding elements.
- Model Workspace – The area where model block diagrams are created and edited.
- Palettes Pane – Tools for creating MapleSim models and managing projects with two tabs:

- Libraries – Sample models and domain-specific components that can be added to models.
- Project – Tools for creating and browsing models, managing simulation results, probes, and documents that can be attached to models.

- Console – Contains panes for Help, progress messages, and diagnostic debugging messages.
- 3D Workspace (Visualization Area) – The area in the window where models being created or edited are displayed.
- Parameters Pane – Provides viewing and changing of parameter values and component names in physical models, specifies simulation values, and configures display options for simulation results graphs/plots. The contents of the parameters pane change depending on the component or subsystem selected in the Model Workspace. This portion of the user interface forms the basis and essence of MapleSim's parametric mathematical modeling capabilities. All engineering parameters can be edited, with the exception of constants (such as pi), where modifications would not make sense.

The MapleSim 5 interface includes many new and improved features to reduce model development time and help manage complex models. For example, enhanced diagnostic tools provide early feedback related to the definition of the model itself, such as identifying inconsistent initial conditions. MapleSim then provides assistance in resolving the problem, so corrections can be made before running the simulation. Other additions include increased control over parameters and initial conditions, easy export of 3D animation simulation results, and streamlined design environments for building both model diagrams and 3D model representations. The improved simulation engine in MapleSim can now generate highly optimized C code for MapleSim models.