System of Three Equations in Three Variables
When designing a product, you may come across a system of equations to be solved. Whatever these
equations (linear or not), CATIA provides you with resolution methods. These methods are the Simulated
Annealing algorithm and the "SetOfEquations" capability.
Can you use either method ?
If your set of equations is purely mathematical, the answer is yes. Otherwise, no. The
SetOfEquations capability cannot solve systems using CATIA functions such as measures.
To solve a system of equations using measures, you must use the Simulated Annealing algorithm.
The Simulated Annealing algorithm is provided with the Product Engineering Optimizer product.
The set of equations is to be specified as constraints and the variables are to be specified as free
parameters. This resolution method is quite good although sometimes a bit long and you can use it to
solve a broad range of cases. The trick about this algorithm is to adjust the precision and the
other algorithm parameters. The example developed below works well with both methods. Just
to illustrate a system that cannot be solved by both methods, you can draw a cube and create two
user parameters: CubeSurface (of Area type) and CubeVolume (of Volume type). To calculate
CubeSurface and CubeVolume, you can write either:
CubeSurface = smartWetarea ( PartBody\Pad.1 )
CubeVolume = smartVolume ( PartBody\Pad.1 )
or
CubeVolume = smartVolume ( PartBody\Pad.1 )
Solving the System of Equations by a Simulated Annealing
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Open a new part document.
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Create six real type parameters by using the f(x) capabilities. Name these
parameters x1, y1, z1 and x2, y2, z2.
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Access the Product Engineering Optimizer product and click the
icon.
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In the Constraints tab, specify the three constraints (enter the constraints
one-by-one)
x1 + y1 - z1 == 0
x1*y1 - z1 == 0
sin(x1*1rad)**2 - y1 - 1 == 0
Specify a precision of 0.01 for all three constraints.
If need be, see the Product Engineering Optimizer User's Guide.
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In the Problem tab, specify x1, y1, z1 as free parameters and 1 as Step value
for all three parameters.
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Run the optimization process in Simulated Annealing mode. You can use the
default termination criteria. After the process has finished running, the x1, y1 and z1 values
are close to the one below:
x1 = 0.454
y1 = -0.807
z1 = -0.363
Keep your document open and proceed to the next task.
Solving the System of Equations by the "SetOfEquations" Capability
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Access the Knowledge Advisor workbench, then click the
icon.
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In the "Set of Equations" editor, enter the set of equations below:
x2 + y2 == z2 ;
x2*y2 == z2;
sin(x2*1rad)**2 == y2 +1
Specify x2, y2 and z2 as Unknown parameters.
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Click OK. The system of equations is solved. The values below are displayed in
the specification tree
x2 = 0.448043478
y2 = -0.812335288
z2 = -0.364229828
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