What is discretization error in the context of approximating a circle with finite elements?
Conditions
- Occurs when a continuous geometry or field is replaced by a finite collection of elements.
- In the example, the comparison is between the approximated circle and the original circle.
Reasoning, step by step
- Calculate the total length of the approximated circle by summing the element lengths.
- Determine the true circumference of the original continuous circle.
- Subtract the approximated circumference from the original circumference.
- The resulting difference is the discretization error.
Example
The speaker defines the difference between the approximated-circle circumference and the original-circle circumference as “discretization error,” also called “numerical error.”
Common misconceptions
- Believing the approximated circle has the exact same circumference as the original circle.
- Thinking discretization error only occurs in 1D problems and not in solid mechanics.
Watch the explanation
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Related questions
In finite element modeling, the continuous solid domain is broken down into smaller pieces called elements. Deformation is then computed at selected points called nodes, which are located at the boundaries or specific locations within these elements.
Conditions: Used for discretizing a continuous solid or domain into finite pieces.; Deformation values are associated with selected nodal points.
Discretization is the process of breaking continuous entities or problems down into smaller, discrete pieces to facilitate approximation and solution. It allows complex continuous problems that lack exact analytical solutions to be approximated numerically.
Conditions: Applies to continuous entities or problems.; Used to facilitate approximation and numerical solution.
The video states this as a global law for the finite element approximation: the total perimeter of the polygon formed by all the straight elements must equal the length of the approximated circle. This is the mathematical statement that follows from assembling the individual element equations.
Conditions: Applies to the assembled set of finite elements forming the approximated circle.; The equality is for the approximated polygon, not automatically the original continuous circle.
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.