Why does the video state that the sum of element lengths equals the length of the approximated circle?
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.
Reasoning, step by step
- Write the element equation for a single chord.
- Assemble all the finite elements to form a polygonal approximation of the circle.
- Sum the lengths of all the individual elements.
- The total sum represents the perimeter of the approximated circle.
Example
The on-screen text reads “Sum of element lengths = Length of the approximated circle.” The speaker states that the sum of all element lengths should equal the circumference of the approximated circle.
Common misconceptions
- Assuming the sum of element lengths equals the circumference of the original continuous circle.
- Thinking the law applies to the exact geometry rather than the discretized approximation.
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Connected concepts
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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.
Discretization error, also called numerical error, is the difference between the circumference of the polygonal approximation (the sum of element lengths) and the true circumference of the original continuous circle. It represents the unavoidable gap introduced when a smooth object is replaced by 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.
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.